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

Explanation

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=1/2I_1omega_1^2+1/2I_2omega_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=2/5mr^2` and `I_2=2/5mr^2+mR^2` (using parallel axis theorem,) (2)
In addition to
`omega_1=v/t` and `omega_2=v/R` (3)
Using (2) and (3) in (1), we get `T^'=(7)/(10)mv^2(1+(2r^2)/(7R^2))`