7. A pendulum consists of a uniform circular disc, of radius \(a\) and mass \(4 m\), whose centre is fixed to the end \(B\) of a uniform \(\operatorname { rod } A B\). The rod has mass \(3 m\) and length \(4 l\), where \(2 l > a\). The rod lies in the same plane as the disc. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the disc. The moment of inertia of the pendulum about \(L\) is \(2 m \left( a ^ { 2 } + 40 l ^ { 2 } \right)\).

1. Find the approximate period of small oscillations of the pendulum about its position of stable equilibrium. The pendulum is held with \(B\) vertically above \(A\) and is then slightly displaced from rest. In the subsequent motion the midpoint of \(A B\) strikes a small peg, which is fixed at the same horizontal level as \(A\), and the pendulum rebounds upwards. Immediately before it strikes the peg, the angular speed of the pendulum is \(\omega\).
2. Show that \(\omega ^ { 2 } = \frac { 22 g l } { \left( a ^ { 2 } + 40 l ^ { 2 } \right) }\) Immediately after it strikes the peg, the angular speed of the pendulum is \(\frac { 1 } { 2 } \omega\).
3. Find, in terms of \(m , g , a\) and \(l\), the magnitude of the impulse exerted on the peg by the pendulum.
4. Show that the size of the angle turned through by the pendulum, between it hitting the peg and it next coming to rest, is \(\arcsin \frac { 1 } { 4 }\).
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