Question

The moment of inertia of the hollow sphere of mass $M$ and radius $R$ about the tangential axis is ……….

Answer 1

The parallel axis theorem is also known as the Huygens Steiner theorem that is used for finding the moment of inertia about a parallel axis. By using the theorem of parallel axis, we will find the centre of mass of the hollow sphere and add the value of centre of mass and $M{R^2}$ .
The parallel axis theorem is given by
$\Rightarrow I = {I_{COM}} + M{R^2}$
Where, $I$ is the moment of inertia about tangential axis, ${I_{COM}}$ is the moment of inertia at the centre of the hollow sphere, $M$ is the mass of the hollow sphere and $R$ is the radius of the sphere.

Complete step by step solution:
It is given that the
Mass of the hollow sphere is $M$
Radius of the hollow sphere about tangential axis is $R$
We know that the moment of inertia at the centre of the hollow sphere is given by:
$\Rightarrow {I_{COM}}$ = $\dfrac{2}{5}M{R^2}$
Now using the formula, we get
$\Rightarrow I = {I_{COM}} + M{R^2}$
Putting the value of ${I_{COM}}$ in the above formula of the parallel axis theorem, we get
$\Rightarrow I = \dfrac{2}{5}M{R^2} + M{R^2}$
By performing the basic arithmetic operation, we get
$\Rightarrow I = \dfrac{7}{3}M{R^2}$

Hence the moment of inertia of the hollow sphere about the tangential axis is given as $\dfrac{7}{3}M{R^2}$.

Note:
The parallel axis theorem is also used for the rigid body by considering its inertia at a parallel axis and the perpendicular distance from the centre of the rigid mass. Where the perpendicular axis theorem is used for calculating moment of inertia of various shapes.