CHAPTER 1 BASIC ARITHMETIC

Transcription

1 CHAPTER 1 BASIC ARITHMETIC EXERCISE 1, Page 4 1. Evaluate 67 kg 8 kg + 4 kg without using a calculator 67 kg 8 kg + 4 kg = 67 kg + 4 kg 8 kg = 101 kg 8 kg = 19 kg. Evaluate 851 mm 7 mm without using a calculator Hence, 851 mm 7 mm = 479 mm. Evaluate without using a calculator Hence, = = - ( ) = Evaluate without using a calculator Hence, = = Evaluate without using a calculator = - (87 647) Hence, = - 5 1

2 6. Evaluate without using a calculator = Hence, = = - ( ) = Evaluate without using a calculator = Hence, = = 107, Evaluate 47 + (- 74) (- ) without using a calculator 47 + (- 74) (- ) = = = = Evaluate 81 - (- 674) without using a calculator 81 - (- 674) = Hence, = Evaluate without using a calculator

3 = - ( ) Hence, = Evaluate $ $8441 without using a calculator Hence, $ $8441 = $15,

4 EXERCISE, Page 6 1. Evaluate without using a calculator: (a) 78 6 (b) 14 7 (a) Hence, 78 6 = 468 (b) Hence, 14 7 = 868. Evaluate without using a calculator: (a) 61 7 (b) 46 9 (a) Hence, 61 7 = 187 (b) Hence, 46 9 = Evaluate without using a calculator: (a) 78 kg 11 (b) 7 kg 8 (a) Hence, 78 kg 11 = 861 kg 4

5 (b) Hence, 7 kg 8 = 584 kg 4. Evaluate without using a calculator: (a) 7 mm 1 (b) 77 mm 1 (a) Hence, 7 mm 1 = 51 mm (b) Hence, 77 mm 1 = 94 mm 5. Evaluate without using a calculator: (a) 88 m 6 (b) 979 m 11 (a) Hence, 88 m 6 = 48 mm (b) Hence, 979 m 11 = 89 mm 6. Evaluate without using a calculator: (a) (b)

6 59 (a) Hence, 181 = = (b) Hence, = = Evaluate without using a calculator: (a) (b) (a) Hence, = = 8067 (b) Hence, = A screw has a mass of 15 grams. Calculate, in kilograms, the mass of 100 such screws. (1 kg = 1000 g) Mass of 100 screws = = g = = 18 kg 6

7 9. Holes are drilled 6 mm apart in a metal plate. If a row of 6 holes is drilled, determine the distance, in centimetres, between the centres of the first and last holes. The number of spaces if 6 holes are drilled = 5 Hence, distance between the centres of the first and last holes = 6 5 = 900 mm Thus, distance in centimetres = = 90 cm 10. A builder needs to clear a site of bricks and top soil. The total weight to be removed is 696 tonnes. Trucks can carry a maximum load of 4 tonnes. Determine the number of truck loads needed to clear the site. Number of truck loads = Hence, number of truck loads needed = 9 7

8 EXERCISE, Page 7 1. Find (a) the HCF and (b) the LCM of the following numbers: 8, 1 (a) 8 = 1 = Hence, HCF = = 4 i.e. 4 is the highest number that will divide into both 8 and 1 (b) LCM = = 4 i.e. 4 is the lowest number that both 8 and 1 will divide into.. Find (a) the HCF and (b) the LCM of the following numbers: 60, 7 (a) 60 = 5 7 = Hence, HCF = = 1 i.e. 1 is the highest number that will divide into both 60 and 7 (b) LCM = 5 = 60 i.e. 60 is the lowest number that both 60 and 7 will divide into.. Find (a) the HCF and (b) the LCM of the following numbers: 50, 70 (a) 50 = = 5 7 Hence, HCF = 5 = 10 i.e. 10 is the highest number that will divide into both 50 and 70 (b) LCM = = 50 i.e. 50 is the lowest number that both 50 and 79 will divide into 4. Find (a) the HCF and (b) the LCM of the following numbers: 70, 900 (a) 70 = = 5 5 Hence, HCF = 5 = 90 i.e. 90 is the highest number that will divide into both 70 and 900 8

9 (b) LCM = 5 5 = 700 i.e. 700 is the lowest number that both 70 and 900 will divide into 5. Find (a) the HCF and (b) the LCM of the following numbers: 6, 10, 14 (a) 6 = 10 = 5 14 = 7 Hence, HCF = i.e. is the highest number that will divide into 6, 10 and 14 (b) LCM = 5 7 = 10 i.e. 10 is the lowest number that 6, 10 and 14 will divide into 6. Find (a) the HCF and (b) the LCM of the following numbers: 1, 0, 45 (a) 1 = 0 = 5 45 = 5 Hence, HCF = i.e. is the highest number that will divide into 1, 0 and 45 (b) LCM = 5 = 180 i.e. 180 is the lowest number that 1, 0 and 45 will divide into 7. Find (a) the HCF and (b) the LCM of the following numbers: 10, 15, 70, 105 (a) 10 = 5 15 = 5 70 = = 5 7 Hence, HCF = 5 i.e. is the highest number that will divide into 10, 15, 70 and 105 (b) LCM = 5 7 = 10 i.e. 10 is the lowest number that 10, 15, 70 and 105 will divide into 9

10 8. Find (a) the HCF and (b) the LCM of the following numbers: 90, 105, 00 (a) 90 = = = 5 5 Hence, HCF = 5 = 15 i.e. 15 is the highest number that will divide into 90, 105 and 00 (b) LCM = = 600 i.e. 600 is the lowest number that 90, 105 and 00 will divide into 10

11 EXERCISE 4, Page 8 1. Evaluate: = (M) = 59 (A). Evaluate: = 17 (D) = 14 (S). Evaluate: (14 - ) (14 - ) = (B) = 86 + (D) = 88 (A) 4. Evaluate: 7( - 18) (1-5) 7( - 18) (1-5) = (B) = 5 (D/M) 5. Evaluate: 6-8(14 ) (14 ) + 6 = (B) = (M) = = (D/M) 11

12 6. Evaluate: ( 7) ( 7) = (B) = (D) = 9 7 = (A/S) 7. Evaluate: (50 14) + 7(16-7) - 7 (50 14) + 7(16-7) - 7 = (B) = (D/M) = 68 (A/D) 8. Evaluate: (7 )(1 6) 4(116) ( 8) (7 )(1 6) 45 4(116) (8) 45 5 = (B) (D) = 0 = 5 (D/M) 4 1

13 CHAPTER 10 TRIGONOMETRY EXERCISE 9, Page Find the length of side x in the diagram below. By Pythagoras, from which, 5 x 7 x 5 7 and x = 5 7 = 4 m. Find the length of side x in the diagram below, correct to significant figures. By Pythagoras, x from which, x = = 9.54 mm. In a triangle ABC, AB = 17 cm, BC = 1 cm and ABC = 90. Determine the length of AC, correct to decimal places. 14

14 By Pythagoras, AC 17 1 from which, AC = 17 1 = 0.81 mm 4. A tent peg is 4.0 m away from a 6.0 m high tent. What length of rope, correct to the nearest centimetre, runs from the top of the tent to the peg? In the side view shown below, AB is the height of the tent and C is the tent peg. By Pythagoras, AC from which, length of rope, AC = = 7.1 m 5. In a triangle ABC, B is a right angle, AB = 6.9 cm and BC = 8.78 cm. Find the length of the hypotenuse. By Pythagoras, AC from which, AC = = cm 144

15 6. In a triangle CDE, D = 90 o, CD = 14.8 mm and CE = 8.1 mm. Determine the length of DE. Triangle CDE is shown below. By Pythagoras, from which, 8.1 DE 14.8 DE and DE = = 4.11 mm 7. A man cycles 4 km due south and then 0 km due east. Another man, starting at the same time as the first man, cycles km due east and then 7 km due south. Find the distance between the two men. With reference to the diagram below, AB = 0 = 1 km and BC = 4 7 = 17 km Hence, distance between the two men, AC = 1 17 = 0.81 km by Pythagoras. 8. A ladder.5 m long is placed against a perpendicular wall with its foot 1.0 m from the wall. How far up the wall (to the nearest centimetre) does the ladder reach? If the foot of the ladder is now 145

16 moved 0 cm further away from the wall, how far does the top of the ladder fall? Distance up the wall, AB = =.5 m by Pythagoras. A 'B A 'C' BC' m Hence, the amount the top of the ladder has moved down the wall, given by AA =.5.5 = 0.10 m or 10 cm 9. Two ships leave a port at the same time. One travels due west at 18.4 knots and the other due south at 7.6 knots. If 1 knot = 1 nautical mile per hour, calculate how far apart the two ships are after 4 hours. After 4 hours, the ship travelling west travels = 7.6 km, and the ship travelling south travels = km, as shown in the diagram below. Hence, distance apart after 4 hours = = 1.7 km by Pythagoras. 146

17 EXERCISE 40, Page Sketch a triangle XYZ such that Y = 90, XY = 9 cm and YZ = 40 cm. Determine sin Z, cos Z, tan X and cos X. Triangle XYZ is shown sketched below. By Pythagoras, XZ 9 40 from which, XZ = sin Z = XY XZ = 9 41 tan X = YZ XY = = 41 cm cos Z = YZ XZ = cos X = XY XZ = In triangle ABC shown below, find sin A, cos A, tan A, sin B, cos B and tan B By Pythagoras theorem, AC = 5 = 4 sin A = opposite BC = hypotenuse AB 5 cos A = adjacent AC = 4 hypotenuse AB 5 tan A = opposite BC adjacent AC 4 opposite AC sin B = = 4 hypotenuse AB 5 cos B = adjacent BC = hypotenuse AB 5 tan B = opposite AC = 4 adjacent BC. If cos A = find sin A and tan A, in fraction form. 147

18 Triangle ABC is shown below with cos A = By Pythagoras, from which, BC BC and BC = = 8 Hence, sin A = BC AC = 8 17 and tan A = BC AB = For the right-angled triangle shown below, find: (a) sin (b) cos (c) tan (a) sin = (b) cos = opposite hypotenuse = adjacent hypotenuse = (c) tan = opposite adjacent = If tan = 7, find sin and cos in fraction form. 4 Triangle ABC is shown below with tan =

19 By Pythagoras, AC 4 7 and AC = 4 7 = 5 Hence, sin = BC AC = 7 5 and cos = AB AC = Point P lies at co-ordinate (-, 1) and point Q at (5, - 4). Determine the distance PQ. From the diagram below, PQ = 5 8 = 9.44 by Pythagoras 149

20 EXERCISE 41, Page Determine, correct to 4 decimal places, sin Using a calculator, sin =.7550, correct to 4 decimal places. Determine, correct to decimal places, 5 cos Using a calculator, 5 cos 14 15= 4.846, correct to decimal places. Determine, correct to 4 significant figures, 7 tan 79 9 Using a calculator, 7 tan 79 9 = 6.5, correct to 4 significant figures 4. Determine (a) cos (b) tan.67 Using a calculator, (a) cos = cos(1.681 rad) = , correct to 4 significant figures (b) tan.67 = tan(.67 rad) = , correct to 4 significant figures 5. Find the acute angle 1 sin in degrees, correct to decimal places Using a calculator, 1 sin = 4.º, correct to decimal places 6. Find the acute angle 1 cos in degrees, correct to decimal places Using a calculator, 1 cos = 15.5º, correct to decimal places 7. Find the acute angle 1 tan.485 in degrees, correct to decimal places Using a calculator, 1 tan.485 = 7.78º, correct to decimal places 150

21 8. Find the acute angle 1 sin in degrees and minutes Using a calculator, 1 sin = 7.94º = 7º56 9. Find the acute angle 1 cos in degrees and minutes Using a calculator, 1 cos = 1.6º = 1º 10. Find the acute angle 1 tan in degrees and minutes Using a calculator, 1 tan = 41.90º = 41º In the triangle shown below, determine angle, correct to decimal places. From trigonometric ratios, tan = 5 9 from which, = 1 5 tan 9 = In the triangle shown, determine angle in degrees and minutes. 151

22 From trigonometric ratios, sin = 8 from which, = 1 8 sin = 0.5 = For the supported beam AB shown in the diagram, determine (a) the angle the supporting stay CD makes with the beam, i.e. θ, correct to the nearest degree, (b) the length of the stay, CD, correct to the nearest centimetre. (a) tan θ = AC 4.6 hence angle θ = AD tan 5.0 = 9.98º = 40º correct to nearest degree (b) By Pythagoras, CD from which, CD = = 6.79 m 15

23 EXERCISE 4, Page 9 1. Calculate the dimensions shown as x in (a) to (f) below, each correct to 4 significant figures. (a) Sin 70 = (b) Sin = (c) Cos 9 = x 1.0 x 15.0 x 17.0 (d) Cos 59 = 4.0 x from which, x = 1.0 sin 70 = 1., correct to 4 significant figures. from which, x = 15.0 sin = 5.619, correct to 4 significant figures. from which, x = 17.0 cos 9 = 14.87, correct to 4 significant figures. from which, x = 4.0 cos59 = 8.49, correct to 4 significant figures. 15

24 x (e) Tan 4 = 6.0 (f) Tan 5 = 7.0 x from which, x = 6.0 tan 4 = 5.595, correct to 4 significant figures. from which, x = 7.0 = 5.75, correct to 4 significant figures. tan 5. Find the unknown sides and angles in the right-angled triangles shown below. The dimensions shown are in centimetres. (a) By Pythagoras, AC = = 5.81 cm Tan C = and C = 1.0 tan 5.0 = 0.96º Hence, A = 180º 90º 0.96º = 59.04º (b) By Pythagoras, DE = = 6.98 cm 154

25 Sin D = and D = sin 8.0 = 0º Hence, F = 180º 90º 0º = 60º (c) J = 180º 90º 8º = 6º sin 8º = HJ 1.0 from which, HJ = 1.0 sin 8º = 5.64 cm By Pythagoras, GH = = cm (d) L = 180º 90º 7º = 6º sin 7º = LM 15.0 from which, LM = 15.0 sin 7º = cm By Pythagoras, KM = = 1.7 cm (e) N = 180º 90º 64º = 6º cos 64º = 4.0 ON from which, ON = 4.0 = 9.15 cm cos64 By Pythagoras, NP = = 8.01 cm (f) S = 180º 90º 41º = 49º cos 41º = 5.0 QS from which, QS = 5.0 = 6.65 cm cos 41 By Pythagoras, RS = = 4.46 cm. A ladder rests against the top of the perpendicular wall of a building and makes an angle of 7 with the ground. If the foot of the ladder is m from the wall, calculate the height of the building. The ladder is shown in the diagram below, where BC is the height of the building. 155

26 Tan 7 = BC from which, height of building, BC = tan 7 = 6.54 m 4. Determine the length x in the diagram below. From triangle ABC in the sketch above, tan 8 = BC 5 from which, x = AB x 5 tan 8 = 9.40 mm 5. A vertical tower stands on level ground. At a point 105 m from the foot of the tower the angle of elevation of the top is 19. Find the height of the tower. A side view is shown below where AB is the tower. 156

27 Tan 19º = AB 105 from which, height of tower, AB = 105 tan 19º = 6.15 m 157

28 EXERCISE 4, Page Use the sine rule to solve triangle ABC and find its area given: A = 9, B = 68, b = 7 mm Triangle ABC is shown below. Since the angles in a triangle add up to 180, then: C = = 8 Applying the sine rule: Using 7 a c sin 68 sin 9 sin8 7 a sin 68 sin 9 and transposing gives: a = 7sin 9 sin 68 = 14.1 mm = BC Using and transposing gives: 7 c sin 68 sin8 c = 7sin8 sin 68 = 8.9 mm = AB Area of triangle XYZ = 1 ab sin C = 1 (14.1)(7) sin 8 = 189 mm. Use the sine rule to solve triangle ABC and find its area given: B = 716', C = 56', b = 8.60 cm Triangle ABC is shown below. 158

29 A = = 5 From the sine rule, 8.60 c sin716' sin56' from which, c = 8.60sin 56' sin 716' = cm Also from the sine rule, a 8.60 sin 5' sin 716' from which, a = 8.60sin 5' sin 716' = 7.15 cm Area = 1 a csin B 1 (7.15)(7.568)sin 71 6' = 5.65cm. Use the sine rule to solve the triangle DEF and find its area given: d = 17 cm, f = cm, F = 6 Triangle DEF is shown below. Applying the sine rule: sin 6 = 17 sin D from which, sin D = 17sin 6 = Hence, D = sin = or Since F = 6, C cannot be 160.0, since is greater than 180. Thus only D = or 1948 is valid. Angle E = = 141. Applying the sine rule: e sin141' sin 6 159

30 from which, e = sin141' sin 6 = 6.0 cm Hence, D = 1948, E = 141 and DF = 6.0 mm Area of triangle ABC = 1 desin F = 1 (17)(6.0)sin 6 = 14 cm 4. Use the sine rule to solve the triangle DEF and find its area given: d =.6 mm, e = 5.4 mm, D = 104' Triangle DEF is shown below. From the sine rule, sin104' sin E from which, sin E = 5.4sin104'.6 = and E = Hence, F = = 68 1 sin = 49.0 or 490 From the sine rule,.6 f sin104' sin 68' from which, f =.6sin 68' sin104' = mm Area = 1 d esin F 1 (.6)(5.4) sin 6 8' = 185.6mm 5. Use the cosine and sine rules to solve triangle PQR and find its area given: q = 1 cm, r = 16 cm, P = 54 Triangle PQR is shown below. 160

31 By the cosine rule, p 1 16 (1)(16) cos 54 = = and p = = 1. cm From the sine rule, 1. 1 sin 54 sin Q from which, sin Q = 1sin 54 = and Q = Q = = sin = 47.5 Area = 1 (1)(16) sin 54 = 77.7cm 6. Use the cosine and sine rules to solve triangle PQR and find its area given: q =.5 m, r = 4.4 m, P = 105 Triangle PQR is shown below. By the cosine rule, p (4.4)(.5) cos105 = (-7.459) = and p = = 6.17 m From the sine rule, sin105 sin R from which, sin R = 4.4sin105 =

32 and R = Q = = sin = Area = 1 (4.4)(.5)sin105 = 6.98m 7. Use the cosine and sine rules to solve triangle XYZ and find its area given: x = 10.0 cm, y = 8.0 cm, z = 7.0 cm Triangle XYZ is shown below. By the cosine rule, (7.0)(8.0) cos X from which, cos X = (7.0)(8.0) and X = 1 cos ( ) = 8. From the sine rule, sin 8. sin Y from which, sin Y = 8.0sin and Y = Hence, Z = = sin = 5.6 Area = 1 (7.0)(8.0)sin8. = 7.8cm 16

33 8. Use the cosine and sine rules to solve triangle XYZ and find its area given: x = 1 mm, y = 4 mm, z = 4 mm Triangle XYZ is shown below. By the cosine rule, (4)(4) cos X from which, cos X = (4)(4) 1 and X = cos ( ) = 9.77 From the sine rule, 1 4 sin 9.77 sin Y from which, sin Y = 4sin and Y = Hence, Z = = sin = 5.50 Area = 1 (1)(4)sin 96.7 = 55mm 16

34 EXERCISE 44, Page A ship P sails at a steady speed of 45 km/h in a direction of W o N (i.e. a bearing of 0 o ) from a port. At the same time another ship Q leaves the port at a steady speed of 5 km/h in a direction N 15 o E (i.e. a bearing of 015 o ). Determine their distance apart after 4 hours. After 4 hours, ship P has travelled 4 45 = 180 km. After 4 hours, ship Q has travelled 4 5 = 140 km. The directions of travel are shown in the diagram below. After 4 hours their distance apart is given by PQ By the cosine rule, (PQ) (180)(140) cos(58 15) = = and PQ = = 19 km. A jib crane is shown below. If the tie rod PR is 8.0 long and PQ is 4.5 m long determine (a) the length of jib RQ, and (b) the angle between the jib and the tie rod. 164

35 (a) Using the cosine rule on triangle PQR shown below gives: RQ (8.0)(4.5) cos10= 10.5 and jib, RQ = 10.5 = 11.4 m = 11.4 m, correct to significant figures (b) From the sine rule, sin R sin10 from which, sin R = 4.5sin and the angle between the jib and the tie rod, R = 1 sin = A building site is in the form of a quadrilateral as shown below, and its area is 1510 m. Determine the length of the perimeter of the site. The quadrilateral is split into two triangles as shown in the diagram below. Area = 1510 = 1 (5.4)(8.5)sin (4.6)(x)sin

36 i.e = x from which, x = = m Hence, perimeter of quadrilateral = = 16.4 m 4. Determine the length of members BF and EB in the roof truss shown below. Using the cosine rule on triangle ABF gives: from which, BF.5 5 (.5)(5) cos 50= BF = =.9 m Using the sine rule on triangle ABF gives:.9.5 sin 50 sin B from which, sin B =.5sin and ABF = 1 sin = 9.41 Assuming ABE = 90, then FBE = = Using the sine rule on triangle BEF gives: 4.9 sin sin E from which, sin E =.9sin and E = 1 sin = Thus, EFB = = 61.7 Using the sine rule on triangle BEF again gives: BE 4 sin 61.7 sin from which, BE = 4sin 61.7 sin = 4.0 m 5. A laboratory 9.0 m wide has a span roof which slopes at 6 o on one side and 44 o on the other. Determine the lengths of the roof slopes. 166

37 A cross-sectional view is shown below. Angle ABC = = 100 Using the sine rule, AB 9.0 sin 44 sin100 from which, AB = 9.0sin 44 sin100 = 6.5 m and BC 9.0 sin 6 sin100 from which, BC = 9.0sin 6 sin100 = 5.7 m 6. PQ and QR are the phasors representing the alternating currents in two branches of a circuit. Phasor PQ is 0.0 A and is horizontal. Phasor QR (which is joined to the end of PQ to form triangle PQR) is 14.0 A and is at an angle of 5 o to the horizontal. Determine the resultant phasor PR and the angle it makes with phasor PQ. The phasors are shown in the diagram below. By the cosine rule, (PR) (0.0)(14.0) cos145 = ( ) = and resultant phasor, PR = =.48 A Using the sine rule, sin145 sin from which, 14.0sin145 sin and ϕ = 1 sin = 14.1º 167

38 7. Calculate, correct to significant figures, the co-ordinates x and y to locate the hole centre at P shown below. Tan ( ) = y x i.e. y = x tan 64 (1) Tan ( ) = y x 100 Equating (1) and () gives: i.e. y = (x + 100) tan 40 () x tan 64 = (x + 100) tan 40 i.e. and x tan 64 - x tan 40 = 100 tan 40 x(tan 64 - tan 40) = 100 tan tan 40 from which, x = tan 64tan40 = mm = 69. mm, correct to significant figures. Substituting in (1) gives: y = x tan 64 = tan 64 = 14 mm holes are equally spaced on a pitch circle of 70 mm diameter. Determine the length of the chord joining the centres of two adjacent holes. If 16 holes are equally spaced around a circle of diameter 70 mm, i.e. radius 5 mm, then the holes are spaced 60 =.5 apart

39 Length x in the diagram is the chord joining the centres of two adjacent holes. Using the cosine rule, from which, x 5 5 (5)(5) cos x = = 1.66 mm 169

40 CHAPTER 11 AREAS OF PLANE FIGURES EXERCISE 45, Page Find the angles p and q in diagram (a) below. p = = 105 (interior opposite angles of a parallelogram are equal) q = = 5. Find the angles r and s in diagram (b) above. r = = 14 (the 8 angle is the alternate angle between parallel lines) s = = 95. Find the angle t in diagram (c) above. t = =

41 EXERCISE 46, Page Name the types of quadrilateral shown in diagrams (i) to (iv) below, and determine for each (a) the area, and (b) the perimeter. (i) Rhombus (a) Area = 4.5 = 14 cm (b) Perimeter = = 16 cm (ii) Parallelogram (a) Area = 0 6 = 180 mm (b) Perimeter = = 80 mm (iii) Rectangle (a) Area = 10 0 = 600 mm (b) Perimeter = ( 10) + ( 0) = 00 mm (iv) Trapezium = 190 (a) Area = cm (b) Perimeter = = = 6.91 cm 171

42 . A rectangular plate is 85 mm long and 4 mm wide. Find its area in square centimetres. 85 mm = 8.5 cm and 4 mm = 4. cm Area of plate = = 5.7cm. A rectangular field has an area of 1. hectares and a length of 150 m. If 1 hectare = m find (a) its width, and (b) the length of a diagonal. Area of field = 1. ha = m = 1000 m (a) Area = length width from which, width = (b) By Pythagoras, length of diagonal = area 1000 = 80 m length 150 = 170 m 4. Find the area of a triangle whose base is 8.5 cm and perpendicular height 6.4 cm. Area of triangle = 1 base perpendicular height = = 7. cm 5. A square has an area of 16 cm. Determine the length of a diagonal. A square ABCD is shown below of side x cm. The diagonal is given by length AC Area of square = x = 16 17

43 By Pythagoras, (AC) x x x = 16 from which, diagonal, AC = x [ 16] = 18 cm 6. A rectangular picture has an area of 0.96 m. If one of the sides has a length of 800 mm, calculate, in millimetres, the length of the other side. Area = 0.96 m = mm and area = length breadth, i.e mm = 800 mm breadth from which, breadth = = 100 mm 7. Determine the area of each of the angle iron sections shown in below. (a) Area = (7 ) + (1 1) = = 9 cm (b) Area = (0 8) + 10(5 8 6) + (6 50) = = 650 mm 8. The diagram below shows a 4 m wide path around the outside of a 41 m by 7 m garden. Calculate the area of the path. 17

44 Area of garden = 41 7 m Area of garden, neglecting the path = (41 8) (7 8) = 9 m Hence, area of path = (41 7) ( 9) = = 560 m 9. The area of a trapezium is 1.5 cm and the perpendicular distance between its parallel sides is cm. If the length of one of the parallel sides is 5.6 cm, find the length of the other parallel side. Area of a trapezium = 1 (sum of parallel sides) (perpendicular distance between the parallel sides) i.e. 1.5 = 1 (5.6 + x) () where x is the unknown parallel side i.e. 7 = (5.6 + x) i.e. from which, 9 = x x = =.4 cm 10. Calculate the area of the steel plate shown below. Area of steel plate = (5 60) + (140 60)(5) (50 5) + = = 6750mm

45 11. Determine the area of an equilateral triangle of side 10.0 cm. An equilateral triangle ABC is shown below. Perpendicular height, AD = by Pythagoras = cm Hence, area of triangle = 1 base perpendicular height = = 4.0 cm 1. If paving slabs are produced in 50 mm by 50 mm squares, determine the number of slabs required to cover an area of m. Number of slabs = 10 mm = 175

46 EXERCISE 47, Page A rectangular garden measures 40 m by 15 m. A 1 m flower border is made round the two shorter sides and one long side. A circular swimming pool of diameter 8 m is constructed in the middle of the garden. Find, correct to the nearest square metre, the area remaining. A sketch of a plan of the garden is shown below. Shaded area = (40 15) [(15 1) + (8 1) + (15 1) + 4 ] = 600 [ ] = = m = 48m, correct to the nearest square metre.. Determine the area of circles having (a) a radius of 4 cm (b) a diameter of 0 mm (c) a circumference of 00 mm. (a) Area = r 4 = 50.7cm (b) Area = d d 0 r 4 4 = 706.9mm (c) Circumference = πr = 00 mm, from which, radius, r = mm Hence, area = 100 r = 18 mm 176

47 . An annulus has an outside diameter of 60 mm and an inside diameter of 0 mm. Determine its area. Area of annulus = d = 51 d mm 4. If the area of a circle is 0 mm, find (a) its diameter, and (b) its circumference. (a) Area of circle, A = d 4 i.e. 0 = d 4 from which, diameter, d = 40 = mm = 0.19 mm correct to decimal places. (b) Circumference of circle = r = d = = 6.41 mm 5. Calculate the areas of the following sectors of circles: (a) radius 9 cm, angle subtended at centre 75 (b) diameter 5 mm, angle subtended at centre 487' (a) Area of sector = r 9 = 5.01cm (b) Area of sector = r 60 = 19.9mm Determine the shaded area of the template shown below. 177

48 Area of template = shaded area = (10 90) = = 577 mm 7. An archway consists of a rectangular opening topped by a semi-circular arch as shown below. Determine the area of the opening if the width is 1 m and the greatest height is m. The semicircle has a diameter of 1 m, i.e. a radius of 0.5 m. Hence, the archway shown is made up of a rectangle of sides 1 m by 1.5 m and a semicircle of radius 0.5 m. Thus, area of opening = (1.5 1) = = 1.89 m 178

49 EXERCISE 48, Page Calculate the area of a regular octagon if each side is 0 mm and the width across the flats is 48. mm. The octagon is shown sketched below and is comprised of 8 triangles of base length 0 mm and perpendicular height 48./ Area of octagon = 8 0 = 19 mm. Determine the area of a regular hexagon which has sides 5 mm. The hexagon is shown sketched below and is comprised of 6 triangles of base length 5 mm and perpendicular height h as shown. Tan 0 = 1.5 h from which, h = 1.5 tan 0 = 1.65 mm Hence, area of hexagon = = 164 mm 179

50 . A plot of land is in the shape shown below. Determine (a) its area in hectares (1 ha = 10 4 m ), and (b) the length of fencing required, to the nearest metre, to completely enclose the plot of land. (a) Area of land = (0 10) = [ ] 9176 = 9176 m = ha = ha (b) Perimeter = = = 456 m, to the nearest metre

51 EXERCISE 49, Page The area of a park on a map is 500 mm. If the scale of the map is 1 to determine the true area of the park in hectares (1 hectare = 10 4 m ) Area of park = m ha = 80 ha A model of a boiler is made having an overall height of 75 mm corresponding to an overall height of the actual boiler of 6 m. If the area of metal required for the model is 1500 mm, determine, in square metres, the area of metal required for the actual boiler. The scale is : 1 i.e. 80 : 1 Area of metal required for actual boiler = m 80 = 80m. The scale of an Ordnance Survey map is 1:500. A circular sports field has a diameter of 8 cm on the map. Calculate its area in hectares, giving your answer correct to significant figures. (1 hectare = 10 4 m ) Area of sports field on map = 8 d 10 m True area of sports field = m 4 ha =.14 ha 181

52 CHAPTER 1 THE CIRCLE EXERCISE 50, Page Calculate the length of the circumference of a circle of radius 7. cm Circumference, c = r = (7.) = 45.4 cm. If the diameter of a circle is 8.6 mm, calculate the circumference of the circle Circumference, c = r = d = (8.6) = 59.5 mm. Determine the radius of a circle whose circumference is 16.5 cm Circumference = πr i.e cm = πr from which, radius, r = 16.5 =.69 cm 4. Find the diameter of a circle whose perimeter is cm If perimeter, or circumference, c = d, then = d and diameter, d = = cm 5. A crank mechanism is shown below, where XY is a tangent to the circle at point X. If the circle radius 0X is 10 cm and length 0Y is 40 cm, determine the length of the connecting rod XY. 18

53 If XY is a tangent to the circle, then 0XY = 90 Thus, by Pythagoras, 0Y 0X XY from which, XY = 0Y 0X = 8.7 cm 6. If the circumference of the earth is km at the equator, calculate its diameter. Circumference, c = r = d from which, diameter, d = c = 17 km = 1,70 km, correct to 4 significant figures. 7. Calculate the length of wire in the paper clip shown below. The dimensions are in millimetres. 1.5 Length of wire = (1.5) + + (.5 6.5) + = = = 97.1 mm (.5 )

54 EXERCISE 51, Page Convert to radians in terms of : (a) 0 (b) 75 (c) 5 (a) 0 = 0 rad = rad (b) 75 = 75 rad = 5 rad (c) 5 = 5 rad = rad rad = 5 rad Convert to radians, correct to decimal places: (a) 48 (b) 8451' (c) 15' (a) 48 = 48 rad = 0.88 rad 180 (b) 8451 = rad = rad (c) 15 =.5 rad = rad 180. Convert to degrees: (a) rad (b) rad (c) rad (a) 7 rad 6 (b) 4 rad 9 (c) 7 rad 1 = = 7 0 = 10 6 = = 4 0 = 80 9 = = 7 15 = Convert to degrees and minutes: (a) rad (b).69 rad (c) 7.41 rad 184

55 (a) rad = = or 04 (b).69 rad = = or 1548 (c) 7.41 rad = = or A car engine speed is 1000 rev/min. Convert this speed into rad/s rev/min = 1000 rev / min rad / rev 60s / min = rad/s 185

56 EXERCISE 5, Page Calculate the area of a circle of radius 6.0 cm, correct to the nearest square centimetre. r 6.0 = 11cm, correct to the nearest square centimetre. Area =. The diameter of a circle is 55.0 mm. Determine its area, correct to the nearest square millimetre. Area of circle = d 55.0 r 4 4 = 76 mm. The perimeter of a circle is 150 mm. Find its area, correct to the nearest square millimetre. Perimeter = circumference = πr i.e. 150 mm = πr from which, radius, r = mm Area = 75 r = 1790 mm, correct to the nearest square millimetre. 4. Find the area of the sector, correct to the nearest square millimetre, of a circle having a radius of 5 mm, with angle subtended at centre of = 80mm Area of sector = r 5 5. An annulus has an outside diameter of 49.0 mm and an inside diameter of 15.0 mm. Find its area correct to 4 significant figures. d d = 1709mm Area of annulus = r1 r d1 d

57 6. Find the area, correct to the nearest square metre, of a m wide path surrounding a circular plot of land 00 m in diameter. = 169 m 4 4 Area of path = d 1 d A rectangular park measures 50 m by 40 m. A m flower bed is made round the two longer sides and one short side. A circular fish pond of diameter 8.0 m is constructed in the centre of the park. It is planned to grass the remaining area. Find, correct to the nearest square metre, the area of grass. Area of grass = (50 40) (50 ) (4 ) = = 1548 m 4 8. With reference to the shape below, determine (a) the perimeter, and (b) the area. (a) Perimeter = r = (π 14) = cm (b) Area = area of rectangle + area of semicircle = (17 8) + 1 r = (14) = π = 78.9cm 187

58 9. Find the area of the shaded portion of the shape below. Shaded portion = area of square area of circle = 10 r = = 100-5π = 1.46m 10. Find the length of an arc of a circle of radius 8. cm when the angle subtended at the centre is.14 radians. Calculate also the area of the minor sector formed. Length of arc, s = rθ = (8.)(.14) = cm Area of minor sector = r (8.) = 74.07cm 11. If the angle subtended at the centre of a circle of diameter 8 mm is 1.46 rad, find the lengths of the (a) minor arc, and (b) major arc. If diameter d = 8 mm, radius r = 8 = 41 mm (a) Minor arc length, s = r = (41)(1.46) = mm (b) Major arc length = circumference minor arc = (41) = = mm 1. A pendulum of length 1.5 m swings through an angle of 10 in a single swing. Find, in centimetres, the length of the arc traced by the pendulum bob. 188

59 Arc length of pendulum bob, s = r = (1.5) = 0.6 m or 6. cm 1. Determine the angle of lap, in degrees and minutes, if 180 mm of a belt drive are in contact with a pulley of diameter 50 mm. Arc length, s = 180 mm, radius, r = 50 = 15 mm Since s = r, the angle of lap, = s = 1.44 rad = 1.44 = 8.5 r Determine the number of complete revolutions a motorcycle wheel will make in travelling km, if the wheel's diameter is 85.1 cm. If wheel diameter = 85.1 cm, then circumference, c = d = (85.1) cm = 67.5 cm =.675 m Hence, number of revolutions of wheel in travelling 000 m = Thus, number of complete revolutions = = The floodlights at a sports ground spread its illumination over an angle of 40 to a distance of 48 m. Determine (a) the angle in radians, and (b) the maximum area that is floodlit. (a) In radians, 40 = 40 rad = rad = rad, correct to decimal places = 804.m (b) Maximum area floodlit = area of sector = r Find the area swept out in 50 minutes by the minute hand of a large floral clock, if the hand is m long = m Area swept out = r 189

60 17. Determine (a) the shaded area shown below, (b) the percentage of the whole sector that the area of the shaded area represents. (a) Shaded area = 1 1 (50) (0.75) (8) (0.75) = 1 (0.75) 50 8 = 96 mm (b) Percentage of whole sector = 96 1 (50) (0.75) 100% = 4.4% 18. Determine the length of steel strip required to make the clip shown below. Angle of sector = = 0 = rad = rad Thus, arc length, s = r = (15)(4.0146) = mm Length of steel strip in clip = = mm 19. A 50 tapered hole is checked with a 40 mm diameter ball as shown below. Determine the length shown as x. 190

61 From the sketch below, tan 5 = 5 AC from which, AC = 5 tan 5 and sin 5 = 0 AB from which, AB = 0 sin 5 = mm = 47. mm i.e. AC = = x AB = x and x = i.e. x = 7.74 mm 191

62 CHAPTER 1 VOLUMES OF COMMON SOLIDS EXERCISE 5, Page Change a volume of 1,00,000 cm to cubic metres. 6 1,00,000 cm = 1,00, m = 1. m. Change a volume of 5000 mm to cubic centimetres mm = cm = 5 cm. A metal cube has a surface area of 4 cm. Determine its volume. A cube has 6 identical faces. Hence each face has an area of 4 6 = 4 cm If each side of each face has length x cm then Hence, volume of cube = x = 8 cm x 4 from which, x = cm. 4. A rectangular block of wood has dimensions of 40 mm by 1 mm by 8 mm. Determine (a) its volume, in cubic millimetres, and (b) its total surface area in square millimetres. (a) Volume = = 840mm (b) Total surface area = ( ) = ( ) = (896) = 179 mm 5. Determine the capacity, in litres, of a fish tank measuring 90 cm by 60 cm by 1.8 m, given 1 litre = 1000 cm. Volume of tank = cm = 97,000 cm 19

63 Capacity, in litres = cm 1000cm / litre = 97 litres 6. A rectangular block of metal has dimensions of 40 mm by 5 mm by 15 mm. Determine its Volume in cm. Find also its mass if the metal has a density of 9 g/cm. Volume = length breadth width = = Mass = density volume = 9 g/cm 15 cm = 15 g = mm 10 cm = 15cm 7. Determine the maximum capacity, in litres, of a fish tank measuring 50 cm by 40 cm by.5 m (1 litre = 1000 cm ) Volume = cm Tank capacity = cm 1000 cm / litre = 500 litre 8. Determine how many cubic metres of concrete are required for a 10 m long path, 150 mm wide and 80 mm deep. Volume of concrete = = 1.44 m 9. A cylinder has a diameter 0 mm and height 50 mm. calculate (a) its volume in cubic centimetres, correct to 1 decimal place, and (b) the total surface area in square centimetres, correct to 1 decimal place. (a) Volume = rh 5 = 5.cm, correct to 1 decimal place. (b) Total surface area = rhr

