Newton's Law of Gravity
Newton's Law of Gravity states that:
FGravity = Gm1m2/r2
where G is the gravitational constant and r is the distance between two objects with masses of m1 and m2.
How is this equation used? Lets look at a sample problem. Geosynchronous orbit means that the object will stay at the same point above the Earth while in orbit. When NASA launches a rocket, they have to know exactly where this orbit is to find the speed with which to launch the rocket. If the satellite is launched too high, it will leave Earth's gravitational pull, wasting millions of dollars. If the orbit is too low, the rocket will come back to Earth, burning up while re-entering the atmosphere.
(For those young'in who do not remember what happened to SkyLab, the US's version of Mir: After SkyLab was launched, it was discovered that it had to be placed in a higher orbit. But due to budget constraints, the procedure to put it in high orbit was delayed, and it just fell back to Earth).
Okay back to the problem....How far should a certain rocket be launched to put a satellite into geosynchronous orbit around the earth?
Okay, let's say there is a satellite that weighs some weight ms and we want to find the height of its geosynchronous orbit. From Newton's law of gravity we know that:
The force of gravity equals the gravitational constant times the mass of the satellite times the mass of the earth divided by the square of the distance between the satellite and the earth
Using the equation for centrifugal force we also know that:
The centrifugal force equals the mass of the satellite times it's velocity squared divided by the distance between the satellite and the earth
What is a centrifugal force? Imagine a person swining a ball attatched to a string. What prevents the ball from dropping (via gravity) or coming closer to the person? That force is centrifugal force. But where is this force coming from? When an object moves in a circular motion, it has constant speed but its velocity is changing its direction at every moment. So contrary to intuition, its acceleration is also changing due to change in direction of the velocity. Finally change in acceleration leads to Force.
Since the satellite is not falling we have to conclude that both forces have to equal each other (imagine two people in a tug of war, if they don't pull with the same force the center of the rope would be pulled to one side. The centrifugal force and force of gravity are the same way, pulling at each other. If they are not equal the satellite wouldn't stay in the same place).
The force of gravity equals the centrifugal force
G msmEarth/r2 = msv2/r
The gravitational constant times the mass of the satellite times the mass of the earth divided by the square of the distance between the satellite and earth EQUALS the mass of the satellite times the square of the velocity divided by the distance between the satellite and earth
We also know that the velocity v has to equal the same speed the Earth rotates on its axis (since the satellite is in geosynchronous orbit, which means always it stays above the same point on the earth) so:
velocity equals 2 times pi times the distance between the satellite and the earth divided by the time of rotation
where T is 1 day = 24 hours = 1440 minutes = 86400 seconds.
Armed with these equations, and using algebra, we can figure out the altitude and we get r = 4.23*107m from the Earth's center, or 36,000 km (subtracting 6380km for the Earth's radius) above the Earth's surface is how high the satellite needs to be launched.
