7. A uniform plane circular disc, of mass \(m\) and radius \(a\), hangs in equilibrium from a point \(B\) on its circumference. The disc is free to rotate about a fixed smooth horizontal axis which is in the plane of the disc and tangential to the disc at \(B\). A particle \(P\), of mass \(m\), is moving horizontally with speed \(u\) in a direction which is perpendicular to the plane of the disc. At time \(t = 0 , P\) strikes the disc at its centre and adheres to the disc.

1. Show that the angular speed of the disc immediately after it has been struck by \(P\) is \(\frac { 4 u } { 9 a }\).
   (6) It is given that \(u ^ { 2 } = \frac { 1 } { 10 } a g\), and that air resistance is negligible.
2. Find the angle through which the disc turns before it first comes to instantaneous rest. The disc first returns to its initial position at time \(t = T\).
3. 1. Write down an equation of motion for the disc.
   2. Hence find \(T\) in terms of \(a , g\) and \(m\), using a suitable approximation which should be justified.