3.03o Advanced connected particles: and pulleys

8 \includegraphics[max width=\textwidth, alt={}, center]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-14_388_1216_264_461} Two particles \(A\) and \(B\), of masses 0.3 kg and 0.5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to a horizontal plane and to the top of an inclined plane. The particles are initially at rest with \(A\) on the horizontal plane and \(B\) on the inclined plane, which makes an angle of \(30 ^ { \circ }\) with the horizontal. The string is taut and \(B\) can move on a line of greatest slope of the inclined plane. A force of magnitude 3.5 N is applied to \(B\) acting down the plane (see diagram).

1. Given that both planes are smooth, find the tension in the string and the acceleration of \(B\).
2. It is given instead that the two planes are rough. When each particle has moved a distance of 0.6 m from rest, the total amount of work done against friction is 1.1 J . Use an energy method to find the speed of \(B\) when it has moved this distance down the plane. [You should assume that the string is sufficiently long so that \(A\) does not hit the pulley when it moves 0.6 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.

6 A car of mass 1600 kg is pulling a caravan of mass 800 kg . The car and the caravan are connected by a light rigid tow-bar. The resistances to the motion of the car and caravan are 400 N and 250 N respectively.

1. The car and caravan are travelling along a straight horizontal road.
   1. Given that the car and caravan have a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the power of the car's engine.
   2. The engine's power is now suddenly increased to 39 kW . Find the instantaneous acceleration of the car and caravan and find the tension in the tow-bar.
2. The car and caravan now travel up a straight hill, inclined at an angle of \(\sin ^ { - 1 } 0.05\) to the horizontal, at a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car's engine is working at 32.5 kW . Find \(v\).

7 \includegraphics[max width=\textwidth, alt={}, center]{ac4bb5a0-c7c0-4e1d-9e76-64f92ae28066-10_214_1461_255_342} As shown in the diagram, particles \(A\) and \(B\) of masses 2 kg and 3 kg respectively are attached to the ends of a light inextensible string. The string passes over a small fixed smooth pulley which is attached to the top of two inclined planes. Particle \(A\) is on plane \(P\), which is inclined at an angle of \(10 ^ { \circ }\) to the horizontal. Particle \(B\) is on plane \(Q\), which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The string is taut, and the two parts of the string are parallel to lines of greatest slope of their respective planes.

1. It is given that plane \(P\) is smooth, plane \(Q\) is rough, and the particles are in limiting equilibrium. Find the coefficient of friction between particle \(B\) and plane \(Q\).
2. It is given instead that both planes are smooth and that the particles are released from rest at the same horizontal level. Find the time taken until the difference in the vertical height of the particles is 1 m . [You should assume that this occurs before \(A\) reaches the pulley or \(B\) reaches the bottom of plane \(Q\).] [6]
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 A particle \(P\) moves in a straight line starting from a point \(O\) and comes to rest 14 s later. At time \(t \mathrm {~s}\) after leaving \(O\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) is given by $$\begin{array} { l l } v = p t ^ { 2 } - q t & 0 \leqslant t \leqslant 6 \\ v = 63 - 4.5 t & 6 \leqslant t \leqslant 14 \end{array}$$ where \(p\) and \(q\) are positive constants.
The acceleration of \(P\) is zero when \(t = 2\).

1. Given that there are no instantaneous changes in velocity, find \(p\) and \(q\).
2. Sketch the velocity-time graph.
3. Find the total distance travelled by \(P\) during the 14 s . \includegraphics[max width=\textwidth, alt={}, center]{e1b91e54-a3ae-436c-a4f7-7095891f7034-10_326_1109_255_520} Two particles \(A\) and \(B\) of masses 2 kg and 3 kg respectively are connected by a light inextensible string. Particle \(B\) is on a smooth fixed plane which is at an angle of \(18 ^ { \circ }\) to horizontal ground. The string passes over a fixed smooth pulley at the top of the plane. Particle \(A\) hangs vertically below the pulley and is 0.45 m above the ground (see diagram). The system is released from rest with the string taut. When \(A\) reaches the ground, the string breaks. Find the total distance travelled by \(B\) before coming to instantaneous rest. You may assume that \(B\) 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.

