Question

A planet has two moons with identical mass. Moon 1 is in a circular orbit of radius r. Moon 2 is in a circular orbit of radius 2r. The magnitude of the gravitational force exerted by the planet on Moon 2 is which of the following compared with the gravitational force exerted by the planet on Moon 1?
half as large
twice as large
one-fourth as large
four times as large
the same

Answer 1

Answer: Half as large

Explanation:

Newton's law of universal gravitation for Moon 1 is:

$F_{1}=G\frac{Mm}{r^{2}}$ (1)

And for Moon 2:

$F_{2}=G\frac{Mm}{(2r)^{2}}$ (2)

Taking into account the mass of Moon 1 is equal to the mass of Moon 2

Where:

$F_{1}$ is the gravitational force exerted by the planet on Moon 1

$F_{2}$ is the gravitational force exerted by the planet on Moon 2

$G=6.674(10)^{-11}\frac{m^{3}}{kgs^{2}}$ is the Gravitational Constant  

$M$ is the mass of the planet

$m$ is the mass of each Moon

$r$ is the distance between the planet and Moon 1

$2r$ is the distance between the planet and Moon 2

Dividing (2) by (1):

$\frac{F_{2}}{F_{1}}=\frac{G\frac{Mm}{(2r)^{2}}}{G\frac{Mm}{(r)^{2}}}$

$\frac{F_{2}}{F_{1}}=\frac{1}{2}$

Isolating $F_{2}$:

$F_{2}=\frac{1}(2) F_{1}$