Physics in School, vol



Physik in der Schule [Physics in School], vol. 31, no. 5, p. 179 ff (1993)

Why does the washer “whir”?

Physical aspects of an interesting toy

H. Joachim Schlichting, University of Essen

The spiral is a spiritualized circle…

In the form of a spiral, the circle has

– uncoiled and unwound – ceased to be vicious;

it has been set free.

Vladimir Nabokov[1]

For me, the spiral is the symbol of life. I believe the spiral is found,

wherever matter ceases to be [inanimate],

and starts to become something living.

Friedensreich Hundertwasser[2]

Automatically rotating washers

Many toys are commonplace objects that are used according to some idea of play. One thinks, for example, of tin can covers or other disks, which one once used as a Frisbee before there even were Frisbees. The Frisbee is a good example of how a commercial toy frequently represents only an improved version of a long known commonplace toy.

Another example is the “Fiddlestick,” which was described several years ago by Jearl Walker [1]. It consists of a rod which is round in cross section and on which plastic washers are made to rotate and revolve around the rod by, for example, putting some washers on the rod and brushing the palm of one’s hand forcefully along the outer rim of the washers, thus setting them into rotation (Fig. 1).

What’s enticing and interesting about this toy is that the washers spin down the rod surprisingly slowly and at a steady rate of rotation, following a spiral path (Fig. 2). Even when set rotating in various ways (albeit not too hard), the washers quickly or over a longer time (that is, just before reaching the end of the rod) almost always arrive at their own state of motion. Also, the pattern of motion persists in the absence of further small disturbances, such as shakes of the rod.

After reading Mr. Walker’s description of the Fiddlestick, I felt recalled to my childhood, in which I counted such a rod among my collection of homemade toys. I recall at that time having used iron wire (for example, welding rod, among others), on which I let washers of various sizes whir down. Prodded by Mr. Walker, I viewed my old toy in new ways. So it came to light that further variations were obvious: using broomsticks and wooden or plastic curtain rings, or using dowels or other round wooden sticks from hardware stores and rings or washers that are sold in craft shops for many different purposes. There are no limits on one’s imagination.

The characteristics of a washer’s rotation can also be varied in many ways. The properties of a washer (its diameter, mass, form (flat or round on the inside), roughness of the surface) and of the rod (its diameter, stiffness, roughness of the surface) lead to different behaviors.

For example:

• The washer winds down the rod through a large angle in a short time.

• The washer rotates at the same height until its rotational energy is nearly exhausted, then it slides down the rod in freefall. According to Nabokov’s quote, this is a “vicious circle.” The quote applies to the unwinding of the circular motion into a spiral motion in order to allow the washer the freedom to exchange energy with its surroundings and thereby to prolong the washer’s rotation along the whole length of the rod. The washer, which first rotates as if caught in an orbit and then drops, does this under certain conditions – particularly when some remnant of rotation (angular momentum) exists – remaining suspended at a lower position and using the potential energy from its fall to recommence an episode of rotation.

• If one utilizes several washers simultaneously, then either they run down the rod simultaneously in beautiful uniformity, or one of the upper washers is faster (i.e., its path is steeper) than the lower ones and runs into them. This can thus lead to a sort of elastic reflection. This process can repeat itself several times.

• Finally, with some practice and with a relatively stable rotating washer, one can prolong the rotation arbitrarily. One inverts the rod shortly before the washer or – even harder – the washers have reached the rod’s end.

From rod to hoop: time becomes cyclic

The idea of (arbitrarily) prolonging the washer’s rotation can be improved further. If one bends the rod into a hoop and joins its ends together, then one can maintain a washer’s rotation indefinitely. One need only take care to turn the hoop so that the washer always remains on the downward sloping part of the hoop. By sliding the hoop through one’s hands so that the hoop rotates like a wheel in the vertical plane, the washer will rotate at a constant height. Now the washer is using the kinetic energy from the rotation of the hoop. From the washer’s viewpoint, time has, in a way, become cyclic: the washer traverses the same points on the hoop again and again.

