A circular disc of radius b has a hole of radius a at its centre (see figure). If the mass per unit area of the disc varies as (σ₀r), then the radius of gyration of the disc about its axis passing through the centre is :

dI=(dm)r²=(σdA)r²
(σ₀r 2πrdr)r²=(σ₀2π)r⁴dr
I=∫dI=∫baσ₀2πr⁴dr=σ₀2π(b⁵-a⁵)/5
m=∫dm=∫σdA=σ₀2π∫bardr
m=σ₀2π(b²-a²)/2
Radius of gyration
k=√(I/m)=√((b⁵-a⁵)/5(b²-a²)/2)=√(2(b⁵-a⁵)/5(b²-a²))
k = √(2(b-a)(b⁴+b³a+b²a²+ba³+a⁴)/5(b-a)(b+a))
k = √(2(b⁴+b³a+b²a²+ba³+a⁴)/5(b+a)(b-a))