Past exam of the mathematics course of the University of Cambridge / 2007 / ia / Paper 4 / 9C / i / 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 givesThe physical mass cancellation also follows from the energy equation in part (ii), since kinetic and gravitational potential energy are both proportional to $m$.