Question

A circular disc of radius R and thickness R/6 has moment of inertia ‘I’ about an axis passing through its centre and perpendicular to its plane. Disc is melted and recast into a solid sphere. The moment of inertia of a sphere about its diameter as axis of rotation is:
A. I/5
B. 2I/5
C. 4I/5
D. I/10

Answer 1

The radius and thickness of the disc is given. Now find the volume of the disc from there. Now we know that the volume of the disc and sphere is the same as the disc is melted and re-casted into the sphere, then find the moment of inertia for both the bodies.

Formula used: $\pi {{R}^{2}}t=\pi {{R}^{2}}\dfrac{R}{6}=\pi \dfrac{{{R}^{3}}}{6}$
$V=\dfrac{4}{3}\pi {{r}^{3}}$
$I=\dfrac{M{{R}^{2}}}{2}$

Complete step by step answer:
Radius of the disc =R (given)
Thickness of the disc = R/6​(given)

This is the diagram of the disc with radius ‘R’
Therefore the volume of the disc will be,
${{V}_{1}}$​=$\pi {{R}^{2}}t=\pi {{R}^{2}}\dfrac{R}{6}=\pi \dfrac{{{R}^{3}}}{6}$ ​
Now, let the radius of the sphere be ${R}'$
Therefore the ,volume of the sphere will be
${{V}_{2}}$​=$\dfrac{4}{3}\pi {{{R}'}^{3}}$
As the disc is melted and re-casted hence volume will remain the same:
$\pi \dfrac{{{R}^{3}}}{6}=\dfrac{4}{3}\pi {{{R}'}^{3}}$
$\Rightarrow {{R}^{3}}=8{{({R}')}^{3}}$
$\Rightarrow R=2{R}'$

In the above diagram, it is a sphere with the axis of rotation.
Now, let I$_{1}$ ​ be the moment of inertia for disc =$\dfrac{M{{R}^{2}}}{2}$
And, let I$_{2}$ be moment of inertia for the sphere = $\dfrac{2}{5}M{{({R}')}^{2}}=\dfrac{2}{5}M{{(\dfrac{R}{2})}^{2}}$
${{I}_{2}}={{I}_{1}}(\dfrac{1}{5})$

So, the correct answer is “Option A”.

Additional Information: Inertia is a property of matter by virtue of which it resists any sort of change to its original state that is either being at rest or in uniform motion unless acted upon by some external force.
When this external force is rather a moment of force, i.e., in reference to rotational motion this quantity expressing the body's tendency to resist change in angular acceleration, which is the sum of the products of the mass of each particle in the body with the square of its distance from the axis of rotation, is known as moment of inertia.

Note: The metal disc was melted and re-casted hence there will be no change in mass. As the volume of the disc and sphere is the same hence with the volume of the disc, we can figure out the volume of the sphere.