 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-2/19/b/i/solution

d V = r d r d ϕ d z

. Integrating around the axis and over the height gives cylindrical-shell volume

2 πh r d r

K = πh \int_{0}^{a} r v (r)^{2} d r = πh \int_{0}^{a} r^{3} ω (r)^{2} d r .

For unit density, this is the kinetic energy of the rotational motion.

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