Chapter 3 Solutions. *3.1 x = r cos θ = (5.50 m) cos 240 = (5.50 m)( 0.5) = 2.75 m. y = r sin θ = (5.50 m) sin 240 = (5.50 m)( 0.866) = 4.

Transcription

1 Chapter 3 Solutions *3.1 = r cos θ = (5.50 m) cos 240 = (5.50 m)( 0.5) = 2.75 m = r sin θ = (5.50 m) sin 240 = (5.50 m)( 0.866) = 4.76 m 3.2 (a) d = ( 2 1 ) 2 + ( 2 1 ) 2 = (2.00 [ 3.00] 2 ) + ( ) 2 d = = 8.60 m (b) r 1 = (2.00) 2 + ( 4.00) 2 = 20.0 = 4.47 m θ 1 = tan = 63.4 r 2 = ( 3.00) 2 + (3.00) 2 = 18.0 = 4.24 m θ 2 = 135 measured from + ais. 3.3 We have 2.00 = r cos r = cos 30.0 = 2.31 and = r sin 30.0 = 2.31 sin 30.0 = (a) = r cos θ and = r sin θ, therefore 1 = (2.50 m) cos 30.0, 1 = (2.50 m) sin 30.0, and ( 1, 1 ) = (2.17, 1.25) m 2 = (3.80 m) cos 120, 2 = (3.80 m) sin 120, and ( 2, 2 ) = ( 1.90, 3.29) m (b) d = ( ) 2 + ( ) 2 = = 4.55 m

2 2 Chapter 3 Solutions 3.5 The distance out to the fl is 2.00 m and the distance up to the fl is 1.00 m. (a) We can use the Pthagorean theorem to find the distance from the origin to the fl, distance = = (2.00 m) 2 + (1.00 m) 2 = 5.00 m 2 = 2.24 m (b) θ = rctan 1 2 = 26.6 ; r = 2.24 m, We have r = and θ = rctan (a) The radius for this new point is ( ) = = r and its angle is rctan = 180 θ ( ) (b) ( 2) 2 + ( 2) 2 = 2r This point is in the third quadrant if (, ) is in the first quadrant or in the fourth quadrant if (, ) is in the second quadrant. It is at angle θ. (c) (3) 2 + ( 3) 2 = 3r This point is in the fourth quadrant if (, ) is in the first quadrant or in the third quadrant if (, ) is in the second quadrant. It is at angle θ C. 3.7 (a) The distance d from to C is d = where = (200) + (300 cos 30.0 ) = 460 km and = 0 + (300 sin 30.0 ) = 150 km 300 km d 30 φ B 200 km d = (460) 2 + (150) 2 = 484 km (b) tan φ = = = φ = tan -1 (0.326) = 18.1 N of W R 13 km 3.8 R 14 km θ = 65 N of E θ 6 km

3 Chapter 3 Solutions tan 35.0 = 100 m = (100 m)(tan 35.0 ) = 70.0 m m 3.10 R = 310 km at 57 S of W B R base E km 200 km 3.11 (a) Using graphical methods, place the tail of vector B at the head of vector. The new vector + B has a magnitude of 6.1 at 112 from the -ais. (b) The vector difference B is found b placing the negative of vector B at the head of vector. The resultant vector B has magnitude 14.8 units at an angle of 22 from the + -ais. B B + B B O

4 4 Chapter 3 Solutions 3.12 Find the resultant F 1 + F 2 graphicall b placing the tail of F 2 at the head of F 1. The resultant force vector F 1 + F 2 is of magnitude 9.5 N and at an angle of 57 above the -ais. F 1 + F 2 F 2 F N 3.13 (a) d = 10.0i = 10.0 m since the displacement is a straight line from point to point B. C (b) The actual distance walked is not equal to the straight-line displacement. The distance follows the curved path of the semi-circle (CB). s = 1 (2π r) = 5π = 15.7 m 2 (c) If the circle is complete, d begins and ends at point. Hence, d = Your sketch should be drawn to scale, and should look somewhat like that pictured below. The angle from the westward direction, θ, can be measured to be 4 N of W, and the distance R from the sketch can be converted according to the scale to be 7.9 m. N 5.00 m B d W θ R 15.0 meters 3.50 meters 8.20 meters 30.0 E S

