Topic 4 Forces. 1. Jan 92 / M1 - Qu 8:

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1 Topic 4 Forces 1. Jan 92 / M1 - Qu 8: A particle of mass m lies on a smooth plane inclined at α. It is held in equilibrium by a string which makes an angle θ with the plane. The tension in the string is T and the reaction force is R (a): Find an expression for T in terms of m, g, θ & α (b): Show that R = mg cos α (1 - tan α tan θ) 4. Jan 93 / M1 - Qu 9: Particle P of mass M is held at point A on a rough plane inclined at θ, where tan θ = 7/24. μ = 11/12. A light inextensible string of length 50 cm is attached to P, passes over a smooth pulley O and is attached at its other end to particle Q, of mass 2M, which hangs freely and is vertically above point B. OA = 32 cm and OB = 43 cm. The particles are released from rest with the string taut. (a): Show that while the string is taut the acceleration of P is 7g/25 (b): Find the speed with which B hits the ground After B hits the ground, A continues to move up the slope a little (c): Determine whether or not A will hit the pulley at O 2. Jun 92 / M1 - Qu 6: Particle A of mass 3M lies on a rough horizontal table and is attached by a light inextensible string passing over a smooth pulley P to particle B of mass 2M, which hangs freely. The system is released from rest with the string taut when A is 1.2 m from P and B is 1 m above the floor. B reaches the floor after 2 s and does not rebound (a): Find the acceleration of a during the first 2 s (b): Find μ (c): Determine if A reaches P 5. Jun 93 / M1 - Qu 10: A small particle P is mass 1.5 kg is placed on a rough plane inclined at 27, where μ = 0.3. Determine the magnitude of the force S when the particle P is in limiting equilibrium and on the point of moving: (a): down the plane (b): up the plane 3. Jan 93 / M1 - Qu 2: Particles A & B of masses 0.3 kg and 0.4 kg are attached by a light inextensible string passing over a smooth pulley placed at the edge of a rough table. A is released from rest with the string taut and moves 0.25 m along the table in 0.5 s while B moves down the smooth plane inclined at 70. Calculate: (a): the tension in the string (to 2 s.f.) (b): μ

2 After A has been moving for 0.5 s, the string suddenly breaks. Given that A subsequently comes to rest on the table (c): Find the total distance travelled by A 6. Jan 94 / M1 - Qu 7: inextensible string of length 3h which passes over a smooth pulley P The coefficient of friction is μ. The system is released from rest (a): Show that, until B hits the floor, the accln of A is ¼g(1-3μ) (b): Find the speed of A immediately before B hits the floor After B hits the floor A comes to rest before reaching P (c): Deduce the range of values of P for this motion to be possible 9. Jan 95 / M1 - Qu 2: A pebble of mass 0.3 kg slides on a rough horizontal concrete path where μ = 3/7. Its initial speed is 12.6 m/s (a): Find the frictional force (b): Find the distance covered by the pebble before coming to rest 10. Jan 95 / M1 - Qu 3: A sledge S of mass 20 kg is placed on an icy track where μ = S is held in limiting equilibrium on the point of moving down the track by a horizontal force H (a): Find H (2 s.f.) H is removed and S moves down the track (b): Find the speed of S after it has been moving for 2 s (to 2 s.f.) 7.Jun 94 / M1 - Qu 5: A small box of mass 20 kg rests on a rough floor where μ = A light inextensible rope is tied to the box and pulled with magnitude P N at 14 to the floor. Given the box is on the point of sliding, find P (to 1 d.p.) 11. Jun 95 / M1(old) - Qu 3: A particle of mass M is at rest on a rough inclined slope. A force of magnitude kmg is applied at an angle of α to the slope and the particle is on the point of moving down the slope Show that: k = μ cos α - sin α cos α - μ sin α 8. Jun 94 / M1 - Qu 9: A force of 400 N acts on a small boat B of mass 100 kg on a slope. The boat is in limiting equilibrium. Find μ to 2 d.p. 12. Jun 95 / M1(old) - Qu 8: Particles A & B of masses 3m and m are attached to the ends of a light

