Question

I looked at the example given with the moon here on cramster and I got 8.5 which is not correct. Can anyone explain how to do this? Thanks

The free-fall acceleration on the surface of Saturn is about equal to that on the surface of the Earth. The radius of Saturn is about 9.25 RE (RE = Earth's radius = 6.4 106 m). Find the ratio of their average densities, ?Saturn/?Earth.

Answer 1

Answer: 0.11 = 11%

Explanation:

We are told that free-fall acceleration (which we call g on the earth’s surface) is about equal both for Saturn and Earth.

This means that we can apply the Universal Law of Gravitation acting upon the same mass on both planets, and solve for the acceleration, as follows:

Fge = m* a = m*ge = G m* me / re²

Fgs = m* a = m*gs = G m* ms / (9.25re)²

Simplifying the common mass m, and replacing the mass of the planets by the product of the density times the volume (assuming that both are perfect spheres), we get the followin equality:

ge = gs → G * δe * 4/3*π* re = G * δs * 4/3*π*(9.25* re)

Simplifying common factors at both sides, we finally have the following equation:

δe = δs* 9.25 → δs/ δe= 1/9.25 = 0.11 = 11%

In order to have approximately the same value of g being much more massive, the only choice is that Saturn must be less dense than Earth (i.e. it is basically a gaseous sphere).