Fake Forces in a Rotating Frame: Derivation of the Centrifugal and Coriolis Forces

Photo: Rhett Allain. Fake forces in a rotating frame

Newton’s second law says that the net force on an object causes an acceleration. These forces are due to real interactions — like the gravitational force, friction, or your hand pushing on an object.

However, this relationship between net force and acceleration only works in an inertial reference frame. You have to measure the positions and velocity with respect to a coordinate system that is not accelerating. But what do you do in a non-inertial reference frame?

Let’s start with a very familiar situation. Suppose you are in an elevator (because you want to go to a higher floor — who knows why, but I’m sure you know why). When you push that “up” button, the elevator accelerates upward. If you were to toss a ball during this time, it would appear to have a greater acceleration than you expect. Also, you seem to “feel” heavier.

It turns out that we can make Newton’s second law work in a non-inertial reference frame by adding a fake force. This fake force will have something to do with the acceleration of the frame.

Linearly Accelerating Frame

I’m going to get to the rotational frames, but let’s start off with a simpler case. Imagine I have two frames, S and S’. At time t = 0, both frames…