3 A particle P of mass 5M rests on a rough plane inclined at α, where sin α = 4/5. It is attached by a light inextensible string over a smooth pulley to particle Q of mass M which hangs freely. The particles are released from rest with the string taut and P moves down the plane with acceleration g/3. Find: (a): the tension in the string in terms of M and g (b): μ (c): the magnitude of the resultant force exerted by the string on the pulley in terms of M and g 13. Jan 96 / M1(old) - Qu 1: A body of mass 5 kg is held in equilibrium under gravity by two light inextensible ropes, one of which is horizontal and the other which has a tension of 72 N, makes an angle α with the horizontal (a): Find α to the nearest degree (b): Find T to the nearest N 14. Jan 96 / M1(old) - Qu 8: (a): Find the accln of A in the period before B hits the ground When B hits the ground, the rope becomes slack and A continues to move up the slope, coming to rest without hitting P (b): Calculate the total distance moved by A before it comes to rest 16. Jan 96 / M1(new) - Qu 9: A sledge of mass 40 kg moves on a horizontal snow surface where μ=0.1 One end of a rope is attached to the sledge and the other end is attached to a towing hook on a snowmobile. The snowmobile pulls the sledge, giving it an acceleration of 0.12 m/s² In an initial model, the rope is assumed to be horizontal (a): Find the tension in the rope A more refined model takes account of the fact that the towing hook is above the level of the sledge. The rope is now assumed to make an angle of 20 to the horizontal (b): Find a revised value for the tension in the rope (1 d.p.) 17. Jun 96 / M1 - Qu 2: The bench is rough and the pulley is smooth. B has mass 0.5 kg and is placed 1.96 m from P. A light inextensible string is attached to B passes over the pulley and is attached to A which has mass 0.2 kg and hangs freely. The system is released from rest with the string taut B moves with acceleration 0.98 m/s², reaching P after t s (a): Find t For the motion until B reaches P, find: (b): the tension in the string (c): μ (d): the resultant force exerted by the string on the pulley 15. Jun 95 / M1(new) - Qu 9: A block A of mass m can move on the face of the rough wedge (μ = 3/4) inclined at α where sin α = 0.6. A light inextensible rope is attached to A, passes over the smooth pulley P and is attached to block B of mass 3m which hangs freely. The system is released from rest with B initially 1 m above the ground Two cart-horses A & B, pull a heavy rock R over rough horizontal ground. A pulls with 800 N and B pulls with 500 N (a): Find α to the nearest degree Given the horses drag the rock slowly at constant speed (b): Find the resistance to motion (c): Suggest why it is acceptable to ignore air resistance (d): Suggest one more refinement, in relation to the ropes that would make the model more accurate 18. Jun 96 / M1 - Qu 4: A parcel A of mass 2 kg rests on a rough slope where tan θ = 3/4. It is connected by a string over a smooth pulley to particle B, of mass 2.2 kg. Particle A is in limiting equilibrium and about to slide up the slope. Find:

4 21. Jun 97 / M1 - Qu 2: (a): the normal contact force on A 19. Jan 97 / M1 - Qu 4: (b): μ A particle P pf mass 1.5 kg slides down the smooth plane. It starts from rest at A and reaches the point B where AB = 3m Find the speed at which it reaches B 22. Jun 97 / M1 - Qu 6: Two ends of a string are attached to points A & B of a horizontal beam A 2 kg mass is attached at C. Find to 3 s.f. (a): the tension in AC (b): the tension in BC 20. Jan 97 / M1 - Qu 8: Particle A of mass m lies on a rough plane (μ = 1/4) inclined at α where tan α = 3/4. It is connected by s string passing over a smooth pulley to particle B of mass 2m, which hangs freely 2 m above the floor. The system is released from rest with the string taut (a): Find the initial acceleration of A (b): State clearly where the modelling assumptions that (i): the string is 'inextensible' (ii): the pulley is 'smooth' have been used in your calculation for part (a) After A has travelled 1 m up the slope, the string breaks. Assuming A does not hit the pulley (c): Find the further distance A travels up the plane before coming to instantaneous rest A box on mass m is placed on the rough slope (μ = 1/2) inclined at α where tan α = 3/4. The force P is the smallest possible value which will enable the box to remain in equilibrium (a): Show all of the forces (b): Find P in terms of m and g If instead P were to have the largest possible value which will keep the box in equilibrium (c): Show the forces and state how they have changed 23. Jan 98 / M1 - Qu 6: A man irons a shirt by pressing down on the iron, applying a force of R N at an angle α to the vertical. He moves the iron slowly at constant speed over the shirt. The iron has mass 1/2 kg and μ = 1/4 The total vertical force exerted on the shirt by the iron must be 20N. Find, to 3 d.p. (a): α (b): R (c): Explain why it is reasonable to ignore air resistance 24. Jun 98 / M1 - Qu 2: A particle P of mass 2 kg is held in equilibrium on a smooth plane (a): Find the normal reaction force (to 3 s.f.) (b): Find F