5 Chapter 3 Solutions To find these vector epressions graphicall, we draw each set of vectors. Measurements of the results are taken using a ruler and protractor. (a) + B = 5.2 m at 60 (b) B = 3.0 m at B B B B a b 0 2 m 4 m 0 2 m 4 m (c) B = 3.0 m at 150 (d) 2B = 5.2 m at 300 B 2B B 2B c d 0 2 m 4 m 0 2 m 4 m *3.16 (a) The large majorit of people are standing or sitting at this hour. Their instantaneous foot-to-head vectors have upward vertical components on the order of 1 m and randoml oriented horizontal components. The citwide sum will be ~10 5 m upward. (b) Most people are ling in bed earl Saturda morning. We suppose their beds are oriented north, south, east, west quite at random. Then the horizontal component of their total vector height is ver nearl zero. If their compressed pillows give their height vectors vertical components averaging 3 cm, and if one-tenth of one percent of the population are on-dut nurses or police officers, we estimate the total vector height as ~ 10 5 (0.03 m) (1 m) ~10 3 m upward.

6 6 Chapter 3 Solutions 3.17 The scale drawing for the graphical solution should be similar to the figure at the right. The magnitude and direction of the final displacement from the starting point are obtained b measuring d and θ on the drawing and appling the scale factor used in making the drawing. The results should be 200 ft d 135 ft 30.0 θ ft d 420 ft and θ 3 N km 3.00 km km R 4.00 km 3.00 km 2.00 km 45.0 R = = 5.24 km φ tθ E θ = tan 1 = 154 or φ = 25.9 N of W 3.19 Call his first direction the direction. R = 10.0 m i m( j) m( i) = 3.00 m i 5.00 m j = (3.00) 2 + (5.00) 2 m at rctan 5 3 to the right R = 5.83 m at 59.0 to the right from his original motion 3.20 Coordinates of super-hero are: = (100 m) cos ( 30.0 ) = 86.6 m = (100 m) sin ( 30.0 ) = 50.0 m

7 Chapter 3 Solutions m

8 8 Chapter 3 Solutions 3.21 The person would have to walk 3.10 sin(25.0 ) = 1.31 km north, and 3.10 cos(25.0 ) = 2.81 km east East, + North Σ = cos 30 = 358 m Σ = sin = 12.5 m d = (Σ) 2 + (Σ) 2 = (358) 2 + ( 12.5) 2 = 358 m tan θ = (Σ) (Σ) = = θ = 2.00 d = 358 m at 2.00 S of E *3.23 Let the positive -direction be eastward, positive -direction be verticall upward, and the positive z-direction be southward. The total displacement is then d = (4.80 cm i cm j) + (3.70 cm j 3.70 cm k) or d = 4.80 cm i cm j 3.70 cm k (a) The magnitude is d = (4.80) 2 + (8.50) 2 + ( 3.70) 2 cm = 10.4 cm (b) Its angle with the -ais follows from cos θ = B = B i + B j + B 2 k B = 4.00i j k, giving θ = B = (4.00) 2 + (6.00) 2 + (3.00) 2 = 7.81 α = cos β = cos = 59.2 = 39.8 γ = cos = 67.4

9 Chapter 3 Solutions = 25.0 = 40.0 = = ( 25.0) 2 + (40.0) 2 = 47.2 units 40.0 From the triangle, we find that φ = 58.0, so that θ = 122 φ 25.0 tθ Goal Solution vector has an component of 25.0 units and a component of 40.0 units. Find the magnitude and direction of this vector. 40 r G: First we should visualize the vector either in our mind or with a sketch. Since the hpotenuse of the right triangle must be greater than either the or components that form the legs, we can estimate the magnitude of the vector to be about 50 units. The direction of the vector appears to be about 120 from the + ais. O: The graphical analsis and visual estimates above ma be sufficient for some situations, but we can use trigonometr to obtain a more precise result. : The magnitude can be found b the Pthagorean theorem: r = r = ( 25.0 units) 2 + (40 units) 2 = 47.2 units We observe that tan φ = (if we consider and to both be positive). φ = tan = tan = tan 1 (1.60) = 58.0 The angle from the + ais can be found b subtracting from 180. = = 122 L: Our calculated results agree with our graphical estimates. We should alwas remember to check that our answers are reasonable and make sense, especiall for problems like this where it is eas to mistakenl calculate the wrong angle b confusing coordinates or overlooking a minus sign. Quite often the direction angle of a vector can be specified in more than one wa, and we must choose a notation that makes the most sense for the given problem. If compass directions were stated in this question, we could have reported the vector angle to be 32.0 west of north or a compass heading of 328.

