3.03k Connected particles: pulleys and equilibrium

6 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 . 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. Given instead that the surface is rough and the coefficient of friction between \(A\) and the surface is 0.1 , find the speed of \(A\) immediately before it reaches the pulley. \(7 \quad\) A particle \(P\) moves in a straight line. The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by $$\begin{array} { l l } v = 5 t ( t - 2 ) & \text { for } 0 \leqslant t \leqslant 4 \\ v = k & \text { for } 4 \leqslant t \leqslant 14 \\ v = 68 - 2 t & \text { for } 14 \leqslant t \leqslant 20 \end{array}$$ where \(k\) is a constant.
3. Find \(k\).
4. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 20\).
5. Find the set of values of \(t\) for which the acceleration of \(P\) is positive.
6. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 20\).

7 \includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-4_601_515_699_815} Two particles \(A\) and \(B\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. Particle \(B\) is held on the horizontal floor and particle \(A\) hangs in equilibrium. Particle \(B\) is released and each particle starts to move vertically with constant acceleration of magnitude \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the value of \(a\). Particle \(A\) hits the floor 1.2 s after it starts to move, and does not rebound upwards.
2. Show that \(A\) hits the floor with a speed of \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the gain in gravitational potential energy by \(B\), from leaving the floor until reaching its greatest height.

5 \includegraphics[max width=\textwidth, alt={}, center]{a4cb105b-55d2-4793-95d2-3d791990a1f6-3_643_481_274_831} Particles \(A\) and \(B\), of masses 0.5 kg and \(m \mathrm {~kg}\) respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. Particle \(B\) is held at rest on the horizontal floor and particle \(A\) hangs in equilibrium (see diagram). \(B\) is released and each particle starts to move vertically. \(A\) hits the floor 2 s after \(B\) is released. The speed of each particle when \(A\) hits the floor is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. For the motion while \(A\) is moving downwards, find
   1. the acceleration of \(A\),
   2. the tension in the string.
   3. Find the value of \(m\).

6 \includegraphics[max width=\textwidth, alt={}, center]{a9f3480e-7a8a-497d-a26a-b2aba9b05512-4_712_529_264_810} Particles \(P\) and \(Q\), of masses 0.55 kg and 0.45 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. The particles are held at rest with the string taut and its straight parts vertical. Both particles are at a height of 5 m above the ground (see diagram). The system is released.

1. Find the acceleration with which \(P\) starts to move. The string breaks after 2 s and in the subsequent motion \(P\) and \(Q\) move vertically under gravity.
2. At the instant that the string breaks, find
   1. the height above the ground of \(P\) and of \(Q\),
   2. the speed of the particles.
   3. Show that \(Q\) reaches the ground 0.8 s later than \(P\). \(7 \quad\) A particle \(P\) starts from rest at the point \(A\) at time \(t = 0\), where \(t\) is in seconds, and moves in a straight line with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 10 s . For \(10 \leqslant t \leqslant 20 , P\) continues to move along the line with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = \frac { 800 } { t ^ { 2 } } - 2\). Find
      1. the speed of \(P\) when \(t = 10\), and the value of \(a\),
      2. the value of \(t\) for which the acceleration of \(P\) is \(- a \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
      3. the displacement of \(P\) from \(A\) when \(t = 20\).

6 \includegraphics[max width=\textwidth, alt={}, center]{efa7175f-832b-4cd3-82ab-52e402115081-4_686_511_269_817} Particles \(A\) and \(B\), of masses 0.3 kg and 0.7 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. Particle \(A\) is held on the horizontal floor and particle \(B\) hangs in equilibrium. Particle \(A\) is released and both particles start to move vertically.

1. Find the acceleration of the particles. The speed of the particles immediately before \(B\) hits the floor is \(1.6 \mathrm {~ms} ^ { - 1 }\). Given that \(B\) does not rebound upwards, find
2. the maximum height above the floor reached by \(A\),
3. the time taken by \(A\), from leaving the floor, to reach this maximum height.

6 \includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-3_476_1305_1519_420} A smooth slide \(A B\) is fixed so that its highest point \(A\) is 3 m above horizontal ground. \(B\) is \(h \mathrm {~m}\) above the ground. A particle \(P\) of mass 0.2 kg is released from rest at a point on the slide. The particle moves down the slide and, after passing \(B\), continues moving until it hits the ground (see diagram). The speed of \(P\) at \(B\) is \(v _ { B }\) and the speed at which \(P\) hits the ground is \(v _ { G }\).

