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
$g= \frac{GM}{r^2}$
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 $r^3$:
$V= \frac{4}{3}\pi r^3$
and we can write the mass as
$M= \frac{4}{3} \pi d r^3$
and then, g becomes
$g= \frac{GM}{r^2}= \frac{4}{3} \frac{G \pi d r^3}{r^2}= \frac{4}{3} G \pi d r$
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:
$W_{new}=mg_{new}=2mg=1159~N$