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 (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 (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

`(1)/(r_(min)^2)=1.11 (1)/(r_(max)^2)`
and so, the ratio between the maximum and the minimum distance is

`(r_(max))/(r_(min))= √( 1.11 )=1.05`