Question

A solid cylinder of radius $\dfrac{R}{3}$ and length $\dfrac{L}{2}$ is removed along the central axis. Find ratio of initial moment of inertia and moment of inertia of removed cylinder. 

• A. $162$
• B. $158$
• C. $138$
• D. $178$

Hint:

When density is uniform, always use volume ratios to find mass ratios.

Answer 1

The Correct Option is A

Step 1: Mass of original solid cylinder.
Let density be $\rho$.
\[ M = \rho \pi R^2 L \]
Step 2: Mass of removed cylinder.
\[ m = \rho \pi \left(\dfrac{R}{3}\right)^2 \left(\dfrac{L}{2}\right) = \dfrac{\rho \pi R^2 L}{18} = \dfrac{M}{18} \]
Step 3: Moment of inertia of original cylinder about its axis.
\[ I = \dfrac{1}{2}MR^2 \]
Step 4: Moment of inertia of removed cylinder.
Radius of removed cylinder = $\dfrac{R}{3}$
\[ I' = \dfrac{1}{2}m\left(\dfrac{R}{3}\right)^2 = \dfrac{1}{2}\cdot\dfrac{M}{18}\cdot\dfrac{R^2}{9} = \dfrac{MR^2}{324} \]
Step 5: Required ratio.
\[ \dfrac{I}{I'} = \dfrac{\dfrac{1}{2}MR^2}{\dfrac{MR^2}{324}} = 162 \]