10 10 Chapter 3 Solutions *3.26 The east and north components of the displacement from Dallas (D) to Chicago (C) are the sums of the east and north components of the displacements from Dallas to tlanta () and from tlanta to Chicago. In equation form: d DCeast = d Deast + d Ceast = 730 cos sin 21.0 = 527 miles. d DCnorth = d Dnorth + d Cnorth = 730 sin cos 21.0 = 586 miles. B the Pthagorean theorem, d = (d DCeast ) 2 + (d DCnorth ) 2 = 788 mi Then tan θ = d DCnorth d DCeast = 1.11 and θ = Thus, Chicago is 788 miles at 48.0 north east of Dallas = d cos θ = (50.0 m)cos(120) = 25.0 m = d sin θ = (50.0 m)sin(120) = 43.3 m d = ( 25.0 m)i + (43.3 m)j 3.28 (a) B B B + B B (b) C = + B = 2.00i j i 2.00j = 5.00i j C = at rctan 4 5 C = 6.40 at 38.7 D = B = 2.00i j 3.00i j = 1.00i j D = ( 1.00) 2 + (8.00) 2 at rctan 8.00 ( 1.00)

11 Chapter 3 Solutions 11 D = 8.06 at ( ) = 8.06 at 97.2

12 12 Chapter 3 Solutions 3.29 d = ( ) 2 + ( ) 2 = ( ) 2 + ( ) 2 = 52.0 = 7.21 m θ = tan = = 8.70i j B = 13.2i 6.60j B + 3C = 0 3C = B = 21.9i 21.6j C = 7.30i 7.20j or C = 7.30 cm C = 7.20 cm 3.31 (a) ( + B) = (3i 2j) + ( i 4j) = 2i 6j (b) ( B) = (3i 2j) ( i 4j) = 4i + 2j (c) + B = = 6.32 (d) B = = 4.47 (e) θ + B = tan = 71.6 = 288 θ B = tan = Let i = east and j = north. R = 3.00b j b cos 45 i b sin 45 j 5.00b i R = 2.17b i b j R = b at rctan N of W = 6.22 blocks at 110 counterclockwise from east

13 Chapter 3 Solutions = r cos θ and = r sin θ, therefore: (a) (b) (c) = 12.8 cos 150, = 12.8 sin 150, and (, ) = ( 11.1i j) m = 3.30 cos 60.0, = 3.30 sin 60.0, and (, ) = (1.65i j) cm = 22.0 cos 215, = 22.0 sin 215, and (, ) = ( 18.0i 12.6j) in 3.34 (a) D = + B + C = 2i + 4j D = = 4.47 m at θ = 63.4 (b) E = B + C = 6i + 6j E = = 8.49 m at θ = d 1 = ( 3.50j) m d 2 = 8.20 cos 45.0 i sin 45.0 j = (5.80i j) m d 3 = ( 15.0i) m R = d 1 + d 2 + d 3 = ( )i + ( )j = ( 9.20i j) m (or 9.20 m west and 2.30 m north) The magnitude of the resultant displacement is R = R 2 + R 2 = ( 9.20) 2 + (2.30) 2 = 9.48 m The direction is θ = rctan = Refer to the sketch R = + B + C = 10.0i 15.0j i = 40.0i 15.0j R = [(40.0) 2 + ( 15.0) 2 ] 1/2 = 42.7 ards = 10.0 B = 15.0 R C = 50.0

