3.03k Connected particles: pulleys and equilibrium

6 A particle travels in a straight line \(P Q\). The velocity of the particle \(t \mathrm {~s}\) after leaving \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 4.5 + 4 t - 0.5 t ^ { 2 }$$

1. Find the velocity of the particle at the instant when its acceleration is zero.
   The particle comes to instantaneous rest at \(Q\).
2. Find the distance \(P Q\). \includegraphics[max width=\textwidth, alt={}, center]{55090630-1413-45cd-8201-4d58662db6bd-10_625_780_260_744} Two particles \(A\) and \(B\), of masses \(3 m \mathrm {~kg}\) and \(2 m \mathrm {~kg}\) respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a plane. The plane is inclined at an angle \(\theta\) to the horizontal. \(A\) lies on the plane and \(B\) hangs vertically, 0.8 m above the floor, which is horizontal. The string between \(A\) and the pulley is parallel to a line of greatest slope of the plane (see diagram). Initially \(A\) and \(B\) are at rest.
   1. Given that the plane is smooth, find the value of \(\theta\) for which \(A\) remains at rest.
      It is given instead that the plane is rough, \(\theta = 30 ^ { \circ }\) and the acceleration of \(A\) up the plane is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   2. Show that the coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 10 } \sqrt { 3 }\).
   3. When \(B\) reaches the floor it comes to rest. Find the length of time after \(B\) reaches the floor for which \(A\) is moving up the plane. [You may assume that \(A\) does not reach the pulley.]
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 \includegraphics[max width=\textwidth, alt={}, center]{e5ee28f2-5876-4149-9a77-18c5792c1bd8-10_551_776_260_689} Two particles \(P\) and \(Q\), of masses 2 kg and 0.25 kg respectively, are connected by a light inextensible string that passes over a fixed smooth pulley. Particle \(P\) is on an inclined plane at an angle of \(30 ^ { \circ }\) to the horizontal. Particle \(Q\) hangs below the pulley. Three points \(A , B\) and \(C\) lie on a line of greatest slope of the plane with \(A B = 0.8 \mathrm {~m}\) and \(B C = 1.2 \mathrm {~m}\) (see diagram). Particle \(P\) is released from rest at \(A\) with the string taut and slides down the plane. During the motion of \(P\) from \(A\) to \(C , Q\) does not reach the pulley. The part of the plane from \(A\) to \(B\) is rough, with coefficient of friction 0.3 between the plane and \(P\). The part of the plane from \(B\) to \(C\) is smooth.

1. 1. Find the acceleration of \(P\) between \(A\) and \(B\).
   2. Hence, find the speed of \(P\) at \(C\).
2. Find the time taken for \(P\) to travel from \(A\) to \(C\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 A car of mass 1500 kg is towing a trailer of mass \(m \mathrm {~kg}\) along a straight horizontal road. The car and the trailer are connected by a tow-bar which is horizontal, light and rigid. There is a resistance force of \(F \mathrm {~N}\) on the car and a resistance force of 200 N on the trailer. The driving force of the car's engine is 3200 N , the acceleration of the car is \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and the tension in the tow-bar is 300 N . Find the value of \(m\) and the value of \(F\).

4 A car of mass 1700 kg is pulling a trailer of mass 300 kg along a straight horizontal road. The car and trailer are connected by a light inextensible cable which is parallel to the road. There are constant resistances to motion of 400 N on the car and 150 N on the trailer. The power of the car's engine is 14000 W . Find the acceleration of the car and the tension in the cable when the speed is \(20 \mathrm {~ms} ^ { - 1 }\).

6 On a straight horizontal test track, driverless vehicles (with no passengers) are being tested. A car of mass 1600 kg is towing a trailer of mass 700 kg along the track. The brakes are applied, resulting in a deceleration of \(12 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The braking force acts on the car only. In addition to the braking force there are constant resistance forces of 600 N on the car and of 200 N on the trailer.

