Two particles over pulley, vertical strings

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.

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.

1 \includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-2_549_775_269_685} A particle \(A\) and a block \(B\) are attached to opposite ends of a light elastic string of natural length 2 m and modulus of elasticity 6 N . The block is at rest on a rough horizontal table. The string passes over a small smooth pulley \(P\) at the edge of the table, with the part \(B P\) of the string horizontal and of length 1.2 m . The frictional force acting on \(B\) is 1.5 N and the system is in equilibrium (see diagram). Find the distance \(P A\).

3. \begin{figure}[h] \end{figure} Two particles \(A\) and \(B\) have masses \(3 m\) and \(k m\) respectively, where \(k > 3\). They are connected by a light inextensible string which passes over a smooth fixed pulley. The system is released from rest with the string taut and the hanging parts of the string vertical, as shown in Fig. 3. While the particles are moving freely, \(A\) has an acceleration of magnitude \(\frac { 2 } { 5 } g\).

1. Find, in terms of \(m\) and g , the tension in the string.
2. State why \(B\) also has an acceleration of magnitude \(\frac { 2 } { 5 } g\).
3. Find the value of \(k\).
4. State how you have used the fact that the string is light.

2 \includegraphics[max width=\textwidth, alt={}, center]{99d30766-9c1b-43a8-986a-112b78b08146-2_643_289_1475_927} Particles \(A\) and \(B\), of masses 0.2 kg and 0.3 kg respectively, are attached to the ends of a light inextensible string. Particle \(A\) is held at rest at a fixed point and \(B\) hangs vertically below \(A\). Particle \(A\) is now released. As the particles fall the air resistance acting on \(A\) is 0.4 N and the air resistance acting on \(B\) is 0.25 N (see diagram). The downward acceleration of each of the particles is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and the tension in the string is \(T \mathrm {~N}\).

1. Write down two equations in \(a\) and \(T\) obtained by applying Newton's second law to \(A\) and to \(B\).
2. Find the values of \(a\) and \(T\).

3 \includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-3_437_846_274_651} A block of mass 50 kg is in equilibrium on smooth horizontal ground with one end of a light wire attached to its upper surface. The other end of the wire is attached to an object of mass mkg . The wire passes over a small smooth pulley, and the object hangs vertically below the pulley. The part of the wire between the block and the pulley makes an angle of \(72 ^ { \circ }\) with the horizontal. A horizontal force of magnitude X N acts on the block in the vertical plane containing the wire (see diagram). The tension in the wire is T N and the contact force exerted by the ground on the block is R N.

1. By resolving forces on the block vertically, find a relationship between T and R . It is given that the block is on the point of lifting off the ground.
2. Show that \(\mathrm { T } = 515\), correct to 3 significant figures, and hence find the value of m .
3. By resolving forces on the block horizontally, write down a relationship between T and X , and hence find the value of \(X\).

2 Fig. 2 shows a light string with an object of mass 4 kg attached at end A . The string passes over a smooth pulley and its other end B is attached to two light strings BC and BD of the same length. The strings BC and BD are attached to horizontal ground and are each inclined at \(20 ^ { \circ }\) to the vertical. The system is in equilibrium. \begin{figure}[h] \end{figure}

1. What information in the question tells you that the tension is the same throughout the string AB ?
2. What is the tension in the string AB ?
3. Calculate the tension in the strings BC and BD .

1 Fig. 1 shows two blocks of masses 3 kg and 5 kg connected by a light string which passes over a smooth, fixed pulley. Initially the blocks are held at rest but then they are released. \begin{figure}[h] \end{figure} Find the acceleration of the blocks when they start to move, and the tension in the string.

2 Fig. 2 shows a sack of rice of weight 250 N hanging in equilibrium supported by a light rope AB . End A of the rope is attached to the sack. The rope passes over a small smooth fixed pulley. \begin{figure}[h] \end{figure} Initially, end B of the rope is attached to a vertical wall as shown in Fig. 2.

