Question

A hollow sphere is cut into two halves and the halves are kept touching each other as shown in the diagram with axis $AA\text{'}$ tangent to both at common point of contact. If original sphere had mass $M$ and radius $R$, then find the moment of inertia of this new assembly about axis $AA\text{'}$.

• A. $\frac{5M{R}^2}{3}$
• B. $\frac{2M{R}^2}{7}$
• C. $\frac{13M{R}^2}{20}$
• D. $\frac{3M{R}^2}{20}$

Answer 1

The correct option is A $\frac{5M{R}^2}{3}$

Mass of the half sphere will become $M/2$.

Now its moment of inertia about centre will be, $\frac{2}{3}\times \frac{M}{2}{R}^2$.

Using parallel axis theorem, its moment of inertia about $A{A}^{\prime }$,

$\frac{2}{3}\times \frac{M}{2}{R}^2+\frac{M}{2}{R}^2=\frac{5}{6}M{R}^2$

Since both hemispheres are symmetrical about $A{A}^{\prime }$
$I=2\times \frac{5}{6}M{R}^2=\frac{5}{3}M{R}^2$

Hence, option $\left(a\right)$ is the correct answer.