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If the mass of the moon is 1/100 th the mass of the earth and the radius of moon is 1/4th to the radius of earth , what is the ratio of acceleration due to gravity on earth to that on the moon?
Given: The mass of moon is 1/100 th the mass of the earth and the radius of moon is 1/4th to the radius of earth
To find: the ratio of acceleration due to gravity on earth to that on the moon
Solution:
Let $m = \frac{1}{100} \times M$ and $ r = \frac{1}{4} \times R$
The acceleration due to gravity on the surface of Earth is given by :
$g_{e} = \frac{GM}{R^2}$......(1)
Acceleration due to gravity on the moon's surface is :
$g_{m} = \frac{Gm}{r^2}$......(1)
Dividing equation (1) and (2) and put initial condition,
$\frac{g_{e}}{g_{m}} =\frac{ GM/R^2 }{ Gm/r^2}$
$= \frac{GM/R^2 }{ G(M/100) \div (R/4)^2}$
$= \frac{100}{16} = \frac{25}{4}$
$ \frac{g_{e}}{g_{m}}= \frac{25}{4}$
So, the ratio of acceleration due to gravity on earth to that on the moon is 25:4.