Question

Imagine a planet $M$ having mass same as the earth and having radius $3$ times that of the earth.
a) Calculate the ratio of acceleration due to gravity on the surface of the planet $M$ to the acceleration due to gravity on the surface of the earth.
b) If the average value of acceleration due to gravity on the surface of the earth is $9.8{ms}^{-2}$, then find the value of average acceleration due to gravity on the surface of planet $M$.

Answer 1

a) Acceleration due to gravity on the earth
$g_E=\frac{G{M}_E}{R_E^2}$
Acceleration due to gravity on the planet $M$
$g_M=\frac{G{M}_M}{R_M^2}$
$g_M=\frac{G{M}_M}{{\left(3R_E\right)}^2}$ $\because R_M=3R_E$
$\frac{g_M}{g_E}=\frac{\frac{G{M}_E}{9R_E^2}}{\frac{G{M}_E}{R_E^2}}$ $\because M_E=M_M$
$\frac{g_M}{g_E}=\frac{G{M}_E}{9R_E^2}\times \frac{R_E^2}{G{M}_E}$ $\because M_E=M_M$
$\frac{g_M}{g_E}=\frac{1}{9}$ $\therefore \frac{g_M}{g_E}=1:9$
b) $\frac{g_M}{g_E}=\frac{1}{9}$ $\therefore g_M=1.08{ms}^{-2}$
$g_M=\frac{g_E}{9}$
$g_M=\frac{9.8}{9}$