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.

6 A car of mass 1750 kg is pulling a caravan of mass 500 kg . The car and the caravan are connected by a light rigid tow-bar. The resistances to the motion of the car and caravan are 650 N and 150 N respectively.

1. The car and caravan are moving along a straight horizontal road at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Find the power of the car's engine.
   2. The engine's power is now suddenly increased to 40 kW . Find the instantaneous acceleration of the car and caravan and find the tension in the tow-bar.
2. The car and caravan now travel up a straight hill, inclined at an angle \(\sin ^ { - 1 } 0.14\) to the horizontal, at a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car's engine is working at 31 kW . The resistances to the motion of the car and caravan are unchanged. Find \(v\).

7 \includegraphics[max width=\textwidth, alt={}, center]{99f20949-471d-4da3-a680-ec24abf6baa5-10_335_937_255_605} Particles \(A\) and \(B\), of masses 2.4 kg and 3.3 kg respectively, are connected by a light inextensible string that passes over a smooth pulley which is fixed to the top of a rough plane. The plane makes an angle of \(\theta ^ { \circ }\) with horizontal ground. Particle \(A\) is on the plane and the section of the string between \(A\) and the pulley is parallel to a line of greatest slope of the plane. Particle \(B\) hangs vertically below the pulley and is 1 m above the ground (see diagram). The coefficient of friction between the plane and \(A\) is \(\mu\).

1. It is given that \(\theta = 30\) and the system is in equilibrium with \(A\) on the point of moving directly up the plane. Show that \(\mu = 1.01\) correct to 3 significant figures.
2. It is given instead that \(\theta = 20\) and \(\mu = 1.01\). 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 and that \(B\) remains at rest after it hits the ground.
   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 A car of mass 1200 kg is pulling a trailer of mass 800 kg up a hill inclined at an angle of \(\sin ^ { - 1 } ( 0.1 )\) to the horizontal. The car and the trailer are connected by a light rigid tow-bar which is parallel to the road. The driving force of the car's engine is 2500 N and the resistances to the car and trailer are 300 N and 100 N respectively.

1. Find the acceleration of the system and the tension in the tow-bar.
2. When the car and trailer are travelling at a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the driving force becomes zero. Find the time, in seconds, before the system comes to rest and the force in the tow-bar during this time.

7 \includegraphics[max width=\textwidth, alt={}, center]{87b42689-791c-4f4e-a36e-bfae3191ca11-12_244_668_264_701} Two particles \(A\) and \(B\), of masses 0.8 kg and 0.2 kg respectively, are connected by a light inextensible string. Particle \(A\) is placed on a horizontal surface. The string passes over a small smooth pulley \(P\) fixed at the edge of the surface, and \(B\) hangs freely. The horizontal section of the string, \(A P\), is of length 2.5 m (see diagram). The particles are released from rest with both sections of the string taut.

1. Given that the surface is smooth, find the time taken for \(A\) to reach the pulley.
2. It is given instead that the surface is rough and that the speed of \(A\) immediately before it reaches the pulley is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done against friction as \(A\) moves from rest to the pulley is 2 J . Use an energy method to find \(v\).

5 \includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-3_314_867_1457_639} A block \(B\) of mass 0.6 kg and a particle \(A\) of mass 0.4 kg are attached to opposite ends of a light inextensible string. The block is held at rest on a rough horizontal table, and the coefficient of friction between the block and the table is 0.5 . The string passes over a small smooth pulley \(C\) at the edge of the table and \(A\) hangs in equilibrium vertically below \(C\). The part of the string between \(B\) and \(C\) is horizontal and the distance \(B C\) is 3 m (see diagram). \(B\) is released and the system starts to move.

1. Find the acceleration of \(B\) and the tension in the string.
2. Find the time taken for \(B\) to reach the pulley.

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.

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.

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 \(A\).

