Question

Two spheres each of mass $M$ and radius $\frac{R}{2}$ are connected with a massless rod of length $R$ as shown in the figure. The moment of inertia of the system about an axis $\left(Y{Y}^{\prime }\right)$ passing through the centre of one of the spheres and perpendicular to the rod is

• A. $\frac{21}{5}M{R}^2$
• B. $\frac{2}{5}M{R}^2$
• C. $\frac{5}{2}M{R}^2$
• D. $\frac{5}{21}M{R}^2$

Answer 1

The correct option is A $\frac{21}{5}M{R}^2$

According to parallel axis theorem, $I=I_o+M{d}^2$
Momemt of inertia of farthest sphere is $I_1=\frac{2}{5}M{\left(\frac{R}{2}\right)}^2+M(2R{)}^2$
Momemt of inertia of nearest sphere is $I_2=\frac{2}{5}M{\left(\frac{R}{2}\right)}^2$
Momemt of inertia of massless rod is $I_3=0$
Moment of inertia of system is given by, $I=I_1+I_2+I_3$
$I=\frac{2}{5}M{\left(\frac{R}{2}\right)}^2+M(2R{)}^2+\frac{2}{5}M{\left(\frac{R}{2}\right)}^2=4M{R}^2+\frac{1}{5}M{R}^2=\frac{21}{5}M{R}^2$.