Dynamics | Part IA, 2003

State the parallel axis theorem and use it to calculate the moment of inertia of a uniform hemisphere of mass mm and radius aa about an axis through its centre of mass and parallel to the base.

[You may assume that the centre of mass is located at a distance 38\frac{3}{8} a from the flat face of the hemisphere, and that the moment of inertia of a full sphere about its centre is 25Ma2\frac{2}{5} M a^{2}, with M=2mM=2 m.]

The hemisphere initially rests on a rough horizontal plane with its base vertical. It is then released from rest and subsequently rolls on the plane without slipping. Let θ\theta be the angle that the base makes with the horizontal at time tt. Express the instantaneous speed of the centre of mass in terms of bb and the rate of change of θ\theta, where bb is the instantaneous distance from the centre of mass to the point of contact with the plane. Hence write down expressions for the kinetic energy and potential energy of the hemisphere and deduce that

(dθdt)2=15gcosθ(2815cosθ)a\left(\frac{d \theta}{d t}\right)^{2}=\frac{15 g \cos \theta}{(28-15 \cos \theta) a}

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