6
\includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-4_368_131_274_1005} A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light inextensible string of length \(L \mathrm {~m}\). The other end of the string is attached to a fixed point \(O\). The particle is held at rest with the string taut and then released. \(P\) starts to move and in the subsequent motion the angular displacement of \(O P\), at time \(t \mathrm {~s}\), is \(\theta\) radians from the downward vertical (see diagram). The initial value of \(\theta\) is 0.05 .

1. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - \frac { g } { L } \sin \theta\).
2. Hence show that the motion of \(P\) is approximately simple harmonic.
3. Given that the period of the approximate simple harmonic motion is \(\frac { 4 } { 7 } \pi \mathrm {~s}\), find the value of \(L\).
4. Find the value of \(\theta\) when \(t = 0.7 \mathrm {~s}\), and the value of \(t\) when \(\theta\) next takes this value.
5. Find the speed of \(P\) when \(t = 0.7 \mathrm {~s}\).
   \includegraphics[max width=\textwidth, alt={}, center]{7a67db39-4934-4808-a56b-c6841950d324-4_422_501_1500_822} A hollow cylinder has internal radius \(a\). The cylinder is fixed with its axis horizontal. A particle \(P\) of mass \(m\) is at rest in contact with the smooth inner surface of the cylinder. \(P\) is given a horizontal velocity \(u\), in a vertical plane perpendicular to the axis of the cylinder, and begins to move in a vertical circle. While \(P\) remains in contact with the surface, \(O P\) makes an angle \(\theta\) with the downward vertical, where \(O\) is the centre of the circle. The speed of \(P\) is \(v\) and the magnitude of the force exerted on \(P\) by the surface is \(R\) (see diagram).
6. Find \(v ^ { 2 }\) in terms of \(u , a , g\) and \(\theta\) and show that \(R = \frac { m u ^ { 2 } } { a } + m g ( 3 \cos \theta - 2 )\).
7. Given that \(P\) just reaches the highest point of the circle, find \(u ^ { 2 }\) in terms of \(a\) and \(g\), and show that in this case the least value of \(v ^ { 2 }\) is \(a g\).
8. Given instead that \(P\) oscillates between \(\theta = \pm \frac { 1 } { 6 } \pi\) radians, find \(u ^ { 2 }\) in terms of \(a\) and \(g\).