Matter & Interactions 2nd ed. Practice Problems
Aaron Titus | High Point University
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1350001     Gravitational field and the weight of an astronaut in the International Space Station.     1350001
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Question

 

The International Space Station orbits Earth at an altitude of about 350 km (from Earth's surface).

  1. What is the magnitude of Earth's gravitational field g at the altitude of the space station?
  2. What is the weight of a 90-kg astronaut who is in the space station? (Note weight in this case is defined as the gravitational force by Earth on the astronaut.)

 



Solution

 

(a) Begin by drawing a picture, showing Earth and the space station (not to scale, of course).


Figure: 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.