AP physics 1 gravitation

Question

What would be the weight of a 65.0 kg astronaut on a planet with the same density as Earth and having twice Earth's radius? (g = 9.81)

Answer 1

Let

m = 65 kg be the mass of the astronaut,

g be gravitational acceleration on a planet; for the Earth g = 9.81 m/s².

The weight of the astronaut is then the product

mg

So we will be able to calculate his weight on any planet, where we can calculate

its gravitational acceleration g.

Consider any planet.

Let

r be the radius of the planet,

V = 4πr³/3 be the volume of the planet,

d be the density of the planet,

M = Vd be the mass of the planet,

G be the gravitational constant.

By Newton's Law of Gravitation the gravitational acceleration on the surface of the planet is

g = GM/r²

= GVd/r²

= G4πr³d/3r²

= r(4Gπd/3)

That means, for planets with the same density d, gravitational acceleration

is linearly proportional to its radius r.

The constant of proportionality is (4Gπd/3).

The planet in the question has radius twice as large as the Earth,

so gravitational acceleration will double to 2×9.81.

And the astronaut's weight will be

2×9.81×65 = 1275.5 N