Question

The ratio of kinetic energies of two spheres rolling with equal center of mass velocities is $2:1$. If their radii are in the ratio $2:1$, then the ratio of their masses will be :

• A. $2:1$
• B. $1:8$
• C. $1:7$
• D. $2\sqrt{2}:1$

Answer 1

The correct option is A $2:1$

$K=K_{tr}+K_{rot}=\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2=\frac{1}{2}m{v}^2+\frac{1}{2}\times \left(\frac{2}{5}m{r}^2\right)\times (\frac{v}{r}{)}^2=\frac{7}{10}m{v}^2$
$\frac{K_1}{K_2}=\frac{m_1}{m_2}\times (\frac{v_1}{v_2}{)}^2$
Given, $v_1=v_2$
$\Rightarrow \frac{K_1}{K_2}=\frac{m_1}{m_2}$
$\Rightarrow \frac{2}{1}=\frac{m_1}{m_2}$
Thus if the ratio of kinetic energy is 2:1 then their ratio of masses will also be 2:1.