1. Find the magnitude of the force in the tow-bar.
2. Find the braking force.
3. At the instant when the brakes are applied, the car has speed \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At this instant the car is 17.5 m away from a stationary van, which is directly in front of the car. Show that the car hits the van at a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. After the collision, the van starts to move with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the car and trailer continue moving in the same direction with speed \(2 \mathrm {~ms} ^ { - 1 }\). Find the mass of the van.

7 \includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-12_439_1095_258_525} Two particles \(P\) and \(Q\) of masses 0.5 kg and \(m \mathrm {~kg}\) respectively are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the top of two inclined planes. The particles are initially at rest with \(P\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal and \(Q\) on a plane inclined at \(45 ^ { \circ }\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes. A force of magnitude 0.8 N is applied to \(P\) acting down the plane, causing \(P\) to move down the plane (see diagram).

1. It is given that \(m = 0.3\), and that the plane on which \(Q\) rests is smooth. Find the tension in the string.
2. It is given instead that the plane on which \(Q\) rests is rough, and that after each particle has moved a distance of 1 m , their speed is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done against friction in this part of the motion is 0.5 J . Use an energy method to find the value of \(m\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 A van of mass 3600 kg is towing a trailer of mass 1200 kg along a straight horizontal road using a light horizontal rope. There are resistance forces of 700 N on the van and 300 N on the trailer.

1. The driving force exerted by the van is 2500 N . Find the tension in the rope.
   The driving force is now removed and the van driver applies a braking force which acts only on the van. The resistance forces remain unchanged.
2. Find the least possible value of the braking force which will cause the rope to become slack.

6 \includegraphics[max width=\textwidth, alt={}, center]{cb2cec83-6f8d-4c13-90a1-03bbf4e4452f-10_451_1315_258_415} The diagram shows a particle of mass 5 kg on a rough horizontal table, and two light inextensible strings attached to it passing over smooth pulleys fixed at the edges of the table. Particles of masses 4 kg and 6 kg hang freely at the ends of the strings. The particle of mass 6 kg is 0.5 m above the ground. The system is in limiting equilibrium.

1. Show that the coefficient of friction between the 5 kg particle and the table is 0.4 .
   The 6 kg particle is now replaced by a particle of mass 8 kg and the system is released from rest.
2. Find the acceleration of the 4 kg particle and the tensions in the strings.
3. In the subsequent motion the 8 kg particle hits the ground and does not rebound. Find the time that elapses after the 8 kg particle hits the ground before the other two particles come to instantaneous rest. (You may assume this occurs before either particle reaches a pulley.)
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 \includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-3_504_387_598_881} \(S _ { 1 }\) and \(S _ { 2 }\) are light inextensible strings, and \(A\) and \(B\) are particles each of mass 0.2 kg . Particle \(A\) is suspended from a fixed point \(O\) by the string \(S _ { 1 }\), and particle \(B\) is suspended from \(A\) by the string \(S _ { 2 }\). The particles hang in equilibrium as shown in the diagram.

1. Find the tensions in \(S _ { 1 }\) and \(S _ { 2 }\). The string \(S _ { 1 }\) is cut and the particles fall. The air resistance acting on \(A\) is 0.4 N and the air resistance acting on \(B\) is 0.2 N .
2. Find the acceleration of the particles and the tension in \(S _ { 2 }\).

5 \includegraphics[max width=\textwidth, alt={}, center]{b5873699-d207-4cad-9518-1321dc429c15-3_305_599_1717_774} Particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string. \(P\) is at rest on a rough horizontal table. The string passes over a small smooth pulley which is fixed at the edge of the table. \(Q\) hangs vertically below the pulley (see diagram). The force exerted on the string by the pulley has magnitude \(4 \sqrt { } 2 \mathrm {~N}\). The coefficient of friction between \(P\) and the table is 0.8 .

1. Show that the tension in the string is 4 N and state the mass of \(Q\).
2. Given that \(P\) is on the point of slipping, find its mass. A particle of mass 0.1 kg is now attached to \(Q\) and the system starts to move.
3. Find the tension in the string while the particles are in motion.

