7. \begin{figure}[h] \end{figure} Figure 3 shows a framework \(A B C\), consisting of two uniform rods rigidly joined together at \(B\) so that \(\angle A B C = 90 ^ { \circ }\). The \(\operatorname { rod } A B\) has length \(2 a\) and mass \(4 m\), and the \(\operatorname { rod } B C\) has length \(a\) and mass \(2 m\). The framework is smoothly hinged at \(A\) to a fixed point, so that the framework can rotate in a fixed vertical plane. One end of a light elastic string, of natural length \(2 a\) and modulus of elasticity \(3 m g\), is attached to \(A\). The string passes through a small smooth ring \(R\) fixed at a distance \(2 a\) from \(A\), on the same horizontal level as \(A\) and in the same vertical plane as the framework. The other end of the string is attached to \(B\). The angle \(A R B\) is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\).

1. Show that the potential energy \(V\) of the system is given by $$V = 8 a m g \sin 2 \theta + 5 a m g \cos 2 \theta + \text { constant }$$
2. Find the value of \(\theta\) for which the system is in equilibrium.
3. Determine the stability of this position of equilibrium. Turn over
   1. A smooth uniform sphere \(S\), of mass \(m\), is moving on a smooth horizontal plane when it collides obliquely with another smooth uniform sphere \(T\), of the same radius as \(S\) but of mass \(2 m\), which is at rest on the plane. Immediately before the collision the velocity of \(S\) makes an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), with the line joining the centres of the spheres. Immediately after the collision the speed of \(T\) is \(V\). The coefficient of restitution between the spheres is \(\frac { 3 } { 4 }\).
   2. Find, in terms of \(V\), the speed of \(S\)
      1. immediately before the collision,
      2. immediately after the collision.
   3. Find the angle through which the direction of motion of \(S\) is deflected as a result of the collision.
      
   1. A \(\operatorname { ship } A\) is moving at a constant speed of \(8 \mathrm {~km} \mathrm {~h} \mathrm {~h} ^ { - 1 }\) on a bearing of \(150 ^ { \circ }\). At noon a second ship \(B\) is 6 km from \(A\), on a bearing of \(210 ^ { \circ }\). Ship \(B\) is moving due east at a constant speed. At a later time, \(B\) is \(2 \sqrt { 3 } \mathrm {~km}\) due south of \(A\).
   Find
   (i) the time at which \(B\) will be due east of \(A\),
   (ii) the distance between the ships at that time.
   1. Two particles, of masses \(m\) and \(2 m\), are connected to the ends of a long light inextensible string. The string passes over a small smooth fixed pulley and hangs vertically on either side. The particles are released from rest with the string taut. Each particle is subject to air resistance of magnitude \(k v ^ { 2 }\), where \(v\) is the speed of each particle after it has moved a distance \(x\) from rest and \(k\) is a positive constant.
   2. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( v ^ { 2 } \right) + \frac { 4 k } { 3 m } v ^ { 2 } = \frac { 2 g } { 3 }\)
   3. Find \(v ^ { 2 }\) in terms of \(x\).
   4. Deduce that the tension in the string, \(T\), satisfies
   $$\frac { 4 m g } { 3 } \leqslant T < \frac { 3 m g } { 2 }$$
   1. A rescue boat, whose maximum speed is \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), receives a signal which indicates that a yacht is in distress near a fixed point \(P\). The rescue boat is 15 km south-west of \(P\). There is a constant current of \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) flowing uniformly from west to east. The rescue boat sets the course needed to get to \(P\) as quickly as possible. Find
   2. the course the rescue boat sets,
   3. the time, to the nearest minute, to get to \(P\).
   When the rescue boat arrives at \(P\), the yacht is just visible 4 km due north of \(P\) and is drifting with the current. Find
4. the course that the rescue boat should set to get to the yacht as quickly as possible,
5. the time taken by the rescue boat to reach the yacht from \(P\). 5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-49_977_1228_205_356} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} A uniform rod \(A B\), of length \(4 a\) and weight \(W\), is free to rotate in a vertical plane about a fixed smooth horizontal axis which passes through the point \(C\) of the rod, where \(A C = 3 a\). One end of a light inextensible string of length \(L\), where \(L > 10 a\), is attached to the end \(A\) of the rod and passes over a small smooth fixed peg at \(P\) and another small smooth fixed peg at \(Q\). The point \(Q\) lies in the same vertical plane as \(P , A\) and \(B\). The point \(P\) is at a distance \(3 a\) vertically above \(C\) and \(P Q\) is horizontal with \(P Q = 4 a\). A particle of weight \(\frac { 1 } { 2 } W\) is attached to the other end of the string and hangs vertically below \(Q\). The rod is inclined at an angle \(2 \theta\) to the vertical, where \(- \pi < 2 \theta < \pi\), as shown in Figure 1.
6. Show that the potential energy of the system is $$W a ( 3 \cos \theta - \cos 2 \theta ) + \text { constant }$$
7. Find the positions of equilibrium and determine their stability.
   1. Two points \(A\) and \(B\) are in a vertical line, with \(A\) above \(B\) and \(A B = 4 a\). One end of a light elastic spring, of natural length \(a\) and modulus of elasticity \(3 m g\), is attached to \(A\). The other end of the spring is attached to a particle \(P\) of mass \(m\). Another light elastic spring, of natural length \(a\) and modulus of elasticity \(m g\), has one end attached to \(B\) and the other end attached to \(P\). The particle \(P\) hangs at rest in equilibrium.
   2. Show that \(A P = \frac { 7 a } { 4 }\)
   The particle \(P\) is now pulled down vertically from its equilibrium position towards \(B\) and at time \(t = 0\) it is released from rest. At time \(t\), the particle \(P\) is moving with speed \(v\) and has displacement \(x\) from its equilibrium position. The particle \(P\) is subject to air resistance of magnitude \(m k v\), where \(k\) is a positive constant.
8. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + k \frac { \mathrm {~d} x } { \mathrm {~d} t } + \frac { 4 g } { a } x = 0$$
9. Find the range of values of \(k\) which would result in the motion of \(P\) being a damped oscillation.