Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / Section A / 1 / Solution

The surface tension has dimensions $[\gamma ]=M{T}^{-2}$, while $[\rho ]=M{L}^{-3}$ and $[g]=L{T}^{-2}$. ConsequentlyMatching the dimension of height gives $2-n=1$, so $n=1$. Dimensional analysis fixes this exponent but cannot determine a dimensionless prefactor or a contact-angle dependence.