DEPARTMENT OF MATHEMATICS

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1 DEPARTMENT OF MATHEMATICS Introduction to A level Mathematics Mechanics course workbook Name:

2 Thank you for choosing to continue to study Mathematics in the sixth form at Caedmon College Whitby. The Mathematics Department is committed to ensuring that you make good progress throughout your A level course. We hope that you will use this workbook to give you an organised set of notes for revision and that it will help you enjoy and benefit from the course more. ORGANISATION Keep this workbook, worksheets, homework and assessments in a file + BOOK SCRUTINY You will be asked to hand this in/bring it along to a progress interview with your teacher or Mr Gower during the year TEXTBOOKS CALCULATOR ATTENDANCE ASSESSMENT REVISION BOOKLET TRIAL EXAM This workbook should be your main resource so we do not use a textbook in lessons. Resources are also available in the Learning Centre, but if you wish to use textbooks at home for homework and revision, then we recommend that you buy the course textbook: Pearson Modular Mathematics for Edexcel AS and A-level, Mechanics 1. The recommended calculator for the course is the Casio FX 85GT PLUS. If you wish to use a graphical calculator, make sure it is an approved one for exams We expect maximum attendance otherwise your place on the course may be at risk. Should you be unable to attend a lesson, for whatever reason, you MUST catch up on the work missed. You will need to copy someone else s notes and find out what exercises have been set. Where possible this should be done before the next lesson. Each chapter has a test and resit test. If you are falling behind in work and tests, a meeting will be called to discuss your progress. We expect all pupils to meet deadlines and maintain a minimum score close to their target grade in all assessments. If you are falling behind in work and tests, a meeting will be called to discuss your progress. We expect all pupils to maintain a minimum of close to their target grade in all assessments. We will produce revision a revision programme once you have completed all or most of the Mechanics course. As last year there will be an expectation that you do one chapter a week/fortnight to prepare for trial and actual exams. There will be one opportunity to resit the trial exam You may also find the following useful: For resources and links to video tutorials; exam papers and solutions. Use the Mathematics department website: ccwmaths.wordpress.com And CCW Maths on Specifications, the formula book given in exams and past papers can be all downloaded from the EdExcel website Mr J Gower Head of KS5 Mathematics

3 MECHANICS 1 FORMULA SHEET Velocity = change of displacement ; time Acceleration = change of velocity time Kinematics: (for constant acceleration) 2 2 v u at v u 2as s ut 1 at 2 2 s vt 2 at s 2 ( u v) t u= initial velocity, v = final velocity, s = distance, a = acceleration, t = time Momentum = mv Impulse, I = mv mu (= change in momentum) Impulse = force time = F t = Ft So Ft = mv mu (where m = mass, u = initial velocity, v = final velocity, F = force, t = time) Principle of conservation of momentum: m 1u1 m2u2 m1v 1 m2v2 OR: total momentum before impact = total momentum after impact Friction, F μr, where: μ = coefficient of friction and R = normal contact force (remember, if the object is moving or about to move, then F = μr, in fact, you can always use this formula for Mechanics 1) Force, F = ma (i.e. force = mass acceleration) Resolving forces: From the diagram on the right, the effect of the mass, m, of the particle parallel to the plane = mg cos, and the effect of the mass of the particle perpendicular to the plane = mg sin. ɵ m mg cos mg sin Vectors: Always of the form: ai + bj = ( a b ) where: i is usually the vector reading horizontally left to right j is usually the vector reading vertically bottom to top Moments: The moment of a force about a point = force perpendicular distance Moment = Fd

5 Contents 1. Kinematics 2. Vectors 3. Statics 4. Dynamics 5. Momentum 6. Moments 7. Revision 8.

7 M1WB: Kinematics Suvat Notes BAT derive the suvat equations BAT apply them to solve problems BAT change between velocity measures

8 M1WB: Kinematics WB1 A train starts from rest at a station and moves with constant acceleration. 35 seconds later it passes a signal box with a speed of 50 ms -1 What is its acceleration in ms -2? WB2 A boat accelerates from 4 ms -1 to 6.5 ms -1 over 8 seconds How far has the boat gone in this time?

