This essay is intended for physics teachers and students taking a standard introductory physics course. It assumes the reader is familiar with the content of such a course, and is intended as a supplementary resource.

## Cause and effect.

Most people think "Cause and effect" is a physical law. It isn't. All it says is that some physical things are connected in time, and the earlier one is called the "cause" and the later one its "effect". This labeling doesn't explain anything or help us predict anything. Also, this does not necessarily apply to the invented (but sometimes useful) concepts we use in physics.

Newton's third law is sometimes carelessly stated in terms of "action" and "reaction". Are these cause and effect? If so, which is the cause and which the effect? The distinction is pointless, for it implies the two are separated in time. But they are always simultaneous.

We also say that when a stone is swung on a string in a circular path, the tension in the string is "caused by centrifugal force". We say that cyclones, anticyclones, tornados, and hurricanes have a vortex rotation "caused by Coriolis force". None of these statements are correct, and they are seriously misleading.

What causes the tension in the string of the stone being swung in a circle? If we analyze it in an inertial frame there is no centrifugal force that might be responsible. If we choose to analyze it in a frame rotating with the stone, all the same physical forces, gravity and tension are still present in the physical system. If we include (add) the centrifugal force to "explain" the tension in the string, the system is now apparently at rest (in our reference frame) so the net force on each part of the system all add to zero. Can we now say the centrifugal force "causes" the tension? How can it, for if we go back to the inertial frame that force disappears and the string still has the same tension. No physical processes change by choosing a different reference frame. Centrifugal force cannot be the cause of any physical process.

In fact, fictitious forces are never the cause of any physical processes. Only physical forces can cause physical processes. Fictitious forces only cause fictitious appearances resulting from our arbitrary choice of reference frame.

Students are often unclear in their minds whether the fictitious forces are the result of rotation of the system or of the rotation of the chosen reference frame. These are not necessarily the same. In introductory physics courses we do usually restrict problems to frames rotating with the system. But if we chose a different frame, say one rotating at twice the rotation rate of the system, the size of the fictitious forces would change, for they depend on the rotation of the reference frame, not the rotation of the system. Think of the messy force analysis of a toy carousel riding on a full size carousel.

Consider a stone moving in a circle, not attached to a string, but to a spring. The spring's extension is a measure of its tension. It is under stress? What is the "cause" of this physical effect—the stress that stretches the spring? We could say, "Rotation causes it". Rotation is a physical process, and can indeed cause physical effects. Increase the rotation rate and the string stretches more and its tension increases. If analyzed in a rotating reference frame, the stretch is the same. But an observer in that frame asks, "What causes the spring to stretch? Oh, it must be that mysterious centrifugal outward force." But the real causes of such physical effects are still the same in any reference frame, and there's only a mystery if you pretend that you and your reference frame are not rotating or are not aware of its rotation.

This leads to problems about the Earth's rotation. The rotation causes an equatorial bulge. The Moon's monthly revolution and its gravitational field raise tidal bulges. If analyzed in an inertial frame, no centrifugal forces are necessary to account for these effects.

## Fictitious can be useful.

I've made the case that in introductory physics courses it is best not to mention fictitious forces like centrifugal, Coriolis or Euler forces. Do fictitious forces have any legitimate uses in physics?

First let's clear the air about terminology. One can get into endless philosophical discussions of the question, "What is reality, really?" Leave that to others. For the purpose of this discussion I will classify forces in two categories: real and fictitious.

Real objects are those accessible to our human senses. Solids, liquids, gases, sun, moon and stars are examples. There are more, but these examples suffice.

Real forces are those that arise when real objects interact: contact forces between bodies touching each other, gravitational, electric and magnetic forces.

Fictitious forces are our invention to facilitate analysis of physical systems when we choose to use an accelerating or rotating reference frame for analysis. A rotating reference frame is an accelerating frame and is of special interest here. Fictitious forces do not arise from real (physical) bodies, but are a convenience trick arising from our mathematical method of analysis. Fictitious forces are never the cause of any physical phenomena.

When we choose to do rotating frame analysis of simple examples from an introductory physics books we soon realize that it doesn't really simplify the analysis. If we analyze the physical system using an inertial frame there are no centrifugal forces. Newton's laws work just fine.

