5 A uniform rod \(A B\) has mass \(2 m\) and length 4a.
- Show by integration that the moment of inertia of the rod about an axis perpendicular to the rod through \(A\) is \(\frac { 32 } { 3 } \mathrm { ma } ^ { 2 }\)
The rod is initially at rest with \(B\) vertically below \(A\) and it is free to rotate in a vertical plane about a smooth fixed horizontal axis through \(A\). A particle of mass \(m\) is moving horizontally in the plane in which the rod is free to rotate. The particle has speed \(v\), and strikes the rod at \(B\). In the subsequent motion the particle adheres to the rod and the combined rigid body \(Q\), consisting of the rod and the particle, starts to rotate.
- Find, in terms of \(v\) and \(a\), the initial angular speed of \(Q\).
At time \(t\) seconds the angle between \(Q\) and the downward vertical is \(\theta\) radians.
- Show that \(\dot { \theta } ^ { 2 } = \mathrm { k } \frac { \mathrm { g } } { \mathrm { a } } ( \cos \theta - 1 ) + \frac { 9 \mathrm { v } ^ { 2 } } { 400 \mathrm { a } ^ { 2 } }\), stating the value of the constant \(k\).
- Find, in terms of \(a\) and \(g\), the set of values of \(v ^ { 2 }\) for which \(Q\) makes complete revolutions.
When \(Q\) is horizontal, the force exerted by the axis on \(Q\) has vertically upwards component \(R\).
- Find \(R\) in terms of \(m\) and \(g\).
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A compound pendulum consists of a uniform rod \(A B\) of length 1 m and mass 3 kg , a particle of mass 1 kg attached to the rod at \(A\) and a circular disc of radius \(\frac { 1 } { 3 } \mathrm {~m}\), mass 6 kg and centre \(C\). The end \(B\) of the rod is rigidly attached to a point on the circumference of the disc in such a way that \(A B C\) is a straight line. The pendulum is initially at rest with \(B\) vertically below \(A\) and it is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through the point \(P\) on the rod where \(\mathrm { AP } = \mathrm { xm }\) and \(\mathrm { x } < \frac { 1 } { 2 }\) (see diagram). - Show that the moment of inertia of the pendulum about the axis of rotation is \(\left( 10 x ^ { 2 } - 19 x + 12 \right) \mathrm { kg } \mathrm { m } ^ { 2 }\).
The pendulum is making small oscillations about the equilibrium position, such that at time \(t\) seconds the angular displacement that the pendulum makes with the downward vertical is \(\theta\) radians.
- Find the angular acceleration of the pendulum, in terms of \(x , g\) and \(\theta\).
- Show that the motion is approximately simple harmonic, and show that the approximate period of oscillations, in seconds, is given by \(2 \pi \sqrt { \frac { 20 x ^ { 2 } - 38 x + 24 } { ( 19 - 20 x ) g } }\).
- Hence find the value of \(x\) for which the approximate period of oscillations is least.