Question

Calculate the moment of inertia of a wheel about its axis which is having a rim of mass 24M and twenty four spokes each of mass M and length l.

Answer 1

Hint: The moment of inertia is dependent on the mass distribution of the body about an axis. Greater the mass and greater the distance of that mass from the axis, larger is the moment of inertia about that axis.
Formula used:

Moment of inertia of a body about an axis is defined as the sum of products of masses and distance of masses from the axis.

$I = \sum\limits_i {{m_i}r_i^2} $

Where I is the moment of inertia, ${m_i}$is the ith mass of the body and ${r_i}$ is distance of ith mass from the axis of rotation.

The moment of inertia of a rod of length l about one of its ends is given as

$I = \dfrac{{M{l^2}}}{3}$

Complete step by step answer:
Moment of inertia is the resistance faced in rotating a body about an axis. It depends on the mass distribution in the body. Greater the mass and distance of that mass from the axis of rotation, larger is the moment of inertia.

We are given a wheel and we need to find a moment of inertia about its axis. Its rim has mass 24M while there are 24 spokes each having mass M and length l. The moment of inertia of the wheel is equal to the sum of moment of inertia of rim and spokes.

Moment of inertia of rim:

Mass = 24M
Distance from axis$ = l$
\[{I_R} = 24M{l^2}\]

Moment of inertia of spokes:

The spokes of the wheel are basically rods which are being rotated about one of their ends. Therefore, we can write the moment of inertia of 24 such spokes of length l as

$\therefore {I_S} = 24 \times \dfrac{{M{l^2}}}{3} = 8M{l^2}$

Now, the total moment of inertia of wheel is given as:

$\begin{gathered}
  {I_{wheel}} = {I_R} + {I_S} \\
   \Rightarrow {I_{wheel}} = 24M{l^2} + 8M{l^2} \\
   \Rightarrow {I_{wheel}} = 32M{l^2} \\
\end{gathered} $

which is the required answer to our question.

Note: Moment of inertia plays a role only when the body is rotating about an axis. That is why, the axis is always related to the moment of inertia. Moment of inertia is analogous to inertia of the body when it is at rest or in translational motion.