Paper 4, Section II, C

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

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

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

for some function $G$.

(ii) Show that $G$ is given by

$G(\beta)=2 \sqrt{2} \int_{-1}^{1}\left(\frac{1+\beta u^{2}}{1-u^{2}}\right)^{\frac{1}{2}} d u$

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

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