64 =15π π = 19.5π = 61.cm 10. Find (a) the volume, and (b) the total surface area of a right-angled triangular prism of length 80 cm and whose triangular end has a base of 1 cm and perpendicular height 5 cm. (a) Volume = Ah = 1 base perpendicular height height of prism = = 400 cm (b) Total surface area = area of each end + area of three sides In triangle ABC, AC AB BC from which, AC = AB BC 5 1 = 1 cm Hence, total surface area = 1 bh + (AC 80) + (BC 80) + (AB 80) = (1 5) + (1 80) + (1 80) + (5 80) = i.e. total surface area = 460cm 194

65 11. A steel ingot whose volume is m is rolled out into a plate which is 0 mm thick and 1.80 m wide. Calculate the length of the plate in metres. Volume = length width thickness i.e. = length from which, length = = 7.04 m 1. Calculate the volume of a metal tube whose outside diameter is 8 cm and whose inside diameter is 6 cm, if the length of the tube is 4 m. Volume of tube = area of an end length of tube R r length = = cm = 800π = 8796 cm 1. The volume of a cylinder is 400 cm. If its radius is 5.0 cm, find its height. Determine also its curved surface area. Volume of cylinder = i.e. 400 = rh (5.0) h from which, height, h = = cm Curved surface area = πrh = π(5.0)(4.709) = 15.9cm 14. A cylinder is cast from a rectangular piece of alloy 5 cm by 7 cm by 1 cm. If the length of the cylinder is to be 60 cm, find its diameter. 195

66 Volume of rectangular piece of alloy = = 40 cm Volume of cylinder = r h Hence, 40 = r (60) from which, 40 7 r (60) and radius, r = 7 = cm and diameter of cylinder, d = r = =.99 cm 15. Find the volume and the total surface area of a regular hexagonal bar of metal of length m if each side of the hexagon is 6 cm. A hexagon is shown below. In triangle 0BC, tan 0 = x from which, x = tan 0 = cm. Hence, area of hexagon = = 9.5 cm and volume of hexagonal bar = = 8060cm Surface area of bar = 4 in metre units = m 16. A block of lead 1.5 m by 90 cm by 750 mm is hammered out to make a square sheet 15 mm thick. Determine the dimensions of the square sheet, correct to the nearest centimetre. Volume of lead = cm Volume of square sheet = (x)(x)(0.015) where x m is the length of each side of the square 196

67 Hence, = x from which, x = and x = = 8. m, correct to the nearest centimetre Hence, the dimensions of the square sheet are 8. m by 8. m 17. A cylinder is cast from a rectangular piece of alloy 5.0 cm by 6.50 cm by 19. cm. If the height of the cylinder is to be 5.0 cm, determine its diameter, correct to the nearest centimetre. Volume = = rh= r (5.0) from which, r 5.0 and radius, r = = cm and diameter = radius = = 4 cm 18. How much concrete is required for the construction of the path shown below, if the path is 1 cm thick? Area of path = (8.5 ) (.1 ) + (.4.) 4 197

68 = 17 + π = 4.0 m If the concrete is 1 cm thick, i.e. 0.1 m thick, then volume of concrete = = 4.08m 198

69 EXERCISE 54, Page If a cone has a diameter of 80 mm and a perpendicular height of 10 mm, calculate its volume in cm and its curved surface area Volume of cone = rh 1 = 01.1 cm Curved surface area = πrl From the diagram, the slant height is calculated using Pythagoras theorem: l = 4 1 = cm Hence, curved surface area = π(4)(1.649) = 159.0cm. A square pyramid has a perpendicular height of 4 cm. If a side of the base is.4 cm long find the volume and total surface area of the pyramid. Volume of pyramid = 1 (area of base) perpendicular height = 1 (.4.4) 4 = 7.68 cm The total surface area consists of a square base and 4 equal triangles. Area of triangle ADE = 1 base perpendicular height = 1.4 AC (see diagram below) The length AC may be calculated using Pythagoras' theorem on triangle ABC, where AB = 4 cm, BC = 1.4 = 1. cm. 199

70 AC = AB BC = 4 1. = cm Hence, area of triangle ADE = = cm Total surface area of pyramid = (.4.4) + 4(5.011) = 5.81 cm. A sphere has a diameter of 6 cm. Determine its volume and surface area. Since diameter = 6 cm, then radius, r = cm. Volume of sphere = 4 4 r = 11.1 cm Surface area of sphere = 4r 4 = 11.1 cm 4. A pyramid having a square base has a perpendicular height of 5 cm and a volume of 75 cm. Determine, in centimetres, the length of each side of the base. Volume of pyramid = 1 area of base perpendicular height i.e. 75 = 1 area of base 5 from which, area of base = 75 = 9 cm 5 If each side of the base is x cm, then x = 9 from which, x = 9= cm Hence, the length of each side of the base is cm 00

71 5. A cone has a base diameter of 16 mm and a perpendicular height of 40 mm. Find its volume correct to the nearest cubic millimetre Volume of cone = rh 40 = 681 mm 6. Determine (a) the volume, and (b) the surface area of a sphere of radius 40 mm. Since diameter = 6 cm, then radius, r = cm. (a) Volume of sphere = 4 4 r 40 = 68,08mm or cm (b) Surface area of sphere = 4r 4 40 = 0,106 mm or cm 7. The volume of a sphere is 5 cm. Determine its diameter. Volume of sphere = 4 r Hence, 5 = 4 r from which, 5 r 4 5 and radius, r = 4 = 4.65 cm Hence, diameter = radius = 4.65 = 8.5 cm 8. Given the radius of the earth is 680 km, calculate, in engineering notation (a) its surface area in km and (b) its volume in km. (a) Surface area of earth (i.e. a sphere) = 4r = km (b) Volume of earth (i.e. a sphere) = km r 680 = 1 01

72 9. An ingot whose volume is 1.5 m is to be made into ball bearings whose radii are 8.0 cm. How many bearings will be produced from the ingot, assuming 5% wastage? If x is the number of ball bearings then = x 8.0 from which, number of bearings, x = = A spherical chemical storage tank has an internal diameter of 5.6 m. Calculate the storage capacity of the tank, correct to the nearest cubic metre. If 1 litre = 1000 cm, determine the tank capacity in litres. Volume of storage tank = r = = 9 m, correct to the nearest cubic metre 6 Volume of tank = 9 10 cm If 1 litre = 1000 cm, then capacity of storage tank = litres = 9,000 litres 0

73 EXERCISE 55, Page Find the total surface area of a hemisphere of diameter 50 mm. Total surface area = 1 r 4 r r r r = 50 = 5890 mm or cm. Find (a) the volume, and (b) the total surface area of a hemisphere of diameter 6 cm. 14 r r 18 (a) Volume of hemisphere = = 56.55cm (b) Total surface area = 1 r 4 r r r r = 6 = 84.8 cm. Determine the mass of a hemispherical copper container whose external and internal radii are 1 cm and 10 cm, assuming that 1 cm of copper weighs 8.9 g. Volume of hemisphere = r 1 10 cm Mass of copper = volume density = 1 10 cm 8.9g / cm = 1570 g = 1.57 kg 4. A metal plumb bob comprises a hemisphere surmounted by a cone. If the diameter of the hemisphere and cone are each 4 cm and the total length is 5 cm, find its total volume. The plumb bob is shown sketched below. 0

74 1 1 Volume of bob = r h r 5 = 16 4 = 9.cm 5. A marquee is in the form of a cylinder surmounted by a cone. The total height is 6 m and the cylindrical portion has a height of.5 m, with a diameter of 15 m. Calculate the surface area of material needed to make the marquee assuming 1% of the material is wasted in the process The marquee is shown sketched below. Surface area of material for marquee = r l r h Hence, surface area = (7.5)(7.9057) + (7.5)(.5) where l = = = m = m If 1% of material is wasted then amount required = = 9.4m 04

75 6. Determine (a) the volume and (b) the total surface area of the following solids: (i) a cone of radius 8.0 cm and perpendicular height 10 cm (ii) a sphere of diameter 7.0 cm (iii) a hemisphere of radius.0 cm (iv) a.5 cm by.5 cm square pyramid of perpendicular height 5.0 cm (v) a 4.0 cm by 6.0 cm rectangular pyramid of perpendicular height 1.0 cm (vi) a 4. cm by 4. cm square pyramid whose sloping edges are each 15.0 cm (vii) a pyramid having an octagonal base of side 5.0 cm and perpendicular height 0 cm (i) A sketch of the cone is shown below. 1 1 = 670cm (a) Volume of cone = r h (b) Total surface area = r rl where l = = cm = = = 5cm (ii) (a) Volume of sphere = = 180 cm (b) Surface area = 7.0 4r 4 = 154 cm r.0 = 56.5cm (iii) (a) Volume of hemisphere = 1 (4 r ) r r.0 = 84.8cm (b) Surface area = (iv) A sketch of the square pyramid is shown below, where AB = 5.0 cm 05

76 1 (a) Volume of pyramid = = 10.4cm (b) In the diagram, AC = AB BC Surface area = 1 = = =.0 cm (v) A sketch of the rectangular pyramid is shown below = 96.0cm (a) Volume of rectangular pyramid = (b) In the diagram, AC = and AD = = 1.69 cm = cm 06

77 1 1 Hence, surface area = = = 146cm (vi) The square pyramid is shown sketched below. Diagonal on base = cm hence, BC = = cm Hence, perpendicular height, h = = cm 1 (a) Volume of pyramid = (b) AD = = = 86.5cm Hence, surface area = = = 14cm (vii) A pyramid having an octagonal base is shown sketched below. 07

78 One sector is shown in diagram (p) below, where from which, x =.5 tan.5 = cm.5 tan.5 x 1 Hence, area of whole base = = cm 1 (a) Volume of pyramid = = 805cm (p) (q) (b) From diagram (q) above, y = = cm 1 Total surface area = = = 59cm 7. A metal sphere weighing 4 kg is melted down and recast into a solid cone of base radius 8.0 cm. If the density of the metal is 8000 kg/m determine (a) the diameter of the metal sphere and (b) the perpendicular height of the cone, assuming that 15% of the metal is lost in the process. Volume of sphere = mass density 4 kg 8000 kg / m m cm = 000 cm (a) Volume of sphere = 4 r i.e. 000 = 4 r 000 and radius, r = 4 = cm Hence, the diameter of the sphere, d = r = = 17.9 cm (b) Volume of cone = = rh 8.0 h cm = 08

79 from which, perpendicular height of cone, h = = 8.0 cm 8. A buoy consists of a hemisphere surmounted by a cone. The diameter of the cone and hemisphere is.5 m and the slant height of the cone is 4.0 m. Determine the volume and surface area of the buoy. The buoy is shown in the sketch below. Height of cone, h = =.80 m. 1 1 Volume of buoy = r r h = = r l 4r Surface area = = = 8.15 = 5.5 m m 9. A petrol container is in the form of a central cylindrical portion 5.0 m long with a hemispherical section surmounted on each end. If the diameters of the hemisphere and cylinder are both 1. m determine the capacity of the tank in litres (1 litre = 1000 cm ). The petrol container is shown sketched below. 09

80 4 Volume of container = r r h = = = m cm and tank capacity = cm cm / litre = 6560 litres 10. The diagram shows a metal rod section. Determine its volume and total surface area. 1 1 r h (lbw) ( ) Volume of rod = 1 1 = = cm Surface area = r h r = (1.0)(100) = 107cm 10

81 11. The cross-section of part of a circular ventilation shaft is shown below, ends AB and CD being open. Calculate (a) the volume of the air, correct to the nearest litre, contained in the part of the system shown, neglecting the sheet metal thickness, (given 1 litre = 1000 cm ), (b) the cross-sectional area of the sheet metal used to make the system, in square metres, and (c) the cost of the sheet metal if the material costs per square metre, assuming that 5% extra metal is required due to wastage. (a) In cm, volume of air = = = cm = cm 1000cm / litre = 1458 litre, correct to the nearest litre (b) In m, cross-sectional area of the sheet metal 1 4 = = =.11 = m = 9.77m correct to significant figures. (c) Sheet metal required = m Cost of sheet metal = =

82 EXERCISE 56, Page The diameter of two spherical bearings are in the ratio : 5. What is the ratio of their volumes? Diameters are in the ratio :5 Hence, ratio of their volumes = :5 i.e. 8:15. An engineering component has a mass of 400 g. If each of its dimensions are reduced by 0% determine its new mass. New mass = = 17. g MULTIPLE CHOICE QUESTIONS ON APPLIED MATHEMATICS EXERCISE 57, Page (b). (d). (b) 4. (b) 5. (a) 6. (c) 7. (c) 8. (d) 9. (b) 10. (c) 11. (a) 1. (d) 1. (a) 14. (b) 15. (a) 16. (d) 17. (a) 18. (b) 19. (d) 0. (d) 1. (a). (a). (a) 4. (a) 5. (a) 6. (d) 7. (b) 8. (c) 9. (d) 0. (b) 1. (a). (c). (b) 4. (b) 5. (c) 6. (c) 7. (b) 8. (d) 9. (c) 40. (c) 41. (a) 4. (a) 4. (c) 44. (b) 45. (d) 46. (b) 47. (d) 48. (b) 49. (d) 50. (a) 51. (a) 5. (b) 5. (a) 54. (a) 55. (d) 56. (d) 57. (c) 58. (b) 59. (c) 60. (c) 61. (c) 6. (d) 6. (a) 64. (a) 65. (c) 66. (b) 67. (b) 68. (d) 69. (d) 70. (a) 1

83 CHAPTER 14 SI UNITS AND DENSITY EXERCISE 58, Page Express (a) a length of 5 mm in metres (b) 0,000 mm in square metres (c) 10,000,000 mm in cubic metres (a) 5 mm = m = 0.05 m (b) 0000 mm = m 6 = 0.0 m (c) mm 9 = m = 0.01 m. A garage measures 5 m by.5 m. Determine the area in (a) m (b) mm (a) Area = 5.5 = 1.5m (b) Area = 1.5 m = mm. The height of the garage in question is m. Determine the volume in (a) m (b) mm (a) Volume = 5.5 = 7.5 m (b) Volume = 7.5 m = mm 4. A bottle contains 6. litres of liquid. Determine the volume in (a) m (b) cm (c) mm 6 (a) 6. litres = cm = m = m (b) 6. litres = cm = 600 cm (c) 6. litres = 600 cm = mm = mm 1

84 EXERCISE 59, Page Determine the density of 00 cm of lead which has a mass of 80 g. Density = mass volume 0010 m 8010 kg 6 = 11,400 kg/ m. The density of iron is 7500 kg/m. If the volume of a piece of iron is 00 cm, determine its mass. If density = mass volume then mass = density volume = 7500 kg/m m = 1.5 kg 6. Determine the volume, in litres, of 14 kg of petrol of density 700 kg/m. If density = mass volume then volume = mass 14 kg = 0.0 m= density 700kg / m cm 1 litre = 1000 cm hence, volume = litres 1000 = 0 litres 4. The density of water is 1000 kg/m. Determine the relative density of a piece of copper of density 8900 kg/m. Relative density of copper = density of copper 8900 = 8.9 density of water A piece of metal 100 mm long, 80 mm wide and 0 mm thick has a mass of 180 g. Determine the density of the metal. 9 Volume of metal = mm = m = m 6 14

85 Density of the metal = mass volume m kg 6 = 8000 kg/ m 6. Some oil has a relative density of Determine (a) the density of the oil, and (b) the volume of kg of oil. Take the density of water as 1000 kg/m. (a) Relative density of oil = density of oil densityof water Hence, density of oil = relative density density of water = kg/m = 800 kg/ m (b) If density = mass volume mass kg then volume = = density 800kg / m m or cm i.e. 500 cm EXERCISE 60, Page 15 Answers found from within the text of the chapter, pages 149 to 15. EXERCISE 61, Page (c). (d). (b) 4. (a) 5. (b) 6. (c) 7. (b) 8. (b) 9. (c) 10. (a) 15

86 CHAPTER 15 ATOMIC STRUCTURE OF MATTER EXERCISE 6, Page State whether the following are elements, compounds or mixtures: (a) town gas, (b) water, (c) oil and water, (d) aluminium (a) Town gas is a mixture (b) Water is a compound (c) Oil and water is a mixture (d) Aluminium is an element. The solubility of sodium chloride is 0.06 kg in 0.1 kg of water. Determine the amount of water required to dissolve 4 g of sodium chloride kg of soldium chloride dissolves in 0.1 kg, i.e. 6 g of soldium chloride dissolves in 0.1 kg Hence, amount of water to dissolve 4 g of sodium chloride = 4 0.1kg of water = 1. kg 6. Describe, with appropriate sketches, a model depicting the structure of the atom. See pages 156 of textbook 4. Explain, with the aid of a sketch, what is meant by a crystal, and give two examples of materials with a crystalline structure. See pages 158 of textbook 16

87 EXERCISE 6, Page 159 Answers found from within the text of the chapter, pages 155 to 159. EXERCISE 64, Page (d). (c). (a) 4. (b) 5. (a) 6. (b) 7. (b) 8. (c) 17

88 CHAPTER 16 SPEED AND VELOCITY EXERCISE 65, Page A train covers a distance of 96 km in 1 h 0 min. Determine the average speed of the train (a) in km/h and (b) in m/s (a) Average speed = dis tan ce travelled 96 km 96 km/h = 7 km/h time taken 0 1 h (b) 7 km/h = 7 km / h 1000 m / km 7 m/s 60 60s / h.6 = 0 m/s. A horse trots at an average speed of 1 km/h for 18 minutes; determine the distance covered by the horse in this time Average speed = dis tan ce travelled time taken from which, distance covered = speed time taken = 1 km/h 18/60 h =.6 km. A ship covers a distance of 165 km at an average speed of 15 km/h. How long does it take to cover this distance? Average speed = dis tan ce travelled time taken from which, time taken = dis tan ce travelled average speed = 165km 91 = 91 hours = days = days 19 hours 15km / h 4 18

89 EXERCISE 66, Page Using the information given in the distance/time graph shown below, determine the average speed when travelling from 0 to A, A to B, B to C, 0 to C and A to C Average speed from 0 to A = Average speed from A to B = Average speed from B to C = Average speed from 0 to C = Average speed from A to C = dis tan ce travelled 0 km = 0 km/h time taken 1.0 h dis tan ce travelled (50 0) km 0km time taken ( ) h 0.5h dis tan ce travelled (60 50) km 10 km time taken (.5 1.5) h 1h dis tan ce travelled 60 km = 4 km/h time taken.5h dis tan ce travelled (60 0) km 0km time taken (.5 1.0) h 1.5h = 40 km/h = 10 km/h = 0 km/h. The distances travelled by an object from point 0 and the corresponding times taken to reach A, B, C and D, respectively, from the start are as shown: Points Start A B C D Distance (m) Time (s) Draw the distance/time graph and hence determine the average speeds from 0 to A, A to B, B to C, C to D and 0 to D 19

90 The distance /time graph is shown below. Average speed from 0 to A = Average speed from A to B = Average speed from B to C = Average speed from C to D = Average speed from 0 to D = dis tan ce travelled 0 m = 4 m/s time taken 5s dis tan ce travelled (40 0) m 0 m time taken (1 5)s 7s dis tan ce travelled (60 40) m 0 m time taken (18 1)s 6s dis tan ce travelled (80 60) m 0m time taken (5 18)s 7s dis tan ce travelled 80 m =. m/s time taken 5s =.86 m/s =. m/s =.86 m/s. A train leaves station A and travels via stations B and C to station D. The times the train passes the various stations are as shown: Station A B C D Times am am 1.15 pm 1.50 pm The average speeds are: A to B, 56 km/h, B to C, 7 km/h, and C to D, 60 km/h Calculate the total distance from A to D. 0

91 Average speed = dis tan ce travelled time taken from which, distance = speed time Distance from A to B = Distance from B to C = Distance from C to D = km h = 4 km h 60 km 5 7 h = 4 km h 60 km 5 60 h = 5 km h 60 Hence, distance from A to D = = 119 km 4. A gun is fired 5 km north of an observer and the sound takes 15 s to reach him. Determine the average velocity of sound waves in air at this place. Average velocity = dis tan ce travelled 5km time taken 15 h = 100 km/h or =. m/s 5. The light from a star takes.5 years to reach an observer. If the velocity of light is m/s, determine the distance of the star from the observer in kilometres, based on a 65 day year Average speed = dis tan ce travelled time taken from which, distance = speed time Hence, distance to the star = m / s s = m or km 1

92 EXERCISE 67, Page The speed/time graph for a car journey is shown below. Determine the distance travelled by the car Distance travelled = area under the speed/time graph. In the speed/time graph shown below, distance travelled = area A + area B + area C + area D + area E = = = 1.5 km 6 6. The motion of an object is as follows: A to B, distance 1 m, time 64 s, B to C, distance 80 m at an average speed of 0 m/s, C to D, time 7 s at an average speed of 14 m/s

93 Determine the overall average speed of the object when travelling from A to D Distance Time A to B, 1 m 64 s B to C, 80 m 80 0 = 4 s C to D, 7 14 m = 98 m 7 s Totals: 00 m 75 s Hence, the overall average speed of the object when travelling from A to D = di tan ce 00 = 4 m/s time 75 EXERCISE 68, Page 166 Answers found from within the text of the chapter, pages 161 to 166. EXERCISE 69, Page (c). (g). (d) 4. (c) 5. (e) 6. (b) 7. (a) 8. (b) 9. (d) 10. (a) 11. (a) 1. (b) 1. (c) 14. (d)

94 CHAPTER 17 ACCELERATION EXERCISE 70, Page A coach increases velocity from 4 km/h to 40 km/h at an average acceleration of 0. m/s. Find the time taken for this increase in velocity. Average acceleration, a = v u t v u (40 4)km / h from which, time taken for increase in velocity, t = a 0.m/s 1000 m / km 6 km / h 6060s / h 0. m / s = 6 m/s.6 0. m / s = = 50 s. A ship changes velocity from 15 km/h to 0 km/h in 5 min. Determine the average acceleration in m/s of the ship during this time. Average acceleration, a = 0 15 m/s v u.6 t (560)s = m / s 4. A cyclist travelling at 15 km/h changes velocity uniformly to 0 km/h in 1 min, maintains this velocity for 5 min and then comes to rest uniformly during the next 15 s. Draw a velocity/time graph and hence determine the accelerations in m/s (a) during the first minute, (b) for the next 5 minutes, and (c) for the last 10 s. The velocity/time graph is shown below. 4

95 (a) Average acceleration, a = m/s m/s v u 0 15km / h.6.6 = 0.01m / s t 60s 1min 60s (b) Since the speed does not change then the average acceleration is zero (c) Average acceleration, a = km / h m/s v u.6 = 0.70m / s t 10s 10s 4. Assuming uniform accelerations between points draw the velocity/time graph for the data given below, and hence determine the accelerations from A to B, B to C and C to D: Point A B C D Speed (m/s) Time (s) The velocity/time graph is shown below. Acceleration from A to B = changein velocity 5 5 m/s time taken 5 15 s = - m/s Acceleration from B to C = changein velocity 0 5 m / s 5 m/s time taken 5 5 s 10 =.5 m/s 5

96 Acceleration from C to D = changein velocity 15 0 m / s 15 m/s time taken 45 5 s 10 = m/s 6

97 EXERCISE 71, Page An object is dropped from the third floor of a building. Find its approximate velocity 1.5 s later if all forces except that of gravity are neglected. Velocity after 1.5 s, v = u + at = 0 + (9.8)(1.5) = 1.5 m/s. During free fall, a ball is dropped from point A and is travelling at 100 m/s when it passes point B. Calculate the time for the ball to travel from A to B if all forces except that of gravity are neglected Velocity, v = u + at i.e. 100 = (t) 100 m / s from which, time to travel from A to B, t = 9.8m / s = 10. s. A piston moves at 10 m/s at the centre of its motion and decelerates uniformly at 0.8 m/s. Determine its velocity s after passing the centre of its motion. Velocity after s, v = u + at = 10 + (- 0.8)() = 7.6 m/s 4. The final velocity of a train after applying its brakes for 1. min is 4 km/h. If its uniform retardation is 0.06 m/s, find its velocity before the brakes are applied. Velocity of train, v = u + at Hence, 4.6 = u + (- 0.06)(1. 60) i.e = u 4. from which, final velocity, u = = m/s 7

98 = km/h = 9.6 km/h 5. A plane in level flight at 400 km/h starts to descend at a uniform acceleration of 0.6 m/s. It levels off when its velocity is 670 km/h. Calculate the time during which it is losing height. Velocity of plane, v = u + at Hence, = (0.6)(t) m/s.6.6 from which, time during which it is losing height, t = 0.6 m / s = 15 s = min 5 s 6. A lift accelerates from rest uniformly at 0.9 m/s for 1.5 s, travels at constant velocity for 7 s and then comes to rest in s. Determine its velocity when travelling at constant speed and its acceleration during the final s of its travel. Velocity, v = u + at from which, acceleration, a = v u t i.e. 0.9 = v from which, velocity at constant speed, v = (0.9)(1.5) = 1.5 m/s and acceleration, a = v u = m/s t 8

99 EXERCISE 7, Page 17 Answers found from within the text of the chapter, pages 169 to 17. EXERCISE 7, Page (b). (a). (c) 4. (e) 5. (i) 6. (g) 7. (d) 8. (c) 9. (a) 10. (d) 9

100 CHAPTER 18 FORCE, MASS AND ACCELERATION EXERCISE 74, Page 178 (Take g as 9.81 m/s, and express answers to three significant figure accuracy) 1. A car initially at rest, accelerates uniformly to a speed of 55 km/h in 14 s. Determine the accelerating force required if the mass of the car is 800 kg. Initial velocity, v 1 = 0 Final velocity, v = Time, t = 14 s Since v = v 1 + at then acceleration, a = Hence, accelerating force, F = ma = km m h = m/s h km 600s v v m / s t 1 800kg 1.09m / s = 87 N. The brakes are applied on the car in question 1 when travelling at 55 km/h and it comes to rest uniformly in a distance of 50 m. Calculate the braking force and the time for the car to come to rest. Initial velocity, v 1 = 55 km/h = = m/s (from above) Final velocity, v = 0 Distance travelled, s = 50 m v v as 1 from which, acceleration, a = v v s 50 1.m / s Hence, braking force, F = ma = 800 kg.4 m/s = 1867 N = 1.87 kn 0

101 v v Since v = v 1 + at then time to come to rest, t = = 6.55 s a.4. The tension in a rope lifting a crate vertically upwards is.8 kn. Determine its acceleration if the mass of the crate is 70 kg. T mg = ma i.e = 70 a from which, acceleration, a = = m/s 4. A ship is travelling at 18 km/h when it stops its engines. It drifts for a distance of 0.6 km and its speed is then 14 km/h. Determine the value of the forces opposing the motion of the ship, assuming the reduction in speed is uniform and the mass of the ship is 000 t. Initial velocity, v 1 = 18 km/h = Final velocity, v = 14.6 km m h = 5 m/s h km 600s.6 =.889 m/s Distance travelled, s = 0.6 km = 600 m v v as 1 from which, acceleration, a = v v s m / s Hence, force opposing motion, F = ma = kg8. 10 m / s = N = kn 5. A cage having a mass of t is being lowered down a mineshaft. It moves from rest with an acceleration of 4 m/s, until it is travelling at 15 m/s. It then travels at constant speed for 700 m and finally comes to rest in 6 s. Calculate the tension in the cable supporting the cage during 1

102 (a) the initial period of acceleration, (b) the period of constant speed travel, (c) the final retardation period. (a) Initial tension in cable, T 1 = mg ma = m(g a) = 000(9.81 4) = 1160 N = 11.6 kn (b) Tension in cable during constant speed, T = mg ma = mg 0 = (c) Tension in retardation period, T = mg m a where a = vu 015.5m / s t 6 Hence, tension, T = mg m a = (-.5) = 460 N = 4.6 kn = 1960 N = 19.6 kn

103 EXERCISE 75, Page Calculate the centripetal force acting on a vehicle of mass 1 tonne when travelling round a bend of radius 15 m at 40 km/h. If this force should not exceed 750 N, determine the reduction in speed of the vehicle to meet this requirement. Centripetal acceleration = v r where v = km m 1h = m/s and r = 15 m h km 600s Hence, centripetal acceleration, a = = m/s Centripetal force = ma = 1000 kg m/s = 988 N If centripetal force 750 N ma then a = 750 N v 0.75m / s 1000 kg r i.e. v and v = 9.68 m/s 9.68 m/s = m 1km 600s 9.68 = 4.86 km/h s 1000 m 1h Hence the speed reduces form 40 km/h to 4.86 km/h. A speed-boat negotiates an S-bend consisting of two circular arcs of radii 100 m and 150 m. If the speed of the boat is constant at 4 km/h, determine the change in acceleration when leaving one arc and entering the other. 4 km/h = km 1000 m 1h 4 = m/s h 1km 600s

104 Acceleration, Acceleration, a a 1 v m/s r v m / s r 150 Change of acceleration = a1 a = = 0.96 m/s i.e. change in acceleration = 0.m/s. An object is suspended by a thread 400 mm long and both object and thread move in a horizontal circle with a constant angular velocity of.0 rad/s. If the tension in the thread is 6 N, determine the mass of the object. Centripetal force (i.e. tension in thread) = mv r = 6 N The angular velocity, =.0 rad/s and radius, r = 400 mm = 0.4 m. Since linear velocity v = r, v = = 1. m/s, and since F = mv r Fr, then m = v 60.4 i.e. mass of object, m = 1. = 10 kg EXERCISE 76, Page 180 Answers found from within the text of the chapter, pages 175 to 180. EXERCISE 77, Page (c). (b). (a) 4. (d) 5. (a) 6. (b) 7. (b) 8. (a) 9. (a) 10. (d) 11. (d) 1. (c) 4

105 CHAPTER 19 FORCES ACTING AT A POINT EXERCISE 78, Page Determine the magnitude and direction of the resultant of the forces 1. kn and.7 kn, having the same line of action and acting in the same direction. The vector diagram of the two forces acting in the same direction is shown in the diagram below, which assumes that the line of action is horizontal. The resultant force F is given by: F = F 1 + F i.e. F = (1. +.7) kn = 4.0 kn in the direction of the original forces.. Determine the magnitude and direction of the resultant of the forces 470 N and 58 N having the same line of action but acting in opposite directions. The vector diagram of the two forces acting in opposite directions is shown in the diagram below. The resultant force F is given by: F = F - F 1 i.e. F = (58-470) N = 68 N in the direction of the 58 N force.. Use the triangle of forces method to determine the magnitude and direction of the resultant of the forces 1 N at 0 and 5 N at 0 With reference to the diagram shown below: (i) ab is drawn 1 units long horizontally (ii) From b, bc is drawn 5 units long, inclined at an angle of 0 to ab. 5

106 (iii) By measurement, the resultant ac is 6.8 units long inclined at an angle of 0 to ab. That is, the resultant force is 6.8 N, inclined at an angle of 0 to the 1 N force. 4. Use the triangle of forces method to determine the magnitude and direction of the resultant of the forces 5 N at 60 and 8 N at 90 With reference to the diagram shown below: (i) ab is drawn 5 units long at 60 to the horizontal (ii) From b, bc is drawn 8 units long, inclined at an angle of 90 to the horizontal. (iii) By measurement, the resultant ac is 1.6 units long inclined at an angle of 79 to the horizontal. That is, the resultant force is 1.6 N, inclined at an angle of 79 to the horizontal. 5. Use the triangle of forces method to determine the magnitude and direction of the resultant of the forces 1. kn at 45 and.8 kn at - 0 With reference to the diagram shown below: 6

107 (i) ab is drawn 1. units long at 45 to the horizontal (ii) From b, bc is drawn.8 units long, inclined at an angle of - 0 to the horizontal (iii) By measurement, the resultant ac is.4 units long inclined at an angle of - 8 to ab. That is, the resultant force is.4 kn, inclined at an angle of - 8 to the horizontal. 7

108 EXERCISE 79, Page Use the parallelogram of forces method to determine the magnitude and direction of the resultant of the forces 1.7 N at 45 and.4 N at - 60 With reference to the diagram below: (i) ab is drawn at an angle of 45 and 1.7 units in length (ii) ac is drawn at an angle of - 60 and.4 units in length (iii) bd and cd are drawn to complete the parallelogram (iv) ad is drawn. By measurement, ad is.6 units long at an angle of - 0. That is, the resultant force is.6 N at an angle of - 0. Use the parallelogram of forces method to determine the magnitude and direction of the resultant of the forces 9 N at 16 and 14 N at With reference to the diagram below: (i) ab is drawn at an angle of 16 and 9 units in length (ii) ac is drawn at an angle of and 14 units in length (iii) bd and cd are drawn to complete the parallelogram 8

109 (iv) ad is drawn. By measurement, ad is 15.7 units long at an angle of That is, the resultant force is 15.7 N at an angle of Use the parallelogram of forces method to determine the magnitude and direction of the resultant of the forces.8 N at - 50 and 14.4 N at 15 With reference to the diagram below: (i) ab is drawn at an angle of - 50 and.8 units in length (ii) ac is drawn at an angle of 15 and 14.4 units in length (iii) bd and cd are drawn to complete the parallelogram (iv) ad is drawn. By measurement, ad is 6.7 units long at an angle of

110 That is, the resultant force is 6.7 N at an angle of Use the parallelogram of forces method to determine the magnitude and direction of the resultant of the forces 0.7 kn at 147 and 1. kn at - 71 With reference to the diagram below: (i) ab is drawn at an angle of 147 and 0.7 units in length (ii) ac is drawn at an angle of - 71 and 1. units in length (iii) bd and cd are drawn to complete the parallelogram (iv) ad is drawn. By measurement, ad is 0.86 units long at an angle of That is, the resultant force is 0.86 kn at an angle of Use the parallelogram of forces method to determine the magnitude and direction of the resultant of the forces 47 N at 79 and 58 N at 47 With reference to the diagram below: (i) ab is drawn at an angle of 79 and 47 units in length (ii) ac is drawn at an angle of 47 and 58 units in length 40

111 (iii) bd and cd are drawn to complete the parallelogram (iv) ad is drawn. By measurement, ad is 15.5 units long at an angle of That is, the resultant force is 15.5 N at an angle of

112 EXERCISE 80, Page Forces of 7.6 kn at and 11.8 kn at 14 act at a point. Use the cosine and sine rules to calculate the magnitude and direction of their resultant. The space diagram is shown in diagram (a) below. A sketch is made of the vector diagram, oa representing the 7.6 kn force in magnitude and direction and ab representing the 11.8 kn force in (a) (b) magnitude and direction as shown in diagram (b). The resultant is given by length ob. By the cosine rule, ob = oa + ab - (oa)(ab) cosoab = (7.6)(11.8) cos(7 + ) = (64.769) = 1.7 Hence, ob = 1.7 = 11.5 kn By the sine rule, sin aob sin 69 from which, sin aob = 11.8sin 69 = Hence aob = sin 1 (0.9567) = 7. Thus angle in Figure 4.11(b) is 7 + = 105 Hence the resultant of the two forces is 11.5 kn acting at an angle of 105 to the horizontal. Calculate the resultant of the forces 1 N at 0 and 5 N at 0 by using the cosine and sine rules. 4

113 The space diagram is shown in diagram (a). A sketch is made of the vector diagram, oa representing the 1 N force in magnitude and direction and ab representing the 5 N force in magnitude and (a) (b) direction as shown in diagram (b). The resultant is given by length ob. By the cosine rule, ob = oa + ab - (oa)(ab) cosoab = (1)(5) cos 150 = (- 56.9) = Hence, ob = = 6.84 N By the sine rule, from which, sin sin150 sin = 5sin150 = Hence, = sin 1 (0.905) = Hence, the resultant of the two forces is 6.84 kn acting at an angle of 19.8 to the horizontal. Calculate the resultant of the forces 1. kn at 45 and.8 kn at - 0 by using the cosine and sine rules. The space diagram is shown in diagram (a). A sketch is made of the vector diagram, oa representing the 1. kn force in magnitude and direction and ab representing the.8 kn force in magnitude and direction as shown in diagram (b). The resultant is given by length ob. By the cosine rule, 0b = (1.)(.8) cosoab 4

114 = (1.)(.8) cos( ) = ( ) = Hence, ob = =.78 kn (a) (b) By the sine rule, from which,.8.78 sin aob sin105 sin aob =.8sin105 = Hence aob = sin 1 (0.8006) = Thus angle in diagram (b) is = 8.19 Hence, the resultant of the two forces is.8 kn acting at an angle of to the horizontal 4. Calculate the resultant of the forces 9 N at 16 and 14 N at by using the cosine and sine rules. The space diagram is shown in diagram (a). A sketch is made of the vector diagram, oa representing the 9 N force in magnitude and direction and ab representing the 14 N force in magnitude and direction as shown in diagram (b). The resultant is given by length ob. By the cosine rule, ob = oa + ab - (oa)(ab) cosoab = (9)(14) cos( ) = (0.711) =

115 Hence, ob = = N (a) (b) By the sine rule, from which, sin aob sin8 sin aob = 14sin8 = Hence, aob = sin 1 ( ) = 6.. Thus angle in diagram (b) is ( ) = Hence, the resultant of the two forces is N acting at an angle of to the horizontal 5. Calculate the resultant of the forces 0.7 kn at 147 and 1. kn at - 71 by using the cosine and sine rules. The space diagram is shown in diagram (a). A sketch is made of the vector diagram, oa representing the 0.7 kn force in magnitude and direction and ab representing the 1. kn force in magnitude and direction as shown in diagram (b). The resultant is given by length ob. By the cosine rule, ob = oa + ab - (oa)(ab) cos0ab = (0.7)(1.) cos 8 45

116 = (1.4418) = Hence, ob = = kn (a) (b) By the sine rule, sin aob sin 8 from which, sin aob = 1.sin 8 = Hence aob = sin 1 (0.9677) = or In this case, the latter answer is seen to be the correct one. Thus angle in diagram (b) is ( ) = Hence, the resultant of the two forces is 0.86 kn acting at an angle of to the horizontal 46

