Question

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
A. `6`
B. `10`
C. `16`
D. `7`

Answer 1

Correct Answer - A
Let density of sphere of radius x be `rho (x)`
Here `deltaV = 4pi x^(2). deltax`

`:. Deltam = 4pix^(2). Deltax. rho (x)`
Moment of Inertia, `deltaI = (2)/(3) (deltam)x^(2)`
`= (2)/(3) xx 4pix^(2) deltax rho (x).x^(2)`
or `I = int deltaI = underset(0)overset(R )int (2)/(3) 4pi x^(4) deltax. rho (x)`
or we can say that `I prop underset(0)overset(R )int x^(4) deltax. rho (x)`
Using `rho_(A) (r) = K ((r)/(R))` and
`rho_(B) (r) = K(r//R)^(5)` and solving,
we get, `I_(B)//I_(A) = (6)/(10) :. n = 6`