Question

The moment of inertia of a solid cylinder about its own axis is the same as its moment of inertia about an axis passing through its centre of gravity and perpendicular to its length. The relation between its length $L$ and radius $R$ is:
A) $L = \sqrt 2 R$
B) $L = \sqrt 3 R$
C) $L = 3R$
D) $L = R$

Answer 1

In this solution, we will use the formula of the moment of inertia of a solid cylinder about its axis. Then we will use the formula of the moment of inertia of a solid cylinder about an axis passing through the centre of mass and perpendicular to its length. Finally, we will compare these two expressions.

Formula used: In this solution, we will use the following formula:
-Moment of inertia of cylinder about its axis $I = \dfrac{{M{R^2}}}{2}$ where $M$ is the mass of the cylinder and $R$ is the radius
-Moment of inertia of cylinder on an axis perpendicular to its length and passing through the centre of mass: $I = M\left( {\dfrac{{{L^2}}}{{12}} + \dfrac{{{R^2}}}{4}} \right)$

Complete answer:
We’ve been told that the moment of inertia of a solid cylinder about its own axis is the same as its moment of inertia about an axis passing through its centre of gravity and perpendicular to its length.
We know that the moment of inertia of the cylinder about its axis is given by: ${I_1} = \dfrac{{M{R^2}}}{2}$
And the moment of inertia of the cylinder an axis passing through the centre of mass and perpendicular to its length: ${I_2} = M\left( {\dfrac{{{L^2}}}{{12}} + \dfrac{{{R^2}}}{4}} \right)$
Since the moment of inertia is the same about both these two axes, we can write
${I_1} = {I_2}$
$\dfrac{{M{R^2}}}{2} = M\left( {\dfrac{{{L^2}}}{{12}} + \dfrac{{{R^2}}}{4}} \right)$
So, dividing both sides by the mass, we can write
${R^2} = \left( {\dfrac{{{L^2}}}{6} + \dfrac{{{R^2}}}{2}} \right)$
Taking the term containing $R$ on both times, we get
$\dfrac{{{R^2}}}{2} = \dfrac{{{L^2}}}{6}$
Which gives us
$L = \sqrt 3 R$ which corresponds to option (B).

Note:
We should know the formula of the moment of inertia of an axis that is perpendicular to the length of the cylinder to answer this question. To not get confused between the two, we can remember that the moment of inertia of the cylinder through its axis does not depend on length while for an axis passing through the length of the cylinder, it will depend on length.