The density of solid sphere A varies as k(r/R), while the density of solid sphere B varies as k(r/R)⁵. Both spheres have the same radius R. If (moment of inertia of B)/(moment of inertia of A) = n/10, find n.
Physics · Rotational Motion · moment of inertia and kinetic energy in pure rotation
JEE Advanced 2015

Question

The density of solid sphere A varies as k(r/R), while the density of solid sphere B varies as k(r/R)⁵. Both spheres have the same radius R. If (moment of inertia of B)/(moment of inertia of A) = n/10, find n.

Physics · Rotational Motion · moment of inertia and kinetic energy in pure rotation

JEE Advanced 2015

G1 Physics · Accepted Answer

For a thin spherical shell at radius r, dI = (2/3)r²dm and dm = 4πr²ρ(r)dr, so I ∝ ∫ρ(r)r⁴dr. For sphere A, I ∝ (k/R)∫₀ᴿr⁵dr = kR⁵/6. For sphere B, I ∝ (k/R⁵)∫₀ᴿr⁹dr = kR⁵/10. Therefore I(B)/I(A) = (1/10)/(1/6) = 3/5 = 6/10, so n = 6.