1. In the case that \(P\) is released at \(A\), it is given that the kinetic energy of \(P\) at \(B\) is 1.6 J . Find
   1. the value of \(h\),
   2. the kinetic energy of the particle immediately before it reaches the ground,
   3. the ratio \(v _ { G } : v _ { B }\).
   4. In the case that \(P\) is released at the point \(X\) of the slide, which is \(H \mathrm {~m}\) above the ground (see diagram), it is given that \(v _ { G } : v _ { B } = 2.55\). Find the value of \(H\) correct to 2 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-4_384_679_258_733} Particles \(P\) and \(Q\), of masses 0.2 kg and 0.5 kg respectively, are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. \(P\) hangs freely and \(Q\) is in contact with the table. A force of magnitude 3.2 N acts on \(Q\), upwards and away from the pulley, at an angle of \(30 ^ { \circ }\) to the horizontal (see diagram).
      1. The system is in limiting equilibrium with \(P\) about to move upwards. Find the coefficient of friction between \(Q\) and the table. The force of magnitude 3.2 N is now removed and \(P\) starts to move downwards.
      2. Find the acceleration of the particles and the tension in the string.

2 Particles \(A\) of mass 0.65 kg and \(B\) of mass 0.35 kg are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. \(B\) is held at rest with the string taut and both of its straight parts vertical. The system is released from rest and the particles move vertically. Find the tension in the string and the magnitude of the resultant force exerted on the pulley by the string.

5 Particles \(A\) and \(B\), of masses 0.9 kg and 0.6 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley. The system is released from rest with the string taut, with its straight parts vertical and with the particles at the same height above the horizontal floor. In the subsequent motion, \(B\) does not reach the pulley.

1. Find the acceleration of \(A\) and the tension in the string during the motion before \(A\) hits the floor. After \(A\) hits the floor, \(B\) continues to move vertically upwards for a further 0.3 s .
2. Find the height of the particles above the floor at the instant that they started to move.

3 Particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string which passes over a fixed smooth pulley. The system is released from rest with the string taut, with its straight parts vertical, and with both particles at a height of 2 m above horizontal ground. \(P\) moves vertically downwards and does not rebound when it hits the ground. At the instant that \(P\) hits the ground, \(Q\) is at the point \(X\), from where it continues to move vertically upwards without reaching the pulley. Given that \(P\) has mass 0.9 kg and that the tension in the string is 7.2 N while \(P\) is moving, find the total distance travelled by \(Q\) from the instant it first reaches \(X\) until it returns to \(X\).

2 \includegraphics[max width=\textwidth, alt={}, center]{2bb3c9bb-60f0-440d-a148-b4db3478ca31-2_212_625_528_761} Particles \(A\) and \(B\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. \(A\) is held at rest on a rough horizontal table with the string passing over a small smooth pulley at the edge of the table. \(B\) hangs vertically below the pulley (see diagram). The system is released and \(B\) starts to move downwards with acceleration \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find

1. the tension in the string after the system is released,
2. the frictional force acting on \(A\).

2 Particles \(A\) and \(B\) of masses \(m \mathrm {~kg}\) and \(( 1 - m ) \mathrm { kg }\) respectively are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. The system is released from rest with the straight parts of the string vertical. \(A\) moves vertically downwards and 0.3 seconds later it has speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the acceleration of \(A\),
2. the value of \(m\) and the tension in the string.

7 \includegraphics[max width=\textwidth, alt={}, center]{631ddcd9-17c0-4a15-8671-40788c3a84d3-3_565_828_1402_660} Particles \(A\) and \(B\) have masses 0.32 kg and 0.48 kg respectively. The particles are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the edge of a smooth horizontal table. Particle \(B\) is held at rest on the table at a distance of 1.4 m from the pulley. \(A\) hangs vertically below the pulley at a height of 0.98 m above the floor (see diagram). \(A , B\), the string and the pulley are all in the same vertical plane. \(B\) is released and \(A\) moves downwards.

1. Find the acceleration of \(A\) and the tension in the string. \(A\) hits the floor and \(B\) continues to move towards the pulley. Find the time taken, from the instant that \(B\) is released, for
2. \(A\) to reach the floor,
3. \(B\) to reach the pulley.

