Rotating spheres of variable densities!
Classical Mechanics
Level
4
The densities of two solid spheres \(A\) and \(B\) of the same radius \(R\) vary with radial distance \(r\) as
\[\rho_A (r) = k \left( \dfrac rR \right) \quad \text{and} \quad \rho_B (r) = k {\left( \dfrac rR \right)}^5,\]
respectively, where \(k\) is a constant.
Let the moments of inertia of the individual spheres about the axes passing through their centers be \(I_A\) and \(I_B\), respectively.
If \(\ \dfrac{I_B}{I_A} = \dfrac{n}{10}\), then find the value of \(n\).
by
Tapas Mazumdar