9 M1WB: Kinematics WB3 A space shuttle travelling at ms -1 slows with a deceleration of 45 ms -2 for 2 minutes. How many km will it have travelled? WB4 A car is following a Lorry going at 60 kmph on the motorway. The car constantly accelerates at 3 ms -2 to overtake the lorry and keeps accelerating for the 120 metres it takes to overtake. How fast is the car travelling once it has overtaken the lorry? Give your answer in kmph

10 M1WB: Kinematics

11 M1WB: Kinematics Vertical motion Notes BAT Set up a model of vertical motion with acceleration due to gravity BAT Find the high point of a thrown object or distance travelled BAT Solve extended problems

12 M1WB: Kinematics WB5 A marble falls off a shelf which is 1.6 m above the floor. Find: a) the time it takes to reach the floor b) The speed with which it will reach the floor WB6 A marble is catapulted vertically upwards with a velocity of 24.5 ms -1. Modelling the marble as a particle moving under gravity alone, find for how long its height exceeds 29.4 m

13 M1WB: Kinematics WB7 A particle P is projected upwards from a point O with a speed of 28 ms -1. Find: a) The greatest height above O reached by P b) The total time before P returns to O c) The total distance travelled by the particle on return to O

14 M1WB: Kinematics

15 M1WB: Kinematics Kinematics graphs notes BAT Draw accurate diagrams for distance-time and velocity time graphs BAT Find Area under a speed-time graph, gradient of a speed-time graph BAT Explore other graphs and links

16 M1WB: Kinematics WB8 A cyclist travels on a straight road over a 11 s period. For the first four seconds they travel at constant speed of 8 ms -1 For the rest of the time they accelerate at a constant rate of 1 ms- 2 Draw a Velocity time graph of their journey and work out the distance they have travelled

17 M1WB: Kinematics WB9 A man is jogging along a straight road at a constant speed of 4ms -1. He passes a friend with a bicycle who is standing at the side of the road and 20s later cycles to catch him up. The cyclist accelerates at a constant rate of 3 ms -2 until he reaches a speed of 12 ms -1. He then maintains a constant speed. a) On the same diagram sketch the speed-time graphs of both cyclist and jogger b) Find the time that elapses before the cyclist meets the jogger

18 M1WB: Kinematics WB10 Acceleration graph The acceleration-time graph models the motion of a particle. At time t = 0 the particle has a velocity of 8 ms -1 in the positive direction a) Find the velocity of the particle when t = 2 b) At what time does the particle start travelling in the negative direction?

19 M1WB: Vectors Vector Geometry Notes BAT apply the rules of vector geometry to solve problems

20 M1WB: Vectors WB1 OPQ is a triangle R is the midpoint of OP S is the midpoint of PQ OP = p and OQ = q R p P S Diagram NOT accurately drawn O q Q a) Find OS in terms of p and q b) Show that RS is parallel to OQ WB2 Q T Diagram NOT accurately drawn b O P a OPQ is a triangle. T is the point on PQ for which PT: TQ is 2: 1 OP = a and OQ = b a) Write down in terms of a and b, an expression for PQ b) Express OT in terms of a and b, Give your answer in its simplest form

21 M1WB: Vectors WB3 ABCD is a straight line O is a point so that OA = a and OB = b B is the midpoint of AC C is the midpoint of AD Express, in terms of a and b, the vectors a) AC A B C D a b) OD b O Diagram NOT accurately drawn WB4 ABCD is a parallelogram. AB is parallel to DC and AD is parallel to BC A p B AB = p and AD = q q a) Express in terms of p and q: i) AC b) Express AT in terms of p and q ii) BD D C

22 M1WB: Vectors WB5 OABC is a parallelogram P is the point on AC such that AP = 2 3 AC 6a A B D a OA = 6a and OC = 6c P a) Find the vector OP b) Prove that OPM is a straight line O 6c C

23 M1WB: Vectors WB6 OPQR is a trapezium with OR parallel to PQ OP = 2b, PQ = 2a and OR = 6a M is the midpoint of PQ and N is the midpoint of OR a) Find the vector and MN in terms of a and b OPQR is a trapezium with PQ parallel to OR. X is the midpoint of MN and Y is the midpoint of QR b) Prove that XY is parallel to OR