14 14 Chapter 3 Solutions 3.37 (a) F = F 1 + F 2 F = 120 cos (60.0 )i sin (60.0 )j 80.0 cos (75.0 )i sin (75.0 )j F = 60.0i + 104j 20.7i j = (39.3i + 181j) N (b) F = (39.3) 2 + (181) 2 = 185 N ; θ = tan F 3 = F = ( 39.3i 181j) N = 77.8 Goal Solution The helicopter view in Figure P3.37 shows two people pulling on a stubborn mule. Find (a) the single force that is equivalent to the two forces shown and (b) the force that a third person would have to eert on the mule to make the resultant force equal to zero. The forces are measured in units of newtons. G: The resultant force will be larger than either of the two individual forces, and since the two people are not pulling in eactl the same direction, the magnitude of the resultant should be less than the sum of the magnitudes of the two forces. Therefore, we should epect 120 N < R < 200 N. The angle of the resultant force appears to be straight ahead and perhaps slightl to the right. If the stubborn mule remains at rest, the ground must be eerting on the animal a force equal to the resultant R but in the opposite direction. 120 N 80 N O: We can find R b adding the components of the two force vectors. : F 1 = (120 cos 60)i N + (120 sin 60)j N = 60.0i N j N F 2 = (80 cos 75)i N + (80 sin 75)j N = 20.7i N j N R = F 1 + F 2 = 39.3i N j N R = R = (39.3) 2 + (181.2) 2 = 185 N The angle can be found from the arctan of the resultant components. θ = tan = tan = tan 1 (4.61) = 77.8 counterclockwise from the + ais The opposing force that the either the ground or a third person must eert on the mule, in order for the overall resultant to be zero, is 185 N at 258 counterclockwise from +. L: The resulting force is indeed between 120 N and 200 N as we epected, and the angle seems reasonable as well. The process applied to solve this problem can be used for other statics problems encountered in phsics and engineering. If another force is added to act on a sstem that is alread in equilibrium (sum of the forces is equal to zero), then the sstem ma accelerate. Such a sstem is now a dnamic one and will be the topic of Chapter 5.

15 Chapter 3 Solutions East North 0 m 4.00 m R = = 4.64 m at 78.6 N of E 3.39 = 3.00 m, θ = 30.0, B = 3.00 m, θ B = 90.0 = cos θ = 3.00 cos 30.0 = 2.60 m, = sin θ = 3.00 sin 30.0 = 1.50 m so, = i + j = (2.60i j) m B = 0, B = 3.00 m so B = 3.00j m + B = (2.60i j) j = (2.60i j) m *3.40 The coordinate of the airplane is constant and equal to m whereas the coordinate is given b = v i t where v i is the constant speed in the horizontal direction. t t = 30.0 s we have = , so v i = 268 m/s. The position vector as a function of time is P = (268 m/s)t i + ( m)j. t t = 45.0 s, P = [ i j] m. The magnitude is P = ( ) 2 + ( ) 2 m = m and the direction is θ = rctan = 32.2 above the horizontal 3.41 We have B = R = 150 cos 120 = 75.0 cm θ B = 150 sin 120 = 130 cm R = 140 cos 35.0 = 115 cm R = 140 sin 35.0 = 80.3 cm Therefore, B = [115 ( 75)]i + [ ]j = (190i 49.7j) cm R = + B 35.0

16 16 Chapter 3 Solutions B = [ (49.7) 2 ] 1/2 = 196 cm, θ = tan = 14.7

17 Chapter 3 Solutions 17 *3.42 Since + B = 6.00j, we have ( + B )i + ( + B )j = 0i j giving + B = 0, or = B (1) and + B = 6.00 (2) Since both vectors have a magnitude of 5.00, we also have: + B = B 2 + B 2 = (5.00) 2 From = B, it is seen that 2 = B 2. Therefore = B 2 + B 2 gives 2 = B 2. Then = B, and Equation (2) gives = B = B t θ φ = 2t θ t θ Defining θ as the angle between either or B and the ais, it is seen that cos θ = and θ = 53.1 = B B = = The angle between and B is then φ = 2θ = (a) = 8.00i j 4.00 k (b) B = /4 = 2.00i j 1.00k (c) C = 3 = 24.0i 36.0j k 3.44 R = 75.0 cos 240 i sin 240 j cos 135 i sin 135 j cos 160 i sin 160 j R = 37.5i 65.0j 88.4i j 94.0i j R = 220i j R = ( 220) at rctan above the -ais R = 227 paces at (a) C = + B = (5.00i 1.00j 3.00k) m C = (5.00) 2 + (1.00) 2 + (3.00) 2 m = 5.92 m (b) D = 2 B = (4.00i 11.0j k) m