When we choose to do this in a rotating reference frame, Newton's laws alone seem to be violated. Of course, for Newton's laws come with a caveat: they apply only to analyses done in an inertial coordinate frame. In a rotating reference frame, the physical system we are analyzing seems to have a missing force. So we supply one by inventing a centrifugal force.

So, if a body is at rest in our rotating reference frame, measurements made in this frame show that the net real forces on each body do not sum to zero. Adding centripetal forces "fixes" that. The required centripetal forces have size and direction, and that tells us immediately how fast our non-inertial frame is rotating. If we measure net physical forces acting on several objects at two or more different positions, their directions will point toward a common point, telling us the location of our frame's rotation axis.

Nothing is gained by this ploy in simple situations. It is only useful for analyzing cases where our measurements are all made in a rotating reference frame, we want answers measured in that frame, and it would be very inconvenient to make (or convert) them to corresponding measurements in an inertial reference frame.

Such cases are found in physics carried out on a rotating earth of physical systems that are affected by the earth's rotation. The convenient reference frame for systems of small size is one with a coordinate frame fixed to the earth, i.e., Cartesian coordinates with length and width at a right angle in a horizontal plane and height perpendicular to them. Polar coordinates or any other coordinates may be used. "Reference frame" is not synonymous with "coordinate frame". On a larger scale, latitude, longitude and altitude are convenient coordinates, but remember that they lie on a curved surface, which adds mathematical complications for problems where motion of bodies span very large distances.

This is the compelling reason for doing analysis in a non-inertial reference frame. The measurements are naturally in such a reference frame, as are the desired results.

Examples of such problems:

• A plumb line would point to the center of a non-rotating earth. But Earth's rotation causes it to deflect toward the equator. How much is this deflection?
• A body dropped would fall straight toward the center of a non-rotating earth. But Earth's rotation cause it to fall slightly eastward. How much?
• A body (like a bullet fired from a gun) is deflected to the right in the Northern hemisphere, whatever direction it is fired. How much? It is deflected to the left in the southern hemisphere. Why?
It is instructive to do these problems qualitatively in an inertial frame, but if you want a quantitative answer the easiest way is to use a rotating frame fixed to the Earth. Either way this requires a "feel" for solid geometry.

Bottom line: The reason for using fictitious forces is when the data and desired results are most easily expressed in a non-inertial (usually rotating) reference frame.

## Summary.

One reason centrifugal force is so abused in elementary physics texts is that they only apply it simple examples, cases where a non-inertial frame is chosen to rotate with the a body moving in a circular arc. In this frame the body is at rest. In such cases there's no advantage whatsoever to doing that. It just adds a force equal opposite to real the centripetal force. The mathematics is otherwise equivalent, In either case, you still must know how fast the physical system is rotating with respect to an inertial frame, ω, for F = mω2R is the size of both centripetal and centrifugal forces. Other fictitious forces, including the Coriolis force, also depend on ω.

Situations that benefit from analysis in a non-inertial reference frame are those where material bodies are moving in that frame. If all physical objects are at rest in that reference frame, there's no advantage to it. It is just an uninteresting problem in statics.

There's a practical reason. In situations where the measured data is referenced to a non-inertial reference frame and the desired results are also measurements made in that same frame. Example: Motions analyzed on the Earth's surface, with a reference frame fixed on the Earth. For small scale events (in the laboratory) a reference frame fixed to the floor is fine and Cartesian coordinates are usually chosen. For larger events, long range projectiles for example, longitude and latitude are convenient coordinates, fixed on the Earth's reference frame.

Many of the common conceptual errors converge here:

• Cause and effect language/
• Newton's third law. The two forces in this law must act on different bodies.
• Confusion between reference frames and the choice of coordinates used in that frame. These are independent choices.
• Confusion between rotation of the physical system being analyzed and the rotation of the reference frame used in the analysis. They aren't always the same. Consider a fly crawling on the roof of a toy carousel sitting and rotating on on a real amusement park carousel, rotating with respect to the rotating earth, which is in turn revolving about the sun, and so on .

Ultimately confusion boils down to careless language and careless logic.

Donald E. Simanek, April 12, 2020.