25 M1WB: Vectors Vector equations Notes BAT Find resultant vectors using vector triangles BAT Use position & velocity vectors to describe an objects motion BAT Find vector equations for position (and velocity) BAT Solve problems with vector equations

26 M1WB: Vectors WB7 Barry is at 4i + 6j relative to the origin Barry walks over to Hannah: his change in position is 8i 12j Where is Hannah? If Barry took 4 seconds what are his velocity and his speed? WB8 Barry starting position is 4i + 6j His velocity is 2i 3j Where will Barry be after: i) 10 seconds ii) 23 seconds Write an expression for his position after t seconds

27 M1WB: Vectors WB9 Barry s position is given by 4i + 6j + t(2i - 3j) A rock has position vector 28i 57j When will Barry be directly North of the Rock? When will Barry be directly East of the Rock? WB10 Barry starts at 4i + 6j relative to the origin and walks with constant velocity 2i - 3j Hannah starts at 10i 6j and walks with constant velocity 0.5i Will they meet each other (at the same place)? If so, at what time?

28 M1WB: Vectors WB11 At t = 0 two skaters John and Nadine have position vectors 40j and 20i respectively, relative to the centre of the ice rink. John has constant velocity 5i ms -1 and Nadine has constant velocity 3i + 4j ms -1 Show that the skaters will collide and find the time of collision

29 M1WB: Vectors WB6 At 12:00 a helicopter A sets out from its base O and flies with speed 120 kmh - 1 in the direction of vector 3i + 4j At 12:20 helicopter B sets out from O and flies with speed 150 kmh -1 in the direction of the vector 24i + 7j a) Find the velocity vectors of A and B b) Find the position vectors of A and B at 13:00 c) Calculate the distance of A from B at 13:00 At 13:30 B makes an emergency landing. A immediately changes direction and flies at 120 kmh -1 in a straight line to B d) Find the position vector of B from A at 13:30 e) Determine the time when A reaches B

31 M1WB: Statics Force Diagrams Notes BAT resolve forces into i and j components BAT add and subtract forces, finding the resultant force in different situation BAT Use W=mg in problems

32 M1WB: Statics WB1 A Truck is pulled by two horizontal ropes with Tension and directions shown Find the size of the resultant force in the form xi + yj 10 N 10 N 40 0 i 8 N 60 0 Plan view j 15 N i, j notation i WB2 A Sign hanging in space with mass 4 kg is pulled by two Cables with Tension and directions shown. Find the resultant force in the form xi + yj 12 N 50 0 W

33 M1WB: Statics WB3 A Sledge of mass 80 kg is being dragged by a force of 200 Newton's at an angle of 30 0 to the horizontal smooth surface such that it is accelerating horizontally forward as shownwhat is the Reaction Force? Find the resultant force in the form xi + yj R 200 N 30 0 W WB4 The same Sledge (mass 80 kg) is now dragged by the same force along a rough surface such that resistance force is 80 Newton s It is still accelerating horizontally forward What is the Reaction Force? Find the resultant force in the form xi + yj

34 M1WB: Statics

35 M1WB: Statics Equilibrium Notes BAT Set up a model of Equilibrium of forces BAT Use equilibrium to resolve forces and find unknowns BAT Solve higher grade problems with algebraic constants, simultaneous equations

36 M1WB: Statics WB5 The force system shown in the diagram is in equilibrium Calculate P and Q Q 150 P 45 8N WB6 The force system shown in the diagram is in equilibrium Calculate P and θ 5N 60 8N θ P

37 M1WB: Statics WB7 A block with Weight = 10 N rests on a sloping surface. The surface is rough and at an angle of 30 0 to the horizontal. Find the friction and reaction force WB8 An object with Weight suspends in the air held by three inelastic strings : T1, T2 and T3 as shown. If the Tension in T3 is 4 Newtons, find the Tension in T1 and T2 T 1 T 2 T W e i g h

38 M1WB: Statics WB9 An object with Weight of 10 Newtons is held at rest on a rough slope of 30 0 by a force of 8 Newtons acting parallel to the slope. Friction is acting down the slope. Find the magnitude of the reaction force and of Friction WB10 An object is suspended in mid-air by two cables T1 and T2. T1 is at an angle of from the horizontal and has magnitude 6 Newton s. T2 is at an angle of 70 0 from the horizontal. Find the magnitude of T2 and hence the magnitude of the weight of the object