5 25. Jun 98 / M1 - Qu 8: 28. Jun 99 / M1 - Qu 7: A slipway for launching boats is a rough track inclined at 10. A boat of mass 300 kg is pulled down the slipway by means of a rope which is parallel to the slipway. When the tension in the rope is 500 N, the boat moves down at constant speed (a): Find μ (to 2 s.f.) The boat returns and the rope is used to pull the boat up the slipway (b): Why is the magnitude of the friction force the same as in part(a) (c): Find the tension (3 s.f.) 29. Jun 99 / M1 - Qu 9: Particle A of mass 3m rests on a rough (μ=1/3) slope where tan α = 4/3 and is connected by a light inextensible string over a smooth pulley to particle B which has mass 2m and rests on a smooth slope. The particles are release with the string taut and A moves down the plane (a): Show the friction on A has magnitude 3mg/5 (b): find (i): accln in terms of g (ii): tension in terms of m & g (c): State where you have used the assumption that (i): the string is inextensible (ii): the pulley is smooth 26. Jan 99 / M1 - Qu 3: Particle A of mass 2m moves on the surface of a rough (μ = 1/3) table It is connected by a string over a smooth pulley to particle B of mass 3m which hangs freely. The system is released from rest with the string taut and B at height h m above the floor (a): Find the tension while the particles are moving (b): Show that B reaches the ground with speed (14gh/15) When B reaches the ground, it does not rebound and A continues to move but does not reach P before coming to rest (c): Show the total length of the string must be at least 12h/5 30. Jan 00 / M1 - Qu 1: A has mass 3 kg and B has mass m kg and they are connected by a light inextensible string passing over a smooth pulley. The system is released from rest and accelerates at 2.2 m/s². Find the possible values of: (a): the tension (b): m (c): the force exerted on the pulley be the string (d): State how you have used the assumption that the pulley is smooth 27. Jan 99 / M1 - Qu 4: A box of mass 50 kg rests on rough horizontal ground where μ = 0.6. A force P is applied and the box is now in limiting equilibrium. Find P to 3 s.f. Particle P of mass 2 kg is attached to one end of a light string the other end of which is attached to a fixed point O. The particle is held in equilibrium with OP horizontal by a 30 N force applied at α Find: (a): α (to the nearest degree) (b): the tension (to 3 s.f.) 31. Jan 00 / M1 - Qu 8: A 1200 kg car is moving at 40 m/s when the driver makes an emergency stop. When the brakes are fully applied they exert a constant force and the car comes to rest after 80 m. The resistance from all factors other than the brakes is assumed to be constant at 500 N (a): Find the braking force

6 A 600 kg trailer with no brakes is now attached to the car with a towbar. When the trailer is moving it experiences a resistance of 420 N. The car and trailer come to a hill inclined at α where sin α = 1/14 which they move down. The driver again makes an emergency stop applying the same force as before. Find the: (a): deceleration (b): force exerted on the car by the trailer when brakes fully applied (c): maximum speed as which they should travel down the hill to ensure they can still stop within 80 m 32. Jun 00 / M1(old) - Qu 2: A particle P of mass M kg slides down a smooth slope. It passes A at 2 m/s. Find the speed it passes B at (a): How do you know accelerations of A & B have the same magnitude (b): Find the accelerations When B hits the ground, it does not rebound and A continues towards P but does not reach P before coming to rest (c): Find in terms of h the distance moved by A after the string becomes slack 36. Spec 00 / M1(new) - Qu 1: 33. Jun 00 / M1(old) - Qu 4: A 10 kg box lies on a rough floor. A rope with tension 40 N is pulling at 30. The box is in limiting equilibrium. Find μ 34. Jan 01 / M1(old) - Qu 2: A tennis ball P is attached to one end of a light inextensible string, the other end being attached to the top of a vertical pole. A girl applies a horizontal force of magnitude 50 N to P and P is in equilibrium under gravity with the string making an angle of 40 with the pole. Find, to 3 s.f: (a): the tension in the string (b): the weight of P 37. Spec 00 / M1(new) - Qu 6: Particle P is held in equilibrium under gravity by two strings. Find (a): T (b): The weight of P 35. Jan 01(old) - Qu 8: Two particles A & B of masses 3m and 7m are connected by a light inextensible string over a smooth pulley. A is placed on a rough table where μ = 1/3. The particles are released from rest with the string taut. For the period before B hits the ground: A particle A of mass m kg rests on a rough plane inclined at α where tan α = 3/4. It is attached by a light string over a smooth pulley to

