Beads Defying Gravity

A science presenter and comic, Steven Mould, not long ago made a video about a chain of beads that seem to defy gravity and levitate from a beaker. Actually this is a normal phenomenon but modeling its dynamics is not trivial. Most explanations do  not take into account the mass of the beads because they are purely kinematic in nature but, obviously, a complete model of such phenomenon should do and whether the observed motion takes place clearly depends on the mass of the beads.

This is the video under consideration:

One of the better kinematic explanations I have seen is given in this article. However, the author, as many other authors, claims that:

 the mass of the beads don’t matter at all. If you look at the equation predicting the speed of the chain, you’ll see that the mass doesn’t show up anywhere. This is because a chain of beads that’s twice as heavy would also have twice the momentum kick. These two effects cancel each other out, and the motion is independent of the chain’s weight. This is a prediction of this model, and we could easily test it out with a chain of plastic beads.

Right… What about 5 kg beads? That should also not matter according to the kinematic model. Let’s be reasonable…

Of course mass matters. This is a dynamics problem and involves multiple modes and equilibrium points. The kinematic approach fits to a single observation, that is why mass does not matter. However, mass affects the type of curvilinear motion and the period of rotation, which in turn determines the radius of the curvature, the magnitude of the centrifugal forces, the height of free fall, etc.

Mass does not show up in kinematics naturally because the motion is assumed to be causeless. This is like the Ptolemaic system of planetary motion. In that system planet motion is taken for granted and corrected with epicycles and the objective is the determine the relationships in the dyad {T,L}, or in simpler words determine time as a function of distance and vice versa. However, in dynamics, we are interested in the triad {M,L,T}, i.e. how forces, that depend on mass M, cause the motion. For example, given that a force is central and attractive, like the gravitational force, will a mass m circle another mass M? This problem has 4 solutions (conic sections) depending on the masses and initial conditions. This is a dynamics problem. The kinematics problem assumes that the motion is already in place and naturally masses are irrelevant.

The physics in the bead chain problem are as follows: Given a chain of length L with beads of suitable mass m, the downward pulling over the rim of the beaker causes a curvature of the motion that in turn gives rise to a centrifugal force on the beads. As the speed increases due to the free fall, the centrifugal force increases according to the equation Fc = mv^2/r. This causes the radius of the curvature to increase and also the height of the free fall to increases, increasing the speed further and thus the radius of motion. The equilibrium point occurs when sum of the centrifugal forces on the beads balance the sum of gravity forces.  If the beads are too light or too heavy, the centrifugal forces cannot suffice to balance gravitational forces and create an arc and the phenomenon is not observed. Also, if the height of free fall is too long or too short the speed is very high or too low and thus, this phenomenon depends on choosing the right bead mass range and the right height. This is a dynamics problem.

Which brings us to the issue of centrifugal force. Some progressive (should be in quotes) physicists call centrifugal forces fictitious when actually this is one of the most important and real forces around responsible for many desirable and undesirable effects, like train derailments (recent accident on Spain) and separation of chemical compounds in centrifuges. The insist of using its counterpart, the centripetal force in problems, greatly confuses students who should not be interested in the philosophical aspects of force and its different interpretations but on how to solve practical problem, like the chain of beads problem, for which the solution becomes very difficult to conceive unless one submits to the reality of a centrifugal force. Since forces can only be measured by their effects and nobody has ever seen, touched or had a conversation with a force, it is just ludicrous to call some of the forces real and some other fictitious. Possibly, the distinction of an “inertial” force may be appropriate to denote that this force is not an impressed one but one that manifests itself in non-inertial reference frames. This abandonment of this simple approach to problems involving non-inertial reference frames has rendered most students and even physicists unable of solving even simple problems for the benefit of a philosophy that requires all phenomena to have the same interpretation in all reference frames. However, this is not even remotely true and the scientists who push those ideas are intellectually dishonest since we all know by a simple experiment that an accelerating reference frame, like an accelerating car, cannot be considered stationary with respect to a uniformly moving car, because in the former the coffee is spilled on the dashboard and this effect is absolute and cannot be transformed away, as Newton showed 300 or more years ago with his bucket experiment. This is of course unless one assumed the ludicrous position that when one rotates a bucket the whole universe rotates the other way like it cares what you do on a small planet inside a galaxy out of the trillion galaxies out there.

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