Physics - Rotation Question with Solution | TestHub

The system consists of two solid spheres, each of mass $M$ and radius $R$. The distance between their centers is given as $d=4R$.

We need to find the moment of inertia of the system about an axis passing through the center of one of the spheres and perpendicular to the rod.

For the first sphere (where the axis passes through its center), the moment of inertia is:
$I_1=\frac{2}{5}M{R}^2$

For the second sphere, the axis of rotation is at a distance $d=4R$ from its center. Using the parallel axis theorem, its moment of inertia is:
$I_2=I_{cm}+M{d}^2$
$I_2=\frac{2}{5}M{R}^2+M(4R{)}^2$
$I_2=\frac{2}{5}M{R}^2+16M{R}^2=\frac{82}{5}M{R}^2$

The total moment of inertia of the system is the sum of the moments of inertia of the two spheres:
$I=I_1+I_2$
$I=\frac{2}{5}M{R}^2+\frac{82}{5}M{R}^2$
$I=\frac{84}{5}M{R}^2$

Answer: $\frac{84}{5}{\mathrm{MR}}^2$