117 EXERCISE 81, Page Determine graphically the magnitude and direction of the resultant of the following coplanar forces given which are acting at a point: Force A, 1 N acting horizontally to the right, force B, 0 N acting at 140 to force A, force C, 16 N acting 90 to force A. The space diagram is shown in diagram (a). The vector diagram shown in diagram (b), is produced as follows: (i) oa represents the 1 N force in magnitude and direction (a) (b) (ii) From the nose of oa, ab is drawn inclined at 140 to oa and 0 units long (iii) From the nose of ab, bc is drawn 16 units long inclined at 90 to oa (i.e. 110 to the horizontal) (iv) oc represents the resultant; by measurement, the resultant is.1 N inclined at = 45 to the horizontal. Thus the resultant of the three forces, F A, horizontal. F B and F C is a force of.1 N at - 45 to the. Determine graphically the magnitude and direction of the resultant of the following coplanar forces given which are acting at a point: Force 1, kn acting at 80 to the horizontal, force, 0 kn acting at 7 to force 1, force, 15 kn acting at 70 to force. 47

118 The space diagram is shown in diagram (a). The vector diagram shown in diagram (b), is produced as follows: (i) oa represents the kn force in magnitude and direction (a) (b) (ii) From the nose of oa, ab is drawn inclined at 117 to oa and 0 units long (iii) From the nose of ab, bc is drawn 15 units long inclined at 187 to oa (i.e. - 7 to the horizontal) (iv) oc represents the resultant; by measurement, the resultant is 5.5 kn inclined at = 7 to force 1, i.e. 117 to the horizontal. Thus the resultant of the three forces, F 1, F and F is a force of 5.5 kn at 117 to the horizontal.. Determine graphically the magnitude and direction of the resultant of the following coplanar forces given which are acting at a point: Force P, 50 kn acting horizontally to the right, force Q, 0 kn at 70 to force P, force R, 40 kn at 170 to force P, force S, 80 kn at 00 to force P. The space diagram is shown in diagram (a). The vector diagram shown in diagram (b), is produced 48

119 as follows: (i) oa represents the 50 kn force in magnitude and direction (a) (b) (ii) From the nose of oa, ab is drawn inclined at 70 to oa and 0 units long (iii) From the nose of ab, bc is drawn 40 units long inclined at 170 to oa (iv) From the nose of bc, cd is drawn 80 units long inclined at 00 to oa (v) od represents the resultant; by measurement, the resultant is 7 kn inclined at = 7 to the horizontal. Thus the resultant of the three forces, F P, F Q, FR and FS is a force of 7 kn at - 7 to the horizontal (i.e. to force P). 4. Four horizontal wires are attached to a telephone pole and exert tensions of 0 N to the south, 0 N to the east, 50 N to the north-east and 40 N to the north-west. Determine the resultant force on the pole and its direction. The four forces are shown in the space diagram of diagram (a). The vector diagram is shown in diagram (b), oa representing the 0 N force, ab representing the 0N force, bc the 50 N force, and cd the 40 N force. The resultant, od, is found by measurement to represent a force of 4. N and angle is 9. 49

120 Thus, the four forces may be represented by a single force of 4. N at 9 east of north. (a) (b) 50

121 EXERCISE 8, Page A load of 1.5 N is lifted by two strings connected to the same point on the load, making angles of and 1 on opposite sides of the vertical. Determine the tensions in the strings. The space diagram is shown in diagram (a). Since the system is in equilibrium, the vector diagram must close. The vector diagram, shown in diagram (b), is drawn as follows: (i) The load of 00 N is drawn vertically as shown by oa (a) (b) (ii) The direction only of force F 1 is known, so from point o, ob is drawn at to the vertical (iii) The direction only of force F is known, so from point a, ab is drawn at 5 to the vertical (iv) Lines ob and ab cross at point b; hence the vector diagram is given by triangle oab. By measurement, ab is 5.9 N and ob is 8 N. 1.5 F1 By calculation, using the sine rule: sin(1801 ) sin 1 from which, 1.5sin 1 F1 sin17 and 1.5 F sin(1801 ) sin = 8.06 N 1.5sin from which, F = 5.86 N sin17 51

122 Thus the tensions in the ropes are F 1 = 8.06 N and F = 5.86 N. A two-legged sling and hoist chain used for lifting machine parts is shown below. Determine the forces in each leg of the sling if parts exerting a downward force of 15 kn are lifted. The space diagram is shown above. Since the system is in equilibrium, the vector diagram must close. The vector diagram, shown below, is drawn as follows: (i) The load of 15 kn is drawn vertically as shown by oa (ii) The direction only of force F 1 is known, so from point a, ad is drawn at 8 to the vertical (iii) The direction only of force F is known, so from point o, oc is drawn at 7 to the vertical (iv) Lines ad and oc cross at point b; hence the vector diagram is given by triangle oab. By measurement, ab is 10 kn and ob is 7.8 kn. 15 F1 By calculation, using the sine rule: sin(18087 ) sin 7 5

123 15sin 7 from which, F1 sin F and sin115 sin 8 15sin 8 from which, F sin115 = 7.77 kn Thus the tensions in the ropes are F 1 = 9.96 kn and F = 7.77 kn = 9.96 kn. Four coplanar forces acting on a body are such that it is in equilibrium. The vector diagram for the forces is such that the 60 N force acts vertically upwards, the 40 N force acts at 65 to the 60 N force, the 100 N force acts from the nose of the 40 N force and the 90 N force acts from the nose of the 100 N force. Determine the direction of the 100 N and 90 N forces relative to the 60 N force. With reference to the diagram below, 0a is drawn 60 units long vertically upwards. From point a, ab is drawn 40 units long at an angle of 65 to the 60 N force. The direction of the 100 N force is un known, thus arc pq is drawn with a compass, with centre at b, radius 100 units. Since the forces are at equilibrium, the polygon of forces must close. Using a compass with centre at 0, arc rs is drawn having a radius 90 units. The point where the arcs intersect is at d. By measurement, the 100 N force is at an angle of 148 to the 60 N force, and the 90 N force is at an angle of 77 to the 60 N force. 5

124 54

125 EXERCISE 8, Page Resolve a force of.0 N at an angle of 64 into its horizontal and vertical components. Horizontal component =.0 cos 64 = N Vertical component =.0 sin 64 = 0.67 N. Forces of 5 N at 1 and 9 N at 16 act at a point. By resolving these forces into horizontal and vertical components, determine their resultant. The horizontal component of the 5 N force = 5 cos 1 = and the vertical component of the 5 N force = 5 sin 1 = The horizontal component of the 9 N force = 9 cos 16 = and the vertical component of the 9 N force = 9 sin 16 = 7.81 Total horizontal component = ( ) = Total vertical component = = The components are shown sketched in the diagram. By Pythagoras' theorem, r = = 9.09, and by trigonometry, angle = tan = from which, = = 9.9 Hence the resultant of the two forces is a force of 9.09 N acting at 9.9 to the horizontal. 55

126 . Determine the magnitude and direction of the resultant of the following coplanar forces which are acting at a point, by resolution of forces: Force A, 1 N acting horizontally to the right, force B, 0 N acting at 140 to force A, force C, 16 N acting 90 to force A. A tabular approach using a calculator may be made as shown below: Horizontal component Force A 1 cos 0 = 1.00 Force B 0 cos 140 = Force C 16 cos 90 = 5.47 Total horizontal component =.15 Vertical component Force A 1 sin 0 = 0 Force B 0 sin 140 = 1.86 Force C 16 sin 90 = Total vertical component = -.18 The total horizontal and vertical components are shown in the diagram. Resultant r = angle = tan 1.15 = =.06, and Thus the resultant of the three forces given is.06 N acting at an angle of to force A. 4. Determine the magnitude and direction of the resultant of the following coplanar forces which are acting at a point, by resolution of forces: Force 1, kn acting at 80 to the horizontal, force 56

127 , 0 kn acting at 7 to force 1, force, 15 kn acting at 70 to force. A tabular approach using a calculator may be made as shown below: Horizontal component Force 1 cos 80 =.994 Force 0 cos 117 = (Note that force is at = 117 to the horizontal) Force 15 cos 187 = (Note that force is at = 187 to the horizontal) Total horizontal component = Vertical component Force 1 sin 80 =.651 Force 0 sin 117 = 6.70 Force 15 sin 187 = Total vertical component = The total horizontal and vertical components are shown in the diagram. Resultant r = angle = tan = 6.7 = 5.50, and from which, = = Thus the resultant of the three forces given is 5.50 kn acting at an angle of to the horizontal. 5. Determine, by resolution of forces, the resultant of the following three coplanar forces acting at a point: 10 kn acting at to the horizontal, 15 kn acting at 170 to the horizontal; 0 kn acting at 40 to the horizontal. 57

128 A tabular approach using a calculator may be made as shown below: Horizontal component Force 1 10 cos = Force 15 cos 170 = Force 0 cos 40 = Total horizontal component = Vertical component Force 1 10 sin = 5.99 Force 15 sin 170 =.605 Force 0 sin 40 = Total vertical component = The total horizontal and vertical components are shown in the diagram. Resultant r = angle = tan = 0.0 = 18.8, and from which, = = Thus the resultant of the three forces given is 18.8 kn acting at an angle of or 10.0 to the horizontal. 6. The following coplanar forces act at a point: force A, 15 N acting horizontally to the right, force B, N at 81 to the horizontal, force C, 7 N at 10 to the horizontal, force D, 9 N at 65 to the horizontal, and force E, 8 N at 4 to the horizontal. Determine the resultant of the five forces by resolution of the forces. 58

129 A tabular approach using a calculator may be made as shown below: Horizontal component Force A 15 cos 0 = Force B cos 81 =.598 Force C 7 cos 10 = Force D 9 cos 65 = Force E 8 cos 4 =.65 Total horizontal component = Vertical component Force A 15 sin 0 = Force B sin 81 =.717 Force C 7 sin 10 = Force D 9 sin 65 = Force E 8 sin 4 = Total vertical component = The total horizontal and vertical components are shown in the diagram. Resultant r = angle = tan = 10. = 4.96, and Thus the resultant of the five forces given is 4.96 N acting at an angle of 10. to force A. 59

130 EXERCISE 84, Page 195 Answers found from within the text of the chapter, pages 18 to N acting horizontally to the right N at N at 45 EXERCISE 85, Page (b). (a). (b) 4. (d) 5. (b) 6. (c) 7. (b) 8. (b) 9. (c) 10. (d) 11. (c) 1. (d) 1. (d) 14. (a) 60

131 CHAPTER FRACTIONS, DECIMALS AND PERCENTAGES EXERCISE 5, Page Change the improper fraction 15 7 into a mixed number = 1 7 as a mixed number. Change the mixed number 4 9 into an improper fraction. 4 9 = and 18 9 hence, 4 9 = = 9 as an improper fraction. A box contains 165 paper clips. 60 clips are removed from the box. Express this as a fraction in its simplest form by dividing numerator and denominator by = 4 11 by dividing numerator and denominator by 5 4. Order the following fractions from the smallest to the largest: 4 9, 5 8, 7, 1, = = = = = Ordering the fractions gives: 7, 4 9, 1, 5, Evaluate, in fraction form: 1 5 1

132 1 (51) () = Evaluate, in fraction form: (55) (4) = Evaluate, in fraction form: (51) () = Evaluate, in fraction form: (17) (41) = Evaluate, in fraction form: 7 11 (11) (7) 1 = Evaluate, in fraction form: = Evaluate, in fraction form:

133 = = = = Evaluate, in fraction form: = Evaluate, in fraction form: = = Evaluate, in fraction form: (4) 1 1 =

134 EXERCISE 6, Page 1 1. Evaluate: = Evaluate: by cancelling = 111 = Evaluate: by cancelling = = Evaluate: by cancelling = = 8 5. Evaluate: = by cancelling 16

135 6. Evaluate: = by cancelling 7. Evaluate: = or 1 1 by cancelling 8. Evaluate: = by cancelling 9. Evaluate: = by cancelling 10. Evaluate: = by cancelling 11. Evaluate: = by cancelling 17

136 1. Evaluate: = by cancelling 1. Evaluate: = Evaluate: = If a storage tank is holding 450 litres when it is three-quarters full, how much will it contain when it is two-thirds full? If 450 litres is 4 full then 1 full would be 450 = 150 litres. 4 Thus, a full tank would have = 600 litres. of the tank will contain 600 = 400 litres 16. A tank contains 4,000 litres of oil. Initially, 7 10 of the contents are removed, then 5 remainder is removed. How much oil is left in the tank? of the If 7 10 is removed then 4,000 litres remains, i.e. 700 litres. 10 If 5 of this is then removed, then 700 litres remains, i.e. 880 litres 5 18

137 EXERCISE 7, Page 1 1. Evaluate: by cancelling = 5 4 (M) 9 = = (S) Evaluate: by cancelling = 1 4 (M) 9 = 4 9 = 1 9 (S). Evaluate: = Evaluate: (D) = by cancelling

138 = (M) = = 4 (A/S) Evaluate: (D) = by cancelling = 5 (M) 5 1 = = (A/S) 6. Evaluate: (B) = 1 1 (D) = 1 by cancelling = (M) 5 15 = 9 15 = 7 15 (S) 7. Evaluate: 1 7 of

139 of (O) = (B/D) 10 5 = by cancelling = 7 5 (M) = = 4 (A/S) Evaluate: by cancelling = = by cancelling = =

140 EXERCISE 8, Page In a box of paper clips, 9 are defective. Express the non-defective paper clips as a ratio of the defective paper clips, in its simplest form. Non-defective paper clips = 9 = 4 Non-defective paper clips as a ratio of the defective paper clips = 4:9 = 6:1 by dividing by 9. A gear wheel having 84 teeth is in mesh with a 4 tooth gear. Determine the gear ratio in its simplest form. Gear ratio = 84:4 = 4:1 = 1:6 = 7: or.5:1 in its simplest form. In a box of 000 nails, 10 are defective. Express the non-defective nails as a ratio of the defective ones, in its simplest form. Non-defective nails = = 1880 Non-defective nails as a ratio of the defective nails = 1880:10 = 188:1 = 94:6 = 47: in its simplest form 4. A metal pipe.6 m long is to be cut into two in the ratio 6 to 15. Calculate the length of each piece. Number of parts = = 1 Length of 1 part =.6 m 1 = 6 cm 1 = 6 11 = 16 cm 1 7 Hence, 6 parts = 6 16 = 96 cm and 15 parts = = 40 cm

141 5. On the instructions for cooking a turkey it says that it needs to be cooked 45 minutes for every kilogram. How long will it take to cook a 7 kg turkey? If 1 kg takes 45 minutes, then 7 kg takes 7 45 = 15 minutes = 5 hours 15 minutes or hours 6. In a will, 6440 is to be divided between three beneficiaries in the ratio 4::1. Calculate the amount each receives. Number of parts = = 7 Amount for each part = = 90 Hence, 4 parts = 4 90 = 680, 4 parts = 90 = 1840 and 1 part = 1 90 = A local map has a scale of 1:,500. The distance between two motorways is.7 km. How far are they apart on the map? Distance apart on map =.7 km 700m cm = 1 cm 8. A machine produces 0 bolts in a day. Calculate the number of bolts produced by 4 machines in 7 days. The machine produces 0 bolts in 1 day If there were 4 machines, then 4 0 bolts would be produced daily, i.e. 180 bolts. In 7 days, number of bolts produced = = 8960 bolts

142 EXERCISE 9, Page Express correct to decimal places = 14.18, correct to decimal places. Express.7846 correct to 4 significant figures.7846 =.785, correct to 4 significant figures. Express correct to decimal places = 65.8, correct to decimal places 4. Express correct to 4 significant figures = 4.7, correct to 4 significant figures 5. Express 1.97 correct to decimal places 1.97 = 1.97, correct to decimal places 6. Express correct to significant figures = , correct to significant figures. 4

143 EXERCISE 10, Page Evaluate , correct to significant figures Hence, = 80.9 = 80., correct to significant figures. Evaluate , correct to 1 decimal place Hence, = 9.5 = 9., correct to 1 decimal place. Evaluate , correct to 4 significant figures Hence, = 7.57 = 7.54, correct to 4 significant figures 4. Evaluate , correct to decimal places = - ( ) Hence, = = , correct to decimal places 5. Evaluate , correct to 4 significant figures

144 Hence, = 95.1 = 95., correct to 4 significant figures 6

145 EXERCISE 11, Page Evaluate without using a calculator: = 4998, hence = Evaluate without using a calculator: = 47544, hence = Evaluate without using a calculator: = The denominator is multiplied by 10 to change it into an integer. The 1. numerator is also multiplied by 10 to keep the fraction the same Thus, The long division is similar to the long division of integers. Hence, =

146 4. Evaluate without using a calculator: , correct to 5 significant figures = The denominator is multiplied by 10 to change it into an integer. The numerator is also multiplied by 10 to keep the fraction the same. Thus, The long division is similar to the long division of integers Hence, = = 44.8, correct to 5 significant figures 5. Evaluate without using a calculator: , correct to 4 significant figures = The denominator is multiplied by 10 to change it into an integer. The numerator is also multiplied by 10 to keep the fraction the same. Thus, The long division is similar to the long division of integers. 8

147 Hence, = = 66.1, correct to 4 significant figures 6. Express 4 9 as a decimal fraction correct to significant figures Hence, 4 = = 0.444, correct to 4 significant figures 9 7. Express 17 7 as a decimal fraction, correct to 5 decimal place

148 Thus, Express = 0.696, correct to 5 decimal places. as a decimal fraction correct to 4 significant figures Thus, 1 = 1.56, correct to 4 significant figures Express as a decimal fraction, correct to decimal places Thus, = 1.87 = 1.84, correct to decimal places 10. Evaluate , (a) correct to 4 significant figures and (b) correct to decimal places (a) = 4.81, correct to 4 significant figures (b) = 4.81, correct to decimal places

149 11. Evaluate , (a) correct to 5 decimal places and (b) correct to significant figures (a) = , correct to 5 decimal places (b) = , correct to significant figures Evaluate (a) (b) 5. 1 (a) Hence, 1.6 = = (b) 5. 1 =

150 Hence, 5. 1 = = 6.6

151 EXERCISE 1, Page Express 0.00 as a percentage 0.00 = % = 0.%. Express 1.74 as a percentage 1.74 = % = 17.4%. Express as a percentage = % = 5.7% 4. Express 0% as a decimal number 0% = = 0.0 as a decimal number 5. Express 1.5% as a decimal number 1.5% = = as a decimal number 6. Express as a percentage % %

152 Hence, = 68.75% 7. Express as percentages, correct to significant figures: (a) 7 (b) 19 4 (c) (a) (b) (c) % = 1.11 % = 1.%, correct to significant figures % % = 79.%, correct to significant figures % = % = 169%, correct to significant figures Place the following in order of size, the smallest first, expressing each as percentages, correct to 1 decimal place: (a) 1 1 (b) 9 17 (c) 5 9 (d) 6 11 (a) 1 1 = = 57.1% (b) = 5.9% (c) 5 9 (d) 6 = = 54.5% 11 Hence, the order is: (b), (d), (c) and (a) = = 55.6% 9. Express 1.5% as a fraction in its simplest form 4

153 1.5% = = Express 56.5% as a fraction in its simplest form. 56.5% = = Calculate 4.6% of 50 kg 4.6% of 50 kg = kg = 1.8 kg Determine 6% of 7 m 6% of 7 m = m = 9.7 m 1. Calculate correct to 4 significant figures: (a) 18% of 758 tonnes (b) 47% of 18.4 grams (c) 147% of 14.1 seconds (a) 18% of 758 = = t = t, correct to 4 significant figures. 100 (b) 47% of 18.4 = = g = g, correct to 4 significant figures. 100 (c) 147% of 14.1 = = 0.77 s, = 0.7 s, correct to 4 significant figures Express: (a) 140 kg as a percentage of 1 t (b) 47 s as a percentage of 5 min (c) 1.4 cm as a percentage of.5 m (a) 140 kg as a percentage of 1 t % = 14%

154 (b) 47 s as a percentage of 5 min = 100% 100% = % (c) 1.4 cm as a percentage of.5 m = 100% = 5.6% Express 5 mm as a percentage of 867 mm, correct to decimal places. 5 mm as a percentage of 867 mm = 5 100% 867 = 7.49% 16. Express 408 g as a percentage of.40 kg. 408 g as a percentage of.40 kg = 408 g as a percentage of 400 g = % 400 = 17% 17. When signing a new contract, a Premiership footballer s pay increases from 15,500 to 1,500 per week. Calculate the percentage pay increase, correct significant figures. Percentage pay increase = % 100% = 8.7% A metal rod 1.80 m long is heated and its length expands by 48.6 mm. Calculate the percentage increase in length mm Percentage increase in length = mm 100% =.7% 19. A machine part has a length of 6 mm. The length is incorrectly measured as 6.9 mm. Determine the percentage error in the measurement. Percentage error in the measurement = % 100% =.5% too high 6 6 6

155 0. A resistor has a value of 80 Ω ± 5%. Determine the range of resistance values expected. 5 5% of 80 = = 41 The lowest value expected is 80 5% of 80 i.e = 779 Ω The highest value expected is % of 80 i.e = 861 Ω Hence, range of values expected is: 779 Ω to 861 Ω 1. For each of the following resistors, determine the (i) minimum value, (ii) maximum value: (a) 680 Ω ± 0% (b) 47 kω ± 5% (a) 0% of 680 Ω = = 16 Ω Hence, (i) minimum value = = 544 Ω (ii) maximum value = = 816 Ω (b) 5% of 47 kω = 5 47 =.5 kω 100 Hence, (i) minimum value = 47.5 = kω (ii) maximum value = = 49.5 kω. An engine speed is 400 rev/min. The speed is increased by 8%. Calculate the new speed. 8% of 400 rev/min = = 19 rev/min 100 New speed = = 59 rev/min 7

156 CHAPTER 0 WORK, ENERGY AND POWER EXERCISE 86, Page 0 1. Determine the work done when a force of 50 N pushes an object 1.5 km in the same direction as the force. Work done = force distance moved in the direction of the force = 50 N 1500 m = J (since 1 J = 1 Nm) i.e. work done = 75 kj. Calculate the work done when a mass of weight 00 N is lifted vertically by a crane to a height of 100 m. When work is done in lifting then: work done = (weight of the body) (vertical distance moved) Weight is the downward force due to the mass of an object. Hence work done = 00 N 100 m = 0000 J = 0 kj. A motor supplies a constant force of kn to move a load 10 m. The force is then changed to a constant 1.5 kn and the load is moved a further 0 m. Draw the force/distance graph for the complete operation, and, from the graph, determine the total work done by the motor. The force/distance graph is shown below. 61

157 Total work done = area under the force/distance graph = (000 10) + (1500 0) = = J = 50 kj 4. A spring, initially relaxed, is extended 80 mm. Draw a work diagram and hence determine the work done if the spring requires a force of 0.5 N/mm of stretch. Force = 0.5 N/mm 80 mm = 40 N The work diagram is shown below. Total work done = area under the diagram = 1 base height = 1 80 mm 40 N = 1 (80 10 m) 40 N = 1.6 J 5. A spring requires a force of 50 N to cause an extension of 100 mm. Determine the work done in extending the spring (a) from 0 to 100 mm, and (b) from 40 mm to 100 mm. The work diagram is shown below. 6

158 (a) The work done in extending the spring from 0 to 100 mm = 1 base height = 1 ( m) 50 N =.5 J (b) The work done in extending the spring from 40 mm to 100 mm = area ABCE = area ABCD + area ADE = (60 10 m)(0 N) + 1 (60 10 m)(0 N) = =.1 J 6. The resistance to a cutting tool varies during the cutting stroke of 800 mm as follows: (i) the resistance increases uniformly from an initial 5000 N to 10,000 N as the tool moves 500 mm, and (ii) the resistance falls uniformly from 10,000 N to 6000 N as the tool moves 00 mm. Draw the work diagram and calculate the work done in one cutting stroke. The work diagram is shown below. Work done in one cutting stroke = area under the diagram = A + B + C + D = ( ) + 1 ( ) + 1 ( ) + ( ) =.5 kj kj kj kj = 6.15 kj 6

159 EXERCISE 87, Page A machine lifts a mass of weight N through a height of 1 m when 7.85 kj of energy is supplied to it. Determine the efficiency of the machine. Work done in lifting mass = force distance moved = weight of body distance moved = N 1 m = 5886 J = useful energy output Energy input = 7.85 kj = 7850 J Efficiency, = usefuloutput energy input energy = = 0.75 or 75%. Determine the output energy of an electric motor which is 60% efficient if it uses kj of electrical energy. Efficiency, = usefuloutput energy input energy thus = output energy 000J from which, output energy = = 100 J = 1. kj. A machine that is used for lifting a particular mass is supplied with 5 kj of energy. If the machine has an efficiency of 65% and exerts a force of 81.5 N to what height will it lift the mass? Efficiency, = usefuloutput energy input energy i.e = output energy 5000J from which, output energy = Work done = force distance moved 5000 = 50 J hence 50 J = 81.5 N height 64

160 from which, height = 50J 81.5 N = 4 m 4. A load is hoisted 4 m and requires a force of 100 N. The efficiency of the hoist gear is 60% and that of the motor is 70%. Determine the input energy to the hoist. Output energy = work done = force distance = 100 N 4 m = 400 J For the gearing, efficiency = output energy input energy i.e = 400 input energy from which, the input energy to the gears = = 7000 J The input energy to the gears is the same as the output energy of the motor. Thus, for the motor, efficiency = output energy input energy i.e = 7000 input energy Hence, input energy to the hoist = = J = 10 kj 65

161 EXERCISE 88, Page The output power of a motor is 10 kw. How much work does it do in 1 minute? Power = work done time taken from which, work done = power time = W 60 s = J = 600 kj. Determine the power required to lift a load through a height of 0 m in 1.5 s if the force required is.5 kn. Work done = force distance moved = 500 N 0 m = J Power = work done time taken = J 1.5s = 4000 W or 4 kw. 5 kj of work is done by a force in moving an object uniformly through 50 m in 40 s. Calculate (a) the value of the force, and (b) the power. (a) Work done = force distance hence 5000 J = force 50 m from which, force = 5000J 50m = 500 N (b) Power = work done time taken = 5000 J 40s = 65 W 4. A car towing another at 54 km/h exerts a steady pull of 800 N. Determine (a) the work done in 1 hr, and (b) the power required. 4 (a) Work done = force distance moved. The distance moved in 15 min, i.e. 4 1 h, at 54 km/h = 54 4 = 1.5 km. 66

162 Hence, work done = 800 N 1500 m = kj or 10.8 MJ (b) Power required = 6 work done time taken = J 1560s = 1000 W or 1 kw 5. To what height will a mass of weight 500 N be raised in 0 s by a motor using 4 kw of power? Work done = force distance. Hence, work done = 500 N height. Power = work done, from which, work done = power time taken time taken = 4000 W 0 s = J Hence, = 500 N height, from which, height = J 500 N = 160 m 6. The output power of a motor is 10 kw. Determine (a) the work done by the motor in hours, and (b) the energy used by the motor if it is 7% efficient. (a) Work done = power time taken = 10 kw h = 0 kwh = Ws = 7 MJ (b) Efficiency = output energy input energy i.e = 7 input energy Hence, energy used by the motor = = 100 MJ 7. A car is travelling at a constant speed of 81 km/h. The frictional resistance to motion is 0.60 kn. Determine the power required to keep the car moving at this speed. Power = force velocity = 0.60 kn 81 km/h = 600 N m 6060 s = 1500 N m/s = 1500 J/s = 1.5 kw 67

163 8. A constant force of.0 kn is required to move the table of a shaping machine when a cut is being made. Determine the power required if the stroke of 1. m is completed in 5.0 s. Work done in each cutting stroke = force distance = 000 N 1. m = 400 J Power required = work done 400 J = 480 W time taken 5s 9. The variation of force with distance for a vehicle that is decelerating is as follows: Distance (m) Force (kn) If the vehicle covers the 600 m in 1. minutes, find the power needed to bring the vehicle to rest. The force/distance graph is shown below. Work done = area under the force/distance graph = 1 base height = m 4 kn = 700 kj Power needed to bring the vehicle to rest = work done 700kJ = 100 kw time taken 1. 60s 68

164 10. A cylindrical bar of steel is turned in a lathe. The tangential cutting force on the tool is 0.5 kn and the cutting speed is 180 mm/s. Determine the power absorbed in cutting the steel. Power absorbed in cutting the steel = force velocity = 0.5 kn 180 mm/s = 500 N m/s = 90 J/s = 90 W 69

165 EXERCISE 89, Page An object of mass 400 g is thrown vertically upwards and its maximum increase in potential energy is.6 J. Determine the maximum height reached, neglecting air resistance. Potential energy = mgh i.e..6 = (0.4 kg)(9.81 m/s )(h) from which, maximum height, h =.6 (0.4)(9.81) = 8.1 m. A ball bearing of mass 100 g rolls down from the top of a chute of length 400 m inclined at an angle of 0 to the horizontal. Determine the decrease in potential energy of the ball bearing as it reaches the bottom of the chute. With reference the above diagram, sin 0 = opposite h hypotenuse 400 from which, h = 400 sin 0 = 00 m Hence, increase in potential energy = mgh = 0.1 kg 9.81 m/s 00 m = 196. J. A vehicle of mass 800 kg is travelling at 54 km/h when its brakes are applied. Find the kinetic energy lost when the car comes to rest. 1 1 Kinetic energy = mv 54 = (800 kg) m/s.6 70

166 i.e. kinetic energy lost = J or 90 kj 4. Supplies of mass 00 kg are dropped from a helicopter flying at an altitude of 60 m. Determine the potential energy of the supplies relative to the ground at the instant of release, and its kinetic energy as it strikes the ground. Potential energy of supplies at release = mgh = (00 kg)(9.81 m/s )(60 m) = J = kj By the principle of conservation of energy, kinetic energy as the supplies strikes the ground = potential energy at release = kj 5. A shell of mass 10 kg is fired vertically upwards with an initial velocity of 00 m/s. Determine its initial kinetic energy and the maximum height reached, correct to the nearest metre, neglecting air resistance. 1 1 Initial kinetic energy = mv = (10 kg)(00 m/s) = 00 kj At the maximum height, the velocity of the canister is zero and all the kinetic energy has been converted into potential energy. Hence, potential energy = initial kinetic energy = J Then, = mgh = (10)(9.81)(h) from which, height h = (10)(9.81) = 09 m i.e. the maximum height reached is 09 m or.09 km 6. The potential energy of a mass is increased by 0.0 kj when it is lifted vertically through a height of 5.0 m. It is now released and allowed to fall freely. Neglecting air resistance, find its 71

167 kinetic energy and its velocity after it has fallen 10.0 m. Potential energy of mass = 0.0 kj = mgh from which, mass, m = = kg gh (9.81)(5.0) Potential energy after falling 10.0 m = mgh = (81.55)(9.81)(10.0) = 8000 J Kinetic energy = potential energy = 8000 J = 8 kj Kinetic energy = 1 mv from which, 1 i.e = (81.55)v 8000 v and velocity after falling 10.0 m, v = = 14.0 m/s 7. A pile-driver of mass 400 kg falls freely through a height of 1. m on to a pile of mass 150 kg. Determine the velocity with which the driver hits the pile. If, at impact,.5 kj of energy are lost due to heat and sound, the remaining energy being possessed by the pile and driver as they are driven together into the ground a distance of 150 mm, determine (a) the common velocity after impact, (b) the average resistance of the ground. The potential energy of the pile-driver is converted into kinetic energy. Thus, potential energy = kinetic energy, i.e. mgh = 1 mv from which, velocity v = gh = ()(9.81)(1.) = 4.85 m/s. Hence, the pile-driver hits the pile at a velocity of 4.85 m/s (a) Before impact, kinetic energy of pile driver = 1 mv = 1 (400)(4.85) = J = kj Kinetic energy after impact = =.05 kj 7

168 Thus the pile-driver and pile together have a mass of = 550 kg and possess kinetic energy of.05 kj Hence, 05 = 1 mv = 1 (550)v 05 from which, velocity v = 550 Thus, the common velocity after impact is.8 m/s =.8 m/s (b) The kinetic energy after impact is absorbed in overcoming the resistance of the ground, in a distance of 150 mm. Kinetic energy = work done = resistance distance i.e. 05 = resistance from which, resistance = = N Hence, the average resistance of the ground is kn EXERCISE 90, Page 11 Answers found from within the text of the chapter, pages 198 to 11. EXERCISE 91, Page 1 1. (b). (c). (c) 4. (a) 5. (d) 6. (c) 7. (a) 8. (d) 9. (c) 10. (b) 11. (b) 1. (a) 1. (d) 14. (a) 15. (d) 7

169 CHAPTER 1 SIMPLY SUPPORTED BEAMS EXERCISE 9, Page Determine the moment of a force of 5 N applied to a spanner at an effective length of 180 mm from the centre of a nut. Moment, M = force distance = 5 N 0.18 m = 4.5 N m. A moment of 7.5 N m is required to turn a wheel. If a force of 7.5 N applied to the rim of the wheel can just turn the wheel, calculate the effective distance from the rim to the hub of the wheel. Moment, M = force distance from which, distance from rim to hub = moment, M 7.5 N m = 0. m = 00 mm force, F 7.5 N. Calculate the force required to produce a moment of 7 N m on a shaft, when the effective distance from the centre of the shaft to the point of application of the force is 180 mm. Moment, M = force distance moment, M 7 N m from which, force = dis tan ce,d m = 150 N 74

170 EXERCISE 9, Page Determine distance d and the force acting at the support A for the force system shown below, when the system is in equilibrium. Clockwise moment = anticlockwise moment Hence,.8 d = i.e. distance, d = = 50 mm Force at support A, R A = =.8 kn. If the 1 kn force shown below is replaced by a force F at a distance of 50 mm to the left of R A, find the value of F for the system to be in equilibrium. Clockwise moment = anticlockwise moment Hence, if d = 50 mm from above, then.8 50 = F 50 and force, F = = 0.56 kn = 560 N 75

171 . Determine the values of the forces acting at A and B for the force system shown below. At equilibrium, RA RB= = 50 N (1) Taking moments about point A gives: clockwise moment = anticlockwise moment Hence, = R B 76 i.e = 76 R B from which, force acting at B, R B = = 5 N From equation (1), from which, R A + 5 = 50 R A = 50 5 = 5 N 4. The forces acting on a beam are as shown below. Neglecting the mass of the beam, find the value of R A and distance d when the beam is in equilibrium. At equilibrium, from which, R A + 60 = R A = = 5 N Taking moments about the 60 N force gives: clockwise moment = anticlockwise moment 76

172 Hence, 5 d + R A 5 = 40 (5 15) i.e. 5d = 40 0 i.e. 5d = 800 i.e. 5d = from which, distance, d = = 5 mm

173 EXERCISE 94, Page 0 1. Calculate the force R A and distance d for the beam shown below. The mass of the beam should be neglected and equilibrium conditions assumed. At equilibrium, = R A + 1. from which, R A = =.0 kn Taking moments about the.7 kn force gives: clockwise moment = anticlockwise moment Hence, 0.4 (d + 15) + R A 10 = 1. d + 0. (1 + 10) i.e. 0.4d = 1.d i.e. 0.4d = 1.d i.e = 1.d 0.4d and 1.6 = 0.9d from which, distance, d = = 4 mm. For the force system shown below, find the values of F and d for the system to be in equilibrium. At equilibrium, F = i.e. force, F = = 1.0 kn 78

174 Taking moments about the 0.7 kn force gives: clockwise moment = anticlockwise moment Hence, F d +. 1 = 0.8 (d + 5) (14 + 1) i.e. 1.0 d = 0.8d i.e. d 0.8d = and 0.d = 1.8 from which, distance, d = = 64 mm. For the force system shown below, determine distance d for the forces R A and R B to be equal, assuming equilibrium conditions. For equilibrium, R A + R B = = 50 N Hence, if R A = R B then R A = R B = 50 = 5 N Taking moments about the R A gives: clockwise moment = anticlockwise moment Hence, i.e. i.e ( ) = R B (0 + 0) + 10 d = d 1800 = d and 10d = = 800 from which, distance, d = = 80 m 79

175 4. A simply supported beam AB is loaded as shown below. Determine the load F in order that the reaction at A is zero. If R 1= 0, then taking moments about R gives: clockwise moment = anticlockwise moment i.e. F = ( + ) i.e. F = + 40 = 7 from which, load, F = 7 = 6 kn 5. A uniform wooden beam, 4.8 m long, is supported at its left-hand end and also at. m from the left-hand end. The mass of the beam is equivalent to 00 N acting vertically downwards at its centre. Determine the reactions at the supports. The beam is shown above. Taking moments about the left-hand support gives: clockwise moment = anticlockwise moment i.e = R B. 80

176 from which, R B = = 150 N For equilibrium, R A + R B = 00 Hence, R A = 00 - R B = = 50 N 6. For the simply supported beam PQ shown below, determine (a) the reaction at each support, (b) the maximum force which can be applied at Q without losing equilibrium. (a) Taking moments about the left-hand support gives: clockwise moment = anticlockwise moment i.e ( ) + 5 ( ) = R ( ) i.e = 7 R from which, R = = 1 kn 7 7 For equilibrium, R1 R= = 15 kn Hence, R 1+ 1 = 15 from which, R 1 = 15 1 = kn (b) Let the force at Q be R Q Taking moments about R gives: clockwise moment = anticlockwise moment i.e. R Q.0 = i.e. R Q = 9 + = 1 81

177 from which, R Q = 1 = 15.5 kn EXERCISE 95, Page 1 Answers found from within the text of the chapter, pages 14 to 0. EXERCISE 96, Page 1. (a). (c). (a) 4. (d) 5. (a) 6. (d) 7. (c) 8. (a) 9. (d) 10. (c) 8

178 CHAPTER LINEAR AND ANGULAR MOTION EXERCISE 97, Page 6 1. A pulley driving a belt has a diameter of 60 mm and is turning at 700/ revolutions per minute. Find the angular velocity of the pulley and the linear velocity of the belt assuming that no slip occurs. Angular velocity = n, where n is the speed of revolution in revolutions per second, i.e. n = 700 revolutions per second. 60 Thus, angular velocity, = 700 = 90 rad/s 60 The linear velocity of a point on the rim, v = r, where r is the radius of the wheel, i.e. r = 60 = 180 mm = 0.18 m Thus, linear velocity, v = r = = 16. m/s. A bicycle is travelling at 6 km/h and the diameter of the wheels of the bicycle is 500 mm. Determine the angular velocity of the wheels of the bicycle and the linear velocity of a point on the rim of one of the wheels Linear velocity, v = 6 km/h = m/s = 10 m/s 600 (Note that changing from km/h to m/s involves dividing by.6) Radius of wheel, r = 500 = 50 mm = 0.5 m Since, v = r, then angular velocity, = v 10 = 40 rad/s r 0.5 8

