Question

The moment of inertia I of a solid sphere having fixed volume depends upon its volume V as:

• A. $I\propto V^{\frac{5}{3}}$
• B. $Ioc{V}^2/3$
• C. $ioc{V}^3/2$
• D. $Ioc{V}^3/3$

Answer 1

The correct option is A $I\propto V^{\frac{5}{3}}$

Given,

Volume of solid sphere, $V=\frac{4}{3}\pi R^3$

Radius of Sphere$=R$

$V\alpha R^3$

$V^{\frac{1}{3}}\alpha R$

Mass of sphere is, $M=density\times Volume=\rho V$

Moment of inertia of sphere about its axis, $I=\frac{2}{3}M{R}^2=\frac{2}{3}\left(\rho V\right)R^2$

$I\alpha V{R}^2$

$I\alpha V{\left(V^{\frac{1}{3}}\right)}^2$

$I\alpha V^{\frac{5}{3}}$

Moment of inertia, $I\alpha V^{\frac{5}{3}}$