Question

$g_{e}$ and $g_{p}$ denote the acceleration due to gravity on the surface of the earth and another planet whose mass and radius are twice to that of the earth, then :

• A. $g_{p}=\frac{g_{e}}{2}$
• B. $g_{p}=g_{e}$
• C. $g_{p}=2 g_{e}$
• D. $g_{p}=\frac{g_{e}}{\sqrt{2}}$

Answer 1

Answer: (A)

Acceleration due to gravity is given by
$g=\frac{G M}{R^{2}}$
where $G$ is gravitational constant.
For earth: $g_{e}=\frac{G M_{e}}{R_{e}^{2}}$
For planet: $g_{p}=\frac{G M_{p}}{R_{p}^{2}}$
Therefore, $\frac{g_{e}}{g_{p}}=\frac{G M_{e} / R_{e}^{2}}{G M_{p} / R_{p}^{2}}$
or $\frac{g_{e}}{g_{p}}=\frac{M_{e}}{M_{p}} \times \frac{R_{p}^{2}}{R_{e}^{2}}$ ...(i)
Given, $M_{p}=2 M_{e},\, R_{p}=2 R_{e}$
Putting the values in the Eq. (i), we obtain
$\frac{g_{e}}{g_{p}} =\frac{M_{e}}{2 M_{e}} \times \frac{\left(2 R_{e}^{2}\right)}{R_{e}^{2}}$
$=\frac{1}{2} \times \frac{4}{1}$
$=2$
$\therefore g_{p} =\frac{g_{e}}{2}$