Question

If both the mass and radius of earth decrease by $1\%$, the value of (I) acceleration due to gravity would increase by $1\%$ (II) acceleration due to gravity would decrease by $1\%$ (III) escape velocity from earth's surface would decrease by $1 \%$ (IV) the gravitational potential energy of a body on earth's surface will remain unchanged

• A. (I) and (III)
• B. (I) and (IV)
• C. (II) and (IV)
• D. (III) and (IV)

Answer 1

The Correct Option is C

If both the mass and radius of the earth decrease by $1 \%$, the value of acceleration due to gravity $g'=\frac{g\left(0.99 M_{e}\right)}{\left(0.99 R_{e}\right)^{2}} \approx 1.01 \frac{G M_{e}}{R_{e}^{2}}=1.01 g$ So, $g$ would increase by $(1.01-1) \times 100 \%=1 \%$ New escape velocity would be $v_{e}'=\sqrt{\frac{2 \times 0.99 M- G_e}{0.99 R_{e}}}$ $=\sqrt{\frac{2 G M_{e}}{R_{e}}}=v_{e}$ So, it will remain unchanged. Also gravitational potential energy $=\frac{G m\left(0.99 m_{e}\right)}{0.99 R_{e}}$ $=-\frac{G M_{e} m}{R_{e}}$ Hence, it also will remain same.