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

Answer 1

`I = int_0^R(dm)r^2`
`:. I = int_0^2 rho xx4pir^2drxxr^2`
`:. I = 4pi int_0^2 rho r^4 dr`
`:. I_A = 4pi int_0^R k(r )/(R )xxr^4dr = (4pik)/(R ) int_0^R r^5 dr`
`=(4piK)/(R )((R^6)/(6)) = 4piK (R^5)/(6)`
`=(4piK)/(R ) ((R^6)/(6)) =4piK (R^5)/(6)`
`I_B =4pi int_0^R K ((r )/(R )^5 r^4 dr = (4piK)/(R^5)xx(R^10)/(10 = 4piK (R^5)/(10)`
`:. (I_b)/(I_A) = (6)/(10) rArrn=6`