39 M1WB: Statics Coefficient of Friction notes BAT Understand the directly proportional relationship between Friction force and Reaction force BAT Solve equilibrium problems using coefficient of friction

40 M1WB: Statics WB11 The same 5 kg sledge is put on two different surfaces pulled by 100 N force at angle of Given the values of Friction, find the resultant force and the coefficient of friction for each surface

41 M1WB: Statics WB12 A Block with mass 3 kg is pushed by a force parallel to the direction of the slope.the block is at a tipping point where it is about to go up the slope. The slope has coefficient of friction of 0.3 and is at 28 0 to the horizontal. Find the values of friction and the Push force WB13 A Block with mass 8 kg is at the point of slipping down a rough slope. The slope is at 28 0 to the horizontal. Find the value of the coefficient of friction of the slope

42 M1WB: Statics WB14 A Block with mass 3 kg is held at rest on a rough slope by a horizontal force (Push). The slope has coefficient of friction of 1/3 and the slope is at 30 0 to the horizontal. Find the value of the Push force WB15 A Block with Weight 10 Newton s is held at rest on a rough slope by a horizontal force (Push). The slope has coefficient of friction of 0.3 and the slope is at 30 0 to the horizontal. Find the value of the Push, Reaction and friction forces

43 M1WB: Statics Connected particles in equilibrium notes BAT Set up a model of two connected particles with uniform tension on an inelastic string/cable BAT Solve problems such as: i) car towing caravan or ii) object on slope connected to hanging object

44 M1WB: Statics WB16 WB17

45 M1WB: Dynamics Dynamics: straight line and vertical motion Notes BAT Understand Newtons 3 laws BAT Resolve forces for moving objects in a straight line BAT Solve problems using Newtons 2nd law BAT solve problems with moving objects using the suvat model BAT Extend model to vertical motion and object on a slope Newton s Laws: 1 st law. A particle will remain at rest or will continue to move with constant velocity in a straight line unless acted on by a resultant force 2 nd law. The force applied to a particle is proportional to the mass m of the particle and the acceleration produced 3 rd law. Every action has an equal and opposite reaction

46 M1WB: Dynamics WB1 Sledge A sledge of mass 600 g is dragged by forces parallel to the ground Find the acceleration of the sledge when forces of (7i + 13j) N, (4i +4j) N and (-2i 5j) act on it. Find also the magnitude and direction of the acceleration

47 M1WB: Dynamics WB2 Car A car travels a distance of 32m along a straight road while uniformly accelerating from rest to 16 ms -1. By modelling the car as a particle find its acceleration. Given that the mass of the car is 640 Kg, find the magnitude of the accelerating force. WB3 Vertical motion A Particle of mass 2 kg is attached to the lower end of a string hanging vertically. The particle is lowered and moves with acceleration 0.2 ms -2 Find the tension in the string

48 M1WB: Dynamics WB4 Lift A lift is accelerating upward at 1.5 ms -2. A child of mass 30 Kg is standing in the lift. Treating the child as a particle find the force between the child and the floor of the lift. WB5 Vertical motion A stone of mass 0.5 kg is released from rest on the surface of the water in a well. It takes 2 seconds to reach the bottom of the well. Given that the water exerts a constant resistance of 2 N, find the depth of the well.

49 M1WB: Dynamics WB6 Object on slope find a A parcel of mass 5 Kg is released from rest on a rough ramp of inclination θ = 30 0 and slides down the ramp. The resistance due to friction is 8 N Treating the parcel as a particle, find the acceleration of the parcel WB7 Car up slope WB7 A car with a constant driving force of 1500 N meets a slope inclined at an angle of elevation of The resistance forces to the car on the slope = 400 N. If the car has weight 1800 N, find the acceleration of the car up the slope

50 M1WB: Dynamics WB8 Car deccelarating A car of mass 500 kg travelling at a constant speed of 25 ms -1 reaches a slope inclined at 30 0 to the horizontal. The resistance forces on the car travelling up the slope total 300 N. The driver takes their foot off the acceleration pedal at the start of the slope Find the distance travelled by the car before it comes to rest

51 M1WB: Dynamics WB9 Find angle slope WB 9 A parcel of mass 3 kg is sliding down a smooth inclined plane with an acceleration of 4 ms-2. Find the angle of inclination of the plane WB10 Find m WB 9 Daisy is sledging down a slope of 30 0 and accelerating at 1 ms -2. The resistance force due to friction is 10 N. Find Daisy s mass.

