Question

A solid sphere rolls without slipping on the roof. The ratio of its rotational kinetic energy and its total kinetic energy is

• A. 2/5
• B. 4/5
• C. 2/7
• D. 3/7

Answer 1

Answer: (C)

Kinetic energy of sphere
$K_{r_0}=\frac{1}{2}I{\omega }^2$
$\therefore$ Moment of inertia of sphere,
$I=\frac{2}{5}M{R}^2$
$\therefore$ Rotational kinetic energy of sphere
$K_{r_0}=\frac{1}{2}M{R}^2{\omega }^2$
Total energy of sphere
$K_{r_0}=\frac{1}{2}I{\omega }^2+\frac{1}{2}M{v}^2$
$=\frac{1}{2}\times \frac{2}{5}M{R}^2{\omega }^2+\frac{1}{2}M{R}^2{\omega }^2$
$=M{R}^2{\omega }^2\left(\frac{1}{5}+\frac{1}{2}\right)$
$=\frac{7}{10}M{R}^2{\omega }^2$
Total energy of sphere $K_{t_0}=\frac{7}{10}M{R}^2{\omega }^2$
$\frac{K_{r_0}}{K_{t_0}}=\frac{\frac{1}{5}M{R}^2{\omega }^2}{\frac{7}{10}M{R}^2{\omega }^2}=\frac{2}{7}$