\(\,\)
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\par
A heavy symmetrical bell and clapper can both swung freely in a vertical
plane about a point \(O\) on a horizontal beam at the apex of the bell.
The mass of the bell is \(M\) and its moment of inertia about the beam
is \(Mk^{2}\). Its centre of mass, \(G\), is a distance \(h\) from \(O\).
The clapper may be regarded as a small heavy ball on a light rod of
length \(l\). Initially the bell is held with its axis vertical and
its mouth above the beam. The clapper ball rests against the side
of the bell, with the rod making an angle \(\beta\) with the axis.
The bell is then released. Show that, at the moment when the clapper
and bell separate, the clapper rod makes an angle \(\alpha\) with the
upwards vertical, where
\[
\cot\alpha=\cot\beta-\frac{k^{2}}{hl}\mathrm{cosec}\beta.
\]