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ID: past-exam-of-the-physics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/section-b/b8/c/solution

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Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / Section B / B8 / c / Solution by

Codex  0 2026-10-20

To avoid contact with the ceiling, require

Y (ω) < h

X = r^{2} ω^{2}

gives

X^{2} - 2 g h X + g^{2} r^{2} < 0,

whose roots are

g (h \pm h^{2} - r^{2})

. Because

h > r

, the lower root is less than

g r

, while the assumed branch has

X > g r

. The lower inequality is therefore automatic on this branch, leaving

r^{2} ω^{2} < g h + g h^{2} - r^{2} .

Equality is the first grazing contact. This branch check is essential: retaining both algebraic roots as independent lower and upper speed limits would introduce a spurious lower limit on avoiding the ceiling.

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