A spherical uniform planet is rotating about its axis. The velocity of a point on its equator is $$v.$$ Due to the rotation of planet about its axis the acceleration due to gravity $$g$$ at equator is $$\frac{1}{2}$$ of $$g$$ at poles. The escape velocity of a particle on the pole of planet in terms of $$v$$ is

Solution :
$$\eqalign{ & v = \omega R \cr & g = {g_0} - {\omega ^2}R\left[ {\;g = {\text{at}}\,{\text{equator,}}\,{g_0} = {\text{at}}\,{\text{poles}}} \right] \cr & \frac{{{g_0}}}{2} = {g_0} - {\omega ^2}R;\,\,{\omega ^2}R = \frac{{{g_0}}}{2}; \cr & {v^2} = \frac{{{g_0}R}}{2} \cr & {v_e} = \sqrt {2\;{g_0}R} = \sqrt {4{v^2}} = 2v \cr} $$