Question

9) What would be the weight of a 59.1-kg astronaut on a planet with the same density as Earth and having twice Earth's radius?

Answer 1

The weight of the astronaut is given by

`W=mg`
where m=59.1 kg is his mass and
`g=9.81~m/s^2` is the gravitational acceleration on Earth.

To solve the problem, we must find the value of g on the new planet. g is given by

`g= (GM)/(r^2)`
where G is the gravitational constant, M the mass of the planet and r its radius.
The mass of the planet can be written as

`M=dV`
where d is the density and V the volume.
We can assume that the planet is a sphere, therefore the volume is proportional to
`r^3`:

`V= (4)/(3)\pi r^3`
and we can write the mass as

`M= (4)/(3) \pi d r^3`
and then, g becomes

`g= (GM)/(r^2)= (4)/(3) (G \pi d r^3)/(r^2)= (4)/(3) G \pi d r`
So, in the end g is proportional to the radius of the planet, r (because the density of the new planet d is the same as the Earth's one. If the radius of the new planet is twice the Earth's radius, g will be twice the value of g on Earth:

`g_(new)=2g=2\cdot9.81~m/s^2=19.62~m/s^2`
And since the mass of the astronaut is always the same, the weight on the new planet will be twice the weight on Earth:

`W_(new)=mg_(new)=2mg=1159~N`