Question 530: Centre of Mass and Rotational Motion

Question

A circular disc of radius ' $R$ ' and thickness $\frac{R}{8}$ has moment of inertia 'I' about an axis passing through its centre and perpendicular to its plane. It is melted and recasted into a solid sphere then moment of inertia of sphere about an axis passing through diameter is

Accepted Answer

To solve the problem of finding the moment of inertia of a solid sphere formed by recasting a circular disc, first, consider the moment of inertia of the disc.
The moment of inertia $ I $ of a circular disc of radius $ R $ and mass $ m $ about an axis through its center and perpendicular to the plane of the disc is given by:
$$ I = \frac{1}{2} m R^2 $$
The volume of the disc is:
$$ V_{	ext{disc}} = \pi R^2 \cdot \frac{R}{8} = \frac{\pi R^3}{8} $$
Assuming the density is uniform, when the disc is melted into a solid sphere, the volume of the sphere is equal to the volume of the original disc:
$$ V_{	ext{sphere}} = \frac{4}{3} \pi r^3 = \frac{\pi R^3}{8} $$
Solving for the radius $ r $ of the sphere:
$$ \frac{4}{3} \pi r^3 = \frac{\pi R^3}{8} $$
Cancelling $ \pi $ and solving for $ r^3 $:
$$ r^3 = \frac{R^3}{32} $$
Thus,
$$ r = \left(\frac{R^3}{32}ight)^{1/3} = \frac{R}{2} $$
The moment of inertia $ I_{	ext{sphere}} $ of a solid sphere of mass $ m $ and radius $ r $ about its diameter is:
$$ I_{	ext{sphere}} = \frac{2}{5} m r^2 $$
Substitute the radius $ r = \frac{R}{2} $:
$$ I_{	ext{sphere}} = \frac{2}{5} m \left(\frac{R}{2}ight)^2 = \frac{2}{5} m \frac{R^2}{4} = \frac{m R^2}{10} $$
We earlier established $ mR^2 = 2I $, where $ I $ is the moment of inertia of the disc. Plug $ mR^2 = 2I $ into the expression:
$$ I_{	ext{sphere}} = \frac{(2I)}{10} = \frac{I}{5} $$
Thus, the correct option is:
Option C: $\frac{I}{5}$

Answer

$I$

Answer

$\frac{21}{3}$

Answer

$\frac{\mathrm{I}}{5}$

Answer

$\frac{\mathrm{I}}{10}$

Centre of Mass and Rotational Motion

A circular disc of radius ' R ' and thickness \frac{R}{8} has moment of inertia 'I' about an axis passing through its centre and perpendicular to its plane. It is melted and recasted into a solid sphere then moment of inertia of sphere about an axis passing through diameter is

Choose the correct answer: