Question

A cylinder of mass M has length L that is $\sqrt{3}$ times its radius. What is the ratio of its moment of ineria about its own axis and that about an axis passing through its centre and perpendicular to its axis ?

• A. $1$
• B. $\frac{1}{\sqrt{3}}$
• C. $\sqrt{3}$
• D. $\frac{\sqrt{3}}{2}$

Answer 1

The correct option is A $1$

Moment of inertia of cylinder about its own axis is same as of disc which is $\frac{M{R}^2}{2}$.

Moment of inertia of a disc of mass dm which constitute cylinder $=\frac{dm{R}^2}{4}$ (as both the axis passing through disc is symmetric and thus equal and their sum is equal to $\frac{dm{R}^2}{2}$)
By parallel axis theorem, moment of inertia about its transverse axis and passing through centre is $\frac{dm{R}^2}{4}+dm{\left(z\right)}^2$
and by integrating it over $-\frac{L}{2}to\frac{L}{2}$ we get, moment of ineria about its transverse axis and passing through centre $=M\left(\frac{R^2}{4}+\frac{L^2}{12}\right)=M\left(\frac{R^2}{4}+\frac{{\left(\sqrt{3}R\right)}^2}{12}\right)=\frac{M{R}^2}{2}$
So their ratio is 1.