The Ball in Liquid columns Coulumns

Classical Mechanics Level 4

A heavy cylinder of radius $R$ lies in equilibrium on a smooth surface, separating two liquids of densities $\rho_l$ on the left and $\rho_r$ on the right, with $\rho_r > \rho_l.$

If the height of the liquid on the right is $R,$ what is the height $h$ of the liquid on the left in terms of $R?$

Neglect the surface tensions of the liquids.

\frac{2\rho_r}{\rho_r + \rho_l}R

e^{\frac{\rho_r}{\rho_l} - 1}R

\frac{\sqrt{\rho_r}}{\sqrt{\rho_l}}R

\left(1 + \log\frac{\sqrt{\rho_r}}{\sqrt{\rho_l}}\right)R