18 18 Chapter 3 Solutions D = (4.00) 2 + (11.0) 2 + (15.0) 2 m = 19.0 m

19 Chapter 3 Solutions 19 *3.46 The displacement from radar station to ship is S = (17.3 sin 136 i cos 136 j) km = (12.0i 12.4j) km From station to plane, the displacement is P = (19.6 sin 153 i cos 153 j k) km, or P = (8.90i 17.5j k) km. (a) From plane to ship the displacement is D = S P = (3.12i j 2.20k) km (b) The distance the plane must travel is D = D = (3.12) 2 + (5.02) 2 + (2.20) 2 km = 6.31 km 3.47 The hurricane's first displacement is 41.0 km (3.00 h) at 60.0 N of W, and its second h displacement is 25.0 km (1.50 h) due North. With i representing east and j representing h north, its total displacement is: 41.0 km h cos 60.0 (3.00 h)( i) km sin 60.0 (3.00 h) j h km h (1.50 h) j = 61.5 km ( i) km j with magnitude (61.5 km) 2 + (144 km) 2 = 157 km *3.48 (a) E = (17.0 cm) cos 27.0 i + (17.0 cm) sin 27.0 j E = (15.1i j) cm (b) F = (17.0 cm) sin 27.0 i + (17.0 cm) cos 27.0 j (c) F = ( 7.72i j) cm G = +(17.0 cm) sin 27.0 i + (17.0 cm) cos 27.0 j F G E 27.0 G = (+7.72i j) cm

20 20 Chapter 3 Solutions 3.49 = 3.00, = 2.00 (a) = i + j = 3.00i j (b) = = ( 3.00) 2 + (2.00) 2 = 3.61 tanθ = 2.00 = ( 3.00) = 0.667, tan 1 ( 0.667) = 33.7 θ is in the 2 nd quadrant, so θ = ( 33.7 ) = 146 (c) R = 0, R = 4.00, R = + B thus B = R and B = R = 0 ( 3.00) = 3.00, B = R = = 6.00 Therefore, B = 3.00i 6.00j 3.50 Let + = East, + = North, (a) θ = tan 1 = 74.6 N of E (b) R = = 470 km 3.51 Refer to Figure P3.51 in the tetbook. (a) R = 40.0 cos cos 45.0 = 49.5 R = 40.0 sin sin = 27.1 R = 49.5i j (b) R = (49.4) 2 + (27.1) 2 = 56.4 θ = tan = 28.7

21 Chapter 3 Solutions Taking components along i and j, we get two equations: 6.00a 8.00b = a b = 0 Solving simultaneousl, a = 5.00, b = 7.00 Therefore, B + C = 0 *3.53 Let θ represent the angle between the directions of and B. Since and B have the same magnitudes,, B, and R = + B form an isosceles triangle in which the angles are 180 θ, θ /2, and θ /2. The magnitude of R is then R = 2 cos(θ /2). [Hint: appl the law of cosines to the isosceles triangle and use the fact that B =.] gain,, B, and D = B form an isosceles triangle with ape angle θ. ppling the law of cosines and the identit (1 cosθ ) = 2 sin 2 (θ /2) gives the magnitude of D as D = 2 sin(θ /2). The problem requires that R = 100D. Thus, 2 cos(θ /2) = 200 sin(θ /2). This gives tan(θ /2) = and θ = θt/2 R t θ B D t θ B *3.54 Let θ represent the angle between the directions of and B. Since and B have the same magnitudes,, B, and R = + B form an isosceles triangle in which the angles are 180 θ, θ /2, and θ /2. The magnitude of R is then R = 2 cos(θ /2). [Hint: appl the law of cosines to the isosceles triangle and use the fact that B =.] gain,, B, and D = B form an isosceles triangle with ape angle θ. ppling the law of cosines and the identit (1 cosθ ) = 2 sin 2 (θ /2) gives the magnitude of D as D = 2 sin(θ /2). The problem requires that R = nd, or cos(θ /2) = nsin(θ /2), giving θ = 2 tan 1 (1/n). θt/2 R tθ B D t θ B

