http://www.ck12.org Chapter 7. Centripetal Forces Version 2
7.4 Gravity as a Centripetal Force
When can Gravity Act as a Centripetal Force?
We saw last chapter that the force of Gravity causes an attraction between two objects of massm 1 andm 2 at a
distancerof
F~G=Gm^1 m^2
r^2.[4]
By Newton’s Third Law, both objects experience the force: equal in magnitude and opposite in direction, and both
will move as an effect of it. If one of the objects is much lighter than the other (like the earth is to the sun, or a
satellite is to earth) we can approximate the situation by saying that the heavier mass (the sun) does not move, since
its acceleration will be far smaller due to its large mass. Then, if the lighter mass remains at a relatively constant
absolute distance from the heavier one (remember, centripetal force needs to be constant in magnitude), we can say
that the lighter mass experiences aneffectivelycentripetal force.
Math of Centripetal Gravity
Gravity is not always a centripetal force. This is a really important point. It only acts as a centripetal force when
conditions approximate those listed above — very much like it isn’t constant near the surface of the earth, but very
close to it.
If gravity provides centripetal force and acceleration, we can set [2] equal to [4]. It’s important to remember that in
[2]mrefers to the lighter mass, since that is the one traveling. Then,
Gm 1 m 27r
r^2=
m 1 v^2
rSo, the relationship between velocity and radius for a circular orbit of a light object around an heavy mass (note the
mass of the lighter object cancels) is:
Gm 2 =vorb^2 rorb