Question

A solid sphere of mass M and radius R is rolling on a horizontal surface without sliding with a velocity v. The ratio of its rotational and linear kinetic energies is

• A. 2 : 5
• B. 5 : 2
• C. 7 : 10
• D. 2 : 7

Answer 1

The correct option is A 2 : 5

Linear kinetic energy fo solid sphere is given by $(K.E.{)}_L=\frac{1}{2}M{v}^2$ (1)
Rotational kinetic energy of solid sphere is given by
$(K.E.{)}_R=\frac{1}{2}I{\omega }^2=\frac{1}{2}\left(\frac{2}{5}M{r}^2\right)\frac{v^2}{r^2}$
$[\because I_{sphere}=\frac{2}{5}M{r}^{2}andv=r\omega \Rightarrow \omega =\frac{v}{r}]$
or $(K.E{)}_R=\frac{2}{5}\left[\frac{1}{2}M{v}^2\right]=\frac{2}{5}(K.E.{)}_R$ (using (1))
$\Rightarrow \frac{(K.E.{)}_R}{(K.E.{)}_L}=\frac{2}{5}=2:5$