52 M1WB: Dynamics

53 M1WB: Dynamics Dynamics friction reintroduced Notes BAT Extend model to reintroduce the coefficient of friction BAT Solve harder problems with moving objects and several steps or simultaneous equations; BAT Solve problems up to exam level

54 M1WB: Dynamics WB11 A parcel of mass 3 kg is sliding down a rough slope of inclination 30 0 The coefficient of friction between the parcel and the slope is Find the acceleration of the particle WB12 A particle rests in limiting equilibrium on a plane inclined at 30 0 to the horizontal. Determine the acceleration with which the particle will slide down the plane when the angle of inclination is increased to 40 0

55 M1WB: Dynamics WB13 Suvat Find coefficient friction A parcel of mass 5 Kg is released from rest on a rough ramp of inclination θ = arcsin 3/5 and slides down the ramp. After 3 secs it has a speed of 4.9 ms -1 Treating the parcel as a particle, find the coefficient of friction between the parcel and the ramp

56 M1WB: Dynamics WB14 Find angle of inclination WB14 A parcel of mass 3 kg is sliding down a rough inclined plane with an acceleration equal to g cos ms -2. Find the angle of inclination of the plane if the coefficient of friction between the parcel and plane is 0.6

57 M1WB: Dynamics Connected Particles notes BAT Solve dynamics problems with connected particles: i) the car and caravan problems (straight line) or towed at an angle ii) problems with pulleys

58 M1WB: Dynamics WB15 moving in same direction Two particles P and Q, of masses 5kg and 3kg respectively, are connected by a light, inextensible string. Particle P is pulled by a horizontal force of 40N along a rough horizontal plane. The coefficient of friction is 0.2 and the string is taut. WB16 Tow bar A car of mass 1000 kg tows a caravan of mass 750 kg along a straight road. The engine of the car exerts a forward force of 2.5 kn The resistances to the motion of the car and caravan are each k their mass where k is a constant. Given that the car accelerates at 1 ms -2 find the Tension in the tow-bar.

59 M1WB: Dynamics WB17 Tow bar at angle This figure shows a lorry of mass 1600 kg towing a car of mass 900 kg along a straight horizontal road. The two vehicles are joined by a light towbar which is at an angle of 15 to the road. The lorry and the car experience constant resistances to motion of magnitude 600 N and 300 N respectively. The lorry s engine produces a constant horizontal force on the lorry of magnitude 1500 N. Find (a) the acceleration of the lorry and the car, (3) b) the tension in the towbar. (4) Taken further When the speed of the vehicles is 6 m s 1, the towbar breaks. Assuming that the resistance to the motion of the car remains of constant magnitude 300 N, c) find the distance moved by the car from the moment the towbar breaks to the moment when the car comes to rest. (4) d) State whether, when the towbar breaks, the normal reaction of the road on the car is increased, decreased or remains constant. Give a reason for your answer (2)

60 M1WB: Dynamics WB18 Tow bar on slope A lorry of mass 900 kg is towing a trailer of mass 500 kg up an inclined road, at angle α, where tan α = ¾.The two vehicles are joined by a light towbar. The lorry and the trailer experience constant resistances to motion of magnitude 1600 N and 600 N respectively. The lorry s engine produces a constant horizontal force on the lorry of magnitude N. Find a) the acceleration of the lorry and car, (3) b) the tension in the towbar. (4) WB19 Tow bar - decceleration A lorry of mass 900 kg is towing a trailer of mass 300 kg along a straight road. The two vehicles are joined by a light towbar. Assume resistance forces are negligible. The driver sees a red light ahead and brakes causing a braking force of 2400 N. Find a) the tension in the towbar, (4) b) the distance travelled before stopping (4)