22 22 Chapter 3 Solutions 3.55 (a) R = 2.00, R = 1.00, R z = 3.00 (b) R = R 2 + R 2 + R 2 Z = = 14.0 = 3.74 (c) R cos θ = R R θ = cos 1 R R cos θ = R R θ = cos 1 R R z cos θ z = R z R θ z = cos 1 R = 57.7 from + = 74.5 from + = 36.7 from + z *3.56 Choose the +-ais in the direction of the first force. The total force, in newtons, is then 12.0i j 8.40i 24.0j = (3.60i) + (7.00j) N The magnitude of the total force is (3.60) 2 + (7.00) 2 N = 7.87 N and the angle it makes with our +-ais is given b tanθ = (7.00) (3.60) counterclockwise from the horizontal is = 97.8, θ = Thus, its angle 31 N 8.4 N R N horizontal 24 N

23 Chapter 3 Solutions d 1 = 100i d 2 = 300j d 3 = 150 cos (30.0 )i 150 sin (30.0 )j = 130i 75.0j d 4 = 200 cos (60.0 )i sin (60.0 )j = 100i + 173j φ tθ R = d 1 + d 2 + d 3 + d 4 = 130i 202j R = [( 130) 2 + ( 202) 2 ] 1/2 = 240 m R φ = tan = 57.2 θ = φ = 237 *3.58 dp/dt = d(4i + 3j 2t j)/dt = j = (2.00 m/s)j The position vector at t = 0 is 4i + 3j. t t = 1 s, the position is 4i + 1j, and so on. The object is moving straight downward at 2 m/s, so dp/dt represents its velocit vector v = v i + v j = ( cos 30.0 )i + (100 sin 30.0 )j v = (387i j) mi/h v = 390 mi/h at 7.37 N of E 3.60 (a) You start at r 1 = r = 30.0 m i 20.0 m j. The displacement to B is r B r = 60.0i j 30.0i j = 30.0i + 100j You cover one half of this, 15.0i j, to move to r 2 = 30.0i 20.0j i j = 45.0i j Now the displacement from our current position to C is r C r 2 = 10.0i 10.0j 45.0i 30.0j = 55.0i 40.0j You cover one third, moving to r 3 = r 2 + r 23 = 45.0i j + 1 ( 55.0i 40.0j) = 26.7i j 3

24 24 Chapter 3 Solutions The displacement from where ou are to D is r D r 3 = 40.0i 30.0j 26.7i 16.7j = 13.3i 46.7j You traverse one quarter of it, moving to r 4 = r (r D r 3 ) = 26.7i j + 1 (13.3i 46.7j) = 30.0i j 4 The displacement from our new location to E is r E r 4 = 70.0i j 30.0i 5.00j = 100i j of which ou cover one fifth, 20.0i j, moving to r 4 + r 45 = 30.0i j 20.0i j = 10.0i j. The treasure is at (10.0 m, 16.0 m) (b) Following the directions brings ou to the average position of the trees. The steps we took numericall in part (a) bring ou to r (r B r ) = r + r B 2 then to then to (r + r B ) 2 + r C (r + r B )/2 3 = (r + r B + r C ) 3 r + r B + r C 3 and at last to + r D (r + r B + r C ) /3 4 = (r + r B + r C + r D ) 4 r + r B + r C + r D 4 + r E (r + r B + r C + r D )/4 5 = r + r B + r C + r D + r E 5 This center of mass of the tree distribution is in the same location whatever order we take the trees in.

25 Chapter 3 Solutions (a) From the picture R 1 = ai + bj and R 1 = a 2 + b 2 (b) R 2 = ai + bj + ck. Its magnitude is R c 2 = a 2 + b 2 + c 2 z a b O R 1 R 2 c 3.62 (a) r 1 + d = r 2 defines the displacement d, so d = r 2 r 1. (b) r 2 d r The displacement of point P is invariant under rotation of the coordinates. Therefore, r = r' and r 2 = (r') 2 or, = (') 2 + (') 2 lso, from the figure, β = θ α P tan 1 ' ' = tan 1 α r αβ ' ' = tanα 1 + tanα O β θ αt Which we simplif b multipling top and bottom b cos α. Then, ' = cosα + sin α, ' = sin α + cosα

26 26 Chapter 3 Solutions