1. Calculate the horizontal and the vertical forces acting on the wall due to the rope. End B of the rope is now detached from the wall and attached instead to the top of the sack. The sack is in equilibrium with both sections of the rope vertical.
2. Calculate the tension in the rope.

3. \begin{figure}[h] \end{figure} Two particles \(A\) and \(B\) have masses \(3 m\) and \(k m\) respectively, where \(k > 3\). They are connected by a light inextensible string which passes over a smooth fixed pulley. The system is released from rest with the string taut and the hanging parts of the string vertical, as shown in Fig. 3. While the particles are moving freely, \(A\) has an acceleration of magnitude \(\frac { 2 } { 5 } g\).

1. Find, in terms of \(m\) and \(g\), the tension in the string.
   (3 marks)
2. State why \(B\) also has an acceleration of magnitude \(\frac { 2 } { 5 } g\).
3. Find the value of \(k\).
4. State how you have used the fact that the string is light.
   (1 mark)

4 Two particles are connected by a string, which passes over a pulley. Model the string as light and inextensible. The particles have masses of 2 kg and 5 kg . The particles are released from rest. \includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-3_392_209_1685_909}

1. State one modelling assumption that you should make about the pulley in order to determine the acceleration of the particles.
2. By forming an equation of motion for each particle, show that the magnitude of the acceleration of each particle is \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. Find the tension in the string.

4 Two particles, \(A\) and \(B\), are connected by a string that passes over a fixed peg, as shown in the diagram. The mass of \(A\) is 9 kg and the mass of \(B\) is 11 kg .
The particles are released from rest in the position shown, where \(B\) is \(d\) metres higher than \(A\). The motion of the particles is to be modelled using simple assumptions.

1. State one assumption that should be made about the peg.
2. State two assumptions that should be made about the string.
3. By forming an equation of motion for each of the particles \(A\) and \(B\), show that the acceleration of each particle has magnitude \(0.98 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
4. When the particles have been moving for 0.5 seconds, they are at the same level.
   1. Find the speed of the particles at this time.
   2. Find \(d\).

4 Two particles, \(A\) of mass 5 kg and \(B\) of mass 9 kg , are attached to the ends of a light inextensible string. The string passes over a light smooth pulley as shown in the diagram. The particles are released from rest at the same height. \includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-3_378_287_1580_872}

1. By forming an equation of motion for each particle, show that the magnitude of the acceleration of each particle is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the tension in the string.
3. When \(B\) has been moving for 0.5 seconds it hits the floor. Find the height of \(A\), above the floor, at this time. Assume that \(A\) is still below the pulley when \(B\) hits the floor.
   (4 marks)

5 Two particles, \(P\) and \(Q\), are connected by a string that passes over a fixed smooth peg, as shown in the diagram. The mass of \(P\) is 5 kg and the mass of \(Q\) is 3 kg . \includegraphics[max width=\textwidth, alt={}, center]{7ac7dfd0-4c3e-4eb7-920f-ce5b24ad1281-3_209_433_1009_808} The particles are released from rest in the position shown.

1. By forming an equation of motion for each particle, show that the magnitude of the acceleration of each particle is \(2.45 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the tension in the string.
3. State two modelling assumptions that you have made about the string.
4. Particle \(P\) hits the floor when it has moved 0.196 metres and \(Q\) has not reached the peg.
   1. Find the time that it takes \(P\) to reach the floor.
   2. Find the speed of \(P\) when it hits the floor.

7 A builder ties two identical buckets, \(P\) and \(Q\), to the ends of a light inextensible rope. He hangs the rope over a smooth beam so that the buckets hang in equilibrium, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-007_360_296_502_904} The buckets are each of mass 0.6 kg .

