Paper 4, Section II, B

(a) State the parallel axis theorem for moments of inertia.

(b) A uniform circular disc $D$ of radius $a$ and total mass $m$ can turn frictionlessly about a fixed horizontal axis that passes through a point $A$ on its circumference and is perpendicular to its plane. Initially the disc hangs at rest (in constant gravity $g$ ) with its centre $O$ being vertically below $A$. Suppose the disc is disturbed and executes free oscillations. Show that the period of small oscillations is $2 \pi \sqrt{\frac{3 a}{2 g}}$.

(c) Suppose now that the disc is released from rest when the radius $O A$ is vertical with $O$ directly above $A$. Find the angular velocity and angular acceleration of $O$ about $A$ when the disc has turned through angle $\theta$. Let $\mathbf{R}$ denote the reaction force at $A$ on the disc. Find the acceleration of the centre of mass of the disc. Hence, or otherwise, show that the component of $\mathbf{R}$ parallel to $O A$ is $m g(7 \cos \theta-4) / 3$.

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