179 EXERCISE 98, Page 7 1. A flywheel rotating with an angular velocity of 00 rad/s is uniformly accelerated at a rate of 5 rad/s for 15 s. Find the final angular velocity of the flywheel both in rad/s and revolutions per minute. Angular velocity, 1= 00 rad/s, angular acceleration, = 5 rad/s and time, t = 15 s. Final angular velocity, = 1+ t = 00 + (5)(15) = = 75 rad/s 60 In revolutions per minute, 75 rad/s = 75 = 850 rev/min or 66 rev/min. A disc accelerates uniformly from 00 revolutions per minute to 600 revolutions per minute in 5 s. Determine its angular acceleration and the linear acceleration of a point on the rim of the disc, if the radius of the disc is 50 mm. Initial angular velocity, 1= and final angular velocity, = = 1+ t from which, 00 10rad/s rad/s angular acceleration, = = 0.4 rad/s or 1.57 rad/s t 5 5 Linear acceleration, a = r = (0.5)(0.4) = 0.1 m/s or 0.14 m/s 84

180 EXERCISE 99, Page 9 1. A grinding wheel makes 00 revolutions when slowing down uniformly from 1000 rad/s to 400 rad/s. Find the time for this reduction in speed. 1 Angle turned through, t hence 00 = t i.e. 600 = 700t from which, time, t = 600 =.69 s 700. Find the angular retardation for the grinding wheel in question 1. = 1+ t from which, angular acceleration, = = -.8 rad/s t i.e. angular retardation is.8 rad/s. A disc accelerates uniformly from 00 revolutions per minute to 600 revolutions per minute in 5 s. Calculate the number of revolutions the disc makes during this accelerating period. Angle turned through, 1 = t = (5) rad However, there are radians in 1 revolution, hence, number of revolutions = = 5 = revolutions 85

181 4. A pulley is accelerated uniformly from rest at a rate of 8 rad/s. After 0 s the acceleration stops and the pulley runs at constant speed for min, and then the pulley comes uniformly to rest after a further 40 s. Calculate: (a) the angular velocity after the period of acceleration, (b) the deceleration, (c) the total number of revolutions made by the pulley. (a) Angular velocity after acceleration period, = 1+ t = 0 + (8)(0) = 160 rad/s (b) = + t from which, angular acceleration, = = - 4 rad/s t 40 i.e. angular deceleration is 4 rad/s 1 (c) Initial angle turned through, 1 = t = 0160 (0) = 1600 rad = 1600 rev At constant speed, angle turned through, = 160 rad/s ( 60)s = 1900 rad = 1900 rev Angle turned through during deceleration, = (40) = 00 rad = 00 rev Hence, total number of revolutions made by the pulley = = = 4000 = 1000 rev or 80 rev 86

182 EXERCISE 100, Page 1 1. A car is moving along a straight horizontal road at 79. km/h and rain is falling vertically downwards at 6.4 km/h. Find the velocity of the rain relative to the driver of the car. The space diagram is shown in diagram (a). The velocity diagram is shown in diagram (b) and the velocity of the rain relative to the driver is given by vector rc where rc = re + ec rc = = 8.5 km/h and tan (a) (b) i.e. the velocity of the rain relative to the driver is 8.5 km/h at 71.6 to the vertical.. Calculate the time needed to swim across a river 14 m wide when the swimmer can swim at km/h in still water and the river is flowing at 1 km/h. At what angle to the bank should the swimmer swim? The swimmer swims at km/h relative to the water, and as he swims the movement of the water carries him downstream. He must therefore aim against the flow of the water at an angle shown in the triangle of velocities shown below where v is the swimmers true speed. v = 1 km/h = 1000 m/min = 8.87 m/min 60 87

183 Hence, if the width of the river is 14 m, the swimmer will take 14 = minutes 8.87 = 4 min 55 s In the above diagram, sin = 1 from which, = 0 Hence, the swimmer needs to swim at an angle of 60 to the bank (shown as angle in the diagram.. A ship is heading in a direction N 60 E at a speed which in still water would be 0 km/h. It is carried off course by a current of 8 km/h in a direction of E 50 S. Calculate the ship s actual speed and direction. In the triangle of velocities shown below (triangle 0AB), 0A represents the velocity of the ship in still water, AB represents the velocity of the water relative to the earth, and 0B is the velocity of the ship relative to the earth. Total horizontal component of v = 0 cos cos 10 =.46 Total vertical component of v = 0 sin sin 10 =.87 Hence, v = =.79 km/h, and 1.87 tan Hence, the ships actual speed is.79 km/h in a direction E 9.78 N 88

184 EXERCISE 101, Page 1 Answers found from within the text of the chapter, pages 4 to 1. EXERCISE 10, Page 1 1. (b). (c). (a) 4. (c) 5. (a) 6. (d) 7. (c) 8. (b) 9. (d) 10. (c) 11. (b) 1. (d) 1. (a) 89

185 CHAPTER FRICTION EXERCISE 10, Page 5 1. The coefficient of friction of a brake pad and a steel disc is 0.8. Determine the normal force between the pad and the disc if the frictional force required is 105 N. Frictional force, F = normal force i.e. Hence, F = N 105 = 0.8 N from which, normal force, N = = 150 N. A force of 0.1 kn is needed to push a bale of cloth along a chute at a constant speed. If the normal force between the bale and the chute is 500 N, determine the dynamic coefficient of friction. As the bale of cloth is moving at constant speed, the force applied must be that required to overcome frictional forces, i.e. frictional force, F = 10 N; the normal force is 500 N, and since F = N, = F N = = 0.4 i.e. the dynamic coefficient of friction is 0.4. The normal force between a belt and its driver wheel is 750 N. If the static coefficient of friction is 0.9 and the dynamic coefficient of friction is 0.87, calculate (a) the maximum force which can be transmitted, and (b) maximum force which can be transmitted when the belt is running at a constant speed. 90

186 (a) Maximum force that can be transmitted = N = (0.9)(750) = 675 N (b) Maximum force which can be transmitted when the belt is running at a constant speed = N = (0.87)(750) = 65.5 N EXERCISE 104, Page 6 Answers found from within the text of the chapter, pages to 6. EXERCISE 105, Page 6 1. (c). (c). (f) 4. (e) 5. (i) 6. (c) 7. (h) 8. (b) 9. (d) 10. (a) 91

187 CHAPTER 4 SIMPLE MACHINES EXERCISE 106, Page A simple machine raises a load of 85 N through a distance of 0. m. The effort is 50 N and moves through a distance of. m. Determine: (a) the force ratio, (b) the movement ratio, (c) the efficiency of the machine at this load. Force ratio = load 85 N =. effort 50 N Movement ratio = dis tan ce moved by theeffort.m = 11 dis tan ce moved by theload 0.m Efficiency = force ratio 100% movement ratio =. 100 = 0% 11. The efficiency of a simple machine is 50%. If a load of 1. kn is raised by an effort of 00 N, determine the movement ratio. Force ratio = load 100 N = 4 effort 00 N Efficiency = force ratio movement ratio force ratio 4 4 from which, movement ratio = = 8 efficiency An effort of 10 N applied to a simple machine moves a load of 40 N through a distance of 100 mm, the efficiency at this load being 80%. Calculate: (a) the movement ratio, (b) the distance moved by the effort. Force ratio = load 40 N = 4 effort 10 N 9

188 (a) Efficiency = force ratio movement ratio (b) Movement ratio = distancemovedbytheeffort dis tan ce moved by theload force ratio 4 4 from which, movement ratio = = 5 efficiency from which, the distance moved by the effort = movement ratio distance moved by the load = = 500 mm 4. The effort required to raise a load using a simple machine, for various values of load is as shown: Load F l (N) Effort F e (N) If the movement ratio for the machine is 0, determine (a) the law of the machine, (b) the limiting force ratio, (c) the limiting efficiency. The load/effort graph is shown below. 9

189 (a) The law of the machine is F e = a F l + b where gradient of curve, a = AB = 0.04 BC and intercept, b = 170. Hence, the law of the machine is: F e = 0.04 F l (b) Limiting force ratio = 1 1 = 5 a 0.04 (c) Limiting efficiency = % a movement ratio = 8.% 5. For the data given in question 4, determine the values of force ratio and efficiency for each value of the load. Hence plot graphs of effort, force ratio and efficiency to a base of load. From the graphs, determine the effort required to raise a load of 6 kn and the efficiency at this load. Load F l (N) Effort F e (N) Efficiency = Force ratio = load effort force ratio movement ratio % 40.4% 5.1% 54.4% 59.% Graphs of load/effort, load/force ratio and load/efficiency are shown below. From the graph, when the load is 6 kn, i.e N effort = 410 N and efficiency = 48% 94

190 95

191 EXERCISE 107, Page 4 1. A pulley system consists of four pulleys in an upper block and three pulleys in a lower block. Make a sketch of this arrangement showing how a movement ratio of 7 may be obtained. If the force ratio is 4., what is the efficiency of the pulley. 96

192 Efficiency = force ratio % = 60% movement ratio 7. A three-pulley lifting system is used to raise a load of 4.5 kn. Determine the effort required to raise this load when losses are neglected. If the actual effort required is 1.6 kn, determine the efficiency of the pulley system at this load. Load = 4.5 kn and movement ratio = n = When losses are neglected, efficiency = 100% = force ratio movement ratio from which, force ratio = load effort = movement ratio i.e. and 4.5kN effort effort = 4.5kN = = 1.5 kn force ratio If the actual effort required is 1.6 kn, efficiency = 100% movement ratio = load 4.5 effort 100% % movement ratio = 9.75% 97

193 EXERCISE 108, Page 4 1. Sketch a simple screw-jack. The single-start screw of such a jack has a lead of 6 mm and the effective length of the operating bar from the centre of the screw is 00 mm. Calculate the load which can be raised by an effort of 150 N if the efficiency at this load is 0%. A simple screw-jack is shown below, where lead, L = 6 mm and radius, r = 00 mm Movement ratio = r (00) 100 L 6 Efficiency = force ratio movement ratio i.e. 0 force ratio from which, force ratio = 0(100 ) Force ratio = load effort from which, load = force ratio effort = N = 000 = 945 N = 9.45 kn. A load of 1.7 kn is lifted by a screw-jack having a single-start screw of lead 5 mm. The effort is applied at the end of an arm of effective length 0 mm from the centre of the screw. Calculate the effort required if the efficiency at this load is 5%. 98

194 Movement ratio = r (0) 18 L 5 Efficiency = force ratio movement ratio i.e. 5 force ratio from which, force ratio = 5(18 ) 100 Force ratio = load effort from which, effort = load 1.7 kn 1700 force ratio = N 99

195 EXERCISE 109, Page The driver gear of a gear system has 8 teeth and meshes with a follower gear having 168 teeth. Determine the movement ratio and the speed of the follower when the driver gear rotates at 60 revolutions per second. Movement ratio = teeth on follower 168 = 6 teethondriver 8 Also, movement ratio = speed of driver speed of follower i.e. 6 = 60 rev / s speed of follower from which, the speed of the follower = 60 6 = 10 rev/s. A compound gear train has a 0-tooth driver gear A, meshing with a 90-tooth follower gear B. Mounted on the same shaft as B and attached to it is a gear C with 60 teeth, meshing with a gear D on the output shaft having 10 teeth. Calculate the movement and force ratios if the overall efficiency of the gears is 7%. The speed of D = T T A C speed of A T B T D speed of A TB TD Movement ratio = = speed of D T T A C = force ratio The efficiency of any simple machine = 100% movement ratio from which, force ratio = efficiency movement ratio = = 4. 00

196 . A compound gear train is as shown on page. The movement ratio is 6 and the numbers of teeth on gears A, C and D are 5, 100 and 60, respectively. Determine the number of teeth on gear B and the force ratio when the efficiency is 60%. speed of A TB TD Movement ratio = speed of D T T A C i.e. B 6 = T from which, number of teeth on B, T B = = 50 Efficiency = force ratio movement ratio i.e. 60 force ratio from which, force ratio = =.6 01

197 EXERCISE 110, Page In a second-order lever system, the force ratio is.5. If the load is at a distance of 0.5 m from the fulcrum, find the distance that the effort acts from the fulcrum if losses are negligible. Force ratio = dis tan ceof effort from fulcrum dis tan ceof load from fulcrum i.e..5 = x 0.5 Hence, the distance that the effort acts from the fulcrum, x = = 1.5 m. A lever AB is m long and the fulcrum is at a point 0.5 m from B. Find the effort to be applied at A to raise a load of 0.75 kn at B when losses are negligible. Clockwise moment = anticlockwise moment F e Hence, effort at A, F e = ( 05. )( 075. ) 15. = 0.5 kn or 50 N. The load on a third-order lever system is at a distance of 750 mm from the fulcrum and the effort required to just move the load is 1 kn when applied at a distance of 50 mm from the fulcrum. Determine the value of the load and the force ratio if losses are negligible. 0

198 Clockwise moment = anticlockwise moment F l i.e. F l = 50 1 kn =. N 750 Force ratio = dis tan ceof effort from fulcrum 50 = 1 dis tan ceof load from fulcrum 750 EXERCISE 111, Page 46 Answers found from within the text of the chapter, pages 8 to 46. EXERCISE 11, Page (b). (f). (c) 4. (d) 5. (b) 6. (a) 7. (b) 8. (d) 9. (c) 10. (d) 11. (d) 1. (b) 0

199 CHAPTER 5 THE EFFECTS OF FORCES ON MATERIALS EXERCISE 11, Page 5 1. A rectangular bar having a cross-sectional area of 80 mm has a tensile force of 0 kn applied to it. Determine the stress in the bar. Stress = force F Pa = 50 MPa 6 area A A circular section cable has a tensile force of 1 kn applied to it and the force produces a stress of 7.8 MPa in the cable. Calculate the diameter of the cable. Stress = force F area A hence, cross-sectional area, A = force F m 6 stress Circular area = r = m 6 6 from which, r = and radius r = m = 6.88 mm and diameter d = r = 6.88 = 1.78 mm. A square-sectioned support of side 1 mm is loaded with a compressive force of 10 kn. Determine the compressive stress in the support. Stress = force F Pa = MPa 6 area A A bolt having a diameter of 5 mm is loaded so that the shear stress in it is 10 MPa. Determine the value of the shear force on the bolt. 04

200 Stress = force F area A hence, force = stress area = stress r = 1010 = 56 N or.56 kn 5. A split pin requires a force of 400 N to shear it. The maximum shear stress before shear occurs is 10 MPa. Determine the minimum diameter of the pin. Stress = force F area A hence, cross-sectional area, A = force F m 6 stress 1010 Circular area = r =. 10 m 6 from which, r =.10 6 and radius r = m = 1.00 mm and diameter d = r = 1.00 =.06 mm 6. A tube of outside diameter 60 mm and inside diameter 40 mm is subjected to a tensile load of 60 kn. Determine the stress in the tube. Area of tube end (annulus) = D d = mm Stress = force F Pa = 8. MPa area A

201 EXERCISE 114, Page A wire of length 4.5 m has a percentage strain of 0.050% when loaded with a tensile force. Determine the extension in the wire. Original length of wire = 4.5 m = 4500 mm and strain = = extension x Strain = hence, extension x = L = ( )(4500) =.5 mm originallength L. A metal bar.5 m long extends by 0.05 mm when a tensile load is applied to it. Determine (a) the strain, (b) the percentage strain. (a) Strain = extension 0.05mm 0.05 original lengh.510 mm 500 = (b) Percentage strain = = 0.00%. An 80 cm long bar contracts axially by 0. mm when a compressive load is applied to it. Determine the strain and the percentage strain. Strain = contraction 0. mm = original lengh 800 mm Percentage strain = = 0.05% 4. A pipe has an outside diameter of 0 mm, an inside diameter of 10 mm and length 0.0 m and it supports a compressive load of 50 kn. The pipe shortens by 0.6 mm when the load is applied. Determine (a) the compressive stress, (b) the compressive strain in the pipe when supporting this load. Compressive force F = 50 kn = N, and cross-sectional area A = D d 4, 06

202 where D = outside diameter = 0 mm and d = inside diameter = 10 mm. Hence, A = (0 10 ) mm (0 10 ) 10 m m F N (a) Compressive stress, = 4 A.5610 m = Pa = 1. MPa (b) Contraction of pipe when loaded, x = 0.6 mm = m, and original length L = 0.0 m. Hence, compressive strain, = x = 0.00 (or 0.0%) L A rectangular block of plastic material 400 mm long by 15 mm wide by 00 mm high has its lower face fixed to a bench and a force of 150 N is applied to the upper face and in line with it. The upper face moves 1 mm relative to the lower face. Determine (a) the shear stress, and (b) the shear strain in the upper face, assuming the deformation is uniform. (a) Shear stress, = force area parallel to the force Area of any face parallel to the force = 400 mm 15 mm 150 N Hence, shear stress, = m = ( ) m = m = 5000 Pa or 5 kpa (b) Shear strain, = x L = 1 00 = 0.04 (or 4%) 07

203 EXERCISE 115, Page A wire is stretched 1.5 mm by a force of 00 N. Determine the force that would stretch the wire 4 mm, assuming the elastic limit of the wire is not exceeded. Hooke's law states that extension x is proportional to force F, provided that the limit of proportionality is not exceeded, i.e. x F or x = kf where k is a constant. When x = 1.5 mm, F = 00 N, thus 1.5 = k(00), from which, constant k = = When x = 4 mm, then 4 = kf i.e. 4 = F from which, force F = = 800 N Thus to stretch the wire 4 mm, a force of 800 N is required.. A rubber band extends 50 mm when a force of 00 N is applied to it. Assuming the band is within the elastic limit, determine the extension produced by a force of 60 N. Hooke's law states that extension x is proportional to force F, provided that the limit of proportionality is not exceeded, i.e. x F or x = kf where k is a constant. When x = 50 mm, F = 00 N, thus 50 = k(00), from which, constant k = When F = 60 N, then x = k(60) i.e. x = 60 = 10 mm Thus, a force of 60 N stretches the wire 10 mm.. A force of 5 kn applied to a piece of steel produces an extension of mm. Assuming the elastic limit is not exceeded, determine (a) the force required to produce an extension of.5 mm, (b) the extension when the applied force is 15 kn. From Hooke s law, extension x is proportional to force F within the limit of proportionality, i.e. 08

204 x F or x = kf, where k is a constant. If a force of 5 kn produces an extension of mm, then = k(5), from which, constant k = 5 = 0.08 (a) When an extension x =.5 mm, then.5 = k(f), i.e..5 = 0.08 F, from which, force F = = 4.75 kn (b) When force F = 15 kn, then extension x = k(15) = (0.08)(15) = 1. mm 4. A test to determine the load/extension graph for a specimen of copper gave the following results: Load (kn) Extension (mm) Plot the load/extension graph, and from the graph determine (a) the load at an extension of 0.09 mm, and (b) the extension corresponding to a load of 1.0 kn. A graph of load/extension is shown below. 09

205 (a) When the extension is 0.09 mm, the load is 19 kn (b) When the load is 1.0 kn, the extension is mm 5. A circular section bar is.5 m long and has a diameter of 60 mm. When subjected to a compressive load of 0 kn it shortens by 0.0 mm. Determine Young's modulus of elasticity for the material of the bar. Force, F = 0 kn = 0000 N and cross-sectional area A = Stress = F 0000 = MPa A m r Bar shortens by 0.0 mm = m Strain = x L = = Modulus of elasticity, E = stress 6 strain = = = 1.6 GPa 6. A bar of thickness 0 mm and having a rectangular cross-section carries a load of 8.5 kn. Determine (a) the minimum width of the bar to limit the maximum stress to 150 MPa, (b) the modulus of elasticity of the material of the bar if the 150 mm long bar extends by 0.8 mm when carrying a load of 00 kn. (a) Force, F = 8.5 kn = 8500 N and cross-sectional area A = (0x)10 6 m, where x is the width of the rectangular bar in millimetres. Stress = F A, from which, A = F 8500 N Pa = m = mm 4 6 Hence, 550 = 0x, from which, width of bar, x = = 7.5 mm = mm = 550 mm 10

206 (b) Stress = F A = 6.64 MPa Extension of bar = 0.8 mm Strain = x L = = Modulus of elasticity, E = stress 6 strain = = = 68. GPa 7. A metal rod of cross-sectional area 100 mm carries a maximum tensile load of 0 kn. The modulus of elasticity for the material of the rod is 00 GPa. Determine the percentage strain when the rod is carrying its maximum load. Stress = F A = 00 MPa Modulus of elasticity, E = stress 6 strain from which, strain = stress E 0010 = Hence, percentage strain, = % = 0.10% EXERCISE 116, Page 59 Answers found from within the text of the chapter, pages 50 to 59. EXERCISE 117, Page (c). (c). (a) 4. (b) 5. (c) 6. (c) 7. (b) 8. (d) 9. (b) 10. (c) 11. (f) 1. (h) 1. (d) 11

207 CHAPTER 6 LINEAR MOMENTUM AND IMPULSE EXERCISE 118, Page Determine the momentum in a mass of 50 kg having a velocity of 5 m/s. Momentum = mass velocity = 50 kg 5 m/s = 50 kg m/s downwards. A milling machine and its component have a combined mass of 400 kg. Determine the momentum of the table and component when the feed rate is 60 mm/min. Momentum = mass velocity = 400 kg m/s =.4 kg m/s downwards. The momentum of a body is 160 kg m/s when the velocity is.5 m/s. Determine the mass of the body. Momentum = mass velocity Hence, 160 = mass.5 from which, mass = 160 = 64 kg.5 4. Calculate the momentum of a car of mass 750 kg moving at a constant velocity of 108 km/h. Momentum = mass velocity Mass = 750 kg and velocity = 108 km/h = m/s = 0 m/s. Hence, momentum = 750 kg 0 m/s =,500 kg m/s 5. A football of mass 00 g has a momentum of 5 kg m/s. What is the velocity of the ball in km/h. 1

208 Momentum = mass velocity Hence, 5 = 0. v from which, velocity, v = 5 = 5 m/s 0. = 5.6 km/h = 90 km/h 6. A wagon of mass 8 t is moving at a speed of 5 m/s and collides with another wagon of mass 1 t, which is stationary. After impact, the wagons are coupled together. Determine the common velocity of the wagons after impact. Mass m 1 = 8 t = 8000 kg, m = 1000 kg and velocity u 1 = 5 m/s, u = 0. Total momentum before impact = m 1 u 1 + m u = (8000 5) + (1000 0) = kg m/s Let the common velocity of the wagons after impact be v m/s Since total momentum before impact = total momentum after impact: = m 1 v + m v = v(m 1 + m ) = v(0000) Hence v = = m/s i.e. the common velocity after impact is m/s in the direction in which the 8 t wagon is initially travelling. 7. A car of mass 800 kg was stationary when hit head-on by a lorry of mass 000 kg travelling at 15 m/s. Assuming no brakes are applied and the car and lorry move as one, determine the speed of the wreckage immediately after collision. Mass m 1 = 800 kg, m = 000 kg and velocity u 1 = 0, u = 15 m/s 1

209 Total momentum before impact = m 1 u 1 + m u = (800 0) + (000 15) = 0000 kg m/s Let the common velocity of the wagons after impact be v m/s Since total momentum before impact = total momentum after impact: 0000 = m 1 v + m v = v(m 1 + m ) = v(800) Hence v = 0000 = m/s 800 i.e. the speed of the wreckage immediately after collision is m/s in the direction in which the lorry is initially travelling. 8. A body has a mass of 5 g and is moving with a velocity of 0 m/s. It collides with a second body which has a mass of 15 g and which is moving with a velocity of 0 m/s. Assuming that the bodies both have the same speed after impact, determine their common velocity (a) when the speeds have the same line of action and the same sense, and (b) when the speeds have the same line of action but are opposite in sense. Mass m 1 = 5 g = 0.05 kg, m = 15 g = kg, velocity u 1 = 0 m/s and u = 0 m/s. (a) When the velocities have the same line of action and the same sense, both u 1 and u are considered as positive values Total momentum before impact = m 1 u 1 + m u = (0.05 0) + ( ) = = 1.05 kg m/s Let the common velocity after impact be v m/s Total momentum before impact = total momentum after impact i.e = m 1 v + m v = v(m 1 + m ) 1.05 = v( ) 14

210 from which, common velocity, v = are initially travelling = 6.5 m/s in the direction in which the bodies (b) When the velocities have the same line of action but are opposite in sense, one is considered as positive and the other negative. Taking the direction of mass m 1 as positive gives: velocity u 1 = +0 m/s and u = - 0 m/s Total momentum before impact = m 1 u 1 + m u = (0.05 0) + ( ) = = kg m/s and since it is positive this indicates a momentum in the same direction as that of mass m 1. If the common velocity after impact is v m/s then from which, common velocity, v = initially travelling = v(m 1 + m ) = v(0.040) = 11.5 m/s in the direction that the 5 g mass is 15

211 EXERCISE 119, Page The sliding member of a machine tool has a mass of 00 kg. Determine the change in momentum when the sliding speed is increased from 10 mm/s to 50 mm/s. Change of linear momentum = mass change of velocity Hence, change in momentum = 00 kg (50 10) 10 m/s = 8 kg m/s. A force of 48 N acts on a body of mass 8 kg for 0.5 s. Determine the change in velocity. Impulse = applied force time = change in linear momentum i.e. 48 N 0.5 s = mass change in velocity = 8 kg change in velocity from which, change in velocity = 48 N 0.5s 8kg = 1.5 m/s (since 1 N = 1 kg m/s ). The speed of a car of mass 800 kg is increased from 54 km/h to 6 km/h in s. Determine the average force in the direction of motion necessary to produce the change in speed. Change of momentum = applied force time i.e. mass change of velocity = applied force time 6 54 i.e. 800 kg 6 6 m/s = applied force s from which, applied force = 6. = 1000 N or 1kN 16

212 4. A 10 kg mass is dropped vertically on to a fixed horizontal plane and has an impact velocity of 15 m/s. The mass rebounds with a velocity of 5 m/s. If the contact time of mass and plane is 0.05 s, calculate (a) the impulse, and (b) the average value of the impulsive force on the plane. (a) Impulse = change in momentum = m(u 1 - v 1 ) where u 1 = impact velocity = 15 m/s and v 1 = rebound velocity = - 5 m/s (v 1 is negative since it acts in the opposite direction to u 1 ) Thus, impulse = m(u 1 - v 1 ) = 10 kg ( ) m/s = 10 0 = 00 kg m/s (b) Impulsive force = impulse 00kg m / s = 8000 N or 8 kn time 0.05s 5. The hammer of a pile driver of mass 1. t falls 1.4 m on to a pile. The blow takes place in 0 ms and the hammer does not rebound. Determine the average applied force exerted on the pile by the hammer. Initial velocity, u = 0, acceleration due to gravity, g = 9.81 m/s and distance, s = 1.4 m. Using the equation of motion: v = u + gs gives: v = 0 + (9.81)(1.4) from which, impact velocity, v = ( )( 9. 81)( 1. 4) = 5.41 m/s Neglecting the small distance moved by the pile and hammer after impact, momentum lost by hammer = the change of momentum = mv = 100 kg 5.41 m/s Rate of change of momentum = changeof momentum changeof time = = N Since the impulsive force is the rate of change of momentum, the average force exerted on the pile is 14.5 kn 17

213 6. A tennis ball of mass 60 g is struck from rest with a racket. The contact time of ball on racket is 10 ms and the ball leaves the racket with a velocity of 5 m/s. Calculate (a) the impulse, and (b) the average force exerted by a racket on the ball. (a) Impulse = change of momentum = mv = (0.060 kg)(5 m/s) = 1.5 kg m/s impulse 1.5kg m / s (b) Impulsive force = time 1010 = 150 N 7. In a press-tool operation, the tool is in contact with the work piece for 40 ms. If the average force exerted on the work piece is 90 kn, determine the change in momentum. Change in momentum = applied force time = N = 600 kg m/s EXERCISE 10, Page 67 Answers found from within the text of the chapter, pages 6 to 67. EXERCISE 11, Page (d). (b). (f) 4. (c) 5. (a) 6. (c) 7. (a) 8. (g) 9. (f) 10. (f) 11. (b) 1. (e) 18

214 CHAPTER 7 TORQUE EXERCISE 1, Page Determine the torque developed when a force of 00 N is applied tangentially to a spanner at a distance of 50 mm from the centre of the nut. Torque T = Fd, where force F = 00 N and distance, d = 50 mm = 0.5 m Hence, torque, T = (00)(0.5) = 70 N m. During a machining test on a lathe, the tangential force on the tool is 150 N. If the torque on the lathe spindle is 1 N m, determine the diameter of the work-piece. Torque T = Fr, where torque T = 1 N m, force F = 150 N at radius r Hence, 1 = (150)(r) from which, radius, r = = 0.08 m = 80 mm Hence, diameter = 80 = 160 mm 19

215 EXERCISE 1, Page 7 1. A constant force of 4 kn is applied tangentially to the rim of a pulley wheel of diameter 1.8 m attached to a shaft. Determine the work done, in joules, in 15 revolutions of the pulley wheel. Torque T = Fr, where F = 4000 N and radius r = 1.8 = 0.9 m Hence, torque T = (4000)(0.9) = 600 N m Work done = T joules, where torque, T = 600 N m and angular displacement, = 15 revolutions = 15 rad = 0 rad. Hence, work done = T = (600)(0) = = 9. kj. A motor connected to a shaft develops a torque of.5 kn m. Determine the number of revolutions made by the shaft if the work done is 11.5 MJ. Work done = T joules, where work done = 11.5 Hence, = J and torque, T = 500 N m from which, angular displacement, = = 91.4 rad and number of revolutions = 91.4 = 5.8 rev. A wheel is turning with an angular velocity of 18 rad/s and develops a power of 810 W at this speed. Determine the torque developed by the wheel. Power P = T, where P = 810 W and angular velocity, = 18 rad/s Hence, 810 = T 18 from which, torque, T = = 45 N m 0

216 4. Calculate the torque provided at the shaft of an electric motor that develops an output power of. hp at 1800 rev/min. Power, P = nt, where power P =. h.p. = = 86.4 W and n = = 0 rev/s Hence, 86.4 = 0 T from which, torque, T = = 1.66 N m 5. Determine the angular velocity of a shaft when the power available is.75 kw and the torque is 00 N m. Power, P = nt, where power P = 750 W and torque T = 00 N m Hence, 750 = n 00 from which, n = =.1884 rev/s Angular velocity, = n =.1884 = 1.75 rad/s 6. The drive shaft of a ship supplies a torque of 400 kn m to its propeller at 400 rev/min. Determine the power delivered by the shaft. Power, P = T = nt = = W = MW 7. A motor is running at 1460 rev/min and produces a torque of 180 N m. Determine the average power developed by the motor. 1

217 Power, P = T = nt = = W = 7.5 kw 8. A wheel is rotating at 170 rev/min and develops a power of 600 W at this speed. Calculate (a) the torque, (b) the work done, in joules, in a quarter of an hour. (a) Power, P = nt hence, 600 = T from which, torque, T = =. N m (b) Work done = T, where torque T =. N m and angular displacement in 15 minutes = (15 170)rev = ( ) rad. Hence, work done = T = (.)( ) = J = 540 kj

218 EXERCISE 14, Page A shaft system has a moment of inertia of 51.4 kg m. Determine the torque required to give it an angular acceleration of 5. rad/s. Torque, T = I, where moment of inertia I = 51.4 kg m and angular acceleration, = 5. rad/s. Hence, torque, T = I = (51.4)(5.) = 7.4 N m. A shaft has an angular acceleration of 0 rad/s and produces an accelerating torque of 600 N m. Determine the moment of inertia of the shaft. Torque, T = I, where torque T = 600 N m and angular acceleration, = 0 rad/s. Hence, 600 = I 0 from which, moment of inertia of the shaft, I = = 0 kg m. A uniform torque of. kn m is applied to a shaft while it turns through 5 revolutions. Assuming no frictional or other resistance s, calculate the increase in kinetic energy of the shaft (i.e. the work done). If the shaft is initially at rest and its moment of inertia is 4.5 kg m, determine its rotational speed, in rev/min, at the end of the 5 revolutions. Work done = T = 00 (5 ) = kj Increase in kinetic energy = J = I 1 where I = 4.5 kg m and 1 = 0 Hence, = from which, and = rad/s

219 Hence, rotational speed = 60s / min 0.565rad / s rad/rev = 194 rev/min 4. An accelerating torque of 0 N m is applied to a motor, while it turns through 10 revolutions. Determine the increase in kinetic energy. If the moment of inertia of the rotor is 15 kg m and its speed at the beginning of the 10 revolutions is 100 rev/min, determine its speed at the end. Increase in kinetic energy = work done = T = 0 (10 ) = 1885 J or kj Increase in kinetic energy = 1885 J = I 1 where I = 15 kg m and 1= 100 = 40 = rad/s Hence, 1885 = 15 from which, = Hence, = and = rad/s 60s / min Hence, final speed = rad / s rad/rev = rev/min 5. A shaft with its associated rotating parts has a moment of inertia of 48 kg m. Determine the uniform torque required to accelerate the shaft from rest to a speed of 1500 rev/min while it turns through 15 revolutions. Work done = increase in kinetic energy = T = I 1 where I = 48 kg m, 1= 0 and 1500 = rad/s 60 4

220 Hence, T = I 1 i.e. from which, torque, T = T(15 ) = 48 = = 68 N m or 6.8 kn m 6. A small body, of mass 8 g, is fastened to a wheel and rotates in a circular path of 456 mm diameter. Calculate the increase in kinetic energy of the body when the speed of the wheel increases from 450 rev/min to 950 rev/min. Increase in kinetic energy = I 1 = = mr (0.08) = (0.08)( )( = 16.6 J 7. A system consists of three small masses rotating at the same speed about the same fixed axis. The masses and their radii of rotation are: 16 g at 56 mm, g at 19 mm and 1 g at 176 mm. Determine (a) the moment of inertia of the system about the given axis, and (b) the kinetic energy in the system if the speed of rotation is 150 rev/min. (a) Moment of inertia, I = mr (0.016)(0.56) (0.0)(0.19) (0.01)(0.176) = = kg m 4 4 5

221 (b) Kinetic energy in the system = I = 4.48 J 6

222 EXERCISE 15, Page A motor has an efficiency of 7% when running at 600 rev/min. If the output torque is 16 N m at this speed, determine the power supplied to the motor. Power output, P = nt = (600/60)(16) = Efficiency = power output power input 100% hence 7 = power input 100 from which, power input = = 6050 W or 6.05 kw 7. The difference in tensions between the two sides of a belt round a driver pulley of radius 40 mm is 00 N. If the driver pulley wheel is on the shaft of an electric motor running at 700 rev/min and the power input to the motor is 5 kw, determine the efficiency of the motor. Determine also the diameter of the driven pulley wheel if its speed is to be 100 rev/min. Power output from motor = (F - F 1 )r x x (F - F 1 ) = 00 N, radius r x = 40 mm = 0.4 m and angular velocity, x = 700 rad/s 60 Hence, power output from motor = (F - F 1 )r x x = (00)(0.4) Power input = 5000 W = W Hence, efficiency of the motor = power output power input = = 70.7% 5000 r r x y n y x x from which, driven pulley wheel radius, r y = n x nr n y = = 0.14 m from which, diameter of driven pulley wheel = radius = 0.14 = 0.8 m or 80 mm 7

223 . A winch is driven by a 4 kw electric motor and is lifting a load of 400 kg to a height of 5.0 m. If the lifting operation takes 8.6 s, calculate the overall efficiency of the winch and motor. The increase in potential energy is the work done and is given by mgh (see Chapter 0), where mass, m = 400 kg, g = 9.81 m/s and height h = 5.0 m. Hence, work done = mgh = (400)(9.81)(5.0) = 19.6 kj. Input power = 4 kw = 4000 W Output power = work done W time taken 8.6 Efficiency = output power input power 100 = = 57.0% 4. A belt and pulley system transmits a power of 5 kw from a driver to a driven shaft. The driver pulley wheel has a diameter of 00 mm and rotates at 600 rev/min. The diameter of the driven wheel is 400 mm. Determine the speed of the driven pulley and the tension in the slack side of the belt when the tension in the tight side of the belt is 1. kn. r x = 100 mm = 0.1 m, n x = 600 rev/min, r y = 00 mm = 0. m r r x y n y x x from which, speed of driven pulley, n y nx y r n = 00 rev/min r 0. Available power = (F - F 1 )r x x i.e = (100 - F 1 )(0.1) i.e. (100 - F 1 ) = = Hence, tension in slack side of belt, F 1 = = 404. N 5. The average force on the cutting tool of a lathe is 750 N and the cutting speed is 400 mm/s. Determine the power input to the motor driving the lathe if the overall efficiency is 55%. 8

224 Force resisting motion = 750 N and velocity = 400 mm/s = 0.4 m/s Output power from motor = resistive force velocity of lathe (from Chapter 0) hence 55 = = = 00 W power output Efficiency = 100 power input 00 power input 100 from which, power input = = W 6. A ship's anchor has a mass of 5 t. Determine the work done in raising the anchor from a depth of 100 m. If the hauling gear is driven by a motor whose output is 80 kw and the efficiency of the haulage is 75%, determine how long the lifting operation takes. The increase in potential energy is the work done and is given by mgh (see Chapter 0), where mass, m = 5 t = 5000 kg, g = 9.81 m/s and height h = 100 m Hence, work done = mgh = (5000)(9.81)(100) = MJ Input power = 80 kw = W output power Efficiency = input power 100 hence 75 = output power from which, output power = 75 work done = W = 100 time taken Thus, time taken for lifting operation = work done output power J W = s = 1 min s to the nearest second. 9

225 EXERCISE 16, Page 78 Answers found from within the text of the chapter, pages 70 to 77. EXERCISE 17, Page (d). (b). (c) 4. (a) 5. (c) 6. (d) 7. (a) 8. (b) 9. (c) 10. (d) 11. (a) 1. (c) 0

226 CHAPTER 8 PRESSURE IN FLUIDS EXERCISE 18, Page A force of 80 N is applied to a piston of a hydraulic system of cross-sectional area m. Determine the pressure produced by the piston in the hydraulic fluid. Pressure, p = force area = 80 N m 8000 Pa = 8 kpa That is, the pressure produced by the piston is 8 kpa. Find the force on the piston of question 1 to produce a pressure of 450 kpa. Pressure, p = 450 kpa = Pa Pressure p = force area hence, force = pressure area = = 4500 N = 4.5 kn. If the area of the piston in question 1 is halved and the force applied is 80 N, determine the new pressure in the hydraulic fluid. New area = m New pressure, p = force area = 80 N 56000Pa = 56 kpa 0.005m 1

