Rotating spheres of variable densities!

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 \(n\).


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