Particle on smooth horizontal surface, particle hanging

1 \includegraphics[max width=\textwidth, alt={}, center]{38ece0f6-1c29-4e7a-9d66-16c3e2b695f9-2_200_529_269_808} Two particles \(P\) and \(Q\), of masses 1.7 kg and 0.3 kg respectively, are connected by a light inextensible string. \(P\) is held on a smooth horizontal table with the string taut and passing over a small smooth pulley fixed at the edge of the table. \(Q\) is at rest vertically below the pulley. \(P\) is released. Find the acceleration of the particles and the tension in the string.

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 \(Q\).

3 Fig. 3.1 shows a block of mass 8 kg on a smooth horizontal table.
This block is connected by a light string passing over a smooth pulley to a block of mass 4 kg which hangs freely. The part of the string between the 8 kg block and the pulley is parallel to the table. The system has acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \begin{figure}[h] \end{figure}

1. Write down two equations of motion, one for each block.
2. Find the value of \(a\). The table is now tilted at an angle of \(\theta\) to the horizontal as shown in Fig. 3.2. The system is set up as before; the 4 kg block still hangs freely. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-3_410_727_1324_669} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
3. The system is now in equilibrium. Find the value of \(\theta\).

11 A block of mass 3 kg is at rest on a smooth horizontal table. It is attached to a light inextensible string which passes over a smooth pulley. This part of the string is horizontal. A sphere of mass 1.2 kg is attached to the other end of the string. The sphere hangs with this part of the string vertical as shown in the diagram. A horizontal force of magnitude \(F\) N is applied to the block to prevent motion. \includegraphics[max width=\textwidth, alt={}, center]{1d1e41f3-a834-4230-b6e1-4b0be9450d30-7_268_718_493_244}

1. Complete the copy of the diagram in the Printed Answer Booklet to show all the forces acting on the block and the sphere.
2. Find the value of \(F\). The force \(F\) N is removed, and the system begins to move.
3. The equation of motion of the block is \(\mathrm { T } = 3 \mathrm { a }\), where \(T \mathrm {~N}\) is the tension in the string and \(a \mathrm {~ms} ^ { - 2 }\) is the acceleration of the block. Write down the equation of motion of the sphere.
4. Find the value of \(T\).

5 A block, of mass 12 kg , lies on a horizontal surface. The block is attached to a particle, of mass 18 kg , by a light inextensible string which passes over a smooth fixed peg. Initially, the block is held at rest so that the string supports the particle, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{828e8db1-efcf-4878-8292-ba5bbd80115c-3_346_716_1557_715} The block is then released.

1. Assuming that the surface is smooth, use two equations of motion to find the magnitude of the acceleration of the block and particle.
2. In reality, the surface is rough and the acceleration of the block is \(3 \mathrm {~ms} ^ { - 2 }\).
   1. Find the tension in the string.
   2. Calculate the magnitude of the normal reaction force acting on the block.
   3. Find the coefficient of friction between the block and the surface.
3. State two modelling assumptions, other than those given, that you have made in answering this question.

\includegraphics{figure_2} Particles \(A\) and \(B\), of masses 0.35 kg and 0.15 kg respectively, are attached to the ends of a light inextensible string. \(A\) is held at rest on a smooth horizontal surface with the string passing over a small smooth pulley fixed at the edge of the surface. \(B\) hangs vertically below the pulley at a distance \(h\) m above the floor (see diagram). \(A\) is released and the particles move. \(B\) reaches the floor and \(A\) subsequently reaches the pulley with a speed of \(3 \text{ m s}^{-1}\).

1. Explain briefly why the speed with which \(B\) reaches the floor is \(3 \text{ m s}^{-1}\). [1]
2. Find the value of \(h\). [4]

\includegraphics{figure_4} A particle \(A\) of mass 0.8 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 1.2 kg which hangs freely below the pulley, as shown in Fig. 4. The system is released from rest with the string taut and with \(B\) at a height of 0.6 m above the ground. In the subsequent motion \(A\) does not reach \(P\) before \(B\) reaches the ground. In an initial model of the situation, the table is assumed to be smooth. Using this model, find

1. the tension in the string before \(B\) reaches the ground, [5]
2. the time taken by \(B\) to reach the ground. [3]

In a refinement of the model, it is assumed that the table is rough and that the coefficient of friction between \(A\) and the table is \(\frac{1}{4}\). Using this refined model,

1. find the time taken by \(B\) to reach the ground. [8]

The diagram shows two objects \(A\) and \(B\), of mass 3 kg and 5 kg respectively, connected by a light inextensible string passing over a light smooth pulley fixed at the end of a smooth horizontal surface. Object \(A\) lies on the horizontal surface and object \(B\) hangs freely below the pulley. \includegraphics{figure_8} Initially, \(B\) is supported so that the objects are at rest with the string just taut. Object \(B\) is then released.

1. Find the magnitude of the acceleration of \(A\) and the tension in the string. [6]
2. State briefly what effect a rough pulley would have on the tension in the string. [1]