Question

A circular disc of radius R and thickness $\dfrac{R}{6}$ ​has moment inertia I about an axis passing through its center perpendicular to its plane. It is melted and re-casted into a solid sphere. The moment of inertia of the sphere about its diameter is
A. $I$
B. $\dfrac{{2I}}{8}$​
C. $\dfrac{I}{5}$​
D. $\dfrac{I}{{10}}$​

Answer 1

In order to solve this numerical we should know the formula of volume of disc and volume of sphere. Then we can calculate the moment of inertia of the sphere. Here two cases can exist and let us calculate one by one.

Complete step by step answer:
According to the data from the question the disc is melted and it is re-casted into a solid sphere, hence it acquired a similar volume.Let $V$ be the volume of the disc, $R$ be the radius of the disc, $M$ be the mass of the disc and $I$ be the moment of inertia of of the sphere.Therefore,
${V_{disc}} = \pi {R^2}_{disc}t \\\ \Rightarrow{V_{sphere}} = \dfrac{4}{3}\pi {R^3}_{sphere}$
$\Rightarrow{V_{disc}} = {V_{sphere}}$
$\Rightarrow\pi {R^2}_{disc}t = \dfrac{4}{3}\pi {R^3}_{sphere}$
We know that, $t = \dfrac{{{R_{disc}}}}{6}$
$\pi {R^2}_{disc}\dfrac{{{R_{disc}}}}{6} = \dfrac{4}{3}\pi {R^3}_{sphere} \\\ \Rightarrow{R^3}_{disc} = 8{R^3}_{sphere} \\\ \Rightarrow{R_{sphere}} = \dfrac{{{R_{disc}}}}{2}$
By calculating the moment of inertia of disc
${I_{disc}} = \dfrac{1}{2}M{R^2}_{disc} = I \\\ \Rightarrow M{({R_{disc}})^2} = 2I$
By calculating the moment of inertia of the sphere,

{I_{sphere}} = \dfrac{2}{5}M{R^2}_{sphere} \\\ \Rightarrow{I_{sphere}} = \dfrac{2}{5}M{(\dfrac{{{R_{disc}}}}{2})^2} \\\ \Rightarrow{I_{sphere}} = \dfrac{M}{{10}}{({R_{disc}})^2} \\\ \Rightarrow{I_{sphere}} = \dfrac{{2I}}{{10}} \\\ \therefore{I_{sphere}} = \dfrac{I}{5}$$ **Hence the correct option is C.** **Note:** The moment of inertia is also known as the mass moment of inertia. The mass moment of inertia is different for different rigid bodies. If greater the moments requiring greater torque to change the body's rate of rotation. Also before melting, the volume of an object which is equal to the volume of an object after melting.