Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 12 / b

Table of contents
- Solution b

Solution

Use cylindrical coordinates

(r, ϕ, z)

about the dumbbell axis. Their Jacobian determinant contributes the volume factor

r

, so the handle's mass is the volume integral

M_{h} = ρ \int_{- γa}^{γa} \int_{0}^{2 π} \int_{0}^{H} r d r d ϕ d z = ρ (2 γa) (2 π) \frac{H ^{2}}{2} .

Inserting the radius from part (a) yields

M_{h} = 2 π ρ γ a^{3} [β^{2} - (α - γ)^{2}] .

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