227 EXERCISE 19, Page 8 1. Determine the pressure acting at the base of a dam, when the surface of the water is 5 m above base level. Take the density of water as 1000 kg/m. Take the gravitational acceleration as 9.8 m/s. Pressure at base of dam, p = gh = 1000 kg/m 9.8 m/s 0.5 m = 4000 Pa = 4 kpa. An uncorked bottle is full of sea water of density 100 kg/m. Calculate, correct to significant figures, the pressures on the side wall of the bottle at depths of (a) 0 mm, and (b) 70 mm below the top of the bottle. Take the gravitational acceleration as 9.8 m/s. Pressure on the side wall of the bottle, p = gh (a) When depth, h = 0 mm = 0 10 m, pressure, p = 100 kg/m 9.8 (b) When depth, h = 70 mm = m, pressure, p = 100 kg/m 9.8 m/s m/s 0 10 m = 0 Pa m = 707 Pa. A U-tube manometer is used to determine the pressure at a depth of 500 mm below the free surface of a fluid. If the pressure at this depth is 6.86 kpa, calculate the density of the liquid used in the manometer. Take the gravitational acceleration as 9.8 m/s Pressure, p = gh hence, Pa = 9.8 m/s m from which, density of liquid, = = 1400 kg/m

228 EXERCISE 10, Page 8 1. The height of a column of mercury in a barometer is 750 mm. Determine the atmospheric pressure, correct to significant figures. Take the gravitational acceleration as 9.8 m/s and the density of mercury as 1600 kg/m. Atmospheric pressure, p = gh = 1600 kg/m 9.8 = Pa = 100 kpa m/s m. A U-tube manometer containing mercury gives a height reading of 50 mm of mercury when connected to a gas cylinder. If the barometer reading at the same time is 756 mm of mercury, calculate the absolute pressure of the gas in the cylinder, correct to significant figures. Take the gravitational acceleration as 9.8 m/s and the density of mercury as 1600 kg/m. Pressure, p 1 = gh = 1600 kg/m 9.8 = 0 Pa =. kpa Pressure, p = gh = 1600 kg/m 9.8 m/s m/s = Pa = kpa m m Absolute pressure = atmospheric pressure + gauge pressure = p + p 1 = = 14 kpa. A water manometer connected to a condenser shows that the pressure in the condenser is 50 mm below atmospheric pressure. If the barometer is reading 760 mm of mercury, determine the absolute pressure in the condenser, correct to significant figures. Take the gravitational acceleration as 9.8 m/s and the density of water as 100 kg/m. Pressure, p 1 = - 1 gh 1 = kg/m 9.8 m/s m

229 = - 40 Pa = -.4 kpa Pressure, p = gh = 1600 kg/m 9.8 = 1019 Pa = 101. kpa m/s m Absolute pressure = atmospheric pressure + gauge pressure = p + p 1 = = 97.9 kpa 4. A Bourdon pressure gauge shows a pressure of MPa. If the absolute pressure is 1.5 MPa, find the atmospheric pressure in millimetres of mercury. Take the gravitational acceleration as 9.8 m/s and the density of mercury as 1600 kg/m. Atmospheric pressure = absolute pressure - gauge pressure = 1.5 MPa MPa = MPa = Atmospheric pressure, p = gh = 1600 kg/m 9.8 m/s h Pa i.e Pa = 1600 kg/m 9.8 m/s h from which, height, h = Pa 1600kg / m 9.8m / s = 0.74 m i.e. atmospheric pressure in millimetres of mercury = 0.74 m 1000mm = 74 mm 1m 4

230 EXERCISE 11, Page A body of volume 0.14 m is completely immersed in water of density 1000 kg/m. What is the apparent loss of weight of the body? Take the gravitational acceleration as 9.8 m/s. Mass, m = density, ρ volume, V = 1000 kg/m 0.14 m = 14 kg Apparent loss of weight of the body, W = ρ V g = 14 kg 9.8 m/s = 115 N = 1.15 kn. A body of weight 7.4 N and volume 140 cm is completely immersed in water of specific weight 9.81 kn/m. What is its apparent weight? Take the gravitational acceleration as 9.8 m/s and the density of water as 1000 kg/m. Body weight, W 1 = 7.4 N Apparent weight, W = ρ V g = 7.4 (1000 kg/m m 9.8 m/s ) = 7.4 N 1.15 N = 15.5 N. A body weighs 51.6 N in air and 56.8 N when completely immersed in oil of density 810 kg/m. What is the volume of the body? Take the gravitational acceleration as 9.8 m/s. W = ρ oil V g i.e. ( ) = ρ oil V g i.e = 810 V 9.8 5

231 from which, volume, V = = 0.0 m or. dm 4. A body weighs 4 N in air and 15 N when completely immersed in water. What will it weigh when completely immersed in oil of relative density 0.8? Take the gravitational acceleration as 9.8 m/s and the density of water as 1000 kg/m. W = ρ V g i.e. (4 15) = ρ water V g i.e. 118 = 1000 V 9.8 from which, volume, V = = m Weight in oil = 4 - ρ oil V g = 4 ( ) = = N 5. A watertight rectangular box, 1. m long and 0.75 m wide, floats with its sides and ends vertical in water of density 1000 kg/m. If the depth of the box in the water is 80 mm, what is its weight? Take the gravitational acceleration as 9.8 m/s. Volume of box, V = 1. m 0.75 m m = 0.5 m Weight of box, W = ρ V g = = N =.47 kn 6. A body weighs 18 N in air and 1.7 N when completely immersed in water of density 1000 kg/m. What is the density and relative density of the body? Take the gravitational acceleration as 9.8 m/s. 6

232 W = ρ V g i.e. (18 1.7) = ρ water V g i.e. 4. = 1000 V 9.8 from which, volume, V = = m mass 18 N 1 Density of body, ρ = 4 volume m 9.8 N / kg = 4186 kg/ m or tonnes/ m Relative density = density 4186kg / m = densityof water 1000kg / m 7. A watertight rectangular box is 660 mm long and 0 mm wide. Its weight is 6 N. If it floats with its sides and ends vertical in water of density 100 kg/m, what will be its depth in the water? Take the gravitational acceleration as 9.8 m/s. Volume of box, V = m 0 10 m D where D = depth of box Weight, W = ρ V g i.e. 6 = 100 kg/m ( m 0 10 m D) 9.8 from which, depth of box, D = = m = 159 mm 8. A watertight drum has a volume of m and a weight of 115 N. It is completely submerged in water of density 100 kg/m, held in position by a single vertical chain attached to the underside of the drum. What is the force in the chain? Take the gravitational acceleration as 9.8 m/s. Weight of drum, W = 115 N 7

233 Upthrust = ρ V g = 100 kg/m m 9.8 m/s = N Hence, the force in the chain = = 1551 N = kn EXERCISE 1, Page 91 Answers found from within the text of the chapter, pages 80 to 91. EXERCISE 1, Page (b). (d). (a) 4. (a) 5. (c) 6. (d) 7. (b) 8. (c) 9. (c) 10. (d) 11. (d) 1. (d) 1. (c) 14. (b) 15. (c) 16. (a) 17. (b) 18. (f) 19. (a) 0. (b) 1. (c) 8

234 CHAPTER 9 HEAT ENERGY AND TRANSFER EXERCISE 14, Page Convert the following temperatures into the Kelvin scale: (a) 51C (b) - 78C (c) 18C Kelvin temperature, K = C + 7 (a) When Celsius temperature = 51C, K = = 4 K (b) When Celsius temperature = - 78C, K = = 195 K (c) When Celsius temperature = 18C, K = = 456 K. Convert the following temperatures into the Celsius scale: (a) 07 K (b) 7 K (c) 415 K If K = C + 7 then C = K - 7 (a) When Kelvin temperature = 07 K, C = 07 7 = 4C (a) When Kelvin temperature = 7 K, C = 7 7 = - 6C (a) When Kelvin temperature = 415 K, C = = 14C 9

235 EXERCISE 15, Page Determine the quantity of heat energy (in megajoules) required to raise the temperature of 10 kg of water from 0C to 50C. Assume the specific heat capacity of water is 400 J/(kg C). Quantity of heat energy, Q = mc(t - t 1 ) = 10 kg 400 J/(kg C) (50-0)C = J or 100 kj or.1 MJ. Some copper, having a mass of 0 kg, cools from a temperature of 10C to 70C. If the specific heat capacity of copper is 90 J/(kg C), how much heat energy is lost by the copper? Quantity of heat energy, Q = mc(t - t 1 ) = 0 kg 90 J/(kg C) (70-10)C = = J or 90 kj Hence, the heat energy lost by the copper = 90 kj. A block of aluminium having a specific heat capacity of 950 J/(kg C) is heated from 60C to its melting point at 660C. If the quantity of heat required is.85 MJ, determine the mass of the aluminium block. Quantity of heat, Q = mc(t - t 1 ), hence, J = m 950 J/(kg C) (660-60)C i.e = m from which, mass, m = kg = 5 kg

236 kj of heat energy is required to raise the temperature of kg of lead from 16C to 96C. Determine the specific heat capacity of lead. Quantity of heat, Q = mc(t - t 1 ), hence: J = kg c (96 16)C where c is the specific heat capacity, i.e = c 80 from which, specific heat capacity of lead, c = = 10 J/(kg C) kj of heat energy is supplied to 10 kg of iron which is initially at a temperature of 15C. If the specific heat capacity of iron is 500 J/(kg C) determine its final temperature. Quantity of heat, Q = mc(t - t 1 ), hence, J = 10 kg 500 J/(kg C) (t - 15)C from which, (t - 15) = = 50 Hence, the final temperature, t = = 65C 41

237 EXERCISE 16, Page Some ice, initially at - 40C, has heat supplied to it at a constant rate until it becomes superheated steam at 150C. Sketch a typical temperature/time graph expected and use it to explain the difference between sensible and latent heat. See Section 9.5 and Figure 9.1 on page 98 of textbook. Just replace the - 0C at A with - 40C and replace 10C at F with 150C. 4

238 EXERCISE 17, Page How much heat is needed to melt completely 5 kg of ice at 0C. Assume the specific latent heat of fusion of ice is 5 kj/kg. Quantity of heat required, Q= ml = 5 kg 5 kj/kg = 875 kj or 8.75 MJ. Determine the heat energy required to change 8 kg of water at 100C to superheated steam at 100C. Assume the specific latent heat of vaporisation of water is 60 kj/kg. Quantity of heat required, Q = ml = 8 kg 60 kj/kg = kj or MJ. Calculate the heat energy required to convert 10 kg of ice initially at - 0C completely into water at 0C. Assume the specific heat capacity of ice is.1 kj/(kg C) and the specific latent heat of fusion of ice is 5 kj/kg. Quantity of heat energy needed, Q = sensible heat + latent heat. The quantity of heat needed to raise the temperature of ice from - 0C to 0C i.e. sensible heat, Q 1 = mc(t - t 1 ) = 10 kg 100 J/(kgC) ( )C = ( ) J = 60 kj The quantity of heat needed to melt 10 kg of ice at 0C, i.e. the latent heat, Q = ml = 10 kg 5 kj/kg = 50 kj Total heat energy needed, Q = Q 1 + Q = = 980 kj =.98 MJ 4

239 4. Determine the heat energy needed to convert completely 5 kg of water at 60C to steam at 100C, given that the specific heat capacity of water is 4. kj/(kg C) and the specific latent heat of vaporisation of water is 60 kj/kg. Quantity of heat required = sensible heat + latent heat. Sensible heat, Q 1 = mc(t - t 1 ) = 5 kg 4. kj/(kg C) (100-60)C = 840 kj Latent heat, Q = ml = 5 kg 60 kj/kg = 1100 kj Total heat energy required, Q = Q 1 + Q = ( ) kj = 1140 kj or 1.14 MJ EXERCISE 18, Page 0 Answers found from within the text of the chapter, pages 94 to 0. EXERCISE 19, Page 0 1. (d). (b). (a) 4. (c) 5. (b) 6. (b) 7. (b) 8. (a) 9. (c) 10. (b) 11. (d) 1. (c) 1. (d) 44

240 CHAPTER INDICES, UNITS, PREFIXES AND ENGINEERING NOTATION EXERCISE 1, Page 5 1. Evaluate without the aid of a calculator = = 7. Evaluate 7 without the aid of a calculator 7 = = 18. Evaluate 5 10 without the aid of a calculator 5 10 = = 100, Evaluate 4 without the aid of a calculator = = Evaluate 1 5 without the aid of a calculator = 5 6. Evaluate without the aid of a calculator by cancelling = = 100 8

241 7. Evaluate without the aid of a calculator = 1 8. Evaluate taking positive square roots only =

242 EXERCISE 14, Page 8 1. Evaluate 4 without the aid of a calculator = 18. Evaluate 5 in index form without the aid of a calculator = 9. Evaluate 7 without the aid of a calculator = Evaluate 5 without the aid of a calculator = Evaluate 0 7 without the aid of a calculator 0 7 = 1 (any real number raised to the power zero is 1) 6. Evaluate 7 6 without the aid of a calculator =

243 7. Evaluate without the aid of a calculator = Evaluate without the aid of a calculator = Evaluate without the aid of a calculator = or Evaluate without the aid of a calculator = Evaluate (7 ) in index form without the aid of a calculator 7 7 = Evaluate ( ) without the aid of a calculator = 6 or 79 41

244 1. Evaluate 4 without the aid of a calculator = 5 1 or Evaluate without the aid of a calculator = 7 = Evaluate without the aid of a calculator = 1 or Evaluate without the aid of a calculator ( 8) 78 0 = Simplify expressing the answer in index form and with positive indices = Simplify expressing the answer in index form and with positive indices 4

245 ( ) = Simplify expressing the answer in index form and with positive indices = 5 0. Evaluate = 9 1. Evaluate =. Evaluate / = 1. Evaluate 4 9 1/ 1 1 1/ = 4

246 EXERCISE 15, Page 0 1. State the SI unit of volume? The SI unit of volume is cubic metres, m. State the SI unit of capacitance? The SI unit of capacitance, is the farad, F. State the SI unit of area? The SI unit of area is square metres, m 4. State the SI unit of velocity? The SI unit of velocity is metres per second, m/s 5. State the SI unit of density? The SI unit of density is kilograms per cubic metre, kg / m 6. State the SI unit of energy? The SI unit of energy is the joule, J 7. State the SI unit of charge? The SI unit of charge is the coulomb, C 8. State the SI unit of power? 44

247 The SI unit of power is the watt, W 9. State the SI unit of electric potential? The SI unit of electric potential is the volt, V 10. State which quantity has the unit kg? The quantity which has the unit kg is mass, m 11. State which quantity has the unit symbol? The quantity which has the unit symbol is electrical resistance, R 1. State which quantity has the unit Hz? The quantity which has the unit Hz is frequency, f 1. State which quantity has the unit m/s? The quantity which has the unit m/s is acceleration, a 14. State which quantity has the unit symbol A? The quantity which has the unit symbol A is electric current 15. State which quantity has the unit symbol H? The quantity which has the unit symbol H is inductance 16. State which quantity has the unit symbol m? 45

248 The quantity which has the unit symbol m is length 17. State which quantity has the unit symbol K? The quantity which has the unit symbol K is thermodynamic temperature 18. State which quantity has the unit rad/s? The quantity which has the unit rad/s is angular velocity 19. What does the prefix G mean? The prefix G means multiply by 1000,000,000 i.e. multiply by What is the symbol and meaning of the prefix milli? The symbol for milli is m and the prefix milli means divide by 1000 i.e. multiply by What does the prefix p mean? The prefix p means divide by 1000,000,000,000 i.e. multiply by What is the symbol and meaning of the prefix mega? The symbol for mega is M and the prefix mega means multiply by 1000,000 i.e. multiply by

249 EXERCISE 16, Page 1. Express in standard form: (a) 7.9 (b) 8.4 (c) (a) 7.9 = (b) 8.4 = (c) = Express in standard form: (a) 748 (b),170 (c) 74,18 (a) 748 =.748 (b) 170 = (c) 7418 = Express in standard form: (a) (b) (c) (a) =.401 (b) = 1.74 (c) = Express in standard form: (a) 170. (b) (c) (a) 170. = 1.70 (b) = (c) = Express in standard form: (a) 1 (b) (c) 10 5 (d) 1 47

250 (a) 1 = 0.5 = (b) 11 7 = = (c) 10 5 = 10.6 = (d) 1 = = Express the following numbers as integers or decimal fractions: (a) (b) (c) (d) (a) = 1010 (b) = 9.7 (c) = (d) = 7 7. Express the following numbers as integers or decimal fractions: (a) (b) (c) (a) = (b) = (c) = Evaluate, stating the answers in standard form: (a) ( )( 10 ) (b) ( ) (a) ( )( 10 ) = = = 15 = 1.5 (b) ( ) = = =

251 9. Evaluate, stating the answers in standard form: (a) (b) (.410 )(10 ) 4 (4.810 ) (a) ( 5) 5 = 00 = 10 (b) (.410 )(10 ) (4.810 ) 4.8 = Write the following statements in standard form. (a) The density of aluminium is 710 kg m (b) Poisson's ratio for gold is 0.44 (c) The impedance of free space is 76.7 (d) The electron rest energy is MeV (e) Proton charge-mass ratio is 95,789,700 C kg -1 (f) The normal volume of a perfect gas is m mol -1 (a) The density of aluminium is 710 kg m = kg m (b) Poisson's ratio for gold is 0.44 = (c) The impedance of free space is 76.7 = (d) The electron rest energy is MeV = MeV (e) Proton charge-mass ratio is C kg -1 = C kg -1 (f) The normal volume of a perfect gas is m mol -1 = m mol -1 49

252 EXERCISE 17, Page 1. Express 60,000 Pa in engineering notation in prefix form 60,000 Pa = Pa = 60 kpa. Express W in engineering notation in prefix form W = 0.15 mw or 150 W. Express V in engineering notation in prefix form V = V = = 50 MV 4. Express F in engineering notation in prefix form F = F = 55 nf Express 100,000 W in engineering notation in prefix form 100,000 W = W = 100 kw 6. Express A in engineering notation in prefix form A = A = 0.54 ma or A = 540 A 7. Express in engineering notation in prefix form = = = 1.5 M 50

253 8. Express V in engineering notation in prefix form V = 0.05 V =.5 10 V =.5 mv 9. Express 5,000,000,000 Hz in engineering notation in prefix form 5,000,000,000 Hz = Hz = 5 GHz 10. Express F in engineering notation in prefix form F = F = 15 pf 11. Express A in engineering notation in prefix form A = A = 17 A 1. Express 4600 in engineering notation in prefix form 4600 = = 46. k 1. Rewrite 0.00 ma in A 0.00 ma = = A 14. Rewrite 05 khz as MHz 05 khz = Hz = hz =.05 MHz 15. Rewrite 650 cm in metres 51

254 650 cm = m = 6.50 m 16. Rewrite 4.6 g in kg 4.6 g = 4.6 kg = kg Use a calculator to evaluate in engineering notation: = Use a calculator to evaluate in engineering notation: ( 6) = The distance from Earth to the moon is around m. State the distance in kilometres. 8 Dis tan ce.810 m = = km 0. The radius of a hydrogen atom is m. State the radius in nanometres m Radius m m 10 nm / m = 0.05 nm 1. The tensile stress acting on a rod is Pa. Write this value in engineering notation. Tensile stress = Pa = = 5.6 MPa 5

255 . The expansion of a rod is m. Write this in engineering notation. Expansion = m = 4.10 m = 4. mm 5

256 CHAPTER 0 THERMAL EXPANSION EXERCISE 140, Page A length of lead piping is 50.0 m long at a temperature of 16C. When hot water flows through it the temperature of the pipe rises to 80C. Determine the length of the hot pipe if the coefficient of linear expansion of lead is K 1. Length L 1 = 50.0 m, temperature t 1 = 16C, t = 80C and = K 1 Length of pipe at 80C is given by: L = L 1 [1 + (t - t 1 )] = 50.0[1 + ( )(80-16)] = 50.0[ ] i.e. an increase in length of m or 9.8 mm = 50.0[ ] = m. A rod of metal is measured at 85 K and is.51 m long. At 7 K the rod is.5 m long. Determine the value of the coefficient of linear expansion for the metal. Length L 1 =.51 m, L =.5 m, temperature t 1 = 85 K and temperature t = 7 K Length L = L 1 [1 + (t - t 1 )] i.e..5 =.51[1 + (7-85)].5 =.51 + (.51)()(88) i.e = (.51)()(88) Hence, the coefficient of linear expansion, = 0.00 (.51)(88) = i.e. coefficient of linear expansion, = K 1 45

257 . A copper overhead transmission line has a length of 40.0 m between its supports at 0C. Determine the increase in length at 50C if the coefficient of linear expansion of copper is K 1. Length L = L 1 [1 + (t - t 1 )] = L 1 + L 1 (t - t 1 ) Hence, increase in length = L 1 (t - t 1 ) = (40.0 m)( K 1 = (40.0)( )(0) = m or 0.4 mm )(50 0)C 4. A brass measuring tape measures.10 m at a temperature of 15C. Determine (a) the increase in length when the temperature has increased to 40C (b) the percentage error in measurement at 40C. Assume the coefficient of linear expansion of brass to be K 1. Length L 1 =.10 m, temperature t 1 = 15C, t = 40C and = K 1 (a) Length L = L 1 [1 + (t - t 1 )] = L 1 + L 1 (t - t 1 ) Hence, increase in length = L 1 (t - t 1 ) = (.10 m)( K 1 = (.10)( )(5) = m or mm )(40 15)C (b) Percentage error in measurement at 40C = increasein length % originallength.10 = 0.045% 46

258 5. A pendulum of a grandfather clock is.0 m long and made of steel. Determine the change in length of the pendulum if the temperature rises by 15 K. Assume the coefficient of linear expansion of steel to be K 1. Length L = L 1 [1 + (t - t 1 )] = L 1 + L 1 (t - t 1 ) Hence, increase in length = L 1 (t - t 1 ) = (.0 m)( K 1 )(15 K) = (.0)( )(15) = m or 0.45 mm 6. A temperature control system is operated by the expansion of a zinc rod which is 00 mm long at 15C. If the system is set so that the source of heat supply is cut off when the rod has expanded by 0.0 mm, determine the temperature to which the system is limited. Assume the coefficient of linear expansion of zinc to be K 1. Length L 1 = 00 mm = 0.0 m, L = mm = 00. mm = 0.00, temperature t 1 = 15C Length L = L 1 [1 + (t - t 1 )] = L 1 + L 1 (t - t 1 ) Hence, increase in length = L 1 (t - t 1 ) i.e = (0.0)( )( t - 15) = (0.0)( )( t - 15) i.e. ( t - 15) = =.6C 6 (0.0)(110 ) i.e. the temperature to which the system is limited, t = = 47.6C 47

259 7. A length of steel railway line is 0.0 m long when the temperature is 88 K. Determine the increase in length of the line when the temperature is raised to 0 K. Assume the coefficient of linear expansion of steel to be K 1. Length L = L 1 [1 + (t - t 1 )] = L 1 + L 1 (t - t 1 ) Hence, increase in length = L 1 (t - t 1 ) = (0.0 m)( K 1 )(0-88)K = (0.0)( )(15) = m or 6.75 mm 8. A brass shaft is 15.0 mm in diameter and has to be inserted in a hole of diameter 15.0 mm. Determine by how much the shaft must be cooled to make this possible, without using force. Take the coefficient of linear expansion of brass as K 1. Length L 1 = 15.0 mm = m, L = 15 mm = m Length L = L 1 [1 + (t - t 1 )] i.e = [1 + ( K 1 )(t - t 1 )] = (0.0150)( )( t - t 1 ) i.e = (0.0150)( )( t - t 1 ) Hence, (t - t 1 ) = (0.0150) = K i.e. the shaft must be cooled by 74 K 48

260 EXERCISE 141, Page 1 1. A silver plate has an area of 800 mm at 15C. Determine the increase in the area of the plate when the temperature is raised to 100C. Assume the coefficient of linear expansion of silver to be K 1. A = A 1 [1 + (t - t 1 )] i.e. A = A 1 [1 + (t - t 1 )] since =, to a very close approximation i.e. A = A 1 + A 1 (t - t 1 ) Hence, area increase = A 1 (t - t 1 ) = m 1910 K C = = m or.584 mm. At 8 K a thermometer contains 440 mm of alcohol. Determine the temperature at which the volume is 480 mm assuming that the coefficient of cubic expansion of the alcohol is K 1. V = V 1 [1 + (t - t 1 )] i.e = [1 + ( )(t - 8)] from which, 480 = ( )(t - 8) and = 440( )(t - 8) 40 from which, (t - 8) = = K and temperature, t = = 58.8 K 49

261 . A zinc sphere has a radius of 0.0 mm at a temperature of 0C. If the temperature of the sphere is raised to 40C, determine the increase in: (a) the radius, (b) the surface area, (c) the volume of the sphere. Assume the coefficient of linear expansion for zinc to be K 1. (a) Initial radius, L 1 = 0.0 mm, initial temperature, t 1 = = 9 K, final temperature, t = = 69 K and = K 1. New radius at 69 K is given by: L = L 1 [1 + (t - t 1 )] i.e. L = 0.0[1 + ( )(69-9)] = 0.0[ ] = 0.7 mm Hence the increase in the radius is 0.7 mm (b) Initial surface area of sphere, A 1 = 4r = = 600 mm New surface area at 69 K is given by: A = A 1 [1 + (t - t 1 )] i.e. A = A 1 [1 + (t - t 1 )] since =, to a very close approximation Thus A = 600[1 + ( )(400)] = 600[ ] = (0.048) Hence increase in surface area = 600(0.048) = 80.5 mm (c) Initial volume of sphere, V 1 = 4 r = mm New volume at 69 K is given by: V = V 1 [1 + (t - t 1 )] i.e. V = V 1 [1 + (t - t 1 )] since =, to a very close approximation 50

262 4 Thus V = (0.0) [1 + ( )(400)] = 4 (0.0) [ ] = 4 (0.0) + 4 (0.0) (0.07) Hence, the increase in volume = 4 (0.0) (0.07) = 407 mm 4. A block of cast iron has dimensions of 50 mm by 0 mm by 10 mm at 15C. Determine the increase in volume when the temperature of the block is raised to 75C. Assume the coefficient of linear expansion of cast iron to be K 1. Initial volume of sphere, V 1 = = mm New volume at 75C is given by: V = V 1 [1 + (t - t 1 )] i.e. V = V 1 [1 + (t - t 1 )] since =, to a very close approximation Thus V = [1 + ( )(75-15)] = 15000[ ] = ( ) Hence, the increase in volume = ( ) = 9.7 mm 5. Two litres of water, initially at 0C, is heated to 40C. Determine the volume of water at 40C if the coefficient of volumetric expansion of water within this range is K 1. New volume at 40C is given by: V = V 1 [1 + (t - t 1 )] = [1 + ( )(40 0)] = [ ] = [1.006] =.01 litres 51

263 6. Determine the increase in volume, in litres, of m of water when heated from 9 K to boiling point if the coefficient of cubic expansion is K 1 (1 litre 10 m ). Initial volume of sphere, V 1 = 10 = 000 litres New volume at boiling point (i.e. 7 K) is given by: V = V 1 [1 + (t - t 1 )] Thus V = 000 [1 + ( )(7-9)] = 000[ ] = (0.0168) Hence, the increase in volume = 000 (0.0168) = 50.4 litres 7. Determine the reduction in volume when the temperature of 0.5 litre of ethyl alcohol is reduced from 40C to - 15C. Take the coefficient of cubic expansion for ethyl alcohol as K 1. New volume at - 15C is given by: V = V 1 [1 + (t - t 1 )] Thus V = 0.5 [1 + ( )( )] = 0.5 [1 + ( )(- 55)] = (0.5)( )(- 55) Hence, the reduction in volume = (0.5)( )(55)= litres 5

264 EXERCISE 14, Page 1 Answers found from within the text of the chapter, pages 06 to 1. EXERCISE 14, Page 1 1. (b). (c). (a) 4. (d) 5. (b) 6. (c) 7. (c) 8. (a) 9. (c) 10. (b) 5

265 CHAPTER 1 IDEAL GAS LAWS EXERCISE 144, Page The pressure of a mass of gas is increased from 150 kpa to 750 kpa at constant temperature. Determine the final volume of the gas, if its initial volume is 1.5 m. Since the change occurs at constant temperature (i.e. an isothermal change), Boyle's law applies, i.e. p 1 V 1 = p V where p 1 = 150 kpa, p = 750 kpa and V 1 = 1.5 m Hence, (150)(1.5) = (750)V from which, volume V = = 0. m. In an isothermal process, a mass of gas has its volume reduced from 50 cm to cm. If the initial pressure of the gas is 80 kpa, determine its final pressure. Since the change occurs at constant temperature (i.e. an isothermal change), Boyle's law applies, i.e. p 1 V 1 = p V where p 1 = 80 kpa, V 1 = 50 cm and V = cm Hence, (80)(50) = (p )() from which, pressure p = 8050 = 15 kpa. The piston of an air compressor compresses air to 1 of its original volume during its stroke. 4 Determine the final pressure of the air if the original pressure is 100 kpa, assuming an isothermal change. Since the change occurs at constant temperature (i.e. an isothermal change), Boyle's law applies, 54

266 i.e. p 1 V 1 = p V where p 1 = 100 kpa, V = 1 4 V 1 Hence, (100)( V 1 ) = (p )( 1 4 V ) V1 100 from which, pressure p = = 400 kpa 1 1 V A quantity of gas in a cylinder occupies a volume of m at a pressure of 00 kpa. A piston slides in the cylinder and compresses the gas, according to Boyle's law, until the volume is 0.5 m. If the area of the piston is 0.0 m, calculate the force on the piston when the gas is compressed. An isothermal process means constant temperature and thus Boyle's law applies, i.e. p 1 V 1 = p V where V 1 = m, V = 0.5 m and p 1 = 00 kpa. Hence, (00)() = p (0.5) from which, pressure, p = = 100 kpa Pressure = force area, from which, force = pressure area. Hence, force on the piston = ( Pa)(0.0 m ) = 4000 N = 4 kn 5. The gas in a simple pump has a pressure of 400 mm of mercury (Hg) and a volume of 10 ml. If the pump is compressed to a volume of ml, calculate the pressure of the gas, assuming that its temperature does not change? Since PV 1 1 PV then (400 mm Hg)(10 ml) = ( P )( ml) 55

267 from which, new pressure, P = 400 mm Hg10 ml ml = 000 mm of mercury 56

268 EXERCISE 145, Page Some gas initially at 16C is heated to 96C at constant pressure. If the initial volume of the gas is 0.8 m, determine the final volume of the gas. Since the change occurs at constant pressure (i.e. an isobaric process), Charles law applies, V1 V i.e. T T 1 where V 1 = 0.8 m, T 1 = 16C = (16 + 7)K = 89 K and T = (96 + 7)K = 69 K. 0.8 V Hence, from which, volume at 96C, V = (0.8)(69) 89 = 1.0 m. A gas is contained in a vessel of volume 0.0 m at a pressure of 00 kpa and a temperature of 15C. The gas is passed into a vessel of volume m. Determine to what temperature the gas must be cooled for the pressure to remain the same. Since the process is isobaric it takes place at constant pressure and hence Charles law applies, V1 V i.e. T T 1 where T 1 = (15 + 7)K = 88 K and V 1 = 0.0 m and V = m Hence T from which, final temperature, T = (0.015)(88) 0.0 = 16 K or (16-7)C i.e. - 57C. In an isobaric process gas at a temperature of 10C has its volume reduced by a sixth. Determine the final temperature of the gas. Since the process is isobaric it takes place at constant pressure and hence Charles law applies, 57

269 V1 V i.e. T T 1 where T 1 = (10 + 7)K = 9 K and V = 5 6 V 1 Hence 5 V V T from which, final temperature, T = 5 V = 7.5 K V 6 1 = (7.5-7)C = 54.5C 4. The volume of a balloon is 0 litres at a temperature of 7ºC. If the balloon is under a constant internal pressure, calculate its volume at a temperature of 1ºC. V1 V Since T T where T 1 = (7 + 7)K = 00 K 1 and T = (1 + 7)K = 85 K 0litres V Hence, 00 K 85K and new volume, V = 0litres 85K = 8.5 litres 00 K 58

270 EXERCISE 146, Page 0 1. Gas, initially at a temperature of 7C and pressure 100 kpa, is heated at constant volume until its temperature is 150C. Assuming no loss of gas, determine the final pressure of the gas. p1 p Since the gas is at constant volume, the pressure law applies, i.e. T T 1 where T 1 = (7 + 7)K = 00 K, T = ( )K = 4 K and p 1 = 100 kpa 100 p Hence, 00 4 from which, final pressure, p = (100)(4) 00 = 141 kpa. A pressure vessel is subjected to a gas pressure of 8 atmospheres at a temperature of 15ºC. The vessel can withstand a maximum pressure of 8 atmospheres. Calculate the gas temperature increase the vessel can withstand. P1 P Since T T where T 1 = (15 + 7)K = 88 K 1 Hence, 8atmospheres 8atmospheres 88K T from which, new temperature, T = 8atmospheres 88K 8atmospheres = 1008 K or (1008 7)ºC = 75ºC Hence, temperature rise = ( )K = 70 K or temperature rise = (75 15)ºC = 70ºC Note that a temperature change of 70 K = a temperature change of 70ºC 59

271 EXERCISE 147, Page 1 1. A gas A in a container exerts a pressure of 10 kpa at a temperature of 0C. Gas B is added to the container and the pressure increases to 00 kpa at the same temperature. Determine the pressure that gas B alone exerts at the same temperature. Initial pressure, p A = 10 kpa, and the pressure of gases A and B together, p = p A + p B = 00 kpa By Dalton's law of partial pressure, the pressure of gas B alone is p B = p - p A = = 180 kpa 60

272 EXERCISE 148, Page 1. A gas occupies a volume of 1.0 m when at a pressure of 10 kpa and a temperature of 90C. Determine the volume of the gas at 0C if the pressure is increased to 0 kpa. pv 1 1 pv Using the combined gas law: T T 1 where V 1 = 1.0 m, p 1 = 10 kpa, p = 0 kpa, T 1 = (90 + 7)K = 6 K and (10)(1.0) (0) V T = (0 + 7)K = 9 K, gives: 6 9 from which, volume at 0C, V = (10)(1.0)(9) (6)(0) = 0.6 m. A given mass of air occupies a volume of 0.5 m at a pressure of 500 kpa and a temperature of 0C. Find the volume of the air at STP. pv 1 1 pv Using the combined gas law: T T 1 where V 1 = 0.5 m, p 1 = 500 kpa, T 1 = (0 + 7)K = 9 K, p = kpa, and (500)(0.5) (101.5) V T = 0C = 7 K, gives: 9 7 from which, volume at STP, V = (500)(0.5)(7) (9)(101.5) =.0 m. A balloon is under an internal pressure of 110 kpa with a volume of 16 litres at a temperature of ºC. If the balloon s internal pressure decreases to 50 kpa, what will be its volume if the temperature also decreases to 1ºC. 61

273 pv 1 1 pv Using the combined gas law: T T 1 where V 1 = 16 litres, p 1 = 110 kpa, p = 50 kpa, T 1 = ( + 7)K = 95 K and (110)(16) (50) V T = (1 + 7)K = 85 K, gives: from which, volume at 1C, V = (110)(16)(85) (95)(50) = 4.0 litres 4. A spherical vessel has a diameter of.0 m and contains hydrogen at a pressure of 00 kpa and a temperature of - 0C. Determine the mass of hydrogen in the vessel. Assume the characteristic gas constant R for hydrogen is 4160 J/(kg K). From the characteristic gas equation, pv = mrt where p = 00 kpa, V = = m, T = ( )K = 4 K, and R = 4160 J/(kg K). Hence (00 10 )(4.1888) = m(4160)(4) from which, mass of air, m = (0010 )(4.1888) (4160)(4) = 1.4 kg 5. A cylinder 00 mm in diameter and 1.5 m long contains oxygen at a pressure of MPa and a temperature of 0C. Determine the mass of oxygen in the cylinder. Assume the characteristic gas constant for oxygen is 60 J/(kg K). From the characteristic gas equation, pv = mrt where p = MPa, V = 0.1 (1.5) = m, T = (0 + 7)K = 9 K, and R = 60 J/(kg K). Hence ( 10 6 )(0.0471) = m(60)(9) 6

274 from which, mass of air, m = 6 (10 )(0.0471) (60)(9) = 1.4 kg 6. A gas is pumped into an empty cylinder of volume 0.1 m until the pressure is 5 MPa. The temperature of the gas is 40C. If the cylinder mass increases by 5. kg when the gas has been added, determine the value of the characteristic gas constant. From the characteristic gas equation, pv = mrt from which, R = pv mt where p = Pa, V = 0.1 m, T = (40 + 7)K = 1 K and m = 5. kg Hence, the characteristic gas constant, R = pv mt 5. 1 = 00 J/(kg K) 7. The mass of a gas is 1. kg and it occupies a volume of 1.45 m at STP. Determine its characteristic gas constant. From the characteristic gas equation, pv = mrt from which, R = pv mt where m = 1. kg, V = 1.45 m, p = kpa,, T = 0C = 7 K Hence, the characteristic gas constant, R = pv mt 1. 7 = 4160 J/(kg K) 8. 0 cm of air initially at a pressure of 500 kpa and temperature 150C is expanded to a volume of 100 cm at a pressure of 00 kpa. Determine the final temperature of the air, assuming no losses during the process. 6

275 pv 1 1 pv Using the combined gas law: T T 1 where V 1 = 0 cm, V = 100 cm, p 1 = 500 kpa, p = 00 kpa, and T 1 = ( )K = 4 K, gives: (500)(0) (00)(100) 4 T from which, final temperature, T = (00)(100)(4) (500)(0) = 564 K or (564 7)C = 91C 9. A quantity of gas in a cylinder occupies a volume of 0.05 m at a pressure of 400 kpa and a temperature of 7C. It is compressed according to Boyle's law until its pressure is 1 MPa, and then expanded according to Charles' law until its volume is 0.0 m. Determine the final temperature of the gas. pv 1 1 pv Using the combined gas law: T T 1 where V 1 = 0.05 m, V = 0.0 m, p 1 = 400 kpa, p = 1 MPa = 1000 kpa, and T 1 = (7 + 7)K = 00 K, gives: (0.05)(400) (0.0)(1000) 00 T from which, final temperature, T = (0.0)(1000)(00) (0.05)(400) = 450 K or 177C 10. Some air at a temperature of 5C and pressure bar occupies a volume of 0.08 m. Determine the mass of the air assuming the characteristic gas constant for air to be 87 J/(kg K). (1 bar = 10 5 Pa) From the characteristic gas equation, pv = mrt where T = (5 + 7)K = 08 K, p = bar = 5 10 Pa, V = 0.08 m and R = 87 J/(kg K). 5 Hence ( 10 )(0.08) = m(87)(08) 64

