Circular and Satellite Motion Legacy Problem #21 Guided Solution

The International Space Station is a satellite which is orbiting the Earth. It orbits in what is nearly a circular orbit. On that space station is an 85-kilogram astronaut. He or she, too, is orbiting the Earth, and orbiting in a circular motion. Like all objects moving in a circle, there must be an inward or centripetal force. In this case, that inward force is supplied by the force of gravity between the Earth and the astronaut. This force of gravitational attraction is directly proportional to the product of the masses and inversely proportional to the separation distance between the two object centers. In this problem, we wish to calculate the acceleration that is caused by this force of gravitational attraction. To calculate the acceleration, you need to take the net force and divide it by the mass of the astronaut. Well, the net force is the gravitational force, and we can express that as big G multiplied by the mass of Earth times the mass of the satellite divided by the separation distance squared. Now, if you divide that force of gravitational attraction by the mass of the astronaut, you will find that the mass of the astronaut is in the numerator and the denominator, and such cancels out. Thus, we have an equation for the acceleration of the astronaut. It is simply this big G value multiplied by the mass of the Earth divided by the square of the separation distance. Now, the mass of the Earth is explicitly stated here. And the big G value in the numerator is 6.673 times 10 to the negative 11 Newtons times meters squared per kilogram squared. The denominator could cause some problems. The denominator is simply the separation distance between the center of the Earth and the center of the astronaut. Now, the center of the astronaut is not a whole lot different. It is pretty much at the same location as the edge of the astronaut, so we don't have to worry about the radius of the astronaut. But what we do have to worry about is the radius of the Earth. I typically draw myself a picture of the Earth. The Earth has a radius of 6.37 times 10 to the 6 meters. So from the center of the Earth to the surface of the Earth, that's the distance. Now, the astronaut is not orbiting at the surface of the Earth, thankfully for the astronaut, but is instead 350 kilometers further from the center of the Earth than the surface is. So I draw a picture of an astronaut a little bit above the surface of the Earth. That's where the astronaut orbits, and the radius of orbit is simply the radius of the Earth plus the height of the astronaut above the Earth. Now, the height of the astronaut above the Earth here is 350 kilometers. Put in meters, that's 3.50 times 10 to the 5th meters. So I need to add this 3.50 times 10 to the 5th meters onto the radius of the Earth in order to get the radius of orbit. When I do, I get a value of 6.72 times 10 to the 6 meters. The astronaut will go into the denominator of my equation and get squared, and when I do that, I get an acceleration of the astronaut of 8.8366 meters per second per second. I can round this to three significant figures such that it becomes 8.84 meters per second per second.