Question

Which of the following has the highest moment of inertia when each of them has the same mass and the same radius.
A. A ring about any of its diameter.
B. A disc about any of its diameter
C. A hollow sphere about any of its diameter
D. A solid sphere about any of its diameter.

Answer 1

Hint:The moment of inertia is the product of the mass and the square of the distance from the axis of rotation. It depends on the distribution of the mass about the axis of rotation. If a body is symmetric then the moment of inertia about an axis lying on the plane and intersecting each other at the centre of mass will be the same.

Complete step by step solution:
Let the mass of each is \[M\]and the radius is \[R\].
The moment of inertia of the ring about its diameter is,
\[{I_{ring}} = \dfrac{{M{R^2}}}{2} = 0.5M{R^2} \\ \]
The moment of inertia of the disc about its diameter is,
\[{I_{disc}} = \dfrac{{M{R^2}}}{4} = 0.25M{R^2} \\ \]
The moment of inertia of the hollow sphere about its diameter is,
\[{I_{h.sphere}} = \dfrac{{2M{R^2}}}{3} \approx 0.67M{R^2} \\ \]
The moment of inertia of the solid sphere about its diameter is,
\[{I_{s.sphere}} = \dfrac{{2M{R^2}}}{5} = 0.4M{R^2} \\ \]
On comparing the numerical coefficients, we find
\[0.67 > 0.50 > 0.40 > 0.25 \\ \]
Hence,
\[{I_{h.sphere}} > {I_{ring}} > {I_{s.sphere}} > {I_{disc}} \\ \]
So, the moment of inertia of the hollow sphere about its diameter is highest out of the given shapes in the option.

Therefore, the correct option is C.

Note: We should be careful about the distribution of the mass of the body while calculating moment of inertia. When the body is made of discrete mass distribution then the moment of inertia of the body will be calculated using arithmetic sum, otherwise we need to use integration of mass over the geometry of the body.