Question

The moment of inertia of hollow sphere $\left(massM\right)$ of inner radius $R$ and outer radius $2R$, having material of uniform density, about a diametric axis is

• A. $31M{R}^2/14$
• B. $43M{R}^2/90$
• C. $19M{R}^2/80$
• D. $None$

Answer 1

Answer: (D)

$None$

$\rho =\frac{M}{\left\{\frac{4}{3}\pi \right(2R{)}^3\}-\left\{\frac{4}{3}\pi R^3\right\}}$
$=\frac{3M}{28\pi R_3}$
$M_{2R}=\rho \left(\frac{4}{3}\right)\left(\pi \right)\left(2R^3\right)$
$\left(\frac{3}{28}\right)\left(\frac{4}{3}\right)\left(8\right)M=\frac{8}{7}M$
$M_R=\rho \left(\frac{4}{3}\pi R^3\right)$
$\left(\frac{3}{28}\right)\left(\frac{4}{3}\right)M=\frac{1}{7}M$
Now, $I=I_{2R}-I_R$
$=\frac{1}{2}{M}_{2R}(2R{)}^2-\frac{1}{2}{M}_R{R}^2$
$=\frac{1}{2}\left(\frac{8}{7}M\right)\left(4R^2\right)-\frac{1}{2}\left(\frac{M}{7}\right)R^2$
$=\frac{31}{14}M{R}^2$