(a) Begin by drawing a picture, showing Earth and the space station (not to scale, of course).
Earth and the space station.
The magnitude of Earth's gravitational field g at a distance r from the center of Earth is
In this case, the space station is at a distance r that is equal to the radius of Earth plus the altitude of the space station h.
The Earth's gravitational field at this height has a magnitude
Note that this is less than g at the surface of Earth which is , which is consistent with the fact that g decreases as distance from Earth increases.
(b) The weight of a 90-kg astronaut in the space station is the gravitational force by Earth on the astronaut. It is most easily found by
where w is the magnitude of the weight of the astronaut, m is the mass of the astronaut, and g is the magnitude of the gravitational field of Earth at the location of the astronaut. Thus, the astronaut's weight is
On the surface of Earth, the astronaut weighs . Thus, the astronaut's weight in the space station is of her weight on Earth. Note that the astronaut is NOT WEIGHTLESS! In fact, it's her weight that actually keeps her in orbit, Without any gravitational force, she would not orbit Earth.