7 particle B of mass 3m kg which hangs freely. The particles are released from rest with the string taut and the acceleration of the system is g/2. Find: (a): the tension in the string (b): μ 38. Mock 00 / M1(new) - Qu 2: A particle attached to the ends of two light inextensible strings AB and BC has mass 2 kg. In equilibrium, AB makes 30 with the vertical. The magnitude of the tension in BC is twice that of AB (a): Find the angle BC makes with the vertical (1 d.p.) (b): Hence find the tension in AB (3 s.f.) 39. Mock 00 / M1(new) - Qu 7: two light inextensible strings, one is horizontal and the other makes an angle α to the horizontal. The tension in the horizontal string is 15 N and the tension in the other string is T N. Find: (a): α (b): T 41. Jan 01 / M1(new) - Qu 3: Particles A & B have masses 3m and km (where k > 3). They are connected by a light inextensible string over a smooth pulley and released from rest with the string taut. A accelerates at 2g/5 m/s² (a): Find the tension in the string (in terms of m and g) (b): State why B also has acceleration 2g/5 m/s² (c): Find k (d): How have you used the assumption that the string is light 42. Jan 01 / M1(new) - Qu 7: A sledge of mass 78 kg is pulled up a rough slope (μ = 0.25) inclined at α (tan α = 5/12) by a light inextensible rope parallel to the slope The sledge is accelerating up the slope at 0.5 m/s² (a): Find the tension in the rope The rope suddenly breaks. The sledge subsequently comes to rest and then starts sliding down the slope (b): Find its acceleration down the slope 43. Jun 01 / M1(new) - Qu 2: A parcel of mass 2 kg moves on a rough plane (μ = 0.4) inclined at 30 The parcel is pulled up by light rope which is at 30 to the slope. Then tension in the rope is 24 N (a): Find the acceleration (2 s.f.) The rope breaks and the parcel slows down and comes to rest (b): Show that immediately after the parcel comes to rest, it starts to move down the slope (c): Find its acceleration down the slope Two forces P & Q act on a particle. P has magnitude 5 N and the force Q has magnitude 3 N. The angle between the forces is 40. The resultant of P and Q is F. Find: (a): the magnitude of F (3 s.f.) (b): the angle between F and P 44. Jun 01 / M1(new) - Qu 4: 40. Jan 01 / M1(new) - Qu 2: A particle P of mass 2 kg is held in equilibrium under gravity by A parcel of mass 3 kg is held in equilibrium on a rough plane inclined at 30 by a 30 N force acting horizontally. The parcel is on the point of moving up the slope (a): Show all the forces acting on the parcel (b): Find the normal reaction force (c): Find μ

8 45. Jun 01 / M1(new) - Qu 6: A van of mass 2000 kg is towing a car of mass 1200 kg along a horizontal road. They are joined by a horizontal tow bar. The van and car experience constant resistances of 800 N and 240 N. There is a constant driving force of 2320 N. Find: (a): the acceleration (b): the tension in the tow bar They come to a hill inclined at α where sin α = 1/20 (c): Find the magnitude of their acceleration up the hill and state whether their speed increases or decreases 48. Jun 01 / M1 - Qu 7: 46. Jan 02 / M1 - Qu 7: A ring of mass 0.3 kg is threaded on a rough horizontal curtain pole A light inextensible string is attached to the ring and pulled with a force of 2.5 N at an angle α (where tan α = 3/4) to the pole. In this position, the ring is in limiting equilibrium (a): Find μ Particles A & B of mass 2m and m are attached by a light inextensible string over a smooth pulley. A rests on a rough table and B rests on a smooth slope. They are released from rest with the string taut (a): Show a = g(1-μ)/6 When each particle has moved a distance h, the string breaks. A comes to rest before reaching the pulley. Given that μ = 0.2 (b): Find the total distance moved by A in terms of h (c): State two physical factors apart from air resistance which could be taken into account to improve the model 49. Nov 02 / M1 - Qu 1: The direction of the string is altered so the ring is pulled upwards. (b): Find the normal reaction (c): State whether the ring is in equilibrium without making further detailed calculations 47. Jun 01 / M1 - Qu 4: A particle P of weight 6 N is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point O. The particle P is in equilibrium under gravity and a horizontal force F is applied to P when the string is at 30 to the vertical. Find to 3 s.f: (a): the tension in the string (b): F 50. Nov 02 / M1 - Q5: A box of mass 6 kg lies on a rough (μ = 0.4) plane inclined at 30. It is held in equilibrium, on the point of moving up the plane by a horizontal force P (a): Find the normal reaction (b): Find P The horizontal force is removed (c): Show the box will now start to move down the plane