61 M1WB: Dynamics WB20 Different directions Two particles of mass 2m and 3m are connected by a light, inextensible string over a smooth pulley Find the acceleration of the particles. Find the tension in the string Find the force exerted by the string on the pulley WB21 Pulley table + suspended weight Two particles P and Q of masses 6 kg and 3 kg are connected by a light inextensible string. Particle P rests on a rough horizontal table. The string passes over a smooth pulley fixed at the edge of the table and Q hangs vertically. The system starts from rest. If the coefficient of friction μ = 1 /3, find a) The acceleration of Q b) The Tension in the string c) The force exerted on the pulley

62 M1WB: Dynamics WB22 Pulley Slope + suspended weight A particle P of mass 5 kg lies on a smooth inclined plane of angle θ = arcsin 3 /5. Particle P is connected to a particle Q of mass 4 kg by a light inextensible string which lies along a line of greatest slope on the plane and passes over a smooth peg. The system is held at rest with Q hanging vertically 2 m above a horizontal plane. The system is now released from rest. Assuming P does not reach the peg, find, to 3sf: a) The acceleration of Q b) How long t takes for Q to hit the horizontal plane c) the total distance that P moves up the plane.

63 M1WB: Dynamics WB23 Pulley table + suspended weight A particle A of mass 0.8 kg rests on a horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley P fixed at the edge of the table. The other end of the string is attached to a particle B of mass 1.2 kg which hangs freely below the pulley, as shown in the diagram above. The system is released from rest with the string taut and with B at a height of 0.6 m above the ground. In the subsequent motion A does not reach P before B reaches the ground. In an initial model of the situation, the table is assumed to be smooth. Using this model, find a) the tension in the string before B reaches the ground, b) the time taken by B to reach the ground In a refinement of the model, it is assumed that the table is rough and that the coefficient of friction between A and the table is 1/5 Using this refined model, (c) find the time taken by B to reach the ground.

64 M1WB: Dynamics WB24 Pulley Slope + suspended weight A particle A of mass 4 kg moves on the inclined face of a smooth wedge. This face is inclined at 30 to the horizontal. The wedge is fixed on horizontal ground. Particle A is connected to a particle B, of mass 3 kg, by a light inextensible string. The string passes over a small light smooth pulley which is fixed at the top of the plane. The section of the string from A to the pulley lies in a line of greatest slope of the wedge. The particle B hangs freely below the pulley, as shown in the diagram above. The system is released from rest with the string taut. For the motion before A reaches the pulley and before B hits the ground, find a) the tension in the string, b) the magnitude of the resultant force exerted by the string on the pulley. c) The string in this question is described as being light. (i) Write down what you understand by this description. (ii) State how you have used the fact that the string is light in your answer to part (a).

65 M1WB: Dynamics WB25 Pulley Slope + suspended weight The diagram above shows two particles A and B, of mass m kg and 0.4 kg respectively, connected by a light inextensible string. Initially A is held at rest on a fixed smooth plane inclined at 30 to the horizontal. The string passes over a small light smooth pulley P fixed at the top of the plane. The section of the string from A to P is parallel to a line of greatest slope of the plane. The particle B hangs freely below P. The system is released from rest with the string taut and B descends with acceleration g. a) Write down an equation of motion for B b) Find the tension in the string 16 c) Prove that m = 35 d) State where in the calculations you have used the information that P is a light smooth pulley. On release, B is at a height of one metre above the ground and AP = 1.4 m. The particle B strikes the ground and does not rebound. e) Calculate the speed of B as it reaches the ground. f) Show that A comes to rest as it reaches P

67 M1WB: Moments Moments on rods and beams - Notes BAT find the moment on object at a pivot; BAT understand notation and units for moments

68 M1WB: Moments WB1 A light rod AB is 4 m long and can rotate in a vertical plane about fixed point C Where AC = 1 m. A vertical force F of 8 N acts on the rod downwards. Find the moment of F about C when F acts a) at A b) at B c) at C