6 \includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-3_518_515_1436_815} Particles \(A\) and \(B\), of masses 0.3 kg and 0.7 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley. \(A\) is held at rest and \(B\) hangs freely, with both straight parts of the string vertical and both particles at a height of 0.52 m above the floor (see diagram). \(A\) is released and both particles start to move.

1. Find the tension in the string. When both particles are moving with speed \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the string breaks.
2. Find the time taken, from the instant that the string breaks, for \(A\) to reach the floor. \(7 \quad\) A particle \(P\) starts from rest at a point \(O\) and moves in a straight line. \(P\) has acceleration \(0.6 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at time \(t\) seconds after leaving \(O\), until \(t = 10\).
3. Find the velocity and displacement from \(O\) of \(P\) when \(t = 10\). After \(t = 10 , P\) has acceleration \(- 0.4 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest at a point \(A\).
4. Find the distance \(O A\).

6 \includegraphics[max width=\textwidth, alt={}, center]{79b90ef5-ef3a-4c59-b662-d0fbfba813ca-3_526_519_902_813} Particles \(A\) of mass 0.4 kg and \(B\) of mass 1.6 kg 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 and both particles at a height of 1.2 m above the floor (see diagram). \(A\) is released and both particles start to move.

1. Find the work done on \(B\) by the tension in the string, as \(B\) moves to the floor. When particle \(B\) reaches the floor it remains at rest. Particle \(A\) continues to move upwards.
2. Find the greatest height above the floor reached by particle \(A\).

2 \includegraphics[max width=\textwidth, alt={}, center]{fd534430-2619-4078-ad0a-2355e656e121-2_569_519_676_813} Particle \(A\) of mass 0.2 kg and particle \(B\) of mass 0.6 kg are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley. \(B\) is held at rest at a height of 1.6 m above the floor. \(A\) hangs freely at a height of \(h \mathrm {~m}\) above the floor. Both straight parts of the string are vertical (see diagram). \(B\) is released and both particles start to move. When \(B\) reaches the floor it remains at rest, but \(A\) continues to move vertically upwards until it reaches a height of 3 m above the floor. Find the speed of \(B\) immediately before it hits the floor, and hence find the value of \(h\).

5 \includegraphics[max width=\textwidth, alt={}, center]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-3_259_828_1288_660} A smooth inclined plane of length 2.5 m is fixed with one end on the horizontal floor and the other end at a height of 0.7 m above the floor. Particles \(P\) and \(Q\), of masses 0.5 kg and 0.1 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 \(Q\) is held at rest on the floor vertically below the pulley. The string is taut and \(P\) is at rest on the plane (see diagram). \(Q\) is released and starts to move vertically upwards towards the pulley and \(P\) moves down the plane.

1. Find the tension in the string and the magnitude of the acceleration of the particles before \(Q\) reaches the pulley. At the instant just before \(Q\) reaches the pulley the string breaks; \(P\) continues to move down the plane and reaches the floor with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the length of the string.

1 \includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-2_446_497_258_826} A small ball \(B\) of mass 4 kg is attached to one end of a light inextensible string. A particle \(P\) of mass 3 kg is attached to the other end of the string. The string passes over a fixed smooth pulley. The system is in equilibrium with the string taut and its straight parts vertical. \(B\) is at rest on a rough plane inclined to the horizontal at an angle of \(\alpha\), where \(\cos \alpha = 0.8\) (see diagram). State the tension in the string and find the normal component of the contact force exerted on \(B\) by the plane.

4 \includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-3_442_495_255_826} Particles \(A\) and \(B\), of masses 0.35 kg and 0.15 kg respectively, are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. The system is at rest with \(B\) held on the horizontal floor, the string taut and its straight parts vertical. \(A\) is at a height of 1.6 m above the floor (see diagram). \(B\) is released and the system begins to move; \(B\) does not reach the pulley. Find

1. the acceleration of the particles and the tension in the string before \(A\) reaches the floor,
2. the greatest height above the floor reached by \(B\).