7 \includegraphics[max width=\textwidth, alt={}, center]{4941e074-2f93-4a0c-80ba-0ca96a48389e-10_374_762_259_688} As shown in the diagram, a particle \(A\) of mass 0.8 kg lies on a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal and a particle \(B\) of mass 1.2 kg lies on a plane inclined at an angle of \(60 ^ { \circ }\) to the horizontal. The particles are connected by a light inextensible string which passes over a small smooth pulley \(P\) fixed at the top of the planes. The parts \(A P\) and \(B P\) of the string are parallel to lines of greatest slope of the respective planes. The particles are released from rest with both parts of the string taut.

1. Given that both planes are smooth, find the acceleration of \(A\) and the tension in the string.
2. It is given instead that both planes are rough, with the same coefficient of friction, \(\mu\), for both particles. Find the value of \(\mu\) for which the system is in limiting equilibrium.

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.

7
As shown in the diagram, a particle \(A\) of mass 1.6 kg lies on a horizontal plane and a particle \(B\) of mass 2.4 kg lies on a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The particles are connected by a light inextensible string which passes over a small smooth pulley \(P\) fixed at the top of the inclined plane. The distance \(A P\) is 2.5 m and the distance of \(B\) from the bottom of the inclined plane is 1 m . There is a barrier at the bottom of the inclined plane preventing any further motion of \(B\). The part \(B P\) of the string is parallel to a line of greatest slope of the inclined plane. The particles are released from rest with both parts of the string taut.

1. Given that both planes are smooth, find the acceleration of \(A\) and the tension in the string.
2. It is given instead that the horizontal plane is rough and that the coefficient of friction between \(A\) and the horizontal plane is 0.2 . The inclined plane is smooth. Find the total distance travelled by \(A\).
   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 A particle \(P\) moves in a straight line from a fixed point \(O\). The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) at time \(t \mathrm {~s}\) is given by $$v = t ^ { 2 } - 8 t + 12 \quad \text { for } 0 \leqslant t \leqslant 8$$

1. Find the minimum velocity of \(P\).
2. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 8\). \includegraphics[max width=\textwidth, alt={}, center]{555678d3-f37d-4822-a005-de8c6094dc50-12_401_1102_260_520} Two particles \(A\) and \(B\), of masses 0.4 kg and 0.2 kg respectively, are connected by a light inextensible string. Particle \(A\) is held on a smooth plane inclined at an angle of \(\theta ^ { \circ }\) to the horizontal. The string passes over a small smooth pulley \(P\) fixed at the top of the plane, and \(B\) hangs freely 0.5 m above horizontal ground (see diagram). The particles are released from rest with both sections of the string taut.
3. Given that the system is in equilibrium, find \(\theta\).
4. It is given instead that \(\theta = 20\). In the subsequent motion particle \(A\) does not reach \(P\) and \(B\) remains at rest after reaching the ground.
   1. Find the tension in the string and the acceleration of the system.
   2. Find the speed of \(A\) at the instant \(B\) reaches the ground.
   3. Use an energy method to find the total distance \(A\) moves up the plane before coming to instantaneous rest.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 A car of mass 1400 kg is travelling up a hill inclined at an angle of \(4 ^ { \circ }\) to the horizontal. There is a constant resistance to motion of magnitude 1550 N acting on the car.

1. Given that the engine of the car is working at 30 kW , find the speed of the car at an instant when its acceleration is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. The greatest possible constant speed at which the car can travel up the hill is \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the maximum possible power of the engine. \includegraphics[max width=\textwidth, alt={}, center]{539be201-7bfc-4ba0-8378-c7aec4473ac7-06_643_419_255_863} Two particles \(A\) and \(B\), of masses 1.3 kg and 0.7 kg respectively, are connected by a light inextensible string which passes over a smooth fixed pulley. Particle \(A\) is 1.75 m above the floor and particle \(B\) is 1 m above the floor (see diagram). The system is released from rest with the string taut, and the particles move vertically. When the particles are at the same height the string breaks.

5 A car of mass 1200 kg is pulling a trailer of mass 800 kg up a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.1\). The system of the car and the trailer is modelled as two particles connected by a light inextensible cable. The driving force of the car's engine is 2500 N and the resistances to the car and trailer are 100 N and 150 N respectively.

1. Find the acceleration of the system and the tension in the cable.
2. When the car and trailer are travelling at a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the driving force becomes zero. The cable remains taut. Find the time, in seconds, before the system comes to rest.