A, B and C are a disc, a solid sphere and a spherical shell respectively, with the same radii and masses. They are placed touching each other as shown in the figure, with the plane of the disc A in the plane of the paper. The axis PQ lies in this plane, passes through the centre of A and through the point where B and C touch. The moment of inertia of the system about PQ is $\dfrac{x}{15}I$, where $I$ is the moment of inertia of the disc about its diameter. The value of $x$ is ______.
Numerical value type. Enter your answer.
Answer: 199
Let each body have mass $M$ and radius $R$.
- Disc A: PQ is a diameter of the disc: $\dfrac{MR^2}{4}$.
- Solid sphere B: centre at distance $R$ from PQ: $\tfrac25MR^2 + MR^2$.
- Shell C: centre at distance $R$ from PQ: $\tfrac23MR^2 + MR^2$.
$$I_{PQ} = MR^2\left(\frac14 + \frac25 + 1 + \frac23 + 1\right) = \frac{199}{60}MR^2 = \frac{199}{15}\cdot\frac{MR^2}{4}$$
Since $I = \dfrac{MR^2}{4}$, $x = 199$.
Solution by Sreeraj P, M.Sc Physics