276 from which, mass of air, m = 5 (10 )(0.08) (87)(08) = kg 11. Determine the characteristic gas constant R of a gas that has a specific volume of 0.67 m /kg at a temperature of 17C and pressure 00 kpa. From the characteristic gas equation, pv = mrt from which, R = pv mt where p = Pa, T = (17 + 7)K = 90 K and specific volume, V/m = 0.67 m /kg. p V 0010 T m 90 = 184 J/(kg K) Hence the characteristic gas constant, R =

277 EXERCISE 149, Page 5 1. A vessel P contains gas at a pressure of 800 kpa at a temperature of 5C. It is connected via a valve to vessel Q that is filled with similar gas at a pressure of 1.5 MPa and a temperature of 5C. The volume of vessel P is 1.5 m and that of vessel Q is.5 m. Determine the final pressure at 5C when the valve is opened and the gases are allowed to mix. Assume R for the gas to be 97 J/(kg K). For vessel P: p P = Pa, T P = (5 + 7)K = 98 K, V P = 1.5 m and R = 97 J/(kg K) From the characteristic gas equation, p P V P = m P RT P Hence ( )(1.5) = m P (97)(98) from which, mass of gas in vessel P, m P = For vessel Q: (80010 )(1.5) (97)(98) = kg p Q = Pa, T Q = (5 + 7)K = 98 K, V Q =.5 m and R = 97 J/(kg K) From the characteristic gas equation, p Q V Q = m Q RT Q Hence ( )(.5) = m Q (97)(98) from which, mass of gas in vessel Q, m Q = 6 (1.510 )(.5) (97)(98) = 4.70 kg When the valve is opened, mass of mixture, m = m P + m Q = = kg. Total volume, V = V P + V Q = = 4.0 m, R = 97 J/(kg K), T = 98 K. From the characteristic gas equation, pv = mrt p(4.0) = (55.98)(97)(98) 66

278 from which, final pressure, p = (55.98)(97)(98) 4.0 = 1.4 MPa. A vessel contains 4 kg of air at a pressure of 600 kpa and a temperature of 40C. The vessel is connected to another by a short pipe and the air exhausts into it. The final pressure in both vessels is 50 kpa and the temperature in both is 15C. If the pressure in the second vessel before the air entered was zero, determine the volume of each vessel. Assume R for air is 87 J/(kg K). For vessel 1: m 1 = 4 kg, p 1 = Pa, T 1 = (40 + 7)K = 1 K and R = 87 J/(kg K) From the characteristic gas equation, p 1 V 1 = m 1 RT 1 Hence ( ) V 1 = (4)(87)(1) (4)(87)(1) from which, volume of vessel 1, V 1 = (60010 ) From the characteristic gas equation, pv = mrt (50 10 ) V Total = (4)(87)(15 + 7) = 0.60 m (4)(87)(88) from which, total volume, V Total = 5010 = 1. m Hence, volume of vessel, V = = 0.7 m. A vessel has a volume of 0.75 m and contains a mixture of air and carbon dioxide at a pressure of 00 kpa and a temperature of 7C. If the mass of air present is 0.5 kg determine (a) the partial pressure of each gas, and (b) the mass of carbon dioxide. Assume the characteristic gas constant for air to be 87 J/(kg K) and for carbon dioxide 184 J/(kg K). (a) V = 0.75 m, p = 00 kpa, T = (7 + 7)K = 00 K, m air = 0.50 kg, R air = 87 J/(kg K). 67

279 If p air is the partial pressure of the air, then using the characteristic gas equation, p air V = m air R air T gives: (p air )(0.75) = (0.50)(87)(00) from which, the partial pressure of the air, p air = (0.50)(87)(00) (0.75) = 57.4 kpa By Dalton's law of partial pressure the total pressure p is given by the sum of the partial pressures, i.e. p = p air + p CO, from which, the partial pressure of the carbon dioxide, p CO (b) From the characteristic gas equation, p V = m R T CO CO CO Hence, ( )(0.75) = m (184)(00) CO = p - p air = = 14.6 kpa from which, mass of hydrogen, m ( )(0.75) (184)(00) = CO = 1.94 kg 4. A mass of gas occupies a volume of 0.0 m when its pressure is 150 kpa and its temperature is 17C. If the gas is compressed until its pressure is 500 kpa and its temperature is 57C, determine (a) the volume it will occupy and (b) its mass, if the characteristic gas constant for the gas is 05 J/(kg K). pv 1 1 pv (a) Using the combined gas law: T T 1 where V 1 = 0.0 m, p 1 = 150 kpa, T 1 = (17 + 7)K = 90 K, p = 500 kpa, and (150)(0.0) (500) V T = (57 + 7)K = 0 K, gives: 90 0 from which, volume at STP, V = (150)(0.0)(0) (90)(500) = m (b) From the characteristic gas equation, pv = mrt where T = 90 K, p = 150 kpa, V = 0.0 m and R = 05 J/(kg K). 68

280 Hence (150000)(0.0) = m(05)(90) from which, mass of air, m = (150000)(0.0) (05)(90) = kg 5. A compressed air cylinder has a volume of 0.6 m and contains air at a pressure of 1. MPa absolute and a temperature of 7C. After use the pressure is 800 kpa absolute and the temperature is 17C. Calculate (a) the mass of air removed from the cylinder, and (b) the volume the mass of air removed would occupy at STP conditions. Take R for air as 87 J/(kg K) and atmospheric pressure as 100 kpa. (a) From the characteristic gas equation, p 1 V 1 = mrt 1 from which, mass of air, m = 6 p1v 1 (1.10 )(0.6) R T (87)(7 7) 1 = kg Also, p V = mrt from which, mass of air, m = pv (80010 )(0.6) R T (87)(17 7) = kg Hence, the mass of air removed from the cylinder = =. kg (b) From the characteristic gas equation, pv = mrt where p = 100 kpa, m =. kg, R = 87 J/(kg K) and T = (0 + 7)K = 7 K Hence, (100000)(V) = (.)(87)(7) from which, the volume the mass of air removed would occupy at STP conditions, V = = 1.8 m 69

281 EXERCISE 150, Page 5 Answers found from within the text of the chapter, pages 15 to 5. EXERCISE 151, Page 5 1. (a). (d). (b) 4. (b) 5. (c) 6. (d) 7. (b) 8. (c) 9. (c) 10. (b) 70

282 CHAPTER THE MEASUREMENT OF TEMPERATURE EXERCISE 15, Page 1 1. A platinum-platinum/rhodium thermocouple generates an e.m.f. of 7.5 mv. If the cold junction is at a temperature of 0C, determine the temperature of the hot junction. Assume the sensitivity of the thermocouple to be 6 V/C Temperature difference for 7.5 mv = V V/ C = 150C Temperature at hot junction = temperature of cold junction + temperature difference = 0C + 150C = 170C 71

283 EXERCISE 15, Page 1. A platinum resistance thermometer has a resistance of 100 at 0C. When measuring the temperature of a heat process a resistance value of 177 is measured using a Wheatstone bridge. Given that the temperature coefficient of resistance of platinum is 0.008/C, determine the temperature of the heat process, correct to the nearest degree. R = R 0 (1 + ), where R 0 = 100, R = 177 and = 0.008/C. Rearranging gives: i.e. R = R 0 (1 + ) = R 0 + R 0 R - R 0 = R 0 R R0 and temperature, = R 0 = (0.008)(100) = 0C EXERCISE 154, Page 6 Answers found from within the text of the chapter, pages 7 to 6. EXERCISE 155, Page 6 1. (c). (b). (d) 4. (b) 5. (i) 6. (a) 7. (e) 8. (d) 9. (e) or (f) 10. (k) 11. (b) 1. (g) 7

284 CHAPTER AN INTRODUCTION TO ELECTRIC CIRCUITS EXERCISE 156, Page In what time would a current of 10 A transfer a charge of 50 C? Q = I t hence, time, t = Q 50 = 5 s I 10. A current of 6 A flows for 10 minutes. What charge is transferred? Charge, Q = I t = 6 (10 60) = 600 C. How long must a current of 100 ma flow so as to transfer a charge of 80 C? Q = I t hence, time, t = Q 80 I = 800 s = min = 1 min 0 s 7

285 EXERCISE 157, Page The current flowing through a heating element is 5 A when a p.d. of 5 V is applied across it. Find the resistance of the element. Resistance, R = V 5 = 7 Ω I 5. An electric light bulb of resistance 960 is connected to a 40 V supply. Determine the current flowing in the bulb. Current, I = V 40 1 = 0.5 A R Graphs of current against voltage for two resistors P and Q are shown below. Determine the value of each resistor. For resistor P, R = 6 V 16V 1610 V I 8mA 810 A = 10 = m For resistor Q, R = 6 V 0V 010 V I 4mA 410 A = 510 = 5 m 74

286 4. Determine the p.d. which must be applied to a 5 k resistor such that a current of 6 ma may flow. P.d., V = I R = = 0 V 75

287 EXERCISE 158, Page The hot resistance of a 50 V filament lamp is 65. Determine the current taken by the lamp and its power rating. Current, I = V 50 = 0.4 A I 65 Power rating, P = V I = = 100 W (or P = V 50 R 65 = 100 W or P = I R = 100 W). Determine the resistance of an electric fire which takes a current of 1 A from a 40 V supply. Find also the power rating of the fire and the energy used in 0 h. Resistance, R = V 40 = 0 I 1 Power rating, P = V I = 40 1 = 880 W or.88 kw Energy = power time =.88 kw 0 h = 57.6 kwh. Determine the power dissipated when a current of 10 ma flows through an appliance having a resistance of 8 k. Power, P = IR = 0.8 W J of energy are converted into heat in nine seconds. What power is dissipated? Power, P = energy 85.5J = 9.5 W time 9s 76

288 5. A current of 4 A flows through a conductor and 10 W is dissipated. What p.d. exists across the ends of the conductor? Power, P = V I from which, p.d., V = P 10 =.5 V I 4 6. Find the power dissipated when: (a) a current of 5 ma flows through a resistance of 0 k (b) a voltage of 400 V is applied across a 10 k resistor (c) a voltage applied to a resistor is 10 kv and the current flow is 4 ma (a) Power, P = IR = 0.5 W (b) Power, P = V 400 R 1010 = 1. W (c) Power, P = V I = = 40 W 7. A d.c. electric motor consumes 7 MJ when connected to 400 V supply for h 0 min. Find the power rating of the motor and the current taken from the supply. Power = 6 energy 710 J time = 8000 W = 8 kw = power rating of motor Power, P = V I, hence, current, I = P 810 = 0 A V A p.d. of 500 V is applied across the winding of an electric motor and the resistance of the winding is 50. Determine the power dissipated by the coil. Power, P = V 500 = 5000 W = 5 kw R 50 77

289 9. In a household during a particular week three kw fires are used on average 5 h each and eight 100 W light bulbs are used on average 5 h each. Determine the cost of electricity for the week if 1 unit of electricity costs 15p Energy in week = ( kw 5 h) + 8( Cost = = 670p = kw 5 h) = = 178 kwh 10. Calculate the power dissipated by the element of an electric fire of resistance 0 when a current of 10 A flows in it. If the fire is on for 0 hours in a week determine the energy used. Determine also the weekly cost of energy if electricity costs 1.50p per unit. Power, P = I R 10 0 = 000 W or kw Energy = power time = kw 0 h = 90 kwh Cost = p = 115p =

290 EXERCISE 159, Page A television set having a power rating of 10 W and electric lawnmower of power rating 1 kw are both connected to a 40 V supply. If A, 5 A and 10 A fuses are available state which is the most appropriate for each appliance. Power, P = V I hence, current, I = P V For the television, I = P 10 = 0.5 A, hence the A fuse is the most appropriate V 40 For the lawnmower, I = P 1000 = 4.17 A, hence the 5 A fuse is the most appropriate V 40 EXERCISE 160, Page 50 Answers found from within the text of the chapter, pages 41 to 50. EXERCISE 161, Page (d). (a). (c) 4. (b) 5. (d) 6. (d) 7. (b) 8. (c) 9. (a) 10. (a) 11. (c) 1. (c) 1. (b) 14. (a) 15. (c) 16. (b) 17. (d) 18. (d) 79

291 CHAPTER 4 RESISTANCE VARIATION EXERCISE 16, Page The resistance of a m length of cable is.5. Determine (a) the resistance of a 7 m length of the same cable and (b) the length of the same wire when the resistance is 6.5. (a) If the resistance of a m length of cable is.5, then a 1 m length of cable is 1.5 Thus, the resistance of a 7 m length of cable is = 8.75 (b) If the resistance of a m length of cable is.5, then a.5 m length of cable is 1 Thus, a resistance of 6.5 corresponds to a length of 6.5 m = 5 m.5. Some wire of cross-sectional area 1 mm has a resistance of 0. Determine (a) the resistance of a wire of the same length and material if the cross-sectional area is 4 mm, and (b) the cross- sectional area of a wire of the same length and material if the resistance is. 1 (a) R thus a wire of cross-sectional area 4 mm has a resistance of 0 a 4 = 5 (b) Since wire of cross-sectional area 1 mm has a resistance of 0, then a c.s.a. of 0 mm has a resistance of 1. Hence, a resistance of corresponds to a c.s.a. of 0 = 0.65 mm. Some wire of length 5 m and cross-sectional area mm has a resistance of If the wire is drawn out until its cross-sectional area is 1 mm, determine the resistance of the wire. l (5) R i.e from which, resistivity, a If c.s.a. = 1 mm (i.e. half the original c.s.a.) then the length will double, i.e. l = 5 = 10 m 80

292 Hence, resistance, l R a = Find the resistance of 800 m of copper cable of cross-sectional area 0 mm. Take the resistivity of copper as 0.0 m Resistance, m800m l R a 010 m 6 = Calculate the cross-sectional area, in mm, of a piece of aluminium wire 100 m long and having 6 a resistance of. Take the resistivity of aluminium as m l Since R then c.s.a., a = a m100 m l m R 6 = 1.5 mm 6. The resistance of 500 m of wire of cross-sectional area.6 mm is 5. Determine the resistivity of the wire in m l Since R then resistivity, ρ = a Ra l m =.6 500m 8 10 m or 0.06 m 7. Find the resistance of 1 km of copper cable having a diameter of 10 mm if the resistivity of copper is m. Resistance, m110 m l l R a r 5 10 m 6 =

293 EXERCISE 16, Page A coil of aluminium wire has a resistance of 50 when its temperature is 0C. Determine its resistance at 100C if the temperature coefficient of resistance of aluminium at 0C is 0.008/C Resistance at 100C, R R 1 (100) (0.008) = A copper cable has a resistance of 0 at a temperature of 50C. Determine its resistance at 0C. Take the temperature coefficient of resistance of copper at 0C as 0.004/C. R R 1 (50) from which, R resistance at 0C, R (0.004) = The temperature coefficient of resistance for carbon at 0C is /C. What is the significance of the minus sign? A carbon resistor has a resistance of 500 at 0C. Determine its resistance at 50C. For carbon, resistance falls with increase of temperature, hence the minus sign. R R 1 (50) ( ) = A coil of copper wire has a resistance of 0 at 18C. If the temperature coefficient of resistance of copper at 18C is 0.004/C, determine the resistance of the coil when the temperature rises to 98 o C Resistance at C, R = R18 [1 + 18( - 18)] Hence, resistance at 98C, R98 = 0 [1 + (0.004)(98-18)] = 0 [1 + (0.004)(80)] 8

294 = 0 [1 + 0.] = 0(1.) = The resistance of a coil of nickel wire at 0C is 100. The temperature of the wire is increased and the resistance rises to 10. If the temperature coefficient of resistance of nickel is 0.006/C at 0C, determine the temperature to which the coil has risen. R R 1 ( 0) 0 0 i.e i.e = 0.6 ( - 0) and ( - 0) = Hence, temperature to which the coil has risen, = = 70C 6. Some aluminium wire has a resistance of 50 at 0C. The wire is heated to a temperature of 100C. Determine the resistance of the wire at 100C, assuming that the temperature coefficient of resistance at 0C is 0.004/C R0 = 50, 0 = 0.004/C and R 1 0 0(0) R 1 (100) Hence, R 100 R = 50[1 100(0.004)] [1 0(0.004)] = 50[1 0.40] [1 0.08] = 50(1.40) (1.08) = 64.8 i.e. the resistance of the wire at 100C is A copper cable is 1. km long and has a cross-sectional area of 5 mm. Find its resistance at 80C if at 0C the resistivity of copper is m and its temperature coefficient of resistance is 0.004/C. 8

295 Resistance at 0C, R l a m1.10 m m = 4.8 Resistance at 80C, R R 1 (80 0) = 5.95 EXERCISE 164, Page 58 Answers found from within the text of the chapter, pages 5 to 58. EXERCISE 165, Page (c). (d). (b) 4. (d) 5. (d) 6. (c) 7. (b) 84

296 CHAPTER 5 BATTERIES AND ALTERNATIVE SOURCES OF ENERGY EXERCISE 166, Page Twelve cells, each with an internal resistance of 0.4 and an e.m.f. of 1.5 V are connected (a) in series, (b) in parallel. Determine the e.m.f. and internal resistance of the batteries so formed. (a) Total e.m.f. in series = = 18 V Total internal resistance in series = =.88 (b) Total e.m.f. in parallel = 1.5 V Total internal resistance in parallel = = A cell has an internal resistance of 0.0 and an e.m.f. of. V. Calculate its terminal p.d. if it delivers (a) 1 A, (b) 0 A, (c) 50 A. (a) Terminal p.d., V = E Ir =. (1)(0.0) =.17 V (b) Terminal p.d., V = E Ir =. (0)(0.0) =. 0.6 = 1.6 V (c) Terminal p.d., V = E Ir =. (50)(0.0) =. 1.5 = 0.7 V. The p.d. at the terminals of a battery is 16 V when no load is connected and 14 V when a load taking 8 A is connected. Determine the internal resistance of the battery. When no load is connected the e.m.f. of the battery, E, is equal to the terminal p.d., V, i.e. E = 16 V When current I = 8 A and terminal p.d. V = 14 V, then V = E - Ir i.e. 14 = 16 - (8)r Hence, rearranging, gives 8r = = and the internal resistance, r = 1 =

297 4. A battery of e.m.f. 0 V and internal resistance 0. supplies a load taking 10 A. Determine the p.d. at the battery terminals and the resistance of the load. P.d. at battery terminals, V = E Ir = 0 10(0.) = 18 V Load resistance, R L V 18 = 1.8 I Ten. V cells, each having an internal resistance of 0.1 are connected in series to a load of 1. Determine (a) the current flowing in the circuit, and (b) the p.d. at the battery terminals. E.m.f., E = 10. = V, and internal resistance, r = = 1 (a) Current, I = E rr 11 L = 1 A (b) P.d. at the battery terminals, V = E Ir = (1)(1) = 1 V (or V = I R L (1)(1) = 1 V) 86

298 6. For the circuits shown below the resistors represent the internal resistance of the batteries. Find, in each case: (i) the total e.m.f. across PQ (ii) the total equivalent internal resistances of the batteries. (a)(i) Total e.m.f. across PQ, E = = 6 V (ii) Total internal resistance, r = = 4 Ω (b)(i) Total e.m.f. across PQ, E = V (ii) Total internal resistance, r is given by = 1 r Hence, r = 1 4 = 0.5 Ω 7. The voltage at the terminals of a battery is 5 V when no load is connected and 48.8 V when a load taking 80 A is connected. Find the internal resistance of the battery. What would be the terminal voltage when a load taking 0 A is connected? 87

299 V = E Ir, hence 48.8 = 5 80 r from which, 80 r = =. and internal resistance, r =. 80 = 0.04 When I = 0 A, terminal voltage, V = 5 0(0.04) = = 51. V EXERCISE 167, Page 71 Answers found from within the text of the chapter, pages 60 to 71. EXERCISE 168, Page 7 1. (d). (a). (b) 4. (c) 5. (b) 6. (d) 7. (d) 8. (b) 9. (c) 10. (d) 11. (c) 1. (a) 1. (c) 88

300 CHAPTER 6 SERIES AND PARALLEL NETWORKS EXERCISE 169, Page The p.d. s measured across three resistors connected in series are 5 V, 7 V and 10 V, and the supply current is A. Determine (a) the supply voltage, (b) the total circuit resistance and (c) the values of the three resistors. (a) Supply voltage, V = = V (b) Total circuit resistance, R T V = 11 I V1 5 V 7 V 10 (c) R1 =.5, R =.5 and R = 5 I I I. For the circuit shown below, determine the value of V1. If the total circuit resistance is 6 determine the supply current and the value of resistors R1, R and R Supply voltage, 18 = V Hence, voltage, V 1 = 18 5 = 10 V Supply current, I = V 18 = 0. 5 A R 6 T 89

301 V1 10 V 5 V R 1 = = 0 Ω R = = 10 Ω R = = 6 Ω I 0.5 I 0.5 I 0.5. When the switch in the circuit shown is closed the reading on voltmeter 1 is 0 V and that on voltmeter is 10 V. Determine the reading on the ammeter and the value of resistor R X Voltage across 5 resistor = V1V 0 10 = 0 V Hence, current in 5 resistor, i.e. reading on the ammeter = V5 0 = 4 A 5 5 Total resistance, V 0, hence R X = =.5 I 4 T RT Calculate the value of voltage V in the diagram below. 5 Voltage, V = (7) = 45 V 5 5. Two resistors are connected in series across an 18 V supply and a current of 5 A flows. If one of the resistors has a value of.4 determine (a) the value of the other resistor and (b) the p.d. across the.4 resistor. 90

302 The circuit is shown above. (a) Total resistance, 18 RT.6, hence R X =.6.4 = 1. 5 (b) V1 5.4 = 1 V 6. An arc lamp takes 9.6 A at 55 V. It is operated from a 10 V supply. Find the value of the stabilising resistor to be connected in series. A circuit diagram is shown below. The purpose of the stabilising resistor R S is to cause a volt drop V S in this case equal to 10 55, i.e. 65 V. Hence, R V 65 I 9.6 S S = An oven takes 15 A at 40 V. It is required to reduce the current to 1 A. Find (a) the resistor which must be connected in series, and (b) the voltage across the resistor. (a) If the oven takes 15 A at 40 V, then resistance of oven, R oven 40 = 16 A 15 91

303 A circuit diagram is shown above. If the current is reduced to 1 A then the total resistance of the circuit, R T V 40 = 0 I 1 and RT RS Roven i.e. 0 RS 16 from which, series resistor, RS 0 16 = 4 (b) Voltage across series resistor, VS IRS 1 4 = 48 9

304 EXERCISE 170, Page 8 1. Resistances of 4 and 1 are connected in parallel across a 9 V battery. Determine (a) the equivalent circuit resistance, (b) the supply current, and (c) the current in each resistor. (a) Equivalent circuit resistance, R T = (or use ) R 4 1 T (b) Supply current, I = V 9 = A R T 9 (c) I 1 =.5 A, I = 0.75 A (or, by current division, I =.5 A and 4 I 41 = 0.75 A). For the circuit shown determine (a) the reading on the ammeter, and (b) the value of resistor R. V 15 (a) V = 5 = 15 V. Hence, ammeter reading, I =.5 A 9

305 V 15 (b) I R = = 6 A hence, R = =.5 I 6. Find the equivalent resistance when the following resistance s are connected (a) in series (b) in Parallel (i) and (ii) 0 k and 40 k (iii) 4, 8 and 16 (iv) 800, 4 k and 1500 (a)(i) Total resistance, (ii) Total resistance, (iii) Total resistance, (iv) Total resistance, R T = + = 5 Ω R T = = 60 kω R T = = 8 Ω R T = = 600 Ω or 6. kω (b)(i) Total resistance, R T is given by: from which, R T = 6 R 6 5 = 1. Ω T (ii) Total resistance, R T is given by: from which, R T = 40 R T = 1. kω (iii) Total resistance, (iv) Total resistance, R T is given by: R T is given by: from which, R T = 16 R =.9 Ω T R T from which, R T = = Ω 4. Find the total resistance between terminals A and B of circuit (a) shown below. 618 Resistance of parallel branches, R p = 4.5 Ω

306 Total circuit resistance, R T = = 8 Ω 5. Find the equivalent resistance between terminals C and D of circuit (b) shown below. Resistance of first parallel branches, R P = 7.5 Ω Resistance of second parallel branches, i.e. R P is given by: R R P P = 5 Ω Total circuit resistance, R T = = 7.5 Ω 6. Resistors of 0, 0 and 0 are connected in parallel. What resistance must be added in series with the combination to obtain a total resistance of 10. If the complete circuit expends a power of 0.6 kw, find the total current flowing. The circuit is shown below. For the parallel branch, from which, RP 7.5 R P Hence, resistance to be added in series, RX RT RP =.5 Power, P = I R hence I (10) 95

307 from which, total current flowing, I = = 6 A 7. (a) Calculate the current flowing in the 0 resistor shown in the circuit below (b) What additional value of resistance would have to be placed in parallel with the 0 and 0 resistors, to change the supply current to 8 A, the supply voltage remaining constant. (a) Total resistance, 00 RT = 16 Hence, total current, I = and, by current division, V 64 = 4 A R 16 T 0 I = 1.6 A (b) If I = 8 A then new total resistance, will be: 8 4 = 4 R T 64 = 8 and the resistance of the parallel branch 8 i.e where R X is the additional resistance to be placed in parallel R X from which, from which, R X = 6 R X 96

308 8. For the circuit shown below, find (a) V1, (b) V, without calculating the current flowing (a) Voltage, V 1 = (b) Voltage, V = 5 (7) = 0 V by voltage division (7) = 4 V Determine the currents and voltages indicated in the circuit below from which, R 6 P 1 R P 1 = 1 R P = 1. Hence, total resistance, R T = = 6. I 1 = 1 6. = 5 A, V 1 = I(4) 1 5 4= 0 V, V = 5 1 = 5 V and V = 5 1. = 6 V V 5 I = =.5 A, I 5 = 1.67 A, I 4 = 0.8 A, 5 V 6 I = = A and I 6 = 6 = A 97

309 10. Find the current I in the circuit below. The circuit is reduced step by step as shown in diagrams (a) to (d) below. (a) (b) (c) (d) From (d), From (b), I T 4 = 6 A 4 6 I =.6 A and from (a), I.6 = 1.8 A 98

310 EXERCISE 171, Page If four identical lamps are connected in parallel and the combined resistance is 100, find the resistance of one lamp. If each lamp has a resistance of R then: and R = = 400 = resistance of a lamp 100 R R R R R. Three identical filament lamps are connected (a) in series, (b) in parallel across a 10 V supply. State for each connection the p.d. across each lamp. (a) In series, p.d. across each lamp = 10 = 70 V (b) In parallel, p.d. across each lamp = 10 V EXERCISE 17, Page 85 Answers found from within the text of the chapter, pages 74to 85. EXERCISE 17, Page (a). (c). (c) 4. (c) 5. (a) 6. (d) 7. (b) 8. (c) 9. (d) 10. (d) 99

311 CHAPTER 7 KIRCHHOFF S LAWS EXERCISE 174, Page 9 1. Find currents I, I 4 and I 6 in the circuit below. By Kirchhoff s law, I1 I I i.e. 4 = + I from which, I = 4 = A Also, I I4 I5 i.e. II4 I5 i.e. + I 4 = 1 from which, I 4 = 1 = - 1 A And I5 I6 I1 i.e. 1I6 4 from which, I 6 = 4 1 = A. For the networks shown below, find the values of the currents marked. (a) + I = from which, I = = - 1 A Also, 10 = 7 + I 1 I i.e. 10 = 7 + I 1 ( 1) from which, I 1 = = 4 A And 7 + I+ 1 = I i.e = I i.e. I = 1 A 400

312 (b) = I i.e. I = 60 A Also, I 1 I = 100 from which, I 100 I i.e. I = 40 A And = I i.e. I = 10 A And I = 0 + I 4 i.e. I 4 = I - 0 = 10 0 i.e. I 4 = 100 A And I 4 + I 5 = 0 i.e I 5 = 0 from which, I 5 = = - 80 A. Use Kirchhoff's laws to find the current flowing in the 6 resistor of the circuit below and the power dissipated in the 4 resistor. The currents are labelled as shown in the diagram below. Kirchhoff's voltage law is now applied to each loop in turn: For loop 1: 40 = 5I 1 + 4I (1) For loop : 0 = 4I - 6(I 1 - I ) () Equation () simplifies to: 0 = - 6I I () 5 equation (1) gives: 00 = 5I 1 + 0I (4) 401

313 equation () gives: 0 = - 1I 1 + 0I (5) Equation (4) - equation (5) gives: 00 = (5I I 1 ) i.e. 00 = 7I 1 Hence, current, I 1 = 00 7 = A Substituting I 1 = into equation (1) gives: 40 = 5(5.405) + 4I 40 = I and = 4I 1 from which, I = = =.48 A Hence, the current flowing in the 6 Ω resistance is i.e. I 1 - I = ( ) =.16 A Power dissipated in the 4 resistor = I R.48 4 = 4.09 W 4. Find the current flowing in the resistor for the network shown in circuit (a) below. Find also the p.d. across the 10 and resistors. The currents are labelled as shown in the diagram below. 40

314 Loop 1: 0 = I 1 + (6 + 10)I i.e. I I = 0 (1) Loop : 0 = I 1 + (I 1 - I )(4 + ) i.e. 9I 1-6I = 0 () equation (1) gives: 9I I = 60 () Equation () () gives: 54I = 40 from which, I = = A Substituting in (1) gives: I (0.7407) = 0 i.e. I 1 = = from which, I 1 = =.716 A i.e. the current flowing in the resistor =.716 A P.d. across the 10 Ω resistor = I 10 = = V P.d. across the Ω resistor = (I 1 - I ) = ( ) = =.951 V 5. For the network shown in circuit (b) below, find: (a) the current in the battery, (b) the current in the 00 resistor, (c) the current in the 90 resistor, and (d) the power dissipated in the 150 resistor. 40

315 The currents are labelled as shown in the diagram below. (a) Loop 1: 8 = 0I 1 + ( )I i.e. 0I I = 8 (1) Loop : 8 = 0I 1 + (I 1 - I )( ) i.e. 470I 1-450I = 8 () equation (1) gives: 60I I = 4 () Equation () + () gives: 50I 1 = from which, I 1 = 50 = A i.e. the current in the battery = 60.8 ma (b) Substituting in (1) gives: 0(0.0608) + 150I = 8 i.e. 150I = =

316 from which, I = = A = 45.8 ma Hence, current in 00 resistor = I 1 - I = = ma (c) The current in the 90 resistor = I = 45.8 ma (d) The power dissipated in the 150 resistor = (I 1 - I ) = = 0.04 W or 4.0 mw 6. For the bridge network shown in circuit (c) below, find the currents I 1 to I 5 From loop 1: 6.6I I I A A B From loop : 0 = 4 I 5I I A B A i.e. 5IA IB 6.6 (1) 405

317 and 7IA 5IB 8 () 5 (1) gives: 5IA 10IB () () gives: 14IA 10IB 16 (4) () (4) gives: 9I A = 49 and I A 49 = 1.56 A 9 Substituting in (1) gives: 5(1.56) + IB 6.6 from which, I B 6.6 5(1.56) = A Hence, correct to decimal places, I 1 = I = 1.6 A A I = - I A = 1.56 = 0.74 A I = I = 0.16 A B I 4 = I + A I B = = 1.4 A I 5 = - I - A I B = = 0.58 A EXERCISE 175, Page 9 Answers found from within the text of the chapter, pages 88 to 9. EXERCISE 176, Page 9 1. (a). (d). (c) 4. (b) 5. (c) 406

318 CHAPTER 8 MAGNETISM AND ELECTROMAGNETISM EXERCISE 177, Page What is the flux density in a magnetic field of cross-sectional area 0 mwb? cm having a flux of Flux density, B = A 010 m 10 Wb 4 = 1.5 T. Determine the total flux emerging from a magnetic pole face having dimensions 5 cm by 6 cm, if the flux density is 0.9 T. B from which, flux, A BA =.7 mwb. The maximum working flux density of a lifting electromagnet is 1.9 T and the effective area of a pole face is circular in cross-section. If the total magnetic flux produced is 611 mwb determine the radius of the pole face. B from which, A r r B and radius, r = B 1.9 = 0. m or cm 4. An electromagnet of square cross-section produces a flux density of 0.45 T. If the magnetic flux is 70 Wb find the dimensions of the electromagnet cross-section. B from which, area, A A m B 0.45 Let the side of the square section = x, then x m and side, x = m = 0.04 m = 4 cm 407

319 i.e. the dimensions of the electromagnet cross-section = 4 cm by 4 cm 408

320 EXERCISE 178, Page A conductor carries a current of 70 A at right-angles to a magnetic field having a a flux density of 1.5 T. if the length of the conductor in the field is 00 mm calculate the force acting on the conductor. What is the force when the conductor and field are at an angle of 45? Force, F = B I l sin = sin 90 = 1.0 N When = 45, F = 1.0 sin 45 = 14.8 N. Calculate the current required in a 40 mm length of conductor of a d.c. motor when the conductor is situated at right-angles to the magnetic field of flux density 1.5 T, if a force of 1.0 N is to be exerted on the conductor. Force, F = B I l sin i.e. 1.0 = 1.5 I sin 90º 1.0 from which, current, I = sin 90 = 4.0 A. A conductor 0 cm long is situated at right-angles to a magnetic field. Calculate the flux density of the magnetic field if a current of 15 A in the conductor produces a force on it of.6 N Force, F = B I l sin from which, flux density, B = F.6 Ilsin 150.0sin90 = 0.80 T 4. A conductor 00 mm long carries a current of 1 A and is at right-angles to a magnetic field between two circular pole faces, each of diameter 80 mm. If the total flux between the pole faces is 0.75 mwb calculate the force exerted on the conductor. When conductor and field are at right angles, force, F = B I l where B = A 409

321 Hence, force, F = Il A = 0.58 N 5. (a) A 400 mm length of conductor carrying a current of 5 A is situated at right-angles to a magnetic field between two poles of an electric motor. The poles have a circular cross-section. If the force exerted on the conductor is 80 N and the total flux between the pole faces is 1.7 mwb, determine the diameter of a pole face. (b) If the conductor in part (a) is vertical, the current flowing downwards and the direction of the magnetic field is from left to right, what is the direction of the 80 N force? (a) Force, F = B I l = Il Il A r from which, radius, r = Il F m = 7.1 mm Hence, diameter = r = 7.1 = 14. mm (b) By Fleming s left hand rule, the direction of the force is towards the viewer. 410

322 EXERCISE 179, Page Calculate the force exerted on a charge of C travelling at 6 10 m/s perpendicular to a field of density 7 10 T. Force, F = Q v B = = N. Determine the speed of a if the force on the charge is C charge travelling perpendicular to a field of flux density 10 T, 0 10 N. Force, F = Q v B from which, speed, v = 0 F 10 QB = 6 10 m/s EXERCISE 180, Page 405 Answers found from within the text of the chapter, pages 95 to 405. EXERCISE 181, Page (d). (d). (a) 4. (a) 5. (b) 6. (b) 7. (d) 8. (c) 9. (d) 10. (a) 11. (c) 1. (c) 411

323 CHAPTER 9 ELECTROMAGNETIC INDUCTION EXERCISE 18, Page A conductor of length 15 cm is moved at 750 mm/s at right-angles to a uniform flux density of 1. T. Determine the e.m.f. induced in the conductor. Length, = 15 cm = 0.15 m and velocity, v = 750 mm/s = 0.75 m/s Induced e.m.f., E = B l v sin = sin 90º = 0.15 V. Find the speed that a conductor of length 10 mm must be moved at right angles to a magnetic field of flux density 0.6 T to induce in it an e.m.f. of 1.8 V. Induced e.m.f., E = B v from which, speed, v = E 1.8 B = 5 m/s. A 5 cm long conductor moves at a uniform speed of 8 m/s through a uniform magnetic field of flux density 1. T. Determine the current flowing in the conductor when (a) its ends are open- circuited, (b) its ends are connected to a load of 15 ohms resistance. Induced e.m.f., E = B v = =.4 V (a) If the conductor is open circuited, then no current will flow. (b) Current, I = E.4 = 0.16 A R A straight conductor 500 mm long is moved with constant velocity at right angles both to its length and to a uniform magnetic field. Given that the e.m.f. induced in the conductor is.5 V and the velocity is 5 m/s, calculate the flux density of the magnetic field. If the conductor forms part of a closed circuit of total resistance 5 ohms, calculate the force on the conductor. 41

324 Induced e.m.f., E = B v i.e..5 = B sin 90º from which, flux density, B = sin 90 = 1 T Force on conductor, F = B I sin = E.5 B sin= (1) (0.500)(sin 00) R 5 = 0.5 N 5. A car is travelling at 80 km/h. Assuming the back axle of the car is 1.76 m in length and the vertical component of the earth s magnetic field is 40 T, find the e.m.f. generated in the axle due to motion. Generated e.m.f, E = B v = = 1.56 mv 6. A conductor moves with a velocity of 0 m/s at an angle of (a) 90 (b) 45 (c) 0, to a magnetic field produced between two square-faced poles of side length.5 cm. If the flux on the pole face is 60 mwb, find the magnitude of the induced e.m.f. in each case. Induced e.m.f., E = B v sin (a) When = 90, E = B v sin 90 = 6010 lvsin sin 90= 48 V 4 A.5 10 (b) When = 45, E = B v sin 45 = 48 sin 45 =.9 V (c) When = 0, E = B v sin 0 = 48 sin 0 = 4 V 7. A conductor 400 mm long is moved at 70 to a 0.85 T magnetic field. If it has a velocity of 115 km/h, calculate (a) the induced voltage, and (b) force acting on the conductor if connected to a 8 resistor. 41

325 (a) Induced voltage, E = B v sin = (0.85)(0.4) (sin 70) = V or 10.1 V (b) Force on conductor, F = B I sin = E B sin= (0.85) (0.4)(sin 70) R 8 = N 414

326 EXERCISE 18, Page The mutual inductance between two coils is 150 mh. Find the magnitude of the e.m.f. induced in one coil when the current in the other is increasing at a rate of 0 A/s. The magnitude of the e.m.f. induced, di 0 dt 1 = 4.5 V 1 E M Determine the mutual inductance between two coils when a current changing at 50 A/s in one coil induces an e.m.f. of 80 mv in the other. E di M dt 1 hence, mutual inductance, M = E 8010 = 1.6 mh di1 50 dt. Two coils have a mutual inductance of 0.75 H. Calculate the magnitude of the e.m.f. induced in one coil when a current of.5 A in the other coil is reversed in 15 ms. 1 Induced e.m.f., E M 0.75 di.5.5 dt = 50 V 4. The mutual inductance between two coils is 40 mh. If the current in one coil changes from 15 A to 6 A in 1 ms, calculate (a) the average e.m.f. induced in the other, (b) the change of flux linked with the other if it is wound with 400 turns. di 15 6 dt 110 = V 1 (a) Induced e.m.f., E M (b) E = N d Edt from which, change of flux, d = 5.4 mwb dt N

