Question

Moment of inertia of a sphere about its diameter is $\dfrac{2}{5}M{R^2}$. What is the moment of inertia about an axis perpendicular to its two diameter and passing through their point of intersection?
(A). $I = \dfrac{2}{5}M{R^2}$
(B). $I = \dfrac{3}{5}M{R^2}$
(C). $I = \dfrac{4}{5}M{R^2}$
(D). $I = \dfrac{5}{5}M{R^2}$

Answer 1

- Hint: You can start by defining what moment of inertia and a diameter is. Then move to describing where the given diameters will intersect (i.e. at the center). Then move on to explain how the axis perpendicular to the point of intersection will also be a diameter and write down the moment of inertia of an axis around a diameter to reach the solution.

Complete step-by-step answer:
Moment of inertia - It is also called mass moment of inertia or the rotational inertia of a body. Moment of inertia is the sum of the products of mass of each particle with the square of the distance of each particle from the axis of rotation. Moment of inertia is based on the concept of center of mass. Center of mass is an imaginary point in a body where all the mass of the body can be considered to be collected. Moment of inertia can also be said to be the tendency of a body to resist rotational motion.
Diameter – Diameter is the longest chord in a circle or sphere. It is any straight line that passes through the geometrical center of a sphere or circle.
In this problem we are given two such diameters, both of which will pass through the center of the sphere and they will intersect at the center.
Moment of inertia of a sphere about its diameter is $\dfrac{2}{5}M{R^2}$
An axis drawn perpendicular to the point of intersection will also pass through the center of the sphere, so this axis is also considered as a diameter.
Hence, the moment of inertia about this perpendicular axis will also be $\dfrac{2}{5}M{R^2}$.
Hence, option A is the correct choice.

Note: In the solution, we saw that any axis that passes through the sphere will have an equal moment of inertia i.e. $\dfrac{2}{5}M{R^2}$. In this case how special of a geometric shape sphere is, the sphere is no doubt the easiest geometrical shape to deal with. Spheres have the least surface area in comparison to all other shapes for the same volumes. Thus bodies tend to adapt to spherical shape, for example – planets (elliptical), atoms, etc.