A thin disc of mass M and radius R has mass per unit area σ r = k r 2 where r is the distance from its centre. Its moment inertia about an axis going through its centre of mass and perpendicular to its plane is:

Step-by-step solution

d m = σ ⋅ d A = σ   2 π r   d r = 2 π k   r 3   d r M = ∫ d m = 2 π k ∫ 0 R r 3 d r = 2 π k R 4 4 = π k R 4 2 d I = d m   r 2 = 2 π k r 5   d r I = 2 π k ∫ 0 R r 5 d r = 2 π k R 6 6 = 2 π k R 6 3 × 2 since, M = πkR 4 2 so, I = 2 M R 2 3