The Shifting-Mass Overbalanced Wheel

Why it isn't perpetual motion.

Bhaskara's first perpetual motion wheel. Mercury in curved spokes.	Bhaskara wheel with mercury in tilted vials.	Overbalanced wheel with weights on articulated arms. 18th century.

Early articulated-mallet wheel.	Articulated arm wheel. Often called the "Arabian Wheel".	Rolling-ball wheel. 17th century.

Quite a number of books tell of this futile quest, citing examples with nice pictures of the failed designs. None of these books gives a simple conceptual explanation of why they don't work and can't work. None of the books expose the flaws of thinking that led people to believe they might work.

The physicist is tempted to cite conservation laws of energy, momentum, and angular momentum, or do an analysis in a rotating system with centripetal force. None of these concepts are well understood or even accepