• # question_answer If g be the acceleration due to gravity and K be the rotational kinetic energy of the earth. If the earth's radius increases by 2% keeping mass constant, then A)  Both decrease by 4%                          B)  g decrease by 4% and K decrease by 2% C)  Both decrease by 2%                          D)  g decrease by 4% and K increase by 4%

Acceleration due to gravity is$g=\frac{GM}{{{R}^{2}}}$ and if L be the angular momentum of the earth, then rotational$KE=\frac{{{L}^{2}}}{2l}$, where l be the moment of inertia. So,   $l=\frac{2}{5}M{{R}^{2}}$ (for sphere) $\therefore$ Rotational KE = $\frac{5{{L}^{2}}}{4M{{R}^{2}}}$ Since, angular momentum remains conserved. So,  rotational KE $(K)\propto \frac{1}{{{R}^{2}}}$ $\therefore$ Both g and K are$\propto {{R}^{-2}}$. $\therefore$$\frac{\Delta g}{g}=\frac{\Delta K}{K}=-2\times \frac{\Delta R}{R}$ $\therefore$Both g and K would decrease by$2\times 2%=4%$