\(\,\)
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A horizontal circular disc of radius \(a\) and centre \(O\) lies on
a horizontal table and is fixed to it so that it cannot rotate. A
light inextensible string of negligible thickness is wrapped round
the disc and attached at its free end to a particle \(P\) of mass \(m\).
When the string is all in contact with the disc, \(P\) is at \(A\).
The string is unwound so that the part not in contact with the disc
is taut and parallel to \(OA\). \(P\) is then at \(B\). The particle
is projected along the table from \(B\) with speed \(V\) perpendicular
to and away from \(OA\). In the general position, the string is tangential
to the disc at \(Q\) and \(\angle AOQ=\theta.\) Show that, in the general
position, the \(x\)-coordinate of \(P\) with respect to the axes shown
in the figure is \(a\cos\theta+a\theta\sin\theta,\) and find \(y\)-coordinate
of \(P\). Hence, or otherwise, show that the acceleration of \(P\) has
components \(a\theta\dot{\theta}^{2}\) and \(a\dot{\theta}^{2}+a\theta\ddot{\theta}\)
along and perpendicular to \(PQ,\) respectively.
The friction force between \(P\) and the table is \(2\lambda mv^{2}/a,\)
where \(v\) is the speed of \(P\) and \(\lambda\) is a constant. Show
that
\[
\frac{\ddot{\theta}}{\dot{\theta}}=-\left(\frac{1}{\theta}+2\lambda\theta\right)\dot{\theta}
\]
and find \(\dot{\theta}\) in terms of \(\theta,\lambda\) and \(a\). Find
also the tension in the string when \(\theta=\pi.\)