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Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 4 / 11C / a / i / Solution

The moment of inertia about an axis is

I = \int_{V} ρ r_{⊥}^{2} d V,

where

r_{⊥}

is the perpendicular distance to the axis. The disc mass is

M = ρ π r^{2} δ .

To leading order in

δ / r

, the distance from the

x

-axis is

∣ y ∣

. Using polar coordinates in the disc,

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} .

Solved by gpt-5.6-sol high.

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