327 EXERCISE 184, Page A transformer has 600 primary turns connected to a 1.5 kv supply. Determine the number of secondary turns for a 40 V output voltage, assuming no losses. N N V V from which, secondary turns, N N V 40 V = 96 turns. An ideal transformer with a turns ratio :9 is fed from a 0 V supply. Determine its output voltage. N1 and V1 0 V N 9 N N V V from which, output voltage, V V N 9 N1 = 990 V. An ideal transformer has a turns ratio of 1:1 and is supplied at 19 V. Calculate the secondary voltage. N N V V from which, secondary voltage, V V N 1 N1 1 = 16 V 4. A transformer primary winding connected across a 415 V supply has 750 turns. Determine how many turns must be wound on the secondary side if an output of 1.66 kv is required. N N V V from which, secondary turns, N N V 1660 = 000 turns V

328 5. An ideal transformer has a turns ratio of 15:1 and is supplied at 180 V when the primary current is 4 A. Calculate the secondary voltage and current. N1 1, V1 0V and I1 4A N 1 N N N N V V from which, output voltage, V V I I N 1 N from which, secondary current, I I 4 1 N 15 N 1 = 1 V = 60 A 6. A step-down transformer having a turns ratio of 0:1 has a primary voltage of 4 kv and a load of 10 kw. Neglecting losses, calculate the value of the secondary current. N1 0 and V1 4000V N 1 N N V V from which, output voltage, V V Secondary power = VI = i.e. 00 I = N 1 N1 0 = 00 V from which, secondary current, I = 50 A A transformer has a primary to secondary turns ratio of 1:15. Calculate the primary voltage necessary to supply a 40 V load. If the load current is A determine the primary current. Neglect any losses. N N V V 1 from which, primary voltage, V V N 1 N 15 = 16 V N N I I from which, primary current, I I N 15 = 45 A N

329 8. A 0 resistance is connected across the secondary winding of a single-phase power transformer whose secondary voltage is 150 V. Calculate the primary voltage and the turns ratio if the supply current is 5 A, neglecting losses. V 150 Secondary current, I = 7.5 A, I1 5A and V = 150 V R 0 N N V V 1 from which, primary voltage, V V V N I 7.5 N I = 5 V N1 I 7.5 Turns ratio, = 1.5 or N I 5 1 or : EXERCISE 185, Page 416 Answers found from within the text of the chapter, pages 408 to 416. EXERCISE 186, Page (c). (b). (c) 4. (a) 5. (d) 6. (a) 7. (b) 8. (c) 9. (d) 10. (a) 11. (b) 1. (a) 1. (d) 14. (b) and (c) 418

330 CHAPTER 4 CALCULATIONS AND EVALUATION OF FORMULAE EXERCISE 18, Page 7 1. Evaluate correct to decimal places Using a calculator, = 5.8 correct to decimal places. Evaluate correct to 5 significant figures Using a calculator, = correct to 5 significant figures. Evaluate correct to decimal places Using a calculator, = 50.0 correct to decimal places 4. Evaluate.5 Using a calculator,.5 = Evaluate 0.06 in engineering form Using a calculator, 0.06 = = Evaluate 1.56 correct to 5 significant figures 54

331 Using a calculator, 1.56 =.440 correct to 5 significant figures 7. Evaluate.14 correct to 4 significant figures Using a calculator,.14 = 0.96 correct to 4 significant figures 8. Evaluate 0.8 correct to 4 decimal places Using a calculator, 0.8 = correct to 4 decimal places 9. Evaluate correct to decimal places Using a calculator, = correct to decimal places 10. Evaluate Using a calculator, = Evaluate correct to 1 decimal place Using a calculator, = 17.9 correct to 1 decimal place 1. Evaluate 1 1 correct to 4 significant figures Using a calculator, 1 1 = correct to 4 significant figures

332 1. Evaluate 4.1 Using a calculator, 4.1 = Evaluate 5 0. correct to 5 significant figures in engineering form Using a calculator, = = correct to 5 significant figures 15. Evaluate correct to 4 decimal places Using a calculator, = correct to 4 decimal places 16. Evaluate correct to 4 significant figures Using a calculator, = 5.70 correct to 4 significant figures 17. Evaluate 1.7 correct to 5 significant figures Using a calculator, 1.7 = 11.1 correct to 5 significant figures 18. Evaluate 0.69 correct to 4 significant figures Using a calculator, 0.69 = correct to 4 significant figures 19. Evaluate 17 correct to decimal places Using a calculator, 17 =.571 correct to decimal places 0. Evaluate 5.1 correct to 4 decimal places 56

333 Using a calculator, 5.1 = 1.56 correct to 4 decimal places 1. Evaluate correct to decimal places Using a calculator, = correct to decimal places. Evaluate in engineering form Using a calculator, = in engineering form. Evaluate in engineering form Using a calculator, = in engineering form 4. Evaluate in engineering form Using a calculator, = in engineering form 5. Evaluate correct to 4 significant figures Using a calculator, = correct to 4 significant figures 57

334 EXERCISE 19, Page 8 1. Evaluate 1 as a fraction 6 7 Using a calculator, 1 = Evaluate as a decimal, correct to 4 significant figures 6 8 Using a calculator, = correct to 4 significant figures Evaluate 1 8 as a fraction 4 1 Using a calculator, 1 8 = Evaluate 8 8 as a mixed number 9 Using a calculator, = Evaluate as a decimal, correct to significant figures Using a calculator, = correct to significant figures

335 6. Evaluate sin correct to 4 decimal places Using a calculator, sin = correct to 4 decimal places 7. Evaluate cos 6.74 correct to 4 decimal places Using a calculator, cos 6.74 = correct to 4 decimal places 8. Evaluate tan sin 5.5 correct to 4 decimal places Using a calculator, tan sin 5.5 = = 0.01 correct to 4 decimal places 9. Evaluate sin(0.47 rad) correct to 4 decimal places Using a calculator, sin(0.47 rad) = 0.4 correct to 4 decimal places 1. Evaluate cos(1.4 rad) correct to 4 decimal places Using a calculator, cos(1.4 rad) = correct to 4 decimal places 11. Evaluate tan(5.67 rad) correct to 4 decimal places Using a calculator, tan(5.67 rad) = correct to 4 decimal places 1. Evaluate sin 4.6 tan8. cos 1.8 correct to 4 decimal places Using a calculator, sin 4.6 tan8. cos 1.8 = correct to 4 decimal places 1. Evaluate 1.59 correct to 4 significant figures 59

336 Using a calculator, 1.59 = correct to 4 significant figures 14. Evaluate 1 1 correct to 4 significant figures Using a calculator, 1 1 = 5.7 correct to 4 significant figures 15. Evaluate 1 e correct to 4 significant figures Using a calculator, 1 e = = correct to 4 significant figures 16. Evaluate e correct to 4 significant figures Using a calculator, e = correct to 4 significant figures 17. Evaluate 5.5 e 6.7 correct to 4 significant figures Using a calculator, 5.5 e 6.7 =.50 correct to 4 significant figures 18. Evaluate e 8.57 correct to 4 significant figures Using a calculator, e = correct to 4 significant figures

337 EXERCISE 0, Page The circumference C of a circle is given by the formula C = r. Determine the circumference given r = 8.40 mm Using a calculator, circumference, C = π 8.40 = 5.78 mm. A formula used in connection with gases is R = PV T. Evaluate R when P = 1500, V = 5 and T = 00 Using a calculator, R = PV = 7.5 T 00. The velocity of a body is given by v = u + at. The initial velocity u is measured when time t is 15 seconds and found to be 1 m/s. If the acceleration a is 9.81 m/s calculate the final velocity v. Using a calculator, final velocity, v = u + at = 1 + (9.81)(15) = = 159 m/s 4. Find the distance s, given that s = 1 gt. Time t = 0.0 seconds and acceleration due to gravity g = 9.81 m/s. Give the answer in millimetres. Using a calculator, distance, s = 1 gt = (0.0) = m = mm = 5.0 mm 5. The energy stored in a capacitor is given by E = 1 CV joules. Determine the energy when capacitance C = farads and voltage V = 40 V. Using a calculator, energy, E = 1 CV = = J 61

338 6. Resistance R is given by R = R1(1 + t). Find R, correct to 4 significant figures, when R1 = 0, = and t = 75.6 Using a calculator, resistance, R = R1(1 + t) = 0( ) = 0( ) = 0(1.0041) = Density = mass volume. Find the density when the mass is.46 kg and the volume is 17 cm. Give the answer in units of kg/m. (Note that 1 cm = 10 m 6 ) mass.46 kg Using a calculator, density = 6 volume 1710 m = 140 kg / m 8. Velocity = frequency wavelength. Find the velocity when the frequency is 185 Hz and the wavelength is m Velocity = frequency wavelength = = 81.1 m/s 9. Evaluate resistance RT, given when R1 = 5.5, R = 7.4 and R = 1.6 R R R R T 1 Using a calculator, = R R R R T 1 from which, resistance, R T = = Power = 4.7 m in 5 s force dis tan ce. Find the power when a force of 760 N raises an object a distance of time forcedis tan ce Using a calculator, power = N m/s = W since 1 W = 1 N m/s time 5 6

339 11. The potential difference, V volts, available at battery terminals is given by V = E - Ir. Evaluate V when E = 5.6, I = 0.70 and R = 4.0 Using a calculator, potential difference, V = E Ir = 5.6 (0.70)(4.0) = =.61 V 1. Given force F = 1 m(v - u ), find F when m = 18., v = 1.7 and u = = Using a calculator, force F = 1 m(v - u ) = 1. Energy, E joules, is given by the formula E = 1 LI. Evaluate the energy when L = 5.5 and I = 1. Using a calculator, energy, E = 1 LI = =.96 J 14. The current I amperes in an a.c. circuit is given by I = V = 50, R = 11.0 and X = 16. V (R X ). Evaluate the current when Using a calculator, current, I = V 50 (R X ) = 1.77 A 15. Distance s metres is given by the formula s = ut + 1 at. If u = 9.50, t = 4.60 and a = -.50, evaluate the distance. Using a calculator, distance, s = ut + 1 at = ( 50. ) 460. = =

340 16. The area, A, of any triangle is given by A = s(s a)(s b)(s c) Evaluate the area, given a =.60 cm, b = 4.00 cm and c = 5.0 cm where s = a b c. a b c Using a calculator, s = = 6.40 Hence, area, A = s(s a)(s b)(s c) 6.40( )( )( = cm 64

341 CHAPTER 40 ALTERNATING VOLTAGES AND CURRENTS EXERCISE 187, Page 4 1. Determine the periodic time for the following frequencies: (a).5 Hz (b) 100 Hz (c) 40 khz (a) Periodic time, T = 1 1 = 0.4 s f.5 (b) Periodic time, T = 1 1 = 0.01 s or 10 ms f 100 (c) Periodic time, T = 1 1 f 4010 = 5 s. Calculate the frequency for the following periodic times: (a) 5 ms (b) 50 s (c) 0. s 1 1 (a) Frequency, f = = 00 Hz or 0. khz T 5 10 (b) Frequency, f = T 5010 = 0 khz (c) Frequency, f = 1 1 = 5 Hz T 0.. An alternating current completes 4 cycles in 5 ms. What is its frequency? Time for one cycle, T = 5 ms = 1.5 ms 4 Hence, frequency, f = 1 1 T = 800 Hz 419

342 EXERCISE 188, Page An alternating current varies with time over half a cycle as follows: Current (A) time (ms) The negative half cycle is similar. Plot the curve and determine: (a) the frequency (b) the instantaneous values at.4 ms and 5.8 ms (c) its mean value, and (d) its r.m.s. value. The graph is shown plotted below. (a) Periodic time, T = 10 ms = 0 ms, hence, frequency, f = 1 1 T 010 = 50 Hz (b) At.4 ms, current, i = 5.5 A and at 5.8 ms, i =.1 A (c) Mean value = area under curve length of base Using the mid-ordinate rule, area under curve = =

343 Hence, mean value = =.8 A (d) r.m.s. value = = =.98 A or 4.0 A, correct to significant figures.. For the waveforms shown below, determine for each (i) the frequency (ii) the average value over half a cycle (iii) the r.m.s. value (iv) the form factor (v) the peak factor. (a) (b) (c) (d) (a) (i) T = 10 ms, hence, frequency, f = 1 1 = 100 Hz T 1010 (ii) Average value = area under curve length of base 510 =.50 A 41

344 (iii) R.m.s. value = i i i i i = =.87 A (iv) Form factor = r.m.s..87 = 1.15 average.50 (v) Peak factor = max imum value 5 = 1.74 r.m.s (b) (i) T = 4 ms, hence, frequency, f = = 50 Hz T 4 10 (ii) Average value = area under curve 0 = 0 V length of base (iii) R.m.s. value = v v v v = = 0 V (iv) Form factor = r.m.s. 0 = 1.0 average 0 (v) Peak factor = max imum value 0 = 1.0 r.m.s (c) (i) T = 8 ms, hence, frequency, f = = 15 Hz T 8 10 (ii) Average value = area under curve 7 = 18 A length of base 4 4 (iii) R.m.s. value = i1 i i i

345 = 8 r.m.s (iv) Form factor = = 1.09 average 18 = A (v) Peak factor = max imum value 4 = 1. r.m.s (d) (i) T = 4 ms, hence, frequency, f = = 50 Hz T 4 10 (ii) Average value = area under curve = 5 V length of base (iii) R.m.s. value = v v v v = = 50 V (iv) Form factor = r.m.s. 50 =.0 average 5 (v) Peak factor = max imum value 100 =.0 r.m.s. 50. An alternating voltage is triangular in shape, rising at a constant rate to a maximum of 00 V in 8 ms and then falling to zero at a constant rate in 4 ms. The negative half cycle is identical in shape to the positive half cycle. Calculate (a) the mean voltage over half a cycle, and (b) the r.m.s. voltage The waveform is shown below. (a) Mean value = s 00 V area under curve length of base 1 10 s = 150 V (b) R.m.s. value = v1 v v v = = 170 V 4

346 4. Calculate the r.m.s. value of a sinusoidal curve of maximum value 00 V. R.m.s. value = peak value = = 1.1 V 5. Find the peak and mean values for a 00 V mains supply. 00 V is the r.m.s. value r.m.s. value = peak value, from which, peak value = Mean value = 0.67 peak value = = 180. V r.m.s. 00 = 8.9 V A sinusoidal voltage has a maximum value of 10 V. Calculate its r.m.s. and average values. R.m.s. value = peak value = = 84.8 V Average value = 0.67 peak value = = 76.4 V 7. A sinusoidal current has a mean value of 15.0 A. Determine its maximum and r.m.s. values. Mean value = 0.67 maximum value, 44

347 from which, maximum value = mean value 15.0 =.55 A R.m.s. value = maximum value = = A EXERCISE 189, Page 46 Answers found from within the text of the chapter, pages 40 to 46. EXERCISE 190, Page (b). (c). (d) 4. (d) 5. (a) 6. (d) 7. (c) 8. (b) 9. (c) 10. (b) 45

348 CHAPTER 41 CAPACITORS AND INDUCTORS EXERCISE 191, Page 4 1. Find the charge on a 10 F capacitor when the applied voltage is 50V. 6 Charge, Q = C V = C =.5 mc. Determine the voltage across a 1000 pf capacitor to charge it with C. Q = CV hence, voltage, V = Q 10 C = 000 V or kv. The charge on the plates of a capacitor is 6 mc when the potential between them is.4 kv. Determine the capacitance of the capacitor. Q = CV hence, capacitance, C = Q V =.5 μf 4. For how long must a charging current of A be fed to a 5 F capacitor to raise the p.d. between its plates by 500 V. Charge Q = I t and Q = C V hence I t = C V from which, time, t = 6 CV = 1.5 ms I 5. A direct current of 10 A flows into a previously uncharged 5 F capacitor for 1 ms. Determine the p.d. between the plates. P.d. between plates, V = Q It C C 510 = 000 V or kv 46

349 6. A capacitor uses a dielectric 0.04 mm thick and operates at 0 V. What is the electric field strength across the dielectric at this voltage? Electric field strength, E = V 0 d kv/m 7. A charge of 1.5 C is carried on two parallel rectangular plates each measuring 60 mm by 80 mm. Calculate the electric flux density. If the plates are spaced 10 mm apart and the voltage between them is 0.5 kv determine the electric field strength. Electric flux density, D = 6 Q A = 1.5 C/m Electric field strength, E = V d 1010 = 50 kv/m 47

350 EXERCISE 19, Page A capacitor consists of two parallel plates each of area 0.01 m, spaced 0.1 mm in air. Calculate the capacitance in picofarads. 1 0 ra Capacitance, C d since, for air, ε r = 1 = = 885 pf. A waxed paper capacitor has two parallel plates, each of effective area 0. m. If the capaci- -tance is 4000 pf determine the effective thickness of the paper if its relative permittivity is. C hence, thickness of the paper, d = d 0 r A 1 0 ra C m = mm. How many plates has a parallel plate capacitor having a capacitance of 5 nf, if each plate is 40 mm by 40 mm and each dielectric is 0.10 mm thick with a relative permittivity of 6. C (n 1) from which, n 1 = d 0 r A 9 C d A r 1 6 = 6 Hence, the number of plates, n = = 7 4. A parallel plate capacitor is made from 5 plates, each 70 mm by 10 mm, interleaved with mica of relative permittivity 5. If the capacitance of the capacitor is 000 pf determine the thickness of the mica. C (n 1) from which, dielectric thickness, d 0 r A d = 1 6 0r A (n 1) (51) = m =.97 mm 1 C

351 5. A capacitor is constructed with parallel plates and has a value of 50 pf. What would be the capacitance of the capacitor if the plate area is doubled and the plate spacing is halved? If the plate area is doubled, so is the capacitance (i.e. direct proportion). If the plate spacing is halved, then the capacitance is doubled (i.e. inverse proportion). Hence, capacitance of capacitor = 4 50 = 00 pf 49

352 EXERCISE 19, Page Capacitors of F and 6 F are connected (a) in parallel and (b) in series. Determine the equivalent capacitance in each case. (a) In parallel, equivalent capacitance, C T = + 6 = 8 μf (b) In series, equivalent capacitance, C T = = 1.5 μf. Find the capacitance to be connected in series with a 10 F capacitor for the equivalent capacitance to be 6 F For series connection, i.e C C C 1 T from which, 10 C C 6 10 and C = = 15 F. What value of capacitance would be obtained if capacitors of 0.15 F and 0.10 F are connected (a) in series and (b) in parallel. (a) In series, equivalent capacitance, C T = = 0.06 μf (b) In parallel, equivalent capacitance, C T = = 0.5 μf 4. Two 6 F capacitors are connected in series with one having a capacitance of 1 F. Find the total equivalent circuit capacitance. What capacitance must be added in series to obtain a capacitance of 1. F? 40

353 Two 6 F capacitors in series has a total capacitance of = F. (Two equal value capacitors in series will have a total capacitance of half the value of one of the capacitors). F in series with 1 F has a total capacitance of 1 1 =.4 F = total circuit capacitance. Let new capacitance be C X then if new total capacitance is to be 1. F then from which 1..4 C X C 1..4 X Hence, capacitance to be added, C X = =.4 F 5. For the arrangement shown below find (a) the equivalent circuit capacitance and (b) the voltage across a 4.5 F capacitor. (a) Three 4.5 F capacitors in series gives 1.5 F and two 1 F capacitors in series gives 0.5 F 1.5 F and 0.5 F capacitors in parallel gives = F F in series with F gives: 6 5 = 1. F = equivalent circuit capacitance (b) The equivalent circuit is shown below where V = 00 V = voltage across three 4.5 F capacitors in series. Hence, voltage across each 4.5 F capacitor = 00/ = 100 V. 41

354 (Alternatively, to find V 1 : Since C T = 1. F then 6 QT CT V C. This is the charge on each capacitor of the circuit shown below. Hence, Q V C 10 6 T = 00 V) 6. In the circuit below, capacitors P, Q and R are identical and the total equivalent capacitance of the circuit is F. Determine the values of P, Q and R..5 F and 4.5 F in parallel gives an equivalent capacitance of = 8 F F in series with 8 F gives = 1.6 F Let the equivalent capacitance of P, Q and R in series be Then C X = from which, C X C X = 1.6 = 1.4 F Thus, i.e (since CP CQ CR) 1.4 C C C C P Q R P CP 1.4= 4. F = CQ CR 4

355 EXERCISE 194, Page When a capacitor is connected across a 00 V supply the charge is 4 C. Find (a) the capacitance and (b) the energy stored. (a) Q = CV from which, capacitance, C = 6 Q 410 = 0 nf or 0.0 F V 00 (b) Energy stored, W = CV = 400 J or 0.4 mj. Find the energy stored in a 10 F capacitor when charged to kv Energy stored, W = CV = 0 J. A 00 pf capacitor is required to store 0.5 mj of energy. Find the p.d. to which the capacitor must be charged. Energy, W = 1 CV from which, p.d., V = W C 0010 = 550 V 4. A bakelite capacitor is to be constructed to have a capacitance of 0.04 F and to have a steady working potential of 1 kv maximum. Allowing a safe value of field stress of 5 MV/m find (a) the thickness of bakelite required, (b) the area of plate required if the relative permittivity of bakelite is 5, (c) the maximum energy stored by the capacitor and (d) the average power developed if this energy is dissipated in a time of 0 s. (a) Field stress, E = V d from which, thickness of dielectric, d = V E m mm = 0.04 mm 4

356 0r (b) Capacitance, C = A d cross-sectional area, A = from which, C d r m = 61.6 cm (c) Maximum energy, W max = CV = 0.0 J energy 0.0 J (d) Energy = power time, hence, power, P = = 1000 W or 1 kw 6 time 010 s 44

357 EXERCISE 195, Page Find the e.m.f. induced in a coil of 00 turns when there is a change of flux of 0 mwb linking with it in 40 ms. Induced e.m.f., E = d 010 N 00 dt 4010 = V. An e.m.f. of 5 V is induced in a coil of 00 turns when the flux linking with it changes by 1 mwb. Find the time, in milliseconds, in which the flux makes the change. E d N from which, time for change, dt = dt N d = s or 144 ms E 5. An ignition coil having turns has an e.m.f. of 8 kv induced in it. What rate of change of flux is required for this to happen? E d N from which, rate of change of flux, d E 810 = = 0.8 Wb/s dt dt N A flux of 5 mwb passing through a 15-turn coil is reversed in 5 ms. Find the magnitude of the average e.m.f. induced. Magnitude of induced e.m.f., E = d N 15 dt 510 =.5 V (Note that since the flux is reversed, it changes from 5 mwb to - 5 mwb, which is a change of , i.e. 70 mwb). 5. Calculate the e.m.f. induced in a coil of inductance 6 H by a current changing at a rate of 15 A/s. 45

358 E.m.f. induced, E = di 15 L 6 dt 1 = - 90 V 46

359 EXERCISE 196, Page An inductor of 0 H has a current of.5 A flowing in it. Find the energy stored in the magnetic field of the inductor. 1 1 = 6.5 J Energy stored, W = LI 0.5. Calculate the value of the energy stored when a current of 0 ma is flowing in a coil of inductance 400 mh. 1 1 = 0.18 mj Energy stored, W = LI The energy stored in the magnetic field of an inductor is 80 J when the current flowing in the inductor is A. Calculate the inductance of the coil. Energy, W = 1 LI from which, inductance, L = W 80 = 40 H I 47

360 EXERCISE 197, Page A flux of 0 mwb links with a 100 turn coil when a current of 5 A is passing through the coil. Calculate (a) the inductance of the coil, (b) the energy stored in the magnetic field, and (c) the average e.m.f. induced if the current is reduced to zero in 0.0 s. (a) Inductance of coil, L = N = 7. H I LI 7. 5 = 90 J (b) Energy stored, W = (c) Induced e.m.f., E = di 5 0 L 7. dt 0.0 = 180 V. An e.m.f. of kv is induced in a coil when a current of 5 A collapses uniformly to zero in 10 ms. Determine the inductance of the coil. Induced e.m.f., E = di L dt from which, inductance, L = E di dt 1010 = 4 H. An average e.m.f. of 60 V is induced in a coil of inductance 160 mh when a current of 7.5 A is reversed. Calculate the time taken for the current to reverse. Induced e.m.f., E = di L dt hence, 60 = t 15 from which, time, t = = 0.04 s or 40 ms A coil of 500 turns has a flux of 10 mwb linking with it when carrying a current of A. Calculate the coil inductance and the e.m.f. induced in the coil when the current collapses to zero in 0 ms. 48

361 Inductance, L = N = 1.5 H I di 0 Induced e.m.f., E = L 1.5 dt 0 10 = 1.5 kv 5. When a current of A flows in a coil, the flux linking with the coil is 80 Wb. If the coil inductance is 0.5 H, calculate the number of turns of the coil. If L = N LI 0.5 then number of turns, N = 6 I 8010 = 1,500 EXERCISE 198, Page 44 Answers found from within the text of the chapter, pages 49 to 44. EXERCISE 199, Page (a). (b). (c) 4. (a) 5. (b) 6. (b) 7. (c) 8. (c) 9. (c) 10. (d) 11. (c) 1. (b) 1. (d) 14. (a) 49

362 CHAPTER 4 ELECTRICAL MEASURING INSTRUMENTS AND MEASUREMENTS EXERCISE 00, Page A 0 1 A ammeter having a resistance of 50 is used to measure the current flowing in a 1 k resistor when the supply voltage is 50 V. Calculate: (a) the approximate value of current (neglecting the ammeter resistance), (b) the actual current in the circuit, (c) the power dissipated in the ammeter, (d) the power dissipated in the 1 k resistor. (a) Approximate value of current = V 50 = 0.50 A R 1000 V 50 (b) Actual current = = 0.8 A R r a (c) Power dissipated in ammeter, P = I r a =.8 W (d) Power dissipated in the 1 k resistor, P = I r a = W. (a) A current of 15 A flows through a load having a resistance of 4. Determine the power dissipated in the load. (b) A wattmeter, whose current coil has a resistance of 0.0, is connected to measure the power in the load. Determine the wattmeter reading assuming the current in the load is still 15 A. (a) Power in load, P = IR 15 4 = 900 W 440

363 (b) Total resistance in circuit, R T Wattmeter reading, P = I R T = W 441

364 EXERCISE 01, Page For the square voltage waveform displayed on a c.r.o. shown below, find (a) its frequency, (b) its peak-to-peak voltage (a) Periodic time, T = 4.8 cm 5 ms/cm = 4 ms Hence, frequency, f = 1 1 T 410 = 41.7 Hz (b) Peak-to-peak voltage = 4.4 cm 40 V/cm = 176 V. For the pulse waveform shown below, find (a) its frequency, (b) the magnitude of the pulse voltage. (a) Time for one cycle, T =.6 cm 500 ms/cm = 1.8 s Hence, frequency, f = 1 1 = 0.56 Hz T 1.8 (b) Magnitude of the pulse voltage = 4. cm V/cm = 8.4 V 44

365 . For the sinusoidal waveform shown below, determine (a) its frequency, (b) the peak-to-peak voltage, (c) the r.m.s. voltage. (a) Periodic time, T =.8 cm 50 ms/cm = 0.14 s Hence, frequency, f = 1 1 = 7.14 Hz T 0.14 (b) Peak-to-peak voltage = 4.4 cm 50 V/cm = 0 V (c) Peak voltage = 0 = 110 V and r.m.s. voltage = = V 44

366 EXERCISE 0, Page In a Wheatstone bridge PQRS, a galvanometer is connected between Q and S and a voltage source between P and R. An unknown resistor R X is connected between P and Q. When the bridge is balanced, the resistance between Q and R is 00, that between R and S is 10 and that between S and P is 150. Calculate the value of R X From the diagram, 10 R X = and unknown resistor, R X = = k. Balance is obtained in a d.c. potentiometer at a length of 1. cm when using a standard cell of volts. Calculate the e.m.f. of a dry cell if balance is obtained with a length of 46.7 cm. E E l hence, l from which, e.m.f. of dry cell, E E = 1.55 V 444

367 EXERCISE 0, Page 459 Answers found from within the text of the chapter, pages 445 to 459. EXERCISE 04, Page (f). (c). (a) 4. (i) 5. (j) 6. (g) 7. (c) 8. (b) 9. (p) 10. (d) 11. (o) 1. (n) 1. (a) 445

368 CHAPTER 4 INTRODUCTION TO ENGINEERING SYSTEMS EXERCISE 05, Page 47 Answers found from within the text of the chapter, pages 465 to 47. EXERCISE 06, Page (d). (a). (c) 4. (d) 5. (b) 6. (a) 7. (b) 8. (c) 446

369 CHAPTER 5 ALGEBRA EXERCISE 1, Page Find the sum of 4a, - a, a, - 8a 4a + - a + a + - 8a = 4a a + a 8a = 4a + a a 8a = 7a 10 a = - a. Find the sum of a, 5b, - c, - a, - b and 7c a + 5b + - c + - a + - b + 7c = a + 5b c a b + 7c = a a + 5b b + 7c c = a + b + 4c. Simplify 5ab 4a + ab + a 5ab 4a + ab + a = 5ab + ab + a 4a = 6ab a 4. Simplify x y + 5z x y + z + 5x x y + 5z x y + z + 5x = x x + 5x y y + 5z + z = 6x 5y + 8z 5. Add x y + to x + 4y 1 (x + 4y 1) + (x y + ) = x + 4y 1 + x y + = x + x + 4y y 1 + = 4x + y + 6. Subtract a b from 4a + b (4a + b) (a b) = 4a + b a + b = 4a a + b + b = a + 5b 65

370 7. From a + b c take a + b 4c (a + b c) (a + b 4c) = a + b c a b + 4c = a a + b b c + 4c = - a b + c 8. Simplify pq pq r pq pq r = ppqq r p q r = pqr 9. Simplify - 4a - a - 4a - a = 8a 10. Simplify - q - q - q - q = 6q 11. Evaluate pq 5qr pqr when p =, q = - and r = 4 When p =, q = - and r = 4, then pq 5qr pqr = ()(- ) 5(- )(4) ()(- )(4) = = If x = 5 and y = 6, evaluate: (x y) yxyx (x y) yxyx = 1 or If a = 4, b =, c = 5 and d = 6, evaluate a b c d 66

371 If a = 4, b =, c = 5 and d = 6, then a b = c d (4) () = 6 (5) (6) Simplify x 14xy x 14xy = x 14xy = 1 7y by cancelling 15. Multiply a b by a + b (a b)(a + b) = a ab ab b = a ab b 16. Simplify a 9ab a 9ab = a 9ab = 1 b by cancelling 67

372 EXERCISE, Page Simplify z z giving the answer as a power z z z = 8 z. Simplify 5 aa a giving the answer as a power aa a a = 8 a. Simplify n 8 5 n giving the answer as a power n n n = n 4. Simplify b b giving the answer as a power b b b = 11 b 5. Simplify b b giving the answer as a power 5 b b b = 5 b 5 b 6. Simplify 5 4 c c c giving the answer as a power c c c c 84 c c c c = c c c 4 c 7. Simplify m m m m giving the answer as a power 68

373 m m m m m m m m m 117 = 4 m 8. Simplify (x )(x) x 6 giving the answer as a power (x )(x) x x x x x 1 6 x = x x or 9. Simplify x 4 giving the answer as a power x 4 x 4 = 1 x 10. Simplify y giving the answer as a power y y = 1 y 6 y or Simplify t t giving the answer as a power tt t t t = 8 t 7 1. Simplify c giving the answer as a power c c = c Simplify a a 5 giving the answer as a power 69

374 a 5 5 a a a = 1 a 9 a or Simplify 4 1 b giving the answer as a power 4 1 b b 4 = 1 b 1 b or Simplify b b 7 giving the answer as a power b b b b b = 10 b Simplify s giving the answer as a power = s s s 9 s Simplify 5 abc abc and evaluate when a =, b = 1 and c = abc abc 5 a b c 5 1 = ab c or ac b ab c ac b = 9 70

375 18. Simplify (abc) (a b c ) 1 abc abc 6 9 a b c = ab c ab c a b c Simplify (a b c )(ab) 1/ 1/ 1/ ( a bc) abc ab abc ab 6 a b c a b c 1 a bc a b c = a b c or a 6 11 c b 71

376 EXERCISE, Page Expand the brackets: (x + )(x + ) (x + )(x + ) = x xx 6= x 5x 6. Expand the brackets: (x + 4)(x + 1) (x + 4)(x + 1) = x x 8x 4 = x 9x 4. Expand the brackets: (x + ) (x + ) = (x + )(x + ) = 4x 6x 6x 9 = 4x 1x 9 4. Expand the brackets: (j 4)(j + ) (j - 4)(j + ) = j 6j4j 1= j j 1 5. Expand the brackets: (x + 6)(x + 5) (x + 6)(x + 5) = 4x 10x 1x 0 = 4x x 0 6. Expand the brackets: (pq + r)(r + pq) (pq + r)(r + pq) = pqr p q r pqr = pqr p q r 7. Expand the brackets: (x + 6) (x + 6) = (x + 6)(x + 6) = x 6x6x 6= x 1x 6 8. Expand the brackets: (5x + ) 7

377 (5x + ) = (5x + )(5x + ) = 5x 15x 15x 9 = 5x 0x 9 9. Expand the brackets: (x 6) (x - 6) = (x - 6)(x - 6) = 4x 1x 1x 6 = 4x 4x Expand the brackets: (x )(x + ) (x )(x + ) = (x - )(x + ) = 4x 6x 6x 9 = 4x Expand the brackets: a(b a) a(b a) = ab 6a 1. Expand the brackets: x(x y) x(x y) = x xy 1. Expand the brackets: (a - 5b)(a + b) (a - 5b)(a + b) = a ab 5ab 5b = a ab 5b 14. Expand the brackets: (p q) (q 4p) (p q) (q 4p) = 9p 6q q + 4p = 1p 7q 15. Expand the brackets: (x 4y) + (y z) (z 4x) (x 4y) + (y z) (z 4x) = x 4y + y z z + 4x = 7x y 4z 7

378 16. Expand the brackets: (a + 5b)(a 5b) (a + 5b)(a 5b) = 4a 10ab 10ab 5b = 4a 5b 17. Expand the brackets: x y (x y) = (x y)(x y) = x xyxy 4y = x 4xy 4y 18. Expand the brackets: x + [y - (x + y)] x + [y - (x + y)] = x + y x y = Expand the brackets: a + [a - (a - )] a + [a - (a - )] = a + a (a ) = a + a 6a + 4 = - a + 4 or 4 - a 0. Expand the brackets: 4a a ba 7ab 4 a a ba 7ab 4 a 6aba 7ab 4 a ab = 8a 4ab or 4ab 8a 74

379 EXERCISE 4, Page Factorise and simplify: x + 4 x + 4 = (x + ). Factorise and simplify: xy 8xz xy 8xz = x(y 4z). Factorise and simplify: pb + pc pb + pc = p(b + c) 4. Factorise and simplify: x + 4xy x + 4xy = x(1 + y) 5. Factorise and simplify: 4d 1df 5 4d 5 1df = 4dd f 5 6. Factorise and simplify: 4x + 8x 4x + 8x = 4x(1 + x) 7. Factorise and simplify: q + 8qn q + 8qn = q(q + 4n) 8. Factorise and simplify: rs + rp +rt 75

380 rs + rp +rt = r( s + p + t) 9. Factorise and simplify: x + x + 5x x + x + 5x = x1 x 5x 10. Factorise and simplify: abc + b c abc + b c = bca b 11. Factorise and simplify: 4 x y 15x y 18x y = xy xy 5y 6 4 x y 15x y 18x y 1. Factorise and simplify: 4p q 10pq 4p q 10pq = pq p 5q 1. Factorise and simplify: 1a b - 8ab 1a b 8ab = 7ab(ab 4) 14. Factorise and simplify: xy + 6x y + 8x y xy 6x y 8x y = xy(y + x + 4x ) 15. Factorise and simplify: 4 x y 4xy 8x y = xy x y 4x y 4 x y 4xy 8x y 76

381 EXERCISE 5, Page Simplify: x + x 4x x x + x 4x x = x 8x x (M) = x 8x (S). Simplify: (y + y) 4y y (y + y) 4y y = y 4y y (B) = 1y y (M). Simplify: 4b + b (b 6b) 4b + b (b 6b) = 4b + b 5b (B) = 4b 15b (M) 4. Simplify: 8a a + 6a a 8a a + 6a a = 8a 6a a a (D) = 4 + 6a a by cancelling = 4 + a (S) 5. Simplify: 6x (x + x) 4x 6x (x + x) 4x = 6x 4x 4x (B) = 6x 4x 4x (D) = 4x by cancelling 77

382 6. Simplify: 4t (5t t + t) 4t (5t t + t) = 4t 4t = 4t 4t (B) (D) = 1 by cancelling 7. Simplify: y + y 5y + y 8y 6y y + y 5y + y 8y 6y = y + y 5y + y 8y 6y (D) = y + y 5y y by cancelling = y + 10y y (M) = 10y y (S) 8. Simplify: (x + x)x + x 6x 4x (x + x)x + x 6x 4x = x x + x 6x 4x (B) = x x + x 6x 4x (D) = x x + 1 4x by cancelling = 9x + 1 4x (M) 9. Simplify: 5a + a a + a (a 9a) 5a + a a + a (a 9a) = 5a + a a + a 7a (B) 78

383 = 5a + a a + a 7a (D) = 5a + a a by cancelling = 5a + 6a (M) = 6a + 5a Simplify: (t + t)(5t + t) (t t) (t + t)(5t + t) (t t) = 5t 6t - t (B) 5t 6t = t = 5t 1 by cancelling (D) (D) = - 15t (M) 79

384 CHAPTER 6 SIMPLE EQUATIONS EXERCISE 6, Page 5 1. Solve the equation: x + 5 = 7 Since x + 5 = 7 then x = 7 5 i.e. x = from which, x = = 1. Solve the equation: 8 - t = Since 8 t = then 8 = t i.e. 6 = t from which, t = 6 =. Solve the equation: c - 1 = Since c - 1 = then c = + 1 i.e. 4 c = 4 from which, c = = 6 4. Solve the equation: x - 1 = 5x + 11 Since x 1 = 5x + 11 then = 5x x i.e. 1 = x from which, x = 1 = Solve the equation: 7-4p = p - 5 Since 7-4p = p 5 then = p + 4p i.e. 1 = 6p from which, p = 1 6 = 80

