Question

A solid sphere and a hollow sphere, both of the same size and same mass roll down an inclined plane .Then

• A. solid sphere reaches the ground first
• B. hollow sphere reaches the ground first
• C. both spheres reach the ground at the same time
• D. the time at which the spheres reach the ground cannot be specified by the data given
• E. the hollow sphere will not roll down

Answer 1

Answer: (A)

Potential energy of fall is converted into kinetic energy of rotation and translation
$=mgh=\frac{1}{2}I{\omega }^2+\frac{1}{2}m{v}^2$
where$\omega =\frac{v}{r}$
=$mgh=\frac{1}{2}I{\left(\frac{v}{r}\right)}^2+\frac{1}{2}m{v}^2$
For sphere $I=\frac{2}{5}m{r}^2$
$mgh=\frac{1}{2}\left(\frac{2}{5}m{r}^2\right).\frac{v^2}{r^2}+\frac{1}{2}m{v}^2$
$\Rightarrow v=\sqrt{\frac{10}{7}gh}$
For hollow sphere $I=\frac{2}{3}m{r}^2$
$\Rightarrow v=\sqrt{\frac{6}{5}gh}$
.'.Since, v of sphere > velocity of hollow sphere