Rotation about fixed axis: impulsive impact and subsequent motion

5. A uniform circular disc has mass \(m\) and radius \(a\). The disc can rotate freely about an axis that is in the same plane as the disc and tangential to the disc at a point \(A\) on its circumference. The disc hangs at rest in equilibrium with its centre \(O\) vertically below \(A\). A particle \(P\) of mass \(m\) is moving horizontally and perpendicular to the disc with speed \(\sqrt { } ( k g a )\), where \(k\) is a constant. The particle then strikes the disc at \(O\) and adheres to it at \(O\). Given that the disc rotates through an angle of \(90 ^ { \circ }\) before first coming to instantaneous rest, find the value of \(k\).
(Total 10 marks)

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.

5. A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\). The rod is hanging in equilibrium with \(B\) below \(A\) when it is hit by a particle of mass \(m\) moving horizontally with speed \(v\) in a vertical plane perpendicular to the axis. The particle strikes the rod at \(B\) and immediately adheres to it.

1. Show that the angular speed of the rod immediately after the impact is \(\frac { 3 v } { 8 a }\). Given that the rod rotates through \(120 ^ { \circ }\) before first coming to instantaneous rest,
2. find \(v\) in terms of \(a\) and \(g\).
3. find, in terms of \(m\) and \(g\), the magnitude of the vertical component of the force acting on the \(\operatorname { rod }\) at \(A\) immediately after the impact.
   (5)