Paper 4, Section I, C

Dynamics and Relativity | Part IA, 2015

Find the moment of inertia of a uniform sphere of mass MM and radius aa about an axis through its centre.

The kinetic energy TT of any rigid body with total mass MM, centre of mass R\mathbf{R}, moment of inertia II about an axis of rotation through R\mathbf{R}, and angular velocity ω\omega about that same axis, is given by T=12MR˙2+12Iω2T=\frac{1}{2} M \dot{\mathbf{R}}^{2}+\frac{1}{2} I \omega^{2}. What physical interpretation can be given to the two parts of this expression?

A spherical marble of uniform density and mass MM rolls without slipping at speed VV along a flat surface. Explaining any relationship that you use between its speed and angular velocity, show that the kinetic energy of the marble is 710MV2\frac{7}{10} M V^{2}.

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