4 \includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-3_702_709_269_719} Particles \(P\) and \(Q\), of masses 0.6 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed peg. The particles are held at rest with the string taut. Both particles are at a height of 0.9 m above the ground (see diagram). The system is released and each of the particles moves vertically. Find

1. the acceleration of \(P\) and the tension in the string before \(P\) reaches the ground,
2. the time taken for \(P\) to reach the ground.

4 \includegraphics[max width=\textwidth, alt={}, center]{dafc271d-a77b-4401-9170-e13e484d6e5f-3_499_567_260_788} The diagram shows a vertical cross-section of a triangular prism which is fixed so that two of its faces are inclined at \(60 ^ { \circ }\) to the horizontal. One of these faces is smooth and one is rough. Particles \(A\) and \(B\), of masses 0.36 kg and 0.24 kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the highest point of the cross-section. \(B\) is held at rest at a point of the cross-section on the rough face and \(A\) hangs freely in contact with the smooth face (see diagram). \(B\) is released and starts to move up the face with acceleration \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. By considering the motion of \(A\), show that the tension in the string is 3.03 N , correct to 3 significant figures.
2. Find the coefficient of friction between \(B\) and the rough face, correct to 2 significant figures.

7 Loads \(A\) and \(B\), of masses 1.2 kg and 2.0 kg respectively, are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. \(A\) is held at rest and \(B\) hangs freely, with both straight parts of the string vertical. \(A\) is released and starts to move upwards. It does not reach the pulley in the subsequent motion.

1. Find the acceleration of \(A\) and the tension in the string.
2. Find, for the first 1.5 metres of \(A\) 's motion,
   1. A's gain in potential energy,
   2. the work done on \(A\) by the tension in the string,
   3. A's gain in kinetic energy. B hits the floor 1.6 seconds after \(A\) is released. \(B\) comes to rest without rebounding and the string becomes slack.
   4. Find the time from the instant the string becomes slack until it becomes taut again.

6 \includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-3_465_849_1475_648} Particles \(P\) and \(Q\), of masses 0.6 kg and 0.4 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a vertical cross-section of a triangular prism. The base of the prism is fixed on horizontal ground and each of the sloping sides is smooth. Each sloping side makes an angle \(\theta\) with the ground, where \(\sin \theta = 0.8\). Initially the particles are held at rest on the sloping sides, with the string taut (see diagram). The particles are released and move along lines of greatest slope.

1. Find the tension in the string and the acceleration of the particles while both are moving. The speed of \(P\) when it reaches the ground is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). On reaching the ground \(P\) comes to rest and remains at rest. \(Q\) continues to move up the slope but does not reach the pulley.
2. Find the time taken from the instant that the particles are released until \(Q\) reaches its greatest height above the ground.

5 \includegraphics[max width=\textwidth, alt={}, center]{fa0e0e0d-b0a6-44e0-8b4f-4923e235c6c6-3_529_195_255_977} A block \(A\) of mass 3 kg is attached to one end of a light inextensible string \(S _ { 1 }\). Another block \(B\) of mass 2 kg is attached to the other end of \(S _ { 1 }\), and is also attached to one end of another light inextensible string \(S _ { 2 }\). The other end of \(S _ { 2 }\) is attached to a fixed point \(O\) and the blocks hang in equilibrium below \(O\) (see diagram).

1. Find the tension in \(S _ { 1 }\) and the tension in \(S _ { 2 }\). The string \(S _ { 2 }\) breaks and the particles fall. The air resistance on \(A\) is 1.6 N and the air resistance on \(B\) is 4 N .
2. Find the acceleration of the particles and the tension in \(S _ { 1 }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{918b65cc-617d-4942-8d96-b02eef21e417-4_506_471_255_836} Two particles \(A\) and \(B\) have masses 0.12 kg and 0.38 kg respectively. The particles are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. \(A\) is held at rest with the string taut and both straight parts of the string vertical. \(A\) and \(B\) are each at a height of 0.65 m above horizontal ground (see diagram). \(A\) is released and \(B\) moves downwards. Find

