Question

Two lead spheres have volumes in the ratio $1:n$. The ratio of their moments of inertia about their respective diameters as axis is:

• A. $1:n^{1/3}$
• B. $1:n^{2/3}$
• C. $1:n^{5/3}$
• D. $1:n^{5/2}$

Answer 1

Answer: (C)

$1:n^{5/3}$

Moment of Inertia of solid sphere $=\frac{2}{5}M{R}^2$
Volumes are in ratio $1:n$
$\Rightarrow \frac{V_1}{V_2}=\frac{1}{n}=\frac{R_1^3}{R_2^3}$
$\Rightarrow \frac{R_1}{R_2}={\left(\frac{1}{n}\right)}^{1/3}$
mass $M=SV$
$\Rightarrow \frac{M_1}{M_2}=\frac{V_1}{V_2}=\frac{1}{n}$
$\therefore ratioofmomentofInertia=\frac{M_1}{M_2}{\left(\frac{R_1}{R_2}\right)}^2$
$=\frac{1}{n}{\left(\frac{1}{n}\right)}^{2/3}$
$=\frac{1}{n^{5/3}}$