9 A suitcase of mass 10 kg slides down a rough ramp inclined at 30. The suitcase leaves A at 10 m/s and passes B at 8 m/s. AB = 5 m Find: (a): the deceleration (b): μ The suitcase reaches the bottom of the ramp (c): Find the greatest possible length of AC the point of moving up the slope by the tension in the string. Find T 51. Nov 02 / M1 - Qu 8: 54. Jan 03 / M1 - Qu 8: Two particles A & B of mass m kg and 3 kg are connected by a light inextensible string over a smooth pulley. A is at rest on a smooth slope of 30 and B hangs freely 0.25 m above the floor. They are released from rest with the string taut. B descends with acceleration 2g/5. Given that A does not reach P, calculate: (a): the tension in the string while B is descending (b): m B strikes the ground and does not rebound. Find: (c): the impulse exerted by B on the ground (c): the time between B striking the ground and A reaching its highest point 52. Jan 03 / M1 - Qu 2: A has mass 0.8 kg and B has mass 1.2 kg. Initially B is 0.6 m above the floor. They are released from rest with the string taut. A does not reach P before B hits the ground. In an initial model, the table is assumed to be smooth. Find: (a): the tension (b): the time taken for B to reach the ground In a refinement of the model, the table is assumed to be rough (μ=1/5) Find: (c): the time taken for B to reach the ground 55. Jun 03 / M1 - Qu 4: Angle BOC = θ. A particle at O is in equilibrium under these 3 forces (a): Find θ (1 d.p.) (b): Find X (2 d.p.) 53. Jan 03 / M1 - Qu 5: A box of mass 1.5 kg is placed on this rough slope (μ=1/3) and held on

10 A parcel of mass 5 kg lies on this rough slope where tan α = 3/4. The 20 N force just prevents the particle from sliding down the plane. Find μ 56. Jun 03 / M1 - Qu 6: A particle P of mass 3 kg is projected up a rough slope (μ = 0.4) inclined at 30 with initial speed 6 m/s. Find: (a): the frictional force (b): the dist moved up the plane before P comes to instantaneous rest 57. Jun 03 / M1 - Qu 8: A truck of mass 1200 kg is towing a 800 kg car using a horizontal towrope. The truck provides a driving force of 2400 N. The resistances to motion of the truck and the car are 600 N and 400 N. Find: (a): the acceleration (b): the tension When they are moving at 20 m/s, the rope breaks (c): Show the truck reaches a speed of 28 m/s approximately 6 s earlier than it would have (c): The string is described as "light" (i): What does this mean (ii): How have you used this assumption 61. Jun 04 / M1 - Qu 1: 58. Jan 04 / M1 - Qu 3: A tiles of mass m kg slides from rest down a rough roof (μ = 0.4) inclined at 30 (a): Find its acceleration The tile moves 3 m before reaching the edge of the roof (b): Find the speed it reaches the edge of the roof at (c): Write down the answer to part (a) if the tile had mass 2m kg 59. Jan 04 / M1 - Qu 4: A particle C of weight W N is attached to the ends of two light inextensible strings AC & BC. The other ends are attached to a horizontal ceiling. The particle hangs in equilibrium when the tension in AC is 50 N. Find: (a): the tension in BC (3 s.f.) (b): W 62. Jun 04 / M1 - Qu 5: Two small rings A & B each of mass 2m are threaded on a rough horizontal pole. The rings are attached to the ends of a light inextensible string. A smooth ring C of mass 3m is threaded on the string. The rings A & B are in limiting equilibrium. The coefficient of friction between each ring and the pole is μ. tan θ = 3/4 (a): Show the tension is 5mg/2 (b): Find μ 60. Jan 04 / M1 - Qu 5: Particle A of mass 4 kg moves up the smooth slope when the system is released from rest with the string taut. Particle B has mass 3 kg Find: (a): the tension (b): the magnitude of the resultant force exerted by the string on the pulley The boat B of mass 400 kg is held at rest on a rough slipway (μ = 0.2) by rope. The boat is on the point of sliding down the slope (a): Find the tension (The boat is 50 m from the bottom of the slipway. The rope breaks (b): Find the time taken for the boat to reach the bottom