Question

A uniform cylinder has a radius $R$ and length $L$. If the moment of inertia of this cylinder about an axis passing through its centre and normal to its circular face is equal to the moment of inertia of the same cylinder about an axis passing through its centre and normal to its length; then

• A. $L=R$
• B. $L=\sqrt{3}R$
• C. $L=\frac{R}{\sqrt{3}}$
• D. $L=0$

Answer 1

The correct option is B $L=\sqrt{3}R$

Moment of inertia of a cylinder about an axis passing through centre and normal to circular face, $I_1=\frac{M{R}^2}{2}$
Moment of inertia of a cylinder about an axis passing through centre and normal to its length,
$I_2=M[\frac{L^2}{12}+\frac{R^2}{4}]$
But $I_1=I_2$
$\frac{M{R}^2}{2}=M[\frac{L^2}{12}+\frac{R^2}{4}]$
$\Rightarrow \frac{R^2}{2}=\frac{L^2}{12}+\frac{R^2}{4}\Rightarrow \frac{R^2}{4}=\frac{L^2}{12}$
$L=\sqrt{3}R$
Hence, the correct answer is option (b).