Q. A satellite is in orbit around the earth. How far away from the center of the earth is it if we know that it makes one complete revolution in two days?

Relevant constants:

G = 6.67 * 10-11 N m2/kg2
ME = 6 * 1024 kg.

Any time an object moves in a circle there must be a centripetal force acting on the object.

Now, here's the really touchy concept: The term "centripetal force" is merely a label. It is used to describe the actual force which causes the object to move in a circle. In this example, the actual force which points to the center of the orbit is the gravitational force. Therefore, gravity is the centripetal force that we place into the above equation!

There are two tricky points here:
  • The mass in the ma part of the equation is the mass of the satellite since that is the object whose motion is caused by the centripetal force, gravity.
  • The value of R is not the radius of the Earth!
    For the gravitational force, R stands for the distance between the centers of the two masses.
    For the v2/R part, R stands for the radius of the circular orbit.
It just so happens, that the value of R is the same in both cases.

We can cancel an R and the m on both sides of the equation:


We're after R so we need to come up with another expression for v. The satellite moves in its orbit around the earth at a constant speed, so we can rewrite v as:
where T is the time it takes to make a complete orbit and 2R is the circumference of the orbital circle.

Putting this in for v:


The goal is to find R, so we then try to get R by itself.


You may have seen this equation before, but let me caution you against memorizing it. It's much easier to start the way we did and work our way through to the end. I dared my physics students one semester to memorize it and over half of them mixed up the powers on an exam.

Make sure you know how to perform cube-roots with your calculator. Not all calculators have a cube-root button, so you may have to do some fancy button pushing to obtain the desired result.


Okay, now we're ready to plug in some numbers, but first...

Always, always, always be careful with the units of the values that you use. Namely, try to ensure that everything is in m, kg, & s.


Sure, we know that the orbital period T is two days, but we'll need to convert that into seconds before we can plug it in.


So finally, we have:

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