Question

An annular disc has a mass m, inner radius r and outer radius r 2 . The disc rolls on a flat surface without slipping. If the velocity of the center of mass is v, the kinetic energy of the disc is

Answer 1

Total kinetic energy = linear kinetic energy + rotational kinetic energy
= $\frac{1}{2}mv^2+\frac{1}{2}I\omega^2$

moment of inertia of annular disc , $I=\frac{1}{2}m[r^2+(2r)^2]$

= $\frac{1}{2}m[r^2+4r^2]=\frac{5}{2}mr^2$

given, velocity of centre of mass is v
in rolling motion,
$v=r\omega$
so, v = 2r$\omega$
hence, $\omega$= v/2r

now, total kinetic energy = 1/2 mv² + 1/2 × 5/2 × mr² × (v/2r)²
= 1/2 mv² + 5/16 mv²

= (13/16) mv²