385 6. Solve the equation: a + 6-5a = 0 Since a + 6 5a = 0 then 6 = 5a a i.e. 6 = a from which, a = 6 = 7. Solve the equation: x - - 5x = x - 4 Since x - - 5x = x 4 then 4 = x x + 5x i.e. = 4x from which, x = 4 = 1 8. Solve the equation: 0d - + d = 11d Since 0d + d = 11d then 0d + d 11d = i.e. 1d = 0 from which, d = 0 9. Solve the equation: (x - 1) = 4 Since (x - 1) = 4 then x = 4 i.e. x = 4 + = 6 from which, x = 6 = 10. Solve the equation: 16 = 4(t + ) Since 16 = 4(t + ) then 16 = 4t + 8 i.e = 4t i.e. 8 = 4t from which, t = 8 4 = 11. Solve the equation: 5(f - ) - (f + 5) + 15 = 0 81

386 Since 5(f ) (f + 5) + 15 = 0 then 5f 10 6f = 0 i.e. 5f 6f = and - f = 10 from which, f = Solve the equation: x = 4(x - ) Since x = 4(x - ) then x = 4x 1 i.e. 1 = 4x x i.e. 1 = x from which, x = 1 = 6 1. Solve the equation: 6( - y) - 4 = - (y - 1) Since 6( y) 4 = - (y 1) then 1 18y 4 = - y + i.e. -18y + y = and - 16y = from which, y = = Solve the equation: (g - 5) - 5 = 0 Since (g - 5) - 5 = 0 then 6g 10 5 = 0 i.e. 6g = i.e. 6g = 15 from which, g = 15 6 = Solve the equation: 4(x + 1) = 7(x + 4) - (x + 5) Since 4(x + 1) = 7(x + 4) (x + 5) then 1x + 4 = 7x + 8 x 10 i.e. 1x 7x + x =

387 and 7x = 14 from which, x = 14 7 = 8

388 EXERCISE 7, Page Solve the equations: 1 5 d + = 4 Since i.e. 1 5 d + = 4 then 1 5 d = d = 1 from which, d = = 5 1. Solve the equations: + 4 y = 1 + y Multiplying each term by 1 (the lowest common denominator of, 4 and 6) gives: 5 (1)() (1) y (1)(1) (1) y (1) 4 6 i.e y = 1 + 8y + 10 and 9y 8y = i.e. y = -. Solve the equations: 1 4 (x - 1) + = 1 Multiplying each term by 4 gives: 1 1 (4) (x 1) (4)() (4) 4 i.e. x = and x = i.e. x = - 9 from which, x = 9 = Solve the equations: 1 5 (f - ) + 1 (f - 4) = 0 84

389 Multiplying each term by 0 gives: 1 1 (0) (f ) (0) (f 4) (0) i.e. 6(f ) + 5(f 4) + 4 = 0 and 1f f = 0 i.e. 1f + 5f = and 17f = 4 from which, f = 4 17 = 5. Solve the equations: x - x 5 = Multiplying each term of x - x 5 = by 15 gives: (15) x - (15) x 5 = (15) i.e. 5x x = 0 i.e. x = 0 from which, x = 0 = Solve the equations: 1 - y = + y - y 6 Multiplying each term by 6 gives: y y y (6)(1) (6) (6)() (6) (6) 6 i.e. 6 y = 18 + y y and y - y + y = 18 6 i.e. -y = 1 from which, y = 1 1 =

390 7. Solve the equations: a = 8 Multiplying each term by 8a gives: (8a) a = (8a) 8 i.e. 8() = a() i.e. 16 = a from which, a = 16 = Solve the equations: 1 n + 1 4n = 7 4 Multiplying each term by 4n gives: (4n) (4n) (4n) n 4n 4 i.e = 7n i.e. 14 = 7n from which, n = 14 7 = 9. Solve the equations: x x = Multiplying each term by 0 gives: x x (0) (0) (0)() 4 5 i.e. 5(x + ) = 4(x ) + 40 i.e. 5x + 15 = 4x and 5x 4x = from which, x = Solve the equations: a = a 1 86

391 Multiplying each term by (a )(a + 1) gives: (a )(a 1) (a )(a 1) a a1 i.e. (a + 1) = (a ) i.e. 4a + = a 9 and 4a a = from which, a = Solve the equations: x 4 - x 6 5 = x Multiplying each term by 0 gives: x x 6 x (0) (0) (0) 4 5 i.e. 5x 4(x + 6) = 10(x + ) i.e. 5x 4x 4 = 10x + 0 and 5x 4x 10x = i.e. 9x = 54 from which, x = = Solve the equations: t = 9 Since t = 9 then t = 9 = Squaring both sides gives: t = = 9 1. Solve the equations: y = 5 Since y = 5 then y = 5 87

392 Squaring both sides gives: y = 5 5 = Solve the equations: 4 = a + Since 4 = i.e. a + then 4 = a a = 1 Squaring both sides gives: a = 1 = 1 from which, = a or a = 15. Solve the equations: 10 = 5 x 1 Dividing both sides by 5 gives: Squaring both sides gives: i.e. x 1 x 4 1 x 41 i.e. 5 = x from which, x = (5)() = Solve the equations: 16 = t 9 Since 16 = t 9 then (16)(9) = t i.e. 144 = and t = 144 = ± 1 t 88

393 EXERCISE 8, Page A formula used for calculating resistance of a cable is R = a = 10-4 find the value of. L a. Given R = 1.5, L = 500 and Since R = l A 500 then 1.5 = 4 10 Multiplying both sides by gives: Dividing both sides by 500 gives: from which, by calculator, = or = Force F newtons is given by F = ma, where m is the mass in kilograms and a is the acceleration in metres per second squared. Find the acceleration when a force of 4 kn is applied to a mass of 500 kg Since F = ma then 4000 = (500)(a) Dividing both sides by 500 gives: = a from which, acceleration, a = 8 m/s. PV = mrt is the characteristic gas equation. Find the value of m when P = , V =.00, R = 88 and T = 00. Dividing both sides of PV = mrt by RT gives: PV RT = m from which, m = 8800 =.47, correct to 4 significant figures. 89

394 4. When three resistors R1, R and R are connected in parallel the total resistance RT is determined from R R R R T 1 (a) Find the total resistance when R1 =, R = 6 amd R = 18 (b) Find the value of R given that RT =, R1 = 5 and R = 10 (a) R R R R T 1 Turning both sides upside down gives: R T = = 1.8 (b) from which, 5 10 R R Turning both sides upside down gives: R = 0 1 = 0 5. Ohm's law may be represented by I = V/R, where I is the current in amperes, V is the voltage in volts and R is the resistance in ohms. A soldering iron takes a current of 0.0 A from a 40 V supply. Find the resistance of the element. Multiplying both sides of I = V R by R gives: IR = V and dividing both sides by I gives: R = V I Thus, resistance, R = = The stress, σ Pascal s, acting on the reinforcing rod in a concrete column is given in the following equation: Find the value of the stress in MPa. 90

395 Since then and stress, σ = = Pa 176 MPa 91

396 EXERCISE 9, Page Given R = R1(1 + t), find given R1 = 5.0, R = 6.0 and t = 51.5 Substituting into R R 1 t gives: 6.0 = 5.0[1 + (51.5)] 1 and (51.5) 5.0 i.e (51.5) and i.e = (51.5) 0.06 = (51.5) from which, = = If v = u + as, find u given v = 4, a = - 40 and s = 4.05 Substituting into v u as gives: 4 u ( 40)(4.05) i.e. 576 = and = u - 4 u i.e. u = 900 and u = 900 = 0. The relationship between the temperature on a Fahrenheit scale and that on a Celsius scale is given by F = 9 5 C +. Express 11o F in degrees Celsius. Since F = 9 C 5 then 11 = 9 C 5 i.e. i.e. 11 = 9 C 5 81 = 9 C 5 9

397 Multiplying both sides by 5 gives: (81)(5) = 9C Dividing both sides by 9 gives: (81)(5) 9 = C from which, C = 45 Hence, 11 Fahrenheit is equivalent to 45 Celsius w 4. If t =, find the value of S given w = 1.19, g = 9.81 and t = 0.1 Sg Since t = w/sg then and S S9.81 from which, S9.81 Thus, S from which, S = = 50 using a calculator 5. A rectangular laboratory has a length equal to one and a half times its width and a perimeter of 40 m. Find its length and width. Let length of laboratory = l and width = w Length, l = 1.5w and perimeter = 40 = l + w Hence, i.e. 40 = (1.5w) + w 40 = w + w = 5w from which, width, w = 40 5 = 8 m and length, l = 1.5w = 1.5(8) = 1 m 9

398 6. Applying the principle of moments to a beam results in the following equation: F = (5 F) 7 where F is the force in Newtons. Determine the value of F. Since F = (5 F) 7 then F = 5 7F i.e. F + 7F = 5 i.e. 10F = 5 from which, F = 5 10 =.5 N 94

399 CHAPTER 7 TRANSPOSITION OF FORMULAE EXERCISE 0, Page Make d the subject of the formula: a + b = c - d - e Since a + b = c - d e then d = c e a - b. Make x the subject of the formula: y = 7x Dividing both sides of y = 7x by 7 gives: x = y 7. Make v the subject of the formula: pv = c Dividing both sides of pv = c by p gives: v = c p 4. Make a the subject of the formula: v = u + at Since v = u + at then v u = at and dividing both sides by t gives: v u t = a or a = v u t 5. Make y the subject of the formula: x + y = t Since x + y = t then y = t x and dividing both sides by gives: y t x 1 or y t x 6. Make r the subject of the formula: c = r 95

400 Dividing both sides of c = r by gives: c r or r = c 7. Make x the subject of the formula: y = mx + c Since y = mx + c then y c = mx y c and dividing both sides by m gives: m = x or x = y c m 8. Make T the subject of the formula: I = PRT Dividing both sides of I = PRT by PR gives: I T PR or T = I PR 9. Make L the subject of the formula: XL f L Dividing both sides of XL XL f L by πf gives: f L or L = XL f 10. Make R the subject of the formula: I = E R Multiplying both sides of I = E R by R gives: I R = E and dividing both sides by I gives: R = E I 11. Make x the subject of the formula: x y a Since x y then y = x a a Multiplying both sides by a gives: a(y ) = x or x = a(y ) 96

401 1. Make C the subject of the formula: F = 9 5 C + Rearranging F = 9 C gives: F = 9 C 5 5 Multiplying both sides by 5 9 gives: F C F C 9 i.e. or C F

402 EXERCISE 1, Page Make r the subject of the formula: S = a 1 r Multiplying both sides of S = i.e. from which, a 1 r by (1 r) gives: S(1 r) = a S Sr = a S a = Sr and dividing both sides by S gives: S a S = r i.e. r = S a S or r = 1 - a S. Make x the subject of the formula: y = (x d) d Multiplying both sides of y = d xd by d gives: yd = (x d) Dividing both sides by gives: yd x d and yd d = x or yd xd Alternatively, from the first step, yd = (x d) i.e. and yd = x - d yd + d = x from which, x = yd d d y d i.e. x = y. Make f the subject of the formula: A = (F f ) L 98

403 Multiplying both sides of A = (F f ) L by L gives: AL = (F f) Dividing both sides by gives: AL F f and f F AL or f = F AL 4. Make D the subject of the formula: y = AB 5CD Multiplying both sides of y = AB 5CD by D gives: yd = AB 5C Dividing both sides by y gives: D = AB 5Cy 5. Make t the subject of the formula: R = R0(1 + t) Removing the bracket in R R 1 t R R R t 0 gives: 0 0 from which, R R0 R0 t R R0 and R = t or t = R R0 R Make R the subject of the formula: 1 R = 1 R R Rearranging gives: R R R R R R R1 R i.e. R R R RR 1 1 RR1 Turning both sides upside down gives: R R R 1 99

404 7. Make R the subject of the formula: I = E e R r Multiplying both sides by (R + r) gives: i.e. and I(R + r) = E e I R + I r = E e I R = E e I r and dividing both sides by I gives: R = E e Ir I or R = E e r I 8. Make b the subject of the formula: y = 4ab c Dividing both sides by 4a c gives: y b b 4ac or y 4ac y Taking the square root of both sides gives: b = 4ac 9. Make x the subject of the formula: a x + b y = 1 Rearranging a x b y 1 gives: a b y b 1 x y y Turning both sides upside down gives: Multiplying both sides by a gives: x y a y b x a y a y y b y b Taking the square root of both sides gives: x = i.e. x = ay ay ay y b y b y b y ay b 100

405 10. Make L the subject of the formula: t = L g Dividing both sides of t = L by gives: g t L g Squaring both sides gives: t L g or L t g Multiplying both sides by g gives: L = g t or L = gt Make u the subject of the formula: v = u + as Since v = u + as then v - as = u or u = v - as Taking the square root of each side gives: u = v as 1. Make a the subject of the formula: N = a x y Squaring both sides of N = a x y gives: a x N y Multiplying both sides by y gives: Nya x or a + x = Ny from which, a = Ny - x 1. The lift force, L, on an aircraft is given by: L = 1 v ac where ρ is the density, v is the velocity, a is the area and c is the lift coefficient. Transpose the equation to make the velocity the subject. 101

406 Since L = 1 v ac then L ac v from which, velocity, v = L ac 10

407 EXERCISE, Page 6 1. Make a the subject of the formula: y = aman x aman Multiplying both sides of y = by x gives: xy = am an x and factorising gives: xy = a m n Dividing both sides by (m n) gives: xy m n a or xy a m n Taking the square root of both sides gives: a xy m n. Make R the subject of the formula: M = (R 4 - r 4 ) Dividing both sides of M = R 4 r 4 and rearranging gives: by gives: M R r M r R or R M r 4 4 Taking the fourth root of both sides gives: R = M 4 r 4. Make r the subject of the formula: x + y = r r Multiplying both sides of x + y = Multiplying the brackets gives: and rearranging gives: r r by ( + r) gives: (x + y)( + r) = r x + xr + y + yr = r xr + yr r = -x y Factorising gives: r(x + y 1) = -( x + y) Dividing both sides by (x + y 1) gives: r = (x y) xy1 10

408 Multiplying numerator and denominator by -1 gives: r = (x y) 1 x y 4. Make L the subject of the formula: m = L L rcr Multiplying both sides of L m by (L + rcr) gives: m(l + rcr) = L L rcr Removing brackets gives: and rearranging gives: ml + mrcr = L mrcr = L - ml Factorising gives: mrcr = L( - m) Dividing both sides by ( - m) gives: L = mrcr m 5. Make b the subject of the formula: a = b c b Multiplying both sides by b gives: ab b c and rearranging gives: Factorising gives: c b a b or b 1 a c b ab c Dividing both sides by 1 a gives: b c 1 a Taking the square root of both sides gives: b = Hence, b = c c 1 a 1 a c 1 a 6. Make r the subject of the formula: x y = 1 r 1 r 104

409 Rearranging by cross-multiplying gives: x1 r y1 r Removing brackets gives: and rearranging gives: xxr y yr xy yr xr or yr xr x y Factorising gives: r x y x y Dividing both sides by (x + y) gives: x y r x y Taking the square root of both sides gives: r = x y x y 7. A formula for the focal length, f, of a convex lens is: 1 f = 1 u + 1. Transpose the formula to make v v the subject and evaluate v when f = 5 and u = 6 Rearranging gives: f u v Turning each side upside down gives: v = u f v f u uf uf u f When f = 5 and u = 6, then v = uf (6)(5) 0 u f = 0 8. The quantity of heat, Q, is given by the formula Q = mc(t - t1). Make t the subject of the formula and evaluate t when m = 10, t1 = 15, c = 4 and Q = 1600 Removing the brackets in Q = mct t gives: Q = mct mct1 1 and rearranging gives: Q + mct1 mct or mct Q mvt1 Q mvt1 Q Q Dividing both sides by mc gives: t or t t1 or t t1 mc mc mc When m = 10, t 1 = 15, c = 4 and Q = 1600, 105

410 Q t = t = 55 mc (10)(4) The velocity, v, of water in a pipe appears in the formula h = 0.0Lv dg of the formula and evaluate v when h = 0.71, L = 150, d = 0.0 and g = Express v as the subject Multiplying both sides of h = 0.0L v dg Dividing both sides by 0.0L gives: by dg gives: dgh = 0.0L v dgh v 0.0L or dgh v 0.0L Taking the square root of each side gives: v = dgh 0.0L When h = 0.71, L = 150, d = 0.0 and g = 9.81, v = dgh (0.0)(9.81)(0.71) = L 0.0(150) 10. The sag S at the centre of a wire is given by the formula: S = d(l d) 8 Make l the subject of the formula and evaluate l when d = 1.75 and S = 0.80 Squaring both sides of S = d(ld) 8 gives: d l d S 8 Multiplying both sides by 8 gives: 8S dl d Removing the bracket gives: Rearranging gives: or 8S dl d 8S d dl dl 8S d Dividing both sides by d gives: l = 8S d 8S d d d d 106

411 i.e. l = 8S d d When d = 1.75 and S = 0.80, l = 8S 8(0.80) d =.75 d (1.75) 11. An approximate relationship between the number of teeth, T, on a milling cutter, the diameter of cutter, D, and the depth of cut, d, is given by: 1.5D T D 4d Determine the value of D when T = 10 and d = 4 mm. Multiplying both sides of 1.5D T D 4d by D + 4d gives: T(D + 4d) = 1.5D Removing brackets gives: TD + 4dT = 1.5D Rearranging gives: or Factorising gives: 4dT = 1.5D TD 1.5D TD = 4dT D(1.5 T) = 4dT Dividing both sides by (1.5 T) gives: D = 4dT 1.5 T When T = 10 and d = 4 mm, then D = 4dT 4(4)(10) T = 64 mm 1. A simply supported beam of length L has a centrally applied load F and a uniformly distributed load of w per metre length of beam. The reaction at the beam support is given by: 1 F wl R = Rearrange the equation to make w the subject. Hence determine the value of w when L = 4 m, F = 8 kn and R = 10 kn Since R = 1 F wl then R = F + wl 107

412 and R F = wl from which, w = R F L When L = 4 m, F = 8 kn and R = 10 kn, w = (10) 8 1 = kn/m The rate of heat conduction through a slab of material, Q, is given by the formula Q ka(t t ) d 1 where 1 t and t are the temperatures of each side of the material, A is the area of the slab, d is the thickness of the slab, and k is the thermal conductivity of the material. Rearrange the formula to obtain an expression for t ka(t1 t ) Since Q then Qd = kat1 t d i.e. Qd ka = t1 t from which, t = t 1 - Qd ka r 14. The slip, s, of a vehicle is given by: s = 1 100% where r is the tyre radius, ω is the v angular velocity and v the velocity. Transpose to make r the subject of the formula. r Since s = 1 100% v and then s 100 = 1 - r v r s = 1 - v 100 from which, r = v 1 s

413 15. The critical load, F newtons, of a steel column may be determined from the formula L F EI n where L is the length, EI is the flexural rigidity, and n is a positive integer. 1 Transpose for F and hence determine the value of F when n = 1, E = N / m, 6 4 I = m and L = 1.1 m Since and L F n EI then F n EI L F n EI L n i.e. F = EI L 1 When n = 1, E = N / m, I = m 6 4 and L = 1.1 m, n 1 L 1.1 = N = 1.61 MN 1 6 load, F = EI The flow of slurry along a pipe on a coal processing plant is given by: Transpose the equation for r 4 pr V 8 Since and 4 pr V then 8 8 Vpr 8V r 4 p 4 8 V from which, r = 4 p 109

414 CHAPTER 8 SIMULTANEOUS EQUATIONS EXERCISE, Page Solve the simultaneous equations: x y 6 x + y = 6 Numbering the equations gives: x y 6 (1) Equation (1) + equation () gives: x = 1 x + y = 6 () from which, x = 1 = 4 Substituting x = 4 in equation () gives: 4 + y = 6 from which, y = 6 4 = (Checking in equation (1): L.H.S. = (4) = 8 = 6 = R.H.S.). Solve the simultaneous equations: x y = x y = - 9 Numbering the equations gives: x y = (1) x y = - 9 () equation (1) gives: 6x y = 6 () Equation () equation () gives: 5x = = 15 from which, x = 15 5 = Substituting x = in equation (1) gives: 6 - y = from which, y = 6 = 4 (Checking in equation (): L.H.S. = (4) = 1 = - 9 = R.H.S.). Solve the simultaneous equations: x 4y = - 4 5x y = 7 110

415 Numbering the equations gives: x 4y = - 4 (1) 5x y = 7 () equation () gives: 10x 4y = 14 () Equation () equation (1) gives: 9x = = 18 from which, x = 18 9 = Substituting x = in equation (1) gives: - 4y = - 4 from which, + 4 = 4y from which, 4y = 6 and y = 6 4 = 1.5 (Checking in equation (): L.H.S. = 5() (1.5) = 10 = 7 = R.H.S.) 4. Solve the simultaneous equations: x y = 10 5x + y = 1 Numbering the equations gives: x y = 10 (1) 5x + y = 1 () equation () gives: 10x + y = 4 () Equation (1) + equation () gives: 1x = 5 from which, x = 5 1 = 4 Substituting x = 4 in equation (1) gives: 1 - y = 10 from which, 1 10 = y from which, y = and y = 1 (Checking in equation (): L.H.S. = 5(4) + 1 = = 1 = R.H.S.) 5. Solve the simultaneous equations: x 7y = - 8 x + 4y = 17 Numbering the equations gives: x 7y = - 8 (1) x + 4y = 17 () 111

416 equation (1) gives: 6x - 1y = - 4 () equation () gives: 6x + 8y = 4 (4) Equation (4) - equation () gives: 8y - - 1y = i.e. 9y = 58 from which, y = 58 9 = Substituting y = in equation (1) gives: x - 14 = - 8 from which, x = 14 8 = 6 from which, x = 6 = (Checking in equation (): L.H.S. = () + 4() = = 17 = R.H.S.) 6. Solve the simultaneous equations: a + b = 8 b a = - Numbering the equations gives: a + b = 8 (1) a + b = - () equation () gives: - 6a + b = - 6 () Equation (1) - equation () gives: a - - 6a = i.e. 7a = 14 from which, a = 14 7 = Substituting a = in equation (1) gives: + b = 8 from which, b = 8 = 6 from which, b = 6 = (Checking in equation (): L.H.S. = - () + = = - = R.H.S.) 7. Solve the simultaneous equations: a + b = 7 a - b = 11

417 Numbering the equations gives: a + b = 7 (1) Equation (1) + equation () gives: a = 10 a - b = () from which, a = 10 = 5 Substituting a = 5 in equation (1) gives: 5 + b = 7 from which, b = 7 5 = (Checking in equation (): L.H.S. = 5 - = = R.H.S.) 8. Solve the simultaneous equations: x + 5y = 7 x + y = 4 x + 5y = 7 (1) x + y = 4 () equation () gives: x + 6y = 8 () () (1) gives: y = 1 Substituting in (1) gives: x + 5 = 7 i.e. x = 7 5 = and x = = 1 Thus, x = 1 and y = 1 and may be checked by substituting into both of the original equations 9. Solve the simultaneous equations: s + t = 1 4s - t = 5 s + t = 1 (1) 4s - t = 5 () equation () gives: 8s t = 10 () (1) + () gives: 11s = from which, s = 11 = Substituting in (1) gives: 6 + t = 1 i.e. t = 1 6 = 6 and t = 6 = 11

418 Thus, s = and t = and may be checked by substituting into both of the original equations 10. Solve the simultaneous equations: x - y = 1 x + 5y = - 4 x y = 1 (1) x + 5y = -4 () equation (1) gives: 6x 4y = 6 () equation () gives: 6x + 15y = -1 (4) () - (4) gives: -19y = 8 from which, y = 8 19 = - Substituting in (1) gives: x + 4 = 1 i.e. x = 1 4 = 9 and x = 9 = Thus, x = and y = - and may be checked by substituting into both of the original equations 11. Solve the simultaneous equations: 5m - n = 11 m + n = 8 5m - n = 11 (1) m + n = 8 () equation () gives: 9m + n = 4 () (1) + () gives: 14m = 5 from which, m = 5 14 =.5 Substituting in (1) gives: 1.5 n = 11 i.e = n i.e. n = 1.5 from which, n = 1.5 = 0.5 Thus, m =.5 and n = 0.5 and may be checked by substituting into both of the original equations 1. Solve the simultaneous equations: 8a - b = 51 a + 4b =

419 8a - b = 51 (1) a + 4b = 14 () 4 equation (1) gives: a 1b = 04 () equation () gives: 9a + 1b = 4 (4) () + (4) gives: 41a = 46 from which, a = = 6 Substituting in (1) gives: 48 b = 51 i.e = b and b = = - 1 Thus, a = 6 and b = - 1 and may be checked by substituting into both of the original equations 115

420 EXERCISE 4, Page Solve the simultaneous equations: 7p q = 0-1 = q - 5p Rearranging gives: 7p + q = -11 (1) 5p q = 1 () equation (1) gives: 1p + 6q = - () equation () gives: 10p - 6q = (4) () + (4) gives: 1p = -1 from which, p = -1 Substituting in (1) gives: -7 + q = -11 i.e. q = = -4 and q = 4 = - Thus, p = - 1 and q = - and may be checked by substituting into both of the original equations.. Solve the simultaneous equations: x + y = 4 x 6 - y 9 = 0 Rearranging gives: and x y (6) (6) (6)(4) i.e. x + y = 4 (1) x y (18) (18) (18)(0) i.e. x y = 0 () 6 9 (1) - () gives: 4y = 4 from which, y = 6 Substituting in (1) gives: x + 1 = 4 i.e. x = 4-1 = 1 and x = 1 = 4 Thus, x = 4 and y = 6 and may be checked by substituting into both of the original equations.. Solve the simultaneous equations: a - 7 = - b 1 = 5a + b 116

421 Rearranging gives: a () ()7 ()(b) i.e. a + 4b = 14 (1) and ()(1) = ()(5a) + () b i.e. 15a + b = 6 () equation () gives: 0a + 4b = 7 () () - (1) gives: 9a = 58 from which, a = Substituting in (1) gives: + 4b = 14 i.e. 4b = 14 - = 1 and b = 1 4 = Thus, a = and b = and may be checked by substituting into both of the original equations. 4. Solve the simultaneous equations: s - t = 8 s 4 + t = - Rearranging gives: () s ()(t) ()(8) i.e. s 4t = 16 (1) and (4) s 4 + (4)(t) = (4)(- ) i.e. s + 1t = - 8 () equation () gives: s + 6t = - 4 () (1) - () gives: - 40t = 40 from which, t = - 1 Substituting in (1) gives: 1.5s + = 8 i.e. 1.5s = 8 - and s = = 4 Thus, s = 4 and t = - 1 and may be checked by substituting into both of the original equations. 5. Solve the simultaneous equations: x 5 + y = x 7 - y = 0 117

422 Rearranging gives: x y 49 (15) (15) (15) i.e. x + 10y = 49 (1) 5 15 and x y 5 (14) (14) (14) 0 i.e. 6x - 7y = -10 () 7 7 equation (1) gives: 6x + 0y = 98 () () - () gives: 7y = 108 from which, y = = 4 Substituting in (1) gives: x + 40 = 49 i.e. x = = 9 and x = 9 = Thus, x = and y = 4 and may be checked by substituting into both of the original equations. 6. Solve the simultaneous equations: v - 1 = u 1 u + v 4-5 = 0 Rearranging gives: (1)v (1)(1) = (1) u 1 i.e. - u + 1v = 1 (1) and (4)u + (4) v 4 - (4) 5 i.e. 4u + v = 50 () 4 equation (1) gives: - 4u + 48v = 48 () () + () gives: 49v = 98 from which, v = = Substituting in (1) gives: - 1 = u 1 i.e. 1 = u 1 and u = 1 Thus, u = 1 and v = and may be checked by substituting into both of the original equations. 118

423 EXERCISE 5, Page In a system of pulleys, the effort P required to raise a load W is given by P = aw + b, where a and b are constants. If W = 40 when P = 1 and W = 90 when P =, find the values of a and b. P = aw + b, hence if W = 40 when P = 1, then: 1 = 40a + b (1) and if W = 90 when P =, then: = 90a + b () Equation () equation (1) gives: 10 = 50a from which, a = Substituting in (1) gives: 1 = b i.e. 1 = 8 + b from which, b = 4 Thus, a = 1 5 or 0. and b = 4 and may be checked by substituting into both of the original equations.. Applying Kirchhoff's laws to an electrical circuit produces the following equations: 5 = 0.I1 + (I1 - I) 1 = I + 0.4I - (I1 - I) Determine the values of currents I1 and I gives: 5 0.I1I1 I i.e..i1i 5 Rearranging 5 0.I I I 1 1 gives: 1 I 0.4I I1 I i.e. I15.4I 1 Rearranging 1 I 0.4I I I 1 Thus,.I1 I 5 (1) and I15.4I 1 () equation (1) gives: 4.4I1 4I 10 (). equation () gives: 4.4 I I 6.4 (4) 119

424 () + (4) gives: 7.88 I 6.4 from which, I Substituting in (1) gives:. I i.e.. I and I1 6.4 = = Thus, I 1 = 6.47 and I = 4.6 and may be checked by substituting into both of the original equations.. Velocity v is given by the formula v = u + at. If v = 0 when t = and v = 40 when t = 7 find the values of u and a. Hence find the velocity when t =.5 v = u + at, hence if v = 0 when t =, then: 0 = u + a (1) and if v = 40 when t = 7, then: 40 = u + 7a () Equation () equation (1) gives: 0 = 5a from which, a = 0 5 = 4 Substituting in (1) gives: 0 = u + 8 from which, u = 1 Thus, a = 4 and u = 1 and may be checked by substituting into both of the original equations. When t =.5, velocity, v = u + at = 1 + (4)(.5) = 6 4. y = mx + c is the equation of a straight line of slope m and y-axis intercept c. If the line passes through the point where x = and y =, and also through the point where x = 5 and y = 0.5, find the slope and y-axis intercept of the straight line. y = mx + c, hence if x = when y =, then: = m + c (1) and if x = 5 when y = 0.5, then: 0.5 = 5m + c () Equation () equation (1) gives: = m from which, m = Substituting in (1) gives: = c i.e. = c 10

425 from which, c = Thus, m = and c = which may be checked by substituting into both of the original equations. 5. The molar heat capacity of a solid compound is given by the equation c = a + bt. When c = 5, T = 100 and when c = 17, T = 400. Find the values of a and b. c = a + bt, hence if c = 5 when T = 100, then: 5 = a + 100b (1) and if c = 17 when T = 400, then: 17 = a + 400b () Equation () equation (1) gives: 10 = 00b from which, b = = 0.40 Substituting in (1) gives: 5 = a + 40 from which, a = 1 Thus, a = 1 and b = 0.40 and may be checked by substituting into both of the original equations. 6. In a system of forces, the relationship between two forces F1 and F is given by: 5F1 + F + 6 = 0 F1 + 5F + 18 = 0 Solve for F1 and F Rearranging gives: 5F1 + F = - 6 (1) F1 + 5F = - 18 () equation (1) gives: 15 F1 + 9F = - 18 () 5 equation () gives: 15 F1 + 5F = - 90 (4) () - (4) gives: - 16F = = 7 from which, F = 7 16 = Substituting in (1) gives: 5F1 1.5 = - 6 i.e. 5F1 = = 7.5 and F 1 = =

426 Thus, F 1 = 1.5 and F = and may be checked by substituting into both of the original equations. 7. For a balanced beam, the equilibrium of forces is given by: R1 R 1.0 kn As a result of taking moments: 0.R R Determine the values of the reaction forces R1and R Rearranging gives: R1 R 1.0 (1) 0.R1 0.8R.9 () 5 () gives: R1 4.0R 19.5 () (1) () gives: 5.0 R = 1.5 from which, R = = 6. kn Substituting in (1) gives: R Hence, R 1 = = 5.7 kn 1

427 CHAPTER 9 STRAIGHT LINE GRAPHS EXERCISE 6, Page Assuming graph paper measuring 0 cm by 0 cm is available, suggest suitable scales for the following ranges of values: (a) Horizontal axis: V to 55 V (b) Horizontal axis: 7 m to 86 m (c) Horizontal axis: 5 N to 150 N Vertical axis: 10 to 180 Vertical axis: 0. V to 1.69 V Vertical axis: 0.6 mm to.4 mm (a) Horizontal scale: 55 = 5V; V Hence, 1 cm = 4 V (or even 1 cm = 5 V) would be the best scale to use Vertical scale: = 170 Ω; = 8.5 Ω Hence, 1 cm = 10 Ω would be the best scale to use (b) Horizontal scale: 86 7 = 79 m; m Hence, 1 cm = 5 m would be the best scale to use Vertical scale: = 1.66 V; V Hence, 1 cm = 0.1 V would be the best scale to use (c) Horizontal scale: = 145 N; N Hence, 1 cm = 10 N would be the best scale to use Vertical scale: =.8 mm; mm Hence, 1 cm = 0. mm would be the best scale to use. Corresponding values obtained experimentally for two quantities are: x y Plot a graph of y (vertically) against x (horizontally) to scales of cm = 1 for the 1

428 horizontal x-axis and 1 cm = 1 for the vertical y-axis. (This graph will need the whole of the graph paper with the origin somewhere in the centre of the paper). From the graph find: (a) the value of y when x = 1 (b) the value of y when x = -.5 (c) the value of x when y = - 6 (d) the value of x when y = 7 14

429 From the above graph: (a) When x = 1, y = - 1 (b) When x = -.5, y = - 8 (c) When y = - 6, x = (d) When y = 7, x = 5. Corresponding values obtained experimentally for two quantities are: x y Use a horizontal scale for x of 1 cm = 1 unit and a vertical scale for y of 1 cm = units and draw a graph of x against y. Label the graph and each of its axes. By interpolation, find from the graph the value of y when x is.5 Graph of y/x 15

430 The graph of y against x is shown plotted above. From the graph, when x =.5, y = Draw a graph of y - x + 5 = 0 over a range of x = - to x = 4. Hence determine (a) the value of y when x = 1. and (b) the value of x when y = - 9. y x + 5 = 0 i.e. y = x 5 x 0 1 y A graph of y = x 5 is shown below. (a) When x = 1., y = (b) When y = - 9., x = The speed n rev/min of a motor changes when the voltage V across the armature is varied. The results are shown in the following table: n (rev/min) V (volts) It is suspected that one of the readings taken of the speed is inaccurate. Plot a graph of 16

431 speed (horizontally) against voltage (vertically) and find this value. Find also (a) the speed at a voltage of 1 V, and (b) the voltage at a speed of 100 rev/min. A graph of V/n is shown below. The 1010 rev/min reading should be closer to 1070 rev/min. 17

432 (a) When the voltage is 1 V, the speed is 1000 rev/min (b) When the speed is 100 rev/min, the voltage is 167 V 18

433 EXERCISE 7, Page The equation of a line is 4y = x + 5. A table of corresponding values is produced and is shown below. Complete the table and plot a graph of y against x. Find the gradient of the graph. x y y = x + 5 from which, y = x i.e. y = 1 x 5 4 Hence, when x = - 4, y = 1 ( 4) = when x = -, y = 1 ( ) = when x = - 1, y = 1 ( 1) = when x = 1, y = 1 (1) = when x =, y = 1 ( ) =.5 4 when x =, y = 1 () =

434 A graph of y = 1 x 5 is shown above. 4 Gradient of graph = AB BC = 1. Determine the gradient and intercept on the y-axis for each of the following equations: (a) y = 4x (b) y = - x (c) y = - x - 4 (d) y = 4 (a) Since y = 4x, then gradient = 4 and y-axis intercept = - (b) Since y = -x, then gradient = - 1 and y-axis intercept = 0 (c) Since y = -x 4, then gradient = - and y-axis intercept = - 4 (d) Since y = 4 i.e. y = 0x + 4, then gradient = 0 and y-axis intercept = 4. Determine the gradient and y-axis intercept for each of the following equations. Sketch the graphs. (a) y = 6x - (b) y = - x + 4 (c) y = x (d) y = 7 (a) Since y = 6x, then gradient = 6 and y-axis intercept = - A sketch of y = 6x is shown below. (b) Since y = - x + 4, then gradient = - and y-axis intercept = 4 A sketch of y = - x + 4 is shown below. 10

435 (c) Since y = x, then gradient = and y-axis intercept = 0 A sketch of y = x is shown below. (d) Since y = 7, then gradient = 0 and y-axis intercept = 7 A sketch of y = 7 is shown below. 4. Determine the gradient of the straight line graphs passing through the co-ordinates: (a) (, 7) and (-, 4) (b) (- 4, - 1) and (- 5, ) (c) 1, 4 4 and 1 5, 8 11

436 y y1 4 7 (a) From page 7 of textbook, gradient = x x 5 = 5 1 y y1 1 4 (b) Gradient = x x 54 1 = y y (c) Gradient = x 1 1 x = State which of the following equations will produce graphs which are parallel to one another: (a) y - 4 = x (b) 4x = - (y + 1) (c) x = 1 (y + 5) (d) y = x (e) x = 1 (7 - y) (a) Since y 4 = x then y = x + 4 (b) Since 4x = - (y + 1) then y = -4x - 1 (c) Since x = 1 y 5 then x = y + 5 and y = x (d) Since 1 y x then + y = x and y = x (e) Since x = 1 7 y then 4x = 7 y and y = -4x + 7 Thus, (a) and (c) are parallel (since their gradients are the same), and (b) and (e) are parallel. 6. Draw on the same axes the graphs of y = x - 5 and y + x = 7. Find the co-ordinates of the point of intersection. Check the result obtained by solving the two simultaneous equations algebraically. The graphs of y = x 5 and y + x = 7, i.e. y = x 7 are shown below. 1

437 The two graphs intersect at x = and y = 1, i.e. the co-ordinate (, 1) Solving simultaneously gives: y = x 5 i.e. y x = -5 (1) y = x 7 i.e. y + x = 7 () (1) gives: y 9x = -15 () () () gives: 11x = from which, x = Substituting in (1) gives: y 6 = -5 from which, y = 1 as obtained graphically above. 7. A piece of elastic is tied to a support so that it hangs vertically, and a pan, on which weights can be placed, is attached to the free end. The length of the elastic is measured as various weights are added to the pan and the results obtained are as follows: Load, W (N) Length, l (cm) Plot a graph of load (horizontally) against length (vertically) and determine: (a) the value length when the load is 17 N, (b) the value of load when the length is 74 cm, (c) its gradient, and (d) the equation of the graph. 1

438 From the graph: (a) When the load is 17 N, the length = 89 cm (b) When the length is 74 cm, the load = 11 N (c) Gradient of graph = AB BC =.4 (d) The vertical axis intercept = 48, the equation of the graph is: l =.4W