1. 1. State the magnitude of the tension in the rope.
   2. State the magnitude and direction of the force exerted on the beam by the rope.
2. The bucket \(Q\) is held at rest while a stone, of mass 0.2 kg , is placed inside it. The system is then released from rest and, in the subsequent motion, bucket \(Q\) moves vertically downwards with the stone inside.
   1. By forming an equation of motion for each bucket, show that the magnitude of the tension in the rope during the motion is 6.72 newtons, correct to three significant figures.
   2. State the magnitude of the force exerted on the beam by the rope while the motion takes place.

7 A builder ties two identical buckets, \(P\) and \(Q\), to the ends of a light inextensible rope. He hangs the rope over a smooth beam so that the buckets hang in equilibrium, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-6_360_296_502_904} The buckets are each of mass 0.6 kg .

1. 1. State the magnitude of the tension in the rope.
   2. State the magnitude and direction of the force exerted on the beam by the rope.
2. The bucket \(Q\) is held at rest while a stone, of mass 0.2 kg , is placed inside it. The system is then released from rest and, in the subsequent motion, bucket \(Q\) moves vertically downwards with the stone inside.
   1. By forming an equation of motion for each bucket, show that the magnitude of the tension in the rope during the motion is 6.72 newtons, correct to three significant figures.
   2. State the magnitude of the force exerted on the beam by the rope while the motion takes place.

Two particles, of masses \(1.8\) kg and \(1.2\) kg, are connected by a light inextensible string that passes over a fixed smooth pulley. The particles hang vertically. The system is released from rest. Find the magnitude of the acceleration of the particles and find the tension in the string. [4]

\includegraphics{figure_1} Particles \(P\) and \(Q\), of masses \(0.3\) kg and \(0.4\) kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is in motion with the string taut and with each of the particles moving vertically. The downward acceleration of \(P\) is \(a\) m s\(^{-2}\) (see diagram).

1. Show that \(a = -1.4\). [4]

Initially \(P\) and \(Q\) are at the same horizontal level. \(P\)'s initial velocity is vertically downwards and has magnitude \(2.8\) m s\(^{-1}\).

1. Assuming that \(P\) does not reach the floor and that \(Q\) does not reach the pulley, find the time taken for \(P\) to return to its initial position. [3]

In this question use \(g = 9.81\,\mathrm{m}\,\mathrm{s}^{-2}\) Two particles, of mass \(1.8\,\mathrm{kg}\) and \(1.2\,\mathrm{kg}\), are connected by a light, inextensible string over a smooth peg. \includegraphics{figure_14}

1. Initially the particles are held at rest \(1.5\,\mathrm{m}\) above horizontal ground and the string between them is taut. The particles are released from rest. Find the time taken for the \(1.8\,\mathrm{kg}\) particle to reach the ground. [5 marks]
2. State one assumption you have made in answering part (a). [1 mark]

Two objects, \(M\) and \(N\), are connected by a light inextensible string that passes over a smooth peg. \(M\) has a mass of 0.6 kilograms. \(N\) has a mass of 0.5 kilograms. \(M\) and \(N\) are initially held at rest, with the string taut, as shown in the diagram below. \includegraphics{figure_19} \(M\) and \(N\) are released at the same instant and begin to move vertically. You may assume that air resistance can be ignored.

1. It is given that \(M\) and \(N\) move with acceleration \(a\) m s\(^{-2}\) By forming two equations of motion show that $$a = \frac{1}{11}g$$ [5 marks]
2. The speed of \(N\), 0.5 seconds after its release, is \(\frac{g}{k}\) m s\(^{-1}\) where \(k\) is a constant. Find the value of \(k\) [2 marks]
3. State one assumption that must be made for the answer in part (b) to be valid. [1 mark]

The diagram below shows two particles \(A\) and \(B\), of mass 2 kg and 5 kg respectively, which are connected by a light inextensible string passing over a fixed smooth pulley. Initially, \(B\) is held at rest with the string just taut. It is then released. \includegraphics{figure_4}

1. Calculate the magnitude of the acceleration of \(A\) and the tension in the string. [6]
2. What assumption does the word 'light' in the description of the string enable you to make in your solution? [1]