# Q & A: Finding centripetal force

Q:
How do you derive the equation F=mv^2/r?
- Beth (age 18)
Norfolk, England
A:
Formulas express how particular things act. Usually, we can't read back from a formula to figure out what the situation was. Here, however, you're most likely referring to a mass m moving at fixed speed v in a circle of radius r. In order to keep going in a circle, the object has to change the direction of its velocity. A changing velocity is the same thing as an acceleration.

So here's an argument. Draw a picture of a circle of radius r. Take an arrow from the circle to a point to represent the position at some time. You know that over one rotation that point just goes around the circle in time T. You can write a relation between the speed and r and T: |v|=2πr/T since the distance around the circle is 2πr.

Now use exactly the same picture but let the points on it be velocity vectors, so the radius is just |v|. You get by exactly the same reasoning that the magnitude of the acceleration, |a|, is 2π|v|/T.
Now you can just use algebra to get |a|= v2/r.

For those who like arguments simple, the point is that the relation between a and v is the same as the relation between v and r: a/v=v/r, so a=v2/r.

Since F=ma, you get F=mv2/r.

Mike W.

(published on 10/04/2012)