Paper 4, Section II, C

Dynamics | Part IA, 2007

A small ring of mass mm is threaded on a smooth rigid wire in the shape of a parabola given by x2=4azx^{2}=4 a z, where xx measures horizontal distance and zz measures distance vertically upwards. The ring is held at height z=hz=h, then released.

(i) Show by dimensional analysis that the period of oscillations, TT, can be written in the form

T=(a/g)1/2G(h/a)T=(a / g)^{1 / 2} G(h / a)

for some function GG.

(ii) Show that GG is given by

G(β)=2211(1+βu21u2)12duG(\beta)=2 \sqrt{2} \int_{-1}^{1}\left(\frac{1+\beta u^{2}}{1-u^{2}}\right)^{\frac{1}{2}} d u

and find, to first order in h/ah / a, the period of small oscillations.

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