Question

As a moon follows its orbit around a planet, the maximum grav- itational force exerted on the moon by the planet exceeds the minimum gravitational force by 11%. find the ratio rmax/rmin, where rmax is the moon's maximum distance from the center of the planet and rmin is the minimum distance.

Answer 1

The gravitational force exerted on the moon by the planet when the moon is at maximum distance $r_{max}$ is
$F_{min}=G \frac{Mm}{r_{max}^2}$
where G is the gravitational constant, M and m are the planet and moon masses, respectively. This is the minimum force, because the planet and the moon are at maximum distance.

Similary, the gravitational force at minimum distance is
$F_{max}=G \frac{Mm}{r_{min}^2}$
And this is the maximum force, since the distance between planet and moon is minimum.

The problem says that $F_{max}$ exceeds $F_{min}$ by 11%. We can rewrite this as
$F_{max}=(1+0.11)F_{min}=1.11 F_{min}$

Substituing the formulas of Fmin and Fmax, this equation translates into
$\frac{1}{r_{min}^2}=1.11 \frac{1}{r_{max}^2}$
and so, the ratio between the maximum and the minimum distance is
$\frac{r_{max}}{r_{min}}= \sqrt{ 1.11 }=1.05$