PYQs

1) A circular cylinder of radius \(a\) and radius of gyration \(k\) rolls without slipping inside a fixed hollow cylinder of radius \(b\). Show that the plane through axes moves in a circular pendulum of length \((b-a)(1+\dfrac{k^2}{a^2})\).

[2019, 20M]

2) The ends of a heavy rod of length \(2a\) are rigidly attached to two light rings which can respectively slide on the thin smooth fixed horizontal and vertical wires \(O\). The rod starts at an angle \(\alpha\) to the horizon with an angular velocity \(\sqrt{[3 g(1-\sin \alpha) / 2 a]}\) and moves downwards. Show that it will strike the horizontal wire at the end of time \(-2 \sqrt{a /(3 g)} \log \left[\tan \left(\dfrac{\pi}{8}-\dfrac{\alpha}{4}\right) \cot \dfrac{\pi}{8}\right]\).

[2011, 30M]

3) A fine circular tube, radius \(c\), lies on a smooth horizontal plane and contains two equal particles connected by an elastic string in the tube, the natural length of which is equal to half the circumference. The particles are in contact and fastened together, the string being stretched the tube. If the particles become disunited, prove that the velocity of the tube when the string has regained its natural length is \(\left \{ \dfrac{2 \pi \hat{\lambda} m c}{M(M+2 m)} \right \}^{1/2}\) where \(M\), \(m\) the masses of the tube and each particle respectively, and \(\lambda\) is the modulus of elasticity.

[2003, 30M]