OCR Further Mechanics AS (Further Mechanics AS) 2023 June

1 Two particles \(A\), of mass \(m \mathrm {~kg}\), and \(B\), of mass \(3 m \mathrm {~kg}\), are connected by a light inextensible string and placed together at rest on a smooth horizontal surface with the string slack. \(A\) is projected along the surface, directly away from \(B\), with a speed of \(2.4 \mathrm {~ms} ^ { - 1 }\).

1. Find the speed of \(B\) immediately after the string becomes taut.
2. Find, in terms of \(m\), the magnitude of the impulse exerted on \(B\) as a result of the string becoming taut.

2
\includegraphics[max width=\textwidth, alt={}, center]{b190b8c9-75b0-4ede-913f-cdecdb58180f-2_337_579_842_246} A small body \(P\) of mass 3 kg is at rest at the lowest point of the inside of a smooth hemispherical shell of radius 3.2 m and centre \(O\).
\(P\) is projected horizontally with a speed of \(u \mathrm {~ms} ^ { - 1 }\). When \(P\) first comes to instantaneous rest \(O P\) makes an angle of \(60 ^ { \circ }\) with the downward vertical through \(O\).

1. Find the value of \(u\).
2. State one assumption made in modelling the motion of \(P\).

3 A crate of mass 45 kg is sliding with a speed of \(0.8 \mathrm {~ms} ^ { - 1 }\) in a straight line across a smooth horizontal floor. One end of a light inextensible rope is attached to the crate. At a certain instant a builder takes the other end of the rope and starts to pull, applying a constant force of 80 N for 5 seconds. While the builder is pulling the crate, the rope makes a constant angle of \(40 ^ { \circ }\) above the horizontal. Both the rope and the velocity of the crate lie in the same vertical plane (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{b190b8c9-75b0-4ede-913f-cdecdb58180f-3_250_817_605_246} It may be assumed that there is no resistance to the motion of the crate.

1. Determine the work done by the builder in pulling the crate.
2. 1. Find the kinetic energy of the crate at the instant when the builder stops pulling the crate.
   2. Explain why the answers to part (a) and part (b)(i) are not equal.
3. Find the average power developed by the builder in pulling the crate.
4. Calculate the total impulse exerted on the crate by the builder.

4 A rower is rowing a boat in a straight line across a lake. The combined mass of the rower, boat and oars is 240 kg . The maximum power that the rower can generate is 450 W . In a model of the motion of the boat it is assumed that the total resistance to the motion of the boat is 150 N at any instant when the boat is in motion.

1. Find the maximum possible acceleration of the boat, according to the model, at an instant when its speed is \(0.5 \mathrm {~ms} ^ { - 1 }\). At one stage in its motion the boat is travelling at a constant speed and the rower is generating power at an average rate of 210 W , which is assumed to be constant. The boat passes a pole and then, after travelling 350 m , a second pole.
2. Determine how long it takes, according to the model, for the boat to travel between the two poles.
3. State a reason why the assumption that the rower's generated power is constant may be unrealistic.

5 Two identical spheres, \(A\) and \(B\), each of mass 4 kg , are moving directly towards each other along the same straight line on a smooth horizontal surface until they collide. Before they collide, the speeds of \(A\) and \(B\) are \(5 \mathrm {~ms} ^ { - 1 }\) and \(3 \mathrm {~ms} ^ { - 1 }\) respectively. Immediately after they collide, the speed of \(A\) is \(2 \mathrm {~ms} ^ { - 1 }\) and its direction of motion has been reversed.

1. 1. Determine the velocity of \(B\) immediately after \(A\) and \(B\) collide.
   2. Show that the coefficient of restitution between \(A\) and \(B\) is \(\frac { 3 } { 4 }\).
   3. Calculate the total loss of kinetic energy due to this collision. Sphere \(B\) goes on to strike a fixed wall directly. As a result of this collision \(B\) moves along the same straight line with a speed of \(4 \mathrm {~ms} ^ { - 1 }\).
2. Find the coefficient of restitution between \(B\) and the wall, stating whether the collision between \(B\) and the wall is perfectly elastic.
3. Determine the magnitude of the impulse that \(B\) exerts on \(A\) the next time that they collide.

6 The physical quantity pressure, denoted by \(P\), can be calculated using the formula \(P = \frac { F } { A }\) where \(F\) is a force and \(A\) is an area.

1. Find the dimensions of \(P\). An object of mass \(m\) is moving on a smooth horizontal surface subject to a system of forces which begin to act at time \(t = 0\). The initial velocity of the object is \(u\) and its velocity and acceleration at time \(t\) are denoted by \(v\) and \(a\) respectively. The object exerts a pressure \(P\) on the surface. The total work done by the forces is denoted by \(W\). A Mathematics class suggests three formulae to model the quantity \(W\).
   The first suggested formula is \(W = \frac { 1 } { 2 } m v ^ { 2 } - \frac { 1 } { 2 } m u ^ { 2 } + m P\).
2. Use dimensional analysis to show that this formula cannot be correct. The second suggested formula is \(W = k u ^ { \alpha } v ^ { \beta } t ^ { \gamma }\) where \(k\) is a dimensionless constant.
3. Use dimensional analysis to show that this formula cannot be correct for any values of \(\alpha , \beta\) and \(\gamma\). The third suggested formula is \(W = k u ^ { \alpha } a ^ { \beta } m ^ { \gamma } t ^ { \delta }\) where \(k\) is a dimensionless constant.
4. 1. Explain why it is not possible to use dimensional analysis to determine the values of \(\alpha\), \(\beta , \gamma\) and \(\delta\) for the third suggested formula.
   2. Given that \(\alpha = 3\), use dimensional analysis to determine the values of \(\beta , \gamma\) and \(\delta\) for the third suggested formula.
   3. By considering what the formula predicts for large values of \(t\), explain why the formula derived in part (d)(ii) is likely to be incorrect.

7 Two identical light, inextensible strings \(S _ { 1 }\) and \(S _ { 2 }\) are each of length 5 m . Two identical particles \(P\) and \(Q\) are each of mass 1.5 kg . One end of \(S _ { 1 }\) is attached to \(P\). The other end of \(S _ { 1 }\) is attached to a fixed point \(A\) on a smooth horizontal plane. \(P\) moves with constant speed in a horizontal circular path with \(A\) as its centre (see Fig. 1). One end of \(S _ { 2 }\) is attached to \(Q\). The other end of \(S _ { 2 }\) is attached to a fixed point \(B\). \(Q\) moves with constant speed in a horizontal circular path around a point \(O\) which is vertically below \(B\). At any instant, \(B Q\) makes an angle of \(\theta\) with the downward vertical through \(B\) (see Fig. 2). \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Given that the angular speed of \(P\) is the same as the angular speed of \(Q\), show that the tensions in \(S _ { 1 }\) and \(S _ { 2 }\) have the same magnitude.
2. You are given instead that the kinetic energy of \(P\) is 39.2 J and that this is the same as the kinetic energy of \(Q\). Determine the difference between the times taken by \(P\) and \(Q\) to complete one revolution. Give your answer in an exact form.