1 \includegraphics[max width=\textwidth, alt={}, center]{a92f97e2-343f-4cac-ae38-f18a4ad49055-2_241_823_264_660} Two particles \(P\) and \(Q\), of masses 0.6 kg and 0.4 kg respectively, are connected by a light inextensible string. The string passes over a small smooth light pulley fixed at the edge of a smooth horizontal table. Initially \(P\) is held at rest on the table and \(Q\) hangs vertically (see diagram). \(P\) is then released. Find the tension in the string and the acceleration of \(Q\).

6 A van of mass 3000 kg is pulling a trailer of mass 500 kg along a straight horizontal road at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The system of the van and the trailer is modelled as two particles connected by a light inextensible cable. There is a constant resistance to motion of 300 N on the van and 100 N on the trailer.

1. Find the power of the van's engine.
2. Write down the tension in the cable. The van reaches the bottom of a hill inclined at \(4 ^ { \circ }\) to the horizontal with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The power of the van's engine is increased to 25000 W .
3. Assuming that the resistance forces remain the same, find the new tension in the cable at the instant when the speed of the van up the hill is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

3 \includegraphics[max width=\textwidth, alt={}, center]{94c11160-a718-4de5-867a-27c755051fa6-2_312_1207_1320_468} Particles \(P\) and \(Q\), of masses 7 kg and 3 kg respectively, are attached to the two ends of a light inextensible string. The string passes over two small smooth pulleys attached to the two ends of a horizontal table. The two particles hang vertically below the two pulleys. The two particles are both initially at rest, 0.5 m below the level of the table, and 0.4 m above the horizontal floor (see diagram).

1. Find the acceleration of the particles and the speed of \(P\) immediately before it reaches the floor.
2. Determine whether \(Q\) comes to instantaneous rest before it reaches the pulley directly above it.

7 \includegraphics[max width=\textwidth, alt={}, center]{db1b5f31-1a41-44dd-ae9a-0c67336997eb-10_212_1029_255_557} Two particles \(A\) and \(B\) of masses 0.9 kg and 0.4 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 \(A\) on a smooth plane inclined at angle \(\theta ^ { \circ }\) to the horizontal and \(B\) on a plane inclined at angle \(25 ^ { \circ }\) 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 2.5 N is applied to \(B\) acting down the plane (see diagram).

1. For the case where \(\theta = 15\) and the plane on which \(B\) rests is smooth, find the acceleration of \(B\).
2. For a different value of \(\theta\), the plane on which \(B\) rests is rough with coefficient of friction between the plane and \(B\) of 0.8 . The system is in limiting equilibrium with \(B\) on the point of moving in the direction of the 2.5 N force. Find the value of \(\theta\).

4 Two particles \(A\) and \(B\) have masses 0.35 kg and 0.45 kg respectively. The particles are attached to the ends of a light inextensible string which passes over a small fixed smooth pulley which is 1 m above horizontal ground. Initially particle \(A\) is held at rest on the ground vertically below the pulley, with the string taut. Particle \(B\) hangs vertically below the pulley at a height of 0.64 m above the ground. Particle \(A\) is released.

1. Find the speed of \(A\) at the instant that \(B\) reaches the ground.
2. Assuming that \(B\) does not bounce after it reaches the ground, find the total distance travelled by \(A\) between the instant that \(B\) reaches the ground and the instant when the string becomes taut again.

4 Two particles \(A\) and \(B\), of masses \(m \mathrm {~kg}\) and 0.3 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley and the particles hang freely below it. The system is released from rest, with both particles 0.8 m above horizontal ground. Particle \(A\) reaches the ground with a speed of \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the tension in the string during the motion before \(A\) reaches the ground.
2. Find the value of \(m\).

5 \includegraphics[max width=\textwidth, alt={}, center]{007ccd92-79ba-409a-97e8-a4cf1f0a6cc5-08_538_414_260_868} Two particles \(P\) and \(Q\), 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 with the particles hanging freely below it. \(Q\) is held at rest with the string taut at a height of \(h \mathrm {~m}\) above a horizontal floor (see diagram). \(Q\) is now released and both particles start to move. The pulley is sufficiently high so that \(P\) does not reach it at any stage. The time taken for \(Q\) to reach the floor is 0.6 s .

1. Find the acceleration of \(Q\) before it reaches the floor and hence find the value of \(h\). \(Q\) remains at rest when it reaches the floor, and \(P\) continues to move upwards.
2. Find the velocity of \(P\) at the instant when \(Q\) reaches the floor and the total time taken from the instant at which \(Q\) is released until the string becomes taut again.