The moment of inertia about an axis is

I = \int_{V} ρ r_{⊥}^{2} d V,

where

r_{⊥}

M = ρ π r^{2} δ .

To leading order in

δ / r

, the distance from the

x

-axis is

∣ y ∣

I_{x} = ρ δ \int_{0}^{r} \int_{0}^{2 π} (s sin ϕ)^{2} s d ϕ d s = ρ δ \frac{r ^{4}}{4} π = \frac{1}{4} M r^{2} .

ii

Solution

The axis

y = 0, z = h

is parallel to the

x

-axis and lies a distance

∣ h ∣

from it. The parallel axis theorem therefore gives

I = \frac{1}{4} M r^{2} + M h^{2} = M (\frac{r ^{2}}{4} + h^{2})

to leading order in the thickness.

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