Question

Two identical spherical balls of mass $M$ and radius $R$ each are stuck on two ends of a rod of length $2R$ and mass $M$ (see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod is

• A. $\frac{137}{15}M{R}^2$
• B. $\frac{17}{15}M{R}^2$
• C. $\frac{209}{15}M{R}^2$
• D. $\frac{152}{15}M{R}^2$

Answer 1

The correct option is A $\frac{137}{15}M{R}^2$

For ball, using parallel axes theorem,for ball moment of inertaia,
$I_{ball}=\frac{2}{5}M{R}^2+M(2R{)}^2=\frac{22}{5}M{R}^2$

For two balls $2I_{ball}=2\times \frac{22}{5}M{R}^2$

For rod
$I_{rod}=\frac{M(2R{)}^2}{12}=\frac{M{R}^2}{3}$

Moment of inertia of the system is
$I_{system}=2I_{ball}+I_{rod}$
$\frac{44}{5}M{R}^2+\frac{M{R}^2}{3}=\frac{137}{15}M{R}^2$