question_answer 34)

                  Which of the following options are correct?                 (a) Acceleration due to gravity decreases with increasing altitude.                 (b) Acceleration due to gravity increases with increasing depth (assume the earth to be a sphere of uniform density).                 (c) Acceleration due to gravity increases with increasing latitude.                 (d) Acceleration due to gravity is independent of the mass of the earth.

Answer:

                  (a, c) \[{{g}_{n}}\,=\,\frac{g}{\left( 1+\frac{h}{R} \right)},\,{{g}_{depth}}=\,g\left( 1-\frac{d}{R} \right)\]                                \[{{g}_{\theta }}=\,g\,\left[ 1-\,\frac{R{{\omega }^{2}}\,{{\cos }^{2}}\theta }{g} \right]\,\Rightarrow \,{{g}_{{{90}^{o}}}}\,=g\]             and \[{{g}_{{{0}^{o}}}}=\,g\,\,\left( 1-\frac{R{{\omega }^{2}}}{g} \right)\] and \[g=\,\frac{GM}{{{R}^{2}}},\] where                 M is the mass of the earth.