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A uniform rod \(A B\), of mass \(m\) and length \(6 a\), is rigidly attached at \(B\) to a point on the circumference of a uniform circular lamina of mass \(m\), radius \(2 a\) and centre \(O\). The lamina and the rod are in the same vertical plane, and \(A B O\) is a straight line (see diagram). Show that the moment of inertia of the system about an axis \(l\) through \(A\) perpendicular to the plane of the lamina is \(78 m a ^ { 2 }\).
A particle of mass \(2 m\) is now attached at \(B\) and the system is free to rotate in a vertical plane about the fixed axis \(l\) which is horizontal. Initially \(A B\) is horizontal, with \(O\) moving downwards and the system having angular velocity \(\frac { 3 } { 5 } \sqrt { } \left( \frac { g } { a } \right)\). At time \(t , A B\) makes an angle \(\theta\) with the downward vertical through \(A\).
- Find, in terms of \(a , g\) and \(\theta\), an expression for \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } }\).
- Find the angular velocity of the system when \(B\) is vertically below \(A\).