Question

An astronaut on the moon measures the acceleration due to gravity as $1.67m/s^2$. He knows that the Earth is $80$ times more massive than the moon. What is the estimate of ratio of the radius of the Earth to that of the moon?

Answer 1

Acceleration due to gravity = $\frac{GM}{r^2}$

where M is the mass of the celestial body on which acceleration due

to gravity is to be measured .

Acceleration due to gravity on the Earth = $10$ $m{s}^{-2}$

Acceleration due to gravity on the Moon = $1.67$ $m{s}^{-2}$

Therefore ,

$\frac{G{M}_{moon}}{r_{moon}^2}$ = $1.67$ $m{s}^{-2}$ ...(i)

$\frac{G{M}_{earth}}{r_{earth}^2}$ =$10$ $m{s}^{-2}$ ...(ii)

Dividing equation (i) by (ii) :

$\frac{M_{moon}{r}_{earth}^2}{M_{earth}{r}_{moon}^2}$ = $\frac{1}{6}$

$\frac{r_{earth}^2}{r_{moon}^2}$ = $\frac{80}{6}$

Hence , $\frac{r_{earth}}{r_{moon}}$ = $3.65$