Past exam of the physics 1 paper of the University of Oxford / 2013 / 9 / b

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- Solution b

Solution

The Euler-Lagrange equation from a is

2 M R^{2} \ddot{θ} + M g R sin θ = 0

. Linearizing about

θ = 0

gives

ω_{fixed} = \frac{g}{2 R}, f_{fixed} = \frac{1}{2 π} \frac{g}{2 R}, T_{fixed} = 2 π \frac{2 R}{g} .

Both angular frequency and ordinary frequency are given to distinguish the two conventions. The stable position places the extra mass below the pivot.

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