Question

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

Answer 1

Hint:
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.