Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / Section A / A6 / Solution

For the focal length of a rotating liquid mirror, consider steady rigid rotation at angular frequency $\Omega =2\pi \nu$. The free surface is an equipotential of $gz-{\Omega }^2{r}^2/2$. HenceA paraboloidal mirror has $z-z_0=r^2/(4f)$, givingThe numerical estimate uses the conventional standard acceleration $g_n=9.80665\mathrm{m}{\mathrm{s}}^{-2}$, rather than an asserted local measurement or a recovered handbook entry.^[ref] The mercury density cancels; the ideal surface shape follows from gravity and rotation.