question_answer

If the radii of two planets be \[{{R}_{1}}\] and \[{{R}_{2}}\] and their mean densities be \[{{\rho }_{1}}\] and \[{{\rho }_{2}}\], then prove that the ratio of accelerations due to gravity on the planets will be \[{{R}_{1}}{{\rho }_{1}}:{{R}_{2}}{{\rho }_{2}}.\]

Answer:

                On the surface of any planet, \[g=\frac{4}{3}\pi GR\rho \] \[\therefore \]  \[\frac{{{g}_{1}}}{{{g}_{2}}}=\frac{\frac{4}{3}\pi G{{R}_{1}}{{\rho }_{1}}}{\frac{4}{3}\pi G{{R}_{2}}{{\rho }_{2}}}={{R}_{1}}{{\rho }_{1}}:{{R}_{2}}{{\rho }_{2}}.\]