Question

Two spheres each of mass $M$ and radius $R/2$ are connected with a massless rod of length $2R$ as shown in the figure. What will be the moment of inertia of the system about an axis passing through the centre of one of the spheres and perpendicular to the rod:

• 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 $\frac{21}{5}M{R}^2$

Moment of Inertia of A about PQ= $I_A=\frac{2}{5}M{\left(\frac{R}{2}\right)}^2=\frac{1}{10}M{R}^2$

Moment of Inertia of B about its own axis= $I_B=\frac{2}{5}M{\left(\frac{R}{2}\right)}^2=\frac{1}{10}M{R}^2$
Using parallel axis theorem
Moment of Inertia of B along PQ=

$\begin{array}{c}{I}_{B_{PQ}}=M{\left(2R\right)}^2+\frac{1}{10}M{R}^2\\ =\frac{41}{10}M{R}^2\end{array}$

Moment of Inertia of system is

$\begin{array}{c}I=I_A+I_{B_{PQ}}\\ =\frac{1}{10}M{R}^2+\frac{41}{10}M{R}^2=\frac{21}{10}M{R}^2\end{array}$