priscillianaw1

2022-09-07

The gravitational force of a star on an orbiting planet 1 is ${F}_{1}$. Planet 2, which is four times as massive as planet 1 and orbits at four times larger distance from the star, experiences gravitational force ${F}_{2}$.
What is the ratio ${F}_{2}/{F}_{1}$?

Jane Reese

Given that
The gravitational force of a star on an orbiting planet 1 is ${F}_{1}$
mass m, orbit R, force ${F}_{1}$
Planet 2, which is four times as massive as planet 1 and orbits at four times larger distance from the star, experiences gravitational force ${F}_{2}$
mass = 4m, orbit =?
we know by teh law of gravitation, the force is given by
$F=G\frac{Mm}{{R}^{2}}$
where
M=mass of star
m= mass of planet
G=gravitationla constant
Gravitational force on first planet
${F}_{1}=\frac{GMm}{{R}^{2}}$
similary
${F}_{2}=\frac{GM\left(4m\right)}{\left(4R{\right)}^{2}}\phantom{\rule{0ex}{0ex}}{F}_{2}=\frac{GM\left(4m\right)}{16{R}^{2}}\phantom{\rule{0ex}{0ex}}{F}_{2}=\frac{GMm}{4{R}^{2}}\phantom{\rule{0ex}{0ex}}{F}_{2}=\frac{{F}_{1}}{4}$
or
$\frac{{F}_{2}}{{F}_{1}}=\frac{1}{4}$

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