Two particles over single pulley

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\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.

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\includegraphics[max width=\textwidth, alt={}, center]{edf90396-5e17-44ef-bf25-e09cbc5785ba-4_451_729_255_708} Particles \(A\) and \(B\), of masses 0.2 kg and 0.45 kg respectively, are connected by a light inextensible string of length 2.8 m . The string passes over a small smooth pulley at the edge of a rough horizontal surface, which is 2 m above the floor. Particle \(A\) is held in contact with the surface at a distance of 2.1 m from the pulley and particle \(B\) hangs freely (see diagram). The coefficient of friction between \(A\) and the surface is 0.3. Particle \(A\) is released and the system begins to move.

1. Find the acceleration of the particles and show that the speed of \(B\) immediately before it hits the floor is \(3.95 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
2. Given that \(B\) remains on the floor, find the speed with which \(A\) reaches the pulley.

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\includegraphics[max width=\textwidth, alt={}, center]{fdf004fa-3726-4726-a0b8-60030812d451-4_451_729_255_708} Particles \(A\) and \(B\), of masses 0.2 kg and 0.45 kg respectively, are connected by a light inextensible string of length 2.8 m . The string passes over a small smooth pulley at the edge of a rough horizontal surface, which is 2 m above the floor. Particle \(A\) is held in contact with the surface at a distance of 2.1 m from the pulley and particle \(B\) hangs freely (see diagram). The coefficient of friction between \(A\) and the surface is 0.3. Particle \(A\) is released and the system begins to move.

1. Find the acceleration of the particles and show that the speed of \(B\) immediately before it hits the floor is \(3.95 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
2. Given that \(B\) remains on the floor, find the speed with which \(A\) reaches the pulley.

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\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 \(A\).

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\includegraphics[max width=\textwidth, alt={}, center]{60a41d3b-62a0-40d9-a30d-0560903429af-12_565_511_260_817} Two particles \(A\) and \(B\) have masses \(m \mathrm {~kg}\) and \(k m \mathrm {~kg}\) respectively, where \(k > 1\). The particles are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley and the particles hang vertically below it. Both particles are at a height of 0.81 m above horizontal ground (see diagram). The system is released from rest and particle \(B\) reaches the ground 0.9 s later. The particle \(A\) does not reach the pulley in its subsequent motion.

1. Find the value of \(k\) and show that the tension in the string before \(B\) reaches the ground is equal to \(12 m \mathrm {~N}\).
   At the instant when \(B\) reaches the ground, the string breaks.
2. Show that the speed of \(A\) when it reaches the ground is \(5.97 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures, and find the time taken, after the string breaks, for \(A\) to reach the ground.
3. Sketch a velocity-time graph for the motion of particle \(A\) from the instant when the system is released until \(A\) reaches 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.

15 In this question, use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A box, B, of mass 4 kg lies at rest on a fixed rough horizontal shelf.
One end of a light string is connected to B .
The string passes over a smooth peg, attached to the end of the shelf.
The other end of the string is connected to particle, P , of mass 1 kg , which hangs freely below the shelf as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{1f887565-4587-4520-99d4-f3635b015525-22_778_910_760_566} B is initially held at rest with the string taut.
B is then released.
B and P both move with constant acceleration \(a \mathrm {~ms} ^ { - 2 }\) As B moves across the shelf it experiences a total resistance force of 5 N
15

1. State one type of force that would be included in the total resistance force. 15
2. Show that \(a = 1\)
   15
3. When B has moved forward exactly 20 cm the string breaks.
   Find how much further B travels before coming to rest.
   15
4. State one assumption you have made when finding your solutions in parts (b) or (c). [1 mark]

19

1. It is given that \(M\) and \(N\) move with acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
   By forming two equations of motion show that $$a = \frac { 1 } { 11 } g$$ 19
2. The speed of \(N , 0.5\) seconds after its release, is \(\frac { g } { k } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) where \(k\) is a constant. Find the value of \(k\)
   19
3. State one assumption that must be made for the answer in part (b) to be valid.

9. \begin{figure}[h] \end{figure} A small ball \(A\) of mass 2.5 kg is held at rest on a rough horizontal table.
The ball is attached to one end of a string.
The string passes over a pulley \(P\) which is fixed at the edge of the table. The other end of the string is attached to a small ball \(B\) of mass 1.5 kg hanging freely, vertically below \(P\) and with \(B\) at a height of 1 m above the horizontal floor. The system is release from rest, with the string taut, as shown in Figure 2.
The resistance to the motion of \(A\) from the rough table is modelled as having constant magnitude 12.7 N . Ball \(B\) reaches the floor before ball \(A\) reaches the pulley. The balls are modelled as particles, the string is modelled as being light and inextensible, the pulley is modelled as being small and smooth and the acceleration due to gravity, \(g\), is modelled as being \(9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. 1. Write down an equation of motion for \(A\).
   2. Write down an equation of motion for \(B\).
2. Hence find the acceleration of \(B\).
3. Using the model, find the time it takes, from release, for \(B\) to reach the floor.
4. Suggest two improvements that could be made in the model.

9. A truck of mass 180 kg runs on smooth horizontal rails. A light inextensible rope is attached to the front of the truck. The rope runs parallel to the rails until it passes over a light smooth pulley. The rest of the rope hangs down a vertical shaft. When the truck is required to move, a load of \(M \mathrm {~kg}\) is attached to the end of the rope in the shaft and the brakes are then released.

1. Find the tension in the rope when the truck and the load move with an acceleration of magnitude \(0.8 \mathrm {~ms} ^ { - 2 }\) and calculate the corresponding value of \(M\).
2. In addition to the assumptions given in the question, write down one further assumption that you have made in your solution to this problem and explain how that assumption affects both of your answers.