A uniform rod \(A B\), of mass \(M\) and length \(2 L\), is free to rotate in a vertical plane about a smooth fixed horizontal axis through \(A\). The rod is hanging vertically at rest, with \(B\) below \(A\), when it is struck at its midpoint by a particle of mass \(\frac { 1 } { 2 } M\). Immediately before this impact, the particle is moving with speed \(u\), in a direction which is horizontal and perpendicular to the axis. The particle is brought to rest by the impact and immediately after the impact the rod moves with angular speed \(\omega\).
Show that \(\omega = \frac { 3 u } { 8 L }\)
Immediately after the impact, the magnitude of the vertical component of the force exerted on the \(\operatorname { rod }\) at \(A\) by the axis is \(\frac { 3 M g } { 2 }\)
Find \(u\) in terms of \(L\) and \(g\).
Show that the magnitude of the horizontal component of the force exerted on the rod at \(A\) by the axis, immediately after the impact, is zero.
The rod first comes to instantaneous rest after it has turned through an angle \(\alpha\).
Find the size of \(\alpha\).
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