(a) 764.4 N
The weight of the astronaut on Earth is given by:
[tex]F=mg[/tex]
where
m is the astronaut's mass
g is the acceleration due to gravity
Here we have
m = 78.0 kg
g = 9.8 m/s^2 at the Earth's surface
So the weight of the astronaut is
[tex]F=(78.0)(9.8)=764.4 N[/tex]
(b) 21.1 N
The spacecraft is located at a distance of
[tex]r=6R[/tex]
from the center of Earth.
The acceleration due to gravity at a generic distance r from the Earth's center is
[tex]g=\frac{GM}{r^2}[/tex]
where G is the gravitational constant and M is the Earth's mass.
We know that at a distance of r = R (at the Earth's surface) the value of g is 9.8 m/s^2, so we can write:
[tex]GM=9.8R^2[/tex] (1)
the acceleration due to gravity at r=6R instead will be
[tex]g'=\frac{GM}{(6R)^2}[/tex]
And substituting (1) into this formula,
[tex]g'=\frac{9.8R^2}{36R^2}=0.27 m/s^2[/tex]
So the weight of the astronaut at the spacecratf location is
[tex]F'=mg'=(78.0 kg)(0.27 m/s^2)=21.1 N[/tex]
