What is a "fictitious force"?

California Institute of Technology theoretical physicist and 2004 Nobel laureate David Politzer helps shed some light on these mysterious influences.

The forces you feel in a moving car—those that push you back into your seat when the driver steps on the gas or throw you side to side when the car makes sharp turns—are everyday examples of fictitious forces. In general, these influences arise for no reason other than that the natural frame of reference for a given situation is itself accelerating.

The term "fictitious force" has a precise meaning within Newtonian mechanics—in fact, it's always proportional to the mass of the object on which it acts.

An elegant example of these types of apparent influences is the fictitious Coriolis force, which is responsible for the stately precession (or circular rotation) of a carefully suspended pendulum's plane of swing. If such a pendulum were suspended directly above the North Pole, it would appear to rotate 360 degrees every day. If you viewed this pendulum from a stationary point in outer space, however, it would appear to swing in a single, fixed plane while the Earth turned under it. From the outer space perspective, there is no sideways force (that is, perpendicular to the plane of swing) deflecting the sway of the pendulum. That is why the somewhat pejorative term "fictitious" is attached to this force. Likewise in the car, there simply is no real force pushing you back into your seat, your senses notwithstanding.

Nevertheless, analyzing a situation in terms of fictitious forces may, in fact, be the most effective way to understand what is actually going on. Take a stirred cup of tea, a charming example of a consequence of the Coriolis force. If a few tea leaves are present in the cup, they end up in a pile at the center of the bottom surface (and not along the edge, as one might expect, as a result of the also fictitious centrifugal force). If you imagine yourself rotating around in sync with the stirred fluid, most of the fluid would appear to be at rest while the cup would be counter-rotating around you. That rotating cup drags some adjacent fluid along with it. Meanwhile, near the bottom, the Coriolis force on that dragged fluid pushes it toward the center of the cup, carrying the leaves along with it.

With general relativity, Einstein managed to blur forever the distinction between real and fictitious forces. General relativity is his theory of gravity, and gravity is certainly the paradigmatic example of a "real" force. The cornerstone of Einstein's theory, however, is the proposition that gravity is itself a fictitious force (or, rather, that it is indistinguishable from a fictitious force). Now, some 90 years later, we have innumerable and daily confirmations that his theory appears to be correct.

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1. 1. paulnelson 05:05 AM 11/15/07

Coriolis force, which drives atmospheric circulation around regions of high and low pressure, is a fine example of a fictitious force. But it is not the explanation for the beautiful tea leaf effect. That honor goes to pressure. As fluid swirls in a container, centrifugal force causes a rise in hydrostatic pressure from the center to the outer edge. Within the boundary layer at the stationary bottom surface, however, fluid angular velocity is reduced, and centrifugal force is insufficient to counteract the pressure gradient. Driven by this pressure gradient, fluid flows inward, then upward at the center, resulting in a toroidal secondary circulation pattern. Tea leaves are carried along for the ride, but as the flow slows down, the upward velocity in the center becomes insufficient to lift the leaves, and they pile up in a neat cone at the center of the bottom surface. Thinking of the system in terms of a rotating frame of reference does not change this explanation.

2. 2. ErkDemon 02:33 AM 12/11/07

A better explanation of the "tealeaves" effect is the difference between the outward forces at the top and bottom of the cup.
The forces push the rotating body of liquid outwards against the cup walls, at both the top and bottom, but because of friction effects with the cup's base, the liquid at the bottom of the cup rotates more slowly, and is pushed outwards more weakly. So the liquid at the top of the cup "wins". Liquid moves outward at the top of the cup, moves downward alongside the walls, and forces itself inward at the bottom of the teacup, while the liquid that was originally at the bottom gets forced inwards and upwards as a rising column of liquid at the centre of the cup. The tealeaves get caught up in this vortex flow and dragged to the centre of the cup, but the upward current isn't strong enough to completely lift them, so they collect as a pile in the centre.
Einstein wrote a nice paper describing and explaining the effect.

3. 3. ErkDemon 02:40 AM 12/11/07

Yep, what paulnelson said! :)
Cheers, Eric

--
Edited by ErkDemon at 12/10/2007 6:41 PM

4. 4. physics01 12:00 AM 5/24/10

dqw

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