Explanation:
Assuming a uniform mass, let's say ρ is the mass per area density.
ρ = M / (πR²)
Let's look at this as the difference of two disks, a large one and a small one.
The moment of inertia of the large disk is:
I = 1/2 MR²
The mass of the small disk is:
m = ρ πr²
m = M / (πR²) πr²
m = M (r/R)²
Using parallel axis theorem, the moment of inertia of the small disk is:
I = 1/2 mr² + md²
I = 1/2 M (r/R)² r² + M (r/R)² d²
I = 1/2 M (r²/R)² + M (rd/R)²
The total moment of inertia is:
I = 1/2 MR² − 1/2 M (r²/R)² − M (rd/R)²
