# Circular Motion

## Accelerated Reference Frames

We have already discussed relative motion and moving reference frames. We found that we could reconcile the kinematic observations of the two systems by

rA = rB + v t

and

vA = vB + v

These transformation equations may be seen in the following schematic diagram:

How do we reconcile the dynamic observations of the two observers? That is, what happens to Newton's Second Law,

F = m a

as seen by observers in the two reference frames that are moving relative to each other? In particular, what happens when there is an acceleration involved? This acceleration can be due to rotation or due to a change in the linear speed (or velocity).

Consider an accelerating railroad car with a weight hanging inside it, suspended from the ceiling, as shown here.

We know that F = m a is true for an observer on the ground. That observer "sees" the forces we have sketched above. The suspended weight does not hang straight down because the tension needs to provide a horizontal component to give it an acceleration. This "observer on the ground" may be referred to as an inertial observer or an observer in an inertial reference frame because the Law of Inertia, Newton's First Law of Motion, is valid in her reference frame.

What will be seen by an observer inside this accelerating railroad car?

The hanging weight still does not hang straight down. It is still suspended by the cord at an angle as shown. But the weight is at rest with respect to this "onboard observer"! If it is at rest, how can it hang suspended like that? We believe so strongly in the Law of Inertia that we "invent" another force. To keep the Law of Inertia true, there must be some additional force pulling this weight out to one side. We call this an "inertial force" or a "fictitious force".

Such fictitious forces -- or inertial forces -- occur in accelerating reference frames. An accelerating reference frame is a noninertial reference frame. That simply means that the Law of Inertia is not true in that frame unless these fictitious forces or inertial forces are introduced.

Remember a rotating reference frame is an accelerating reference frame. Consider a rotating turntable with a mass on it, held in place by a spring, as shown below.

To an "observer on the ground" -- that is, to an inertial observer -- the stretched spring provides a centripetal force which keeps this mass moving in a circle.

However, the mass is at rest to an "onboard observer" -- an observer sitting on the rotating turntable. Why should the spring be stretched?

This "onboard observer" will see the spring stretched -- which means it exerts a force -- and conclude that there is some outward force acting on it which the stretched spring just balances. In this noninertial reference frame there is a "fictitious force" of m v2/r, acting outward. This is often called a "centrifugal force".