In a recent note, Martin Lieberherr1 appears to argue in favor of studying the notion of “centrifugal force” in elementary mechanics courses, effectively countering Wörner’s argument2 against it. I appreciate the discussion this topic has generated, and while I respect Lieberherr’s perspective, I support Wörner’s view. Over a dozen years of teaching introductory physics courses, I have observed that understanding the true essence of the laws of dynamics is one of the most challenging topics for students. It is particularly difficult for them to understand why the first law is not simply a special case of the second law, which introduces the concept of inertial reference frames and effectively distinguishes true forces from fictitious ones.
For example, understanding why the force experienced by a passenger in a turning vehicle is not a true force but simply a manifestation of the passenger’s inertia is fundamental. If we explain this in terms of fictitious forces, we risk confusing students about the elusive concept of inertia. Conveying these nuances is challenging enough, and introducing the notion of fictitious forces in an elementary mechanics course can be pedagogically undesirable. Experience shows that students best understand even the fictitious Coriolis force and the related phenomena of tropical cyclones from the perspective of an inertial reference frame.3
Regarding the specific arguments in Lieberherr’s note, the conclusions drawn from the example of riding a bicycle at a constant velocity are inaccurate and misleading. From the bystander’s perspective on the road, the motion of the tire can be treated as the superposition of pure rotation and pure translation.4 Given this, a point on the rim of the tire would have a velocity v = vtr + vt, where vtr = const is the translational velocity, and vt is the tangential velocity with constant magnitude. Consequently, the net acceleration of this point is a = dv/dt = dvtr/dt + dvt/dt. Now, since dvtr/dt = 0 and dvt/dt = 0, the acceleration is only due to the change in the direction of vt, which means the acceleration is purely centripetal. Therefore, the corresponding force points toward the axle from both the perspective of the bystander and that of the bicycle rider. Thus, contrary to the note’s claim, the centripetal force is Galilean invariant with respect to both inertial reference frames.
Furthermore, the terminology used in the note is also misleading. Stating that acceleration changes the velocity is contrary to Newton’s first and second laws. Acceleration is not a cause but the result of the action of a force, and velocity can only change due to a force, not acceleration.
Finally, the misconception that centripetal force is like electric or gravitational forces has always been prevalent. While not mentioning it might seem like eliminating the misconception, it does not remove its omnipresence from the relevant texts. That is why it is best to explicitly address it when presenting the related concepts. I use the suggestion of Bill Wedemeyer5 to great effect in addressing this misconception, which is that the “centripetal force” tells us how much inward force of any nature is required to make an object move in a circle with a constant speed.