1. the acceleration of \(B\) while it is moving downwards,
2. the speed with which \(B\) reaches the ground and the time taken for it to reach the ground. \(B\) remains on the ground while \(A\) continues to move with the string slack, without reaching the pulley. The string remains slack until \(A\) is at a height of 1.3 m above the ground for a second time. At this instant \(A\) has been in motion for a total time of \(T \mathrm {~s}\).
3. Find the value of \(T\) and sketch the velocity-time graph for \(A\) for the first \(T \mathrm {~s}\) of its motion.
4. Find the total distance travelled by \(A\) in the first \(T\) s of its motion.

5 \includegraphics[max width=\textwidth, alt={}, center]{2c628138-0729-4160-a95c-d6ab0f199cc5-3_275_663_258_742} A light inextensible string has a particle \(A\) of mass 0.26 kg attached to one end and a particle \(B\) of mass 0.54 kg attached to the other end. The particle \(A\) is held at rest on a rough plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 5 } { 13 }\). The string is taut and parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley at the top of the plane. Particle \(B\) hangs at rest vertically below the pulley (see diagram). The coefficient of friction between \(A\) and the plane is 0.2 . Particle \(A\) is released and the particles start to move.

1. Find the magnitude of the acceleration of the particles and the tension in the string. Particle \(A\) reaches the pulley 0.4 s after starting to move.
2. Find the distance moved by each of the particles.

7 \includegraphics[max width=\textwidth, alt={}, center]{ceb367ee-4e12-4cb2-9020-078ea5724d6e-3_430_860_1585_641} Particle \(A\) of mass 1.26 kg and particle \(B\) of mass 0.9 kg are attached to the ends of a light inextensible string. The string passes over a small smooth pulley \(P\) which is fixed at the edge of a rough horizontal table. \(A\) is held at rest at a point 0.48 m from \(P\), and \(B\) hangs vertically below \(P\), at a height of 0.45 m above the floor (see diagram). The coefficient of friction between \(A\) and the table is \(\frac { 2 } { 7 } . A\) is released and the particles start to move.

1. Show that the magnitude of the acceleration of the particles is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the tension in the string.
2. Find the speed with which \(B\) reaches the floor.
3. Find the speed with which \(A\) reaches the pulley.

7 \includegraphics[max width=\textwidth, alt={}, center]{77976dad-c055-45fd-93fe-e37fa8e9ae22-4_333_1001_264_573} A light inextensible string of length 5.28 m has particles \(A\) and \(B\), of masses 0.25 kg and 0.75 kg respectively, attached to its ends. Another particle \(P\), of mass 0.5 kg , is attached to the mid-point of the string. Two small smooth pulleys \(P _ { 1 }\) and \(P _ { 2 }\) are fixed at opposite ends of a rough horizontal table of length 4 m and height 1 m . The string passes over \(P _ { 1 }\) and \(P _ { 2 }\) with particle \(A\) held at rest vertically below \(P _ { 1 }\), the string taut and \(B\) hanging freely below \(P _ { 2 }\). Particle \(P\) is in contact with the table halfway between \(P _ { 1 }\) and \(P _ { 2 }\) (see diagram). The coefficient of friction between \(P\) and the table is 0.4 . Particle \(A\) is released and the system starts to move with constant acceleration of magnitude \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The tension in the part \(A P\) of the string is \(T _ { A } \mathrm {~N}\) and the tension in the part \(P B\) of the string is \(T _ { B } \mathrm {~N}\).

