CAIE FP2 (Further Pure Mathematics 2) 2011 November

5
\includegraphics[max width=\textwidth, alt={}, center]{0d4a352c-4eda-45b4-9284-60c6fc680f02-2_529_493_1667_826} A uniform solid sphere with centre \(C\), radius \(2 a\) and mass \(3 M\), is pivoted about a smooth horizontal axis and hangs at rest. The point \(O\) on the axis is vertically above \(C\) and \(O C = a\). A particle \(P\) of mass \(M\) is attached to the sphere at its lowest point (see diagram). Show that the moment of inertia of the system about the axis through \(O\) is \(\frac { 84 } { 5 } M a ^ { 2 }\). The system is released from rest with \(O P\) making a small angle \(\alpha\) with the downward vertical. Find

1. the period of small oscillations,
2. the time from release until \(O P\) makes an angle \(\frac { 1 } { 2 } \alpha\) with the downward vertical for the first time.

A uniform rod \(A B\), of weight \(W\) and length \(2 a\), rests with the end \(A\) on a rough horizontal plane. A light inextensible string \(B C\) is attached to the rod at \(B\) and passes over a small smooth fixed peg \(P\), which is at a distance \(h\) vertically above \(A\). A particle is attached at \(C\) and hangs vertically. The points \(A , B\) and \(C\) are all in the same vertical plane. In equilibrium the rod is inclined at an angle \(\theta\) to the horizontal (see diagram). The coefficient of friction between the rod and the plane is \(\mu\). Show that $$\mu \geqslant \frac { 2 a \cos \theta } { h + 2 a \sin \theta }$$ Given that the particle attached at \(C\) has weight \(k W\), angle \(A B P = 90 ^ { \circ }\) and \(h = 3 a\), find

1. the value of \(k\),
2. the horizontal component of the force on \(P\), in terms of \(W\).