69 M1WB: Moments WB2 The diagram shows forces acting on a lamina. Find the sum of the moments acting about point O and state the sense of the moment (cw or acw) A = 12 N O B = 8 N 2 m. 2 m 1.5 m LAMINA WB3 A uniform rod AB of weight 20 N is 4 m long and can rotate in a vertical plane about fixed point C. Where AC = 1 m. A vertical force F of 8 N acts on the rod downwards at A and a vertical force of 12 N acts upwards at B. Find the sum of moments about C

70 M1WB: Moments

71 M1WB: Moments Moments on rods and beams - Notes BAT find the moment on a rod or beam at a pivot; BAT use equilibrium to solve problems with moments and forces

72 M1WB: Moments WB4 A uniform rod AB of length 4 m and mass 5 kg is pivoted at C where AC = 1.5 m Calculate the mass of the particle which must be attached at A to maintain equilibrium with the rod horizontal. WB5 A uniform beam AB of length 5 m and mass 30 kg rests horizontally on supports at C and D. Where AC = BD = 1 m. A man of mass 75 kg stands on the beam at E where AE = 2 m. Calculate the magnitude of the reaction at each of the supports C and D.

73 M1WB: Moments WB6 A uniform plank AB has mass 40 kg and length 4 m. It is supported in a horizontal position by two smooth pivots, one at the end A, the other at the point C of the plank where AC = 3 m, as shown in the diagram above. A man of mass 80 kg stands on the plank which remains in equilibrium. The magnitudes of the reactions at the two pivots are each equal to R newtons. By modelling the plank as a rod and the man as a particle, find a) the value of R, b) the distance of the man from A

74 M1WB: Moments

75 M1WB: Moments Non uniform rods and Tilting Notes BAT include idea of a tilting point in the model BAT extend model to include non-uniform rods

76 M1WB: Moments WB7 A plank AE, of length 6 m and mass 10 kg, rests in a horizontal position on supports at B and D, where AB = 1 m and DE = 2 m. A child of mass 20 kg stands at C, the mid-point of BD, as shown in the diagram. The child is modeled as a particle and the plank as a uniform rod. The child and the plank are in equilibrium. Calculate: a) the magnitude of the force exerted by the support on the plank at B, b) the magnitude of the force exerted by the support on the plank at D. The child now stands at a point F on the plank. The plank is in equilibrium and on the point of tilting about D. c) Calculate the distance DF.

77 M1WB: Moments WB8

78 M1WB: Moments WB9 A plank AB has mass 40 kg and length 3 m. A load of mass 20 kg is attached to the plank at B. The loaded plank is held in equilibrium, with AB horizontal, by two vertical ropes attached at A and C, as shown in the diagram. The plank is modelled as a uniform rod and the load as a particle. Given that the tension in the rope at C is three times the tension in the rope at A, calculate a) the tension in the rope at C, b) the distance CB

79 M1WB: Moments WB10 A steel girder AB has weight 210 N. It is held in equilibrium in a horizontal position by two vertical cables. One cable is attached to the end A. The other cable is attached to the point C on the girder, where AC = 90 cm, as shown in the figure above. The girder is modelled as a uniform rod, and the cables as light inextensible strings. Given that the tension in the cable at C is twice the tension in the cable at A, find a) the tension in the cable at A, b) show that AB = 120 cm A small load of weight W newtons is attached to the girder at B. The load is modelled as a particle. The girder remains in equilibrium in a horizontal position. The tension in the cable at C is now three times the tension in the cable at A. (c) Find the value of W.

80 M1WB: Moments WB11 A non-uniform rod AB has length 5 m and weight 200 N. The rod rests horizontally in equilibrium on two smooth supports C & D. The centre of mass of AB is x metres from A. A particle of weight W newtons is placed on the rod at A. The rod remains in equilibrium and the magnitude of the reaction of C on the rod is 160N (a) Show that 50x W = 100. The particle is now removed from A and placed on the rod at B. The rod remains in equilibrium and the reaction of C on the rod now has magnitude 50N. (b) Obtain another equation connecting W and x. c) Calculate the value of x and the value of W.

