# If the ratio between the weight of an object on a planet to that on earth is 1:2 ,find the ratio between the mass of the planet to the mass of the earth knowing that the diameter of the planet is half that of the earth?

## if the ratio between the weight of an object on a planet to that on earth is 1:2 ,find the ratio between the mass of the planet to the mass of the earth knowing that the diameter of the planet is half that of the earth

Apr 11, 2018

This utilises $F = \frac{G {m}_{1} {m}_{2}}{r} ^ 2$ and $F = m a$

#### Explanation:

For the object on Earth:

${F}_{e} = \frac{G {m}_{o} {m}_{e}}{r} _ {e}^{2}$

Its weight:

${F}_{e} = {m}_{o} g$

${m}_{o} g = \frac{G {m}_{o} {m}_{e}}{r} _ {e}^{2}$

$g = \frac{G {m}_{e}}{r} _ {e}^{2}$ ---(1)

For the object on the Planet:

${F}_{p} = \frac{G {m}_{o} {m}_{p}}{{r}_{e} / 2} ^ 2$
(if the diameter of the planet is half of Earth's, the radius is half of Earth's)

Its weight:

${F}_{p} = {F}_{e} / 2$

${F}_{p} = \frac{{m}_{o} g}{2}$

$\frac{{m}_{o} g}{2} = \frac{G {m}_{o} {m}_{p}}{{r}_{e} / 2} ^ 2$

$g = \frac{8 G {m}_{p}}{r} _ {e}^{2}$ ---(2)

Now from equations (1) and (2) we can find the ratio.

$\frac{G {m}_{e}}{r} _ {e}^{2} = \frac{8 G {m}_{p}}{r} _ {e}^{2}$

${m}_{e} = 8 {m}_{p}$

Ratio of mass of earth to mass of planet is $8 : 1$