Circular and Satellite Motion Legacy Problem #26 Guided Solution

A good problem solver will read the problem carefully and identify the known and the unknown quantities and then think about physics concepts and mathematical relationships in order to plot out a strategy for getting from the known to the unknown quantities. Here we read about geosynchronous or geostationary satellites which have an orbital period of 24 hours, t equal 24 hours, not a great unit, we can convert that to seconds, we'll do that later. We're also told that this geosynchronous or geostationary satellite is orbiting the Earth and we're given the mass of the Earth. One of the outcomes of Newton's Law of Universal Gravitation is that we can relate the orbital period and the mass of the planet or sun or set or whatever object is being orbited to the radius of orbit. And here we have the first question being find the orbital radius of our geosynchronous geostationary satellite. We would calculate that by using the equation that you'd find on the overview page for this set of problems. The equation goes something like this, period squared divided by radius cubed is equal to four times pi squared divided by g, big G, divided by the mass of the object being orbited. Here that's the mass of the Earth. If we rearrange that equation we could get r by itself on one side and we'd have r cubed equal big G times the mass of the Earth times the period squared divided by four divided by pi squared. Now if we substitute in the mass of the Earth given here and the period of orbit which would be 24 hours multiplied by 60 to get to minutes and by 60 again to get to seconds squared, the whole thing squared, divided by four divided by pi squared. This would give us the value of r cubed and I get 7.455 times 10 to the 22nd. If I take the cube root of that I get 4.225 times 10 to the 7th meters and I can round that to three significant digits. It becomes 4.23 times 10 to the 7th meters. I'm going to use the complete value of r in subsequent calculations so you probably want to write the whole thing down so that you can use it in later calculations. In part B we're asked to determine the orbital speed of this geostationary satellite. Now there's two approaches here. One of them, perhaps the easiest, is to think of this geostationary satellite orbiting in a circle and the speed is simply the two pi times the radius you've just calculated divided by the period. So I can take the number on my calculator or that I've stored or written down, the radius, and multiply by two and multiply by pi and then again divide by the period which is that 24 hours multiplied by 60 multiplied by 60. Now when I do that I get the speed in meters per second and I get about 3,073 meters per second when I do that. I can again round that to three significant digits and the answer is 3.07 times 10 to the third meters per second. Alternatively, you could calculate the speed by using the equation you'd find on the overview page that the speed of any orbiting satellite is the square root of big G times the mass of the object being orbited, the Earth here, divided by the period of orbit. You should get the same answer. Now in part C I'm asked to determine the acceleration of this geostationary satellite and once more there's a couple of ways to approach it. Perhaps one of the easiest is to take the speed value that's still on your calculator and to say the acceleration is the speed squared over r. That's true of any object that's orbiting in a circle. It doesn't matter whether it's a satellite or an object on Earth. Alternatively, the other way you could calculate A, the acceleration, is to know that this is the acceleration caused by gravity. This is little g as in the equation g equal, little g equal acceleration of gravity is equal to big G times the mass of the object being orbited divided by the square of the radius of orbit. So you could do it either way. You should get about 0.223 meters per second per second.