Paper 4 , Section II, B

Dynamics and Relativity | Part IA, 2010

A sphere of uniform density has mass mm and radius aa. Find its moment of inertia about an axis through its centre.

A marble of uniform density is released from rest on a plane inclined at an angle α\alpha to the horizontal. Let the time taken for the marble to travel a distance \ell down the plane be: (i) t1t_{1} if the plane is perfectly smooth; or (ii) t2t_{2} if the plane is rough and the marble rolls without slipping.

Explain, with a clear discussion of the forces acting on the marble, whether or not its energy is conserved in each of the cases (i) and (ii). Show that t1/t2=5/7t_{1} / t_{2}=\sqrt{5 / 7}.

Suppose that the original marble is replaced by a new one with the same mass and radius but with a hollow centre, so that its moment of inertia is λma2\lambda m a^{2} for some constant λ\lambda. What is the new value for t1/t2t_{1} / t_{2} ?

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