# Geosynchronous Orbits

1. Jan 21, 2010

### kppc1407

1. The problem statement, all variables and given/known data

Communications satellites are placed in a circular orbit where they stay directly over a fixed point on the equator as the earth rotates. The radius of the earth is 6.37 x 106m, and the altitude of a geosynchronous orbit is 3.58 x 107m. What are (a) the speed and (b) the magnitude of the acceleration of a satellite in a geosynchronous orbit?

2. Relevant equations

M (mass of the earth) = 5.98 x 1024
G (gravitational constant) = 6.67 x10-11
GMm = mv2
r2........ r

3. The attempt at a solution

I understand the problem and how to get the answer. I just do not understand where the units go on the above formula and where it is derived from.

2. Jan 21, 2010

### Redbelly98

Staff Emeritus
Use MKS (SI) units, so units used are N, kg, and m. You may need to look up the units that go with G; it should be in your textbook.

The formula is a result of Newton's 2nd law, F = ma.

Hope that helps.

3. Jan 21, 2010

### Andrew Mason

Just elaborating on what Redbelly has said, the correct equation is better written as:

$$\frac{GMm}{r^2} = m\omega_{e}^2r$$

where $\omega_e$ is the rotational (angular) speed of the earth. Since we know the rotational speed of the earth and m cancels out, the only unknown is r.

The left side is the force of gravity on a body of mass m at a distance r from the earth's center. The right side is mass x the (centripetal) acceleration on the body. So the equation is simply an application of F=ma, as Redbelly has stated.

AM

Last edited: Jan 21, 2010