1. Find \(T _ { A }\) and \(T _ { B }\) in terms of \(a\).
2. Show by considering the motion of \(P\) that \(a = 2\).
3. Find the speed of the particles immediately before \(B\) reaches the floor.
4. Find the deceleration of \(P\) immediately after \(B\) reaches the floor. \end{document}

7 \includegraphics[max width=\textwidth, alt={}, center]{139371b7-e142-4ed6-bff3-3ca4c32b9c6b-4_342_1257_255_445} A smooth inclined plane of length 160 cm is fixed with one end at a height of 40 cm above the other end, which is on horizontal ground. Particles \(P\) and \(Q\), of masses 0.76 kg and 0.49 kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the top of the plane. Particle \(P\) is held at rest on the same line of greatest slope as the pulley and \(Q\) hangs vertically below the pulley at a height of 30 cm above the ground (see diagram). \(P\) is released from rest. It starts to move up the plane and does not reach the pulley. Find

1. the acceleration of the particles and the tension in the string before \(Q\) reaches the ground,
2. the speed with which \(Q\) reaches the ground,
3. the total distance travelled by \(P\) before it comes to instantaneous rest.

7 \includegraphics[max width=\textwidth, alt={}, center]{f4f2996b-5382-4b0d-9804-b5f5945946b3-3_376_1052_1171_548} Particles \(A\) and \(B\), of masses 0.3 kg and 0.7 kg respectively, are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a rough horizontal table with the string passing over a smooth pulley fixed at the edge of the table. The coefficient of friction between \(A\) and the table is 0.2 . Particle \(B\) hangs vertically below the pulley at a height of 0.5 m above the floor (see diagram). The system is released from rest and 0.25 s later the string breaks. A does not reach the pulley in the subsequent motion. Find

1. the speed of \(B\) immediately before it hits the floor,
2. the total distance travelled by \(A\).

5 \includegraphics[max width=\textwidth, alt={}, center]{099c81e0-a95a-4f98-801c-32d905ef7c7d-3_432_710_258_721} Two particles of masses 5 kg and 10 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The 5 kg particle is on a rough fixed slope which is at an angle of \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The 10 kg particle hangs below the pulley (see diagram). The coefficient of friction between the slope and the 5 kg particle is \(\frac { 1 } { 2 }\). The particles are released from rest. Find the acceleration of the particles and the tension in the string.

7 \includegraphics[max width=\textwidth, alt={}, center]{fd2fbf13-912c-46c5-a470-306b2269aa0b-3_378_1001_1672_573} A particle \(A\) of mass 1.6 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 2.4 kg which hangs freely below the pulley. The system is released from rest with the string taut and with \(B\) at a height of 0.5 m above the ground, as shown in the diagram. In the subsequent motion \(A\) does not reach \(P\) before \(B\) reaches the ground.

1. Given that the table is smooth, find the time taken by \(B\) to reach the ground.
2. Given instead that the table is rough and that the coefficient of friction between \(A\) and the table is \(\frac { 3 } { 8 }\), find the total distance travelled by \(A\). You may assume that \(A\) does not reach the pulley.

6 Two particles of masses 1.3 kg and 0.7 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The particles are held at the same vertical height with the string taut. The distance of each particle above a horizontal plane is 2 m , and the distance of each particle below the pulley is 4 m . The particles are released from rest.

1. Find
   1. the tension in the string before the particle of mass 1.3 kg reaches the plane,
   2. the time taken for the particle of mass 1.3 kg to reach the plane.
   3. Find the greatest height of the particle of mass 0.7 kg above the plane.

7 \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-10_336_803_258_671} Two particles \(A\) and \(B\) of masses \(m \mathrm {~kg}\) and 4 kg respectively are connected by a light inextensible string that passes over a fixed smooth pulley. Particle \(A\) is on a rough fixed slope which is at an angle of \(30 ^ { \circ }\) to the horizontal ground. Particle \(B\) hangs vertically below the pulley and is 0.5 m above the ground (see diagram). The coefficient of friction between the slope and particle \(A\) is 0.2 .

1. In the case where the system is in equilibrium with particle \(A\) on the point of moving directly up the slope, show that \(m = 5.94\), correct to 3 significant figures.
2. In the case where \(m = 3\), the system is released from rest with the string taut. Find the total distance travelled by \(A\) before coming to instantaneous rest. You may assume that \(A\) does not reach the pulley.