= Solution
The relevant parameters are $a,h,g$ and the ring mass $m$, with dimensions length, length, length/time$^2$ and mass. The wire is fixed and smooth, so there is no additional friction parameter. Because $m$ is the only parameter carrying a mass dimension, it cannot enter a dimensionless group governing the period. Using $a$ and $g$ as the length and time scales, the independent dimensionless variables are $T\sqrt{g/a}$ and $h/a$. <Dimensional analysis> therefore gives
$$
\boxed{T=\sqrt{\frac ag}\,G(h/a).}
$$
The physical mass cancellation also follows from the energy equation in part (ii), since kinetic and gravitational <potential energy> are both proportional to $m$.
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