\(\ \)\vspace{-1cm}
\noindent
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\rput[tl](-2.5,0.14){\(R\)}
\rput[tl](0.2,0.2){\(O\)}
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\rput[tl](-2.1,-4.26){\(S\)}
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A uniform circular disc with radius \(a\), mass \(4m\) and centre \(O\) is freely
mounted on a fixed horizontal axis which is
perpendicular to its plane and passes through \(O\). A uniform heavy chain
\(PS\) of length \((4+\pi)a\), mass \((4+\pi)m\) and negligible thickness is
hung over the rim of the disc as shown in the diagram: \(Q\) and \(R\) are
the points of the chain at the same level as \(O\). The contact between the
chain and the rim of the disc is sufficiently rough to prevent slipping.
Initially, the system is at rest with \(PQ=RS =2a\). A particle of mass
\(m\) is attached to the chain at \(P\) and the system is released.
By considering the energy of the system, show that when \(P\) has descended
a distance \(x\), its speed \(v\) is given by
$$
(\pi+7)av^2 = 2g(x^2+ax).
$$
By considering the part \(PQ\) of the chain as a body of variable mass, show
that when \(S\) reaches \(R\) the tension in the chain at \(Q\) is
$$
{5\pi -2 \over \pi +7} mg.
$$