 Solution

ID: 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/solution

(r, ϕ, z)

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

r

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}] .

 New to topics? Read the docs here!