Write down the moment of inertia of a uniform circular disc of mass \(m\) and radius \(2 a\) about a diameter.
A uniform solid cylinder has mass \(M\), radius \(2 r\) and height \(h\).
Show by integration, and using the result from part (i), that the moment of inertia of the cylinder about a diameter of an end face is
$$M \left( r ^ { 2 } + \frac { 1 } { 3 } h ^ { 2 } \right)$$
and hence find the moment of inertia of the cylinder about a diameter through the centre of the cylinder.
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A smooth circular wire hoop, with centre \(O\) and radius \(r\), is fixed in a vertical plane. The highest point on the wire is \(H\). A small bead \(B\) of mass \(m\) is free to move along the wire. A light inextensible string of length \(a\), where \(a > 2 r\), has one end attached to the bead. The other end of the string passes over a small smooth pulley at \(H\) and carries at its end a particle \(P\) of mass \(\lambda m\), where \(\lambda\) is a positive constant. The part of the string \(H P\) is vertical and the part of the string \(B H\) makes an angle \(\theta\) radians with the downward vertical where \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\) (see diagram). You may assume that \(P\) remains above the lowest point of the wire.