Question

If a sphere is rolling, the ratio of its rational energy to total kinetic energy is given by:-

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

Answer 1

The correct option is D $2:7$

$\begin{array}{c}Forarollingsphere,therotationkineticenergyandthetotalkineticenergyrespectivelyaare\\ K_r=\frac{1}{2}I{\omega }^{2}and{K}_t=\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2\\ Themomentofinertiaofasphereofmassmandradiusraboutitscentreis\\ I=\frac{2}{5}m{r}^2\\ also,\omega =\frac{v}{r}\\ therefore,I{\omega }^2=\frac{2}{5}m{r}^2\times \frac{v^2}{r^2}=\frac{2}{5}m{v}^2\\ Thus,K_r=\frac{1}{5}m{v}^{2}and{K}_t=\frac{1}{2}m{v}^2+\frac{1}{5}m{v}^2=\frac{7}{10}m{v}^2\\ Hence,\frac{K_r}{K_t}=\frac{\frac{1}{5}m{v}^2}{\frac{7}{10}m{v}^2}=\frac{2}{7}=2:7\end{array}$

Hence,

option $\left(D\right)$ is correct answer.