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

Answer

6

Explanation

Let us select a segment in the form of a spherical shell of radius x and radial thickness dx.

Moment of inertia of this spherical segment can be written as follows:
`(dI)=(2)/(3)(dm)x^(2)`
`impliesdI=(2)/(3)(4pix^(2)dxxxrho(x))x^(2)=(8)/(3)pix^(4)rho(x)dx`
`impliesI=(8)/(3)piint_(0)^(R)x^(4)rho(x)dx`
`(I_(B))/(I_(A))=((1)/(R^(5))int_(0)^(R)x^(4).dx)/((1)/(R)int_(0)^(R)x^(5).dx)=((1)/(10).R^(5))/((1)/(6)R^(5))=(6)/(10)`
On comparing with the given result we get n-6.