8 A point \(O\) is situated a distance \(h\) above a smooth horizontal plane, and a particle \(A\) of mass \(m\) is attached to \(O\) by a light inextensible string of length \(h\). A particle \(B\) of mass \(2 m\) is at rest on the plane, directly below \(O\), and is attached to a point \(C\) on the plane, where \(B C = l\), by a light inextensible string of length \(l . A\) is released from rest with the string \(O A\) taut and making an acute angle \(\theta\) with the downward vertical (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{19c3a9d0-15b6-4dd0-a00b-577c3fd2cf52-5_604_1137_486_552}
\(A\) moves in a vertical plane perpendicular to \(C B\) and collides directly with \(B\). As a result of this collision, \(A\) is brought to rest and \(B\) moves on the plane in a horizontal circle with centre \(C\). After \(B\) has made one complete revolution the particles collide again.

1. Show that, on the next occasion that \(A\) comes to rest, the string \(O A\) makes an angle \(\phi\) with the downward vertical through \(O\), where \(\cos \phi = \frac { 3 + \cos \theta } { 4 }\).
   \(A\) and \(B\) collide again when \(A O\) is next vertical.
2. Find the percentage of the original energy of the system that remains immediately after this collision.
3. Explain why the total momentum of the particles immediately before the first collision is the same as the total momentum of the particles immediately after the second collision.
4. Explain why the total momentum of the particles immediately before the first collision is different from the total momentum of the particles immediately after the third collision. \section*{OCR} \section*{Oxford Cambridge and RSA}