Question

A circular disc of radius $R$ and thickness $\frac{R}{6}$ has moment of inertia $I$ about an axis passing through its centre and perpendicular to its plane. The disc is melted and recasted into a sphere. The moment of inertia of the sphere about a tangent as axis of rotation is :

• A. $\frac{10I}{7}$
• B. $\frac{2I}{3}$
• C. $\frac{7I}{10}$
• D. $\frac{5I}{10}$

Answer 1

The correct option is C $\frac{7I}{10}$

Let the density of material be $\rho$

So $m=\pi r^2\rho r/6$

So $I=\frac{m{r}^2}{2}$

Mass will remain same , so $m=\frac{4\pi r_s^3\rho }{3}=\pi r^2\rho r/6$

So $r_s=r/2$

MI of sphere $I_{cm}=\frac{2m{r}_s^2}{5}$

The moment of inertia of the sphere about a tangent as axis of rotation is

$I_t=\frac{2m{r}_s^2}{5}+m{r}_s^2=\frac{7m{r}_s^2}{5}=\frac{7m{r}^2}{20}=\frac{7I}{10}$