Paper 4, Section I, C

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

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

A spherical marble of uniform density and mass $M$ rolls without slipping at speed $V$ 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 $\frac{7}{10} M V^{2}$.

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