The so-called “whir-ring,” which is currently obtainable as a toy, [3] may have originated as a successful realization of this idea [2]. This hoop has a diameter of 28 cm and consists of stiff wire that’s 6.4 mm thick. It comes with five plastic washers, each of which has a hole about twice as wide as the hoop is thick (Fig. 3). The washers on the hoop are set into motion as in the case of the rod (Fig. 3). The title “whir-ring” might remind one of a hummingbird, which remains in position by rapid wing beats and which thereby produces a uniform noise.

On the physics of the spiral descent

As far as the hoop can be viewed as a curved rod – in the following we assume that – the behavior of washers on the “whir-ring” can be explained in the same way as washers on a rod. Therefore, for the sake of simplicity, we limit ourselves in the following to a consideration of washers on a rod.

If one started a washer spinning in the manner described above, for example, then it arrives relatively quickly at a steady rotational state. Since it is bound to the rod purely topologically and therefore its range of motion is restricted, a washer can’t continue in the tangential direction in which it is “impelled.” Its continual effort to follow a straight, uniform path as a result of inertia leads it instead to revolve around the rod. Thereby the washer presses with its inner side more or less strongly against the rod and tends with its outer side radially away from the rod. This (accelerated) circular motion is generated by the elastic force that acts at the point of contact between the washer and the rod.

One further consequence is that this force hinders a washer’s fall under the influence of gravity. At the point of contact between the washer and the rod (point A’ in Fig. 4), the elastic force produces static friction, which is large enough to compensate for gravity. (Therefore a washer begins to move only above a minimum starting impulse.) Contrary to expectations, pupils frequently express at this point difficulties in comprehension. They don’t immediately comprehend that the friction that’s thus produced is not affected by the revolution of the point of contact. Perhaps an allusion here to the more familiar phenomenon of propulsion and tracking during driving may help. In this case, the point of contact (between tire and street), where friction acts, also rotates.

Since the friction is not exerted at the washer’s center of gravity (as is gravity), a tilting torque operates on the washer (force couple A and A’ in Fig. 4). Consequently the washer leans somewhat to the side that lies opposite the point of contact. Since the washer’s state of motion is changed by this tilt, there arises an inertial “resistance” to this change, which however acts at right angles to the force instead of acting along the direction of the force, as happens when a body moves linearly. Thus the washer’s downward inclination at the point of contact is continually opposed by a corresponding upward deflection that is phase-shifted 90( (force couple B and B’ in Fig. 4).[4] Thus one is dealing with precession, in which the (hypothetical) gyroscope axis, which is rotating around the rod, sweeps out a conical surface around the rod.

Strictly speaking only in the friction-free case would the downward inclination of the washer be completely cancelled by the inertial upward deflection of the washer and lead to a circular path around the rod at point B’. On account of the dissipation of energy through friction, the downward inclination is not completely cancelled. The washer continues to be tilted downward somewhat, and the circular orbit around the rod is opened into a spiral path down the rod.

This symmetry breaking is the decisive event. Only thus does the revolving washer acquire, in the sense of Nabokov’s quote, the capacity for self-organization, by which that so fascinating phenomenon is caused and stabilized against ever-present external disturbances.

It’s interesting in this connection, that the energy dissipation leads not to a slower rate of revolution and thus to a reduction of the static friction, with which the washer’s sudden fall would become inevitable. The downward motion, which is caused by the dissipation, makes it possible for the washer – like a cylinder rolling down a tilted plane – to use the potential energy that’s released by its fall in order to raise its rate of rotation; i.e., to use the potential energy for its own acceleration.

