 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2014-resit/8/a/solution

Measure each rod angle from the downward vertical, as in the held diagram. The upper mass rises by

2 l (1 - cos θ_{1})

; the lower mass rises by

2 l (1 - cos θ_{1}) + l (1 - cos θ_{2})

. With the hanging equilibrium as the zero of gravitational potential energy,

U = m g l [4 (1 - cos θ_{1}) + (1 - cos θ_{2})] ≃ \frac{m g l}{2} (4 θ_{1}^{2} + θ_{2}^{2}) .

The retained expression is quadratic in the small displacements. Both masses contribute to the upper rod's gravitational term.

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