Physics - Rotation Question with Solution | TestHub

The distance between the centres of the two spheres $A$ and $B$ is $d=R+r$.

Let $I_{A,\text{cm}}$ and $I_{B,\text{cm}}$ be the moments of inertia of spheres $A$ and $B$ about their respective vertical centroidal axes. For solid spheres, $I_{A,\text{cm}}=\frac{2}{5}M{R}^2$ and $I_{B,\text{cm}}=\frac{2}{5}m{r}^2$.

The moment of inertia of the system about a vertical axis passing through the centre of $A$ is given by the sum of the moment of inertia of $A$ about its own centre and the moment of inertia of $B$ about the centre of $A$. Using the parallel axis theorem:
$I_A=I_{A,\text{cm}}+\left[I_{B,\text{cm}}+m(R+r{)}^2\right]$

Similarly, the moment of inertia of the system about a vertical axis passing through the centre of $B$ is:
$I_B=\left[I_{A,\text{cm}}+M(R+r{)}^2\right]+I_{B,\text{cm}}$

Now, finding the difference $I_A-I_B$:
$I_A-I_B=\left(I_{A,\text{cm}}+I_{B,\text{cm}}+m(R+r{)}^2\right)-\left(I_{A,\text{cm}}+M(R+r{)}^2+I_{B,\text{cm}}\right)$

$I_A-I_B=m(R+r{)}^2-M(R+r{)}^2$

$I_A-I_B=(m-M)(R+r{)}^2$

Answer: $(m-M)(R+r{)}^2$