Paper 4, Section II, B

Dynamics and Relativity | Part IA, 2012

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

(b) A uniform circular disc DD of radius aa and total mass mm can turn frictionlessly about a fixed horizontal axis that passes through a point AA on its circumference and is perpendicular to its plane. Initially the disc hangs at rest (in constant gravity gg ) with its centre OO being vertically below AA. Suppose the disc is disturbed and executes free oscillations. Show that the period of small oscillations is 2π3a2g2 \pi \sqrt{\frac{3 a}{2 g}}.

(c) Suppose now that the disc is released from rest when the radius OAO A is vertical with OO directly above AA. Find the angular velocity and angular acceleration of OO about AA when the disc has turned through angle θ\theta. Let R\mathbf{R} denote the reaction force at AA on the disc. Find the acceleration of the centre of mass of the disc. Hence, or otherwise, show that the component of R\mathbf{R} parallel to OAO A is mg(7cosθ4)/3m g(7 \cos \theta-4) / 3.

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