81 M1WB: Moments WB12 A uniform plank AB has mass 40 kg and length 3 m. A load of mass 20 kg is attached to the plank at B. The loaded plank is held in equilibrium, with AB horizontal, by two vertical ropes attached at A and C, as shown in the diagram. The plank is modelled as a uniform rod and the load as a particle. Given that the tension in the rope at C is three times the tension in the rope at A, calculate a) the tension in the rope at C, b) the distance CB

82 M1WB: Moments WB13 A plank AB has length 4 m. It lies on a horizontal platform, with the end A lying on the platform and the end B projecting over the edge, as shown below. The edge of the platform is at the point C. Jack and Jill are experimenting with the plank. Jack has mass 40 kg and Jill has mass 25 kg. They discover that, if Jack stands at B and Jill stands at A and BC = 1.6 m, the plank is in equilibrium and on the point of tilting about C. By modelling the plank as a uniform rod, and Jack and Jill as particles, (a) Find the mass of the plank. They now alter the position of the plank in relation to the platform so that, when Jill stands at B and Jack stands at A, the plank is again in equilibrium and on the point of tilting about C. (b) Find the distance BC in this position. c) State how you have used the modelling assumptions that (i) the plank is uniform, (ii) the plank is a rod, (iii) Jack and Jill are particles.

83 M1WB: Moments WB14 A steel girder AB has weight 210 N. It is held in equilibrium in a horizontal position by two vertical cables. One cable is attached to the end A. The other cable is attached to the point C on the girder, where AC = 90 cm, as shown in the figure above. The girder is modelled as a uniform rod, and the cables as light inextensible strings. Given that the tension in the cable at C is twice the tension in the cable at A, find a) the tension in the cable at A, b) show that AB = 120 cm A small load of weight W newton s is attached to the girder at B. The load is modelled as a particle. The girder remains in equilibrium in a horizontal position. The tension in the cable at C is now three times the tension in the cable at A. c) Find the value of W.

84 M1WB: Moments

85 M1WB: Momentum Notes BAT Find momentum of objects and change in their momentum BAT Understand impulse and apply in solving problems

86 M1WB: Momentum WB1 A particle of mass 6 kg is at rest on a smooth surface. A force of 12 N acts on the particle for 7 seconds Find the final speed of the particle WB2 A ball of mass 0.6 kg rebounds against a wall. The impulse of the ball on the wall is 17 Ns and its speed is 15 ms -1 immediately before hitting the wall Find the speed of the ball immediately after the rebound.

87 M1WB: Momentum Notes BAT Find momentum of objects and change in their momentum BAT Understand impulse and apply in solving problems BAT Understand conservation of momentum and apply to solve different types of problems involving collisions

88 M1WB: Momentum WB3 Particle A with mass 500 kg moving with velocity 10 ms -1 hits particle B with mass 800 kg which is at rest. Particle A rebounds with velocity 4 ms -1. i) What is the velocity of particle B after the collision? ii) What is the change in momentum of each particle? WB4 A snooker ball P moving with speed 4 ms -1 hits a stationary ball Q of equal mass. After the impact both balls move in the same direction along the same straight line, but the speed of Q is twice that of P. By modelling the balls as particles moving on a smooth horizontal surface find the speeds of the balls

89 M1WB: Momentum WB5 Particle A with mass 500 kg moving with velocity 10 ms -1 hits particle B with mass 800 kg which is moving in the same direction but with velocity 4 ms -1 The two Particles join together after the collision i) What is the velocity of the joint particles after the collision? ii) What is the change in momentum of A? WB6 Two particles P and Q of masses 3 kg and 6 kg respectively are connected by a light inextensible string. Initially they are at rest on a smooth table with the string slack. Q is projected directly away from P with a speed of 3 ms-1 Find their common speed when the string becomes taught

90 M1WB: Momentum WB7 A missile of mass 400 kg travelling at 200 ms -1 separates into parts A and B. As shown in the diagram A has a mass of 50 kg and B a mass of 350 kg. After separation the speed of B is 250 ms -1 in the original direction of motion. a) Calculate the impulse acting on B in the separation b) Calculate the velocity of A after the separation

MOMENTUM, IMPULSE & MOMENTS

the Further Mathematics network www.fmnetwork.org.uk V 07 1 3 REVISION SHEET MECHANICS 1 MOMENTUM, IMPULSE & MOMENTS The main ideas are AQA Momentum If an object of mass m has velocity v, then the momentum