The densitis of two solid spheres A and B of the same radii R very with radial distance `r as p_A(r ) = k ((r )/(R )) and p_B(r ) = k((r )/(R )^5,` respectively, where k is a constant . The moments of inertia of the inividual spheres about axes passing throgh their centres are `I_A and I_B` respectively. if `(I_B)/(I_A) = (n)/(10),` the value of n is

Explanation

To solve the problem, we need to find the ratio of the moments of inertia of two solid spheres A and B with varying densities. The densities are given as:

- For sphere A: \( p_A(r) = k \left(\frac{r}{R}\right) \)
- For sphere B: \( p_B(r) = k \left(\frac{r}{R^5}\right) \)

### Step 1: Calculate the mass of a thin shell for both spheres

For a thin spherical shell of radius \( r \) and thickness \( dr \):
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