The feedback mechanism: the faster, the slower

However, instead of revolving ever faster, the washer assumes a steady state of motion immediately after a successful start. That is possible only when the driving force that accelerates the washer is balanced by a braking force that likewise increases with the washer’s speed. Indeed, the braking force must increase even faster with the speed than does the driving force, so that the former can overtake and consequently limit the latter. In a steady state of motion, both forces must be equally large. From an energy viewpoint, this condition of constant kinetic energy can be maintained only by completely dissipating the driving energy that flows to the washer from outside; i.e., by giving off the driving energy as heat to the environment. Or expressed otherwise: averaged over time, the rotating washer must receive precisely as much energy from the environment as it loses by dissipation.

Yet how does the washer “know” how much energy it must take from the available reservoir of potential energy in order to maintain a steady pattern of motion, and how does it know how much energy to expend against ever-present external disturbances that can manifest themselves as changes of speed?

For the answer to this question we investigate first what happens when the speed of rotation rises. In this case the perpendicular upward deflection, which increases with speed, and thus the upward tilt in the washer’s trajectory, become more strongly pronounced, and the downward slope of the washer’s path is correspondingly decreased. Consequently the influx of energy is reduced and no longer suffices to compensate for the losses to friction. The washer slows. However, a decrease of the rotational speed causes a reduction of the perpendicular upward deflection and thus an increase in the downward inclination of the washer’s path. As a result, this leads to a downward acceleration and hence an associated increase in energy uptake, and so on.

In this cyclic interplay of acceleration and deceleration (i.e., of increase in the path’s slope and reduction in the path’s slope), one easily discerns the basic control mechanism underlying both the steadiness and stability of the washer’s spiral motion down the rod. Owing to this mechanism, the speed oscillates –although it’s normally hardly noticeable – around the value of the steady speed that characterizes the steady-state equilibrium between propulsion and dissipation. In other words, the control process can ultimately be reduced to this statement: an increase in the speed leads to a reduction of the speed, which in turn leads to an increase in the speed, and so on. Thus we are dealing with a feedback cycle, which we also find in similar form in other systems (see, for example, [3]).

[1] J. Walker, The Flying Circus of Physics [New York, N.Y, U.S.A.: Wiley, 1975], p. 32.

[2] Physik Boutique, Stark Verlag, Freising. [5]

[3] H. J. Schlichting, “Komplexes Verhalten modeliert anhand einfacher Spielzeuge” [Complex behavior modeled by means of simple toys], Physik und Didaktik, vol. 17, no. 3, pp. 231 ff (1989).

Figures accompanying “Why does the washer ‘whir’?”

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[1] Vladimir Nabokov (1899-1977), Russian-American author. The quote is from his memoir, Speak, Memory.

[2] Friedensreich Hundertwasser (1928-2000), Austrian artist.

[3] The “RotaDyn Vibrations-Ring,” which is distributed by Pearl, Inc. of Buggingen, Germany. (See: .)

[4] The washer behaves like a gyroscope. Hence, the upward force of friction that the rod exerts on the washer creates a torque, causing the washer to tilt around that diameter of the washer which intersects the washer’s point of contact with the rod. This tilt causes the washer’s path to incline upward at the point of contact. For a discussion of the basic dynamics involved, see, for example: Daniel Kleppner and Robert Kolenkow, An Introduction to Mechanics, [N.Y., N.Y.: McGraw-Hill, 1973], p. 299 (“Example 7.8: Why a gyroscope precesses”).

[5] The “Physik Boutique” was a retail service provided by Stark Publishers of Freising, Germany. The “Boutique” sold materials for demonstrations during physics lectures.

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Fig. 4. The forces acting on the washer.

Fig. 4. The forces acting on the washer.

Fig. 4. The forces acting on the washer.

Fig. 3. In the left-hand illustration, one sets the washers rotating with one’s hand. In the right-hand illustration, one then turns the hoop to the extent that the washers always rotate downwards.

Fig. 2, The washer runs down the rod along a spiral path.

Fig. 1. The whir-ring is set in motion with one’s hand.

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