A uniform sphere of mass $m$ and radius $r$ rolls without sliding over a horizontal plane,rotating about a horizontal axle OA (figure). In the process, the centre of the sphere moves with velocity $v$ along a circle of radius $R$ . Find the kinetic energy of the sphere.

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Solution

The sphere has two types of motion, one is the rotation about its own axis and the other is motion in a circle of radius R.
Hence the sought kinetic energy

$T = \frac{1}{2} I_{1} {ω_{1}}^{2} + \frac{1}{2} I_{2} {ω_{2}}^{2} (1)$

where $I_{1}$ is the moment of inertia about its own axis, and $I_{2}$ is the moment of inertia about the vertical axis, passing through O,

But, $I_{1} = \frac{2}{5} m r^{2} a n d I_{2} = \frac{2}{5} m r^{2} + m R^{2}$ (using parallel axis theorem,) (2)

In addition to

$ω_{1} = \frac{v}{r} a n d ω_{2} = \frac{v}{R}$ (3)

Using (2) and (3) in (1), we get

$T^{'} = \frac{7}{10} m v^{2} (1 + \frac{2 r^{2}}{7 R^{2}})$

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A uniform sphere of mass m and radius r rolls without sliding over a horizontal plane,rotating about a horizontal axle OA (figure). In the process, the centre of the sphere moves with velocity v along a circle of radius R. Find the kinetic energy of the sphere.

A uniform sphere of mass $m$ and radius $r$ rolls without sliding over a horizontal plane,rotating about a horizontal axle OA (figure). In the process, the centre of the sphere moves with velocity $v$ along a circle of radius $R$ . Find the kinetic energy of the sphere.