Problems

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A smooth hemispherical bowl of mass \(2m\) is rigidly mounted on a light carriage which slides freely on a horizontal table as shown in the diagram. The rim of the bowl is horizontal and has centre \(O\). A particle \(P\) of mass \(m\) is free to slide on the inner surface of the bowl. Initially, \(P\) is in contact with the rim of the bowl and the system is at rest. The system is released and when \(OP\) makes an angle \(\theta\) with the horizontal the velocity of the bowl is \(v\)? Show that \[3v=a\dot{\theta}\sin\theta \] and that \[ v^{2}=\frac{2ga\sin^{3}\theta}{3(3-\sin^{2}\theta)}, \] where \(a\) is the interior radius of the bowl. Find, in terms of \(m,g\) and \(\theta,\) the reaction between the bowl and the particle.