# What is centrifugal force

## Main Question or Discussion Point

Definition/Summary

A non-inertial observer measures the same "real" (physical) forces as an inertial observer does, but if he wants to apply the inertial laws of motion, he must add "fictitious" (non-physical) forces.

One of these "fictitious" forces is a position-dependent centrifugal force ($m\omega^2r$), which must be used by any rotating observer.

For a body stationary relative to a uniformly rotating observer, it is the only "fictitious" force; for a relatively moving body, there is also a velocity-dependent Coriolis force; for a non-uniform rotating observer, there is also a position-dependent Euler force.

These "fictitious" forces are confusingly also called inertial forces (even though they only appear in non-inertial frames) because they are proportional to the mass (the "inertia") of the body.

(By comparison, centripetal force is a "real" force. It is not a separate force, it is another name for an existing physical force, such as tension or friction, which makes a body move in a circle.)

Equations

CENTRIFUGAL FORCE (position-dependent and radially outward) at distance $r$ from axis of rotation of an observer with instantaneous angular speed $\omega$:
$$m\omega^2r$$

CORIOLIS FORCE (velocity-dependent and "magnetic") on velocity $\mathbf{v}_{rel}$ relative to the rotating frame of an observer with instantaneous angular velocity $\mathbf{\omega}$:
$$-2m\mathbf{\omega}\times\mathbf{v}_{rel}$$

EULER FORCE (position-dependent and tangential) at distance $r$ from axis of angular acceleration of an observer with instantaneous angular acceleration $\alpha$:
$$m\alpha r$$

Extended explanation

The Principle of Equivalence (the basis of Einstein's General Theory of Relativity) says that anyone can be a valid observer, but that the inertial equations of motion may have to be adjusted to introduce imaginary (non-physical) forces.

Centrifugal force on a body is such a non-physical force.

In particular, a rotating observer invents an imaginary (non-physical) centrifugal force to explain why objects appear to move round him.

"Centrifugal" means "away from the centre" … it comes from the Latin word fugo (I flee) … as does "refugee". It is the opposite of "centripetal", which means "toward the centre" (and comes from the Latin word peto, I seek … as does "petition").

On an object moving in a circular path, there is no centrifugal force as viewed by an inertial observer.

Centrifugal force on such an object only exists for non-inertial observers.

However, both inertial and non-inertial observers recognise a centrifugal force from such an object, on whatever is keeping it in the circle.

Two different meanings:

Most standard physics textbooks use the "modern" meaning of centrifugal force as a pseudoforce, existing only as an artefact of viewing things in a non-inertial frame.

It is not a "real" (physical) force, since it has no agent.

The "old-fashioned" meaning of centrifugal force as the Newton's-3rd-law pair ("reaction force") of the centripetal force is completely "real", in any frame.

These two different types of centrifugal force act on different bodies.

Whirling on a string:

An observer holding onto a string which is whirling him in a circle feels a force along his arm toward the centre of the circle.

However, he knows that he is not moving toward the centre.

So he also feels a force in the opposite direction, balancing the force along his arm.

In that sense, he genuinely feels a centrifugal force.

In a turning car:

The driver of a car turning sharply left notices that unsecured objects slide to the right … away from the centre of the turn.

In the driver's rotating (non-inertial) frame of reference, that can only be explained by a force to the right.

It is a centrifugal force, acting on everything in the car, but nothing physical is causing it.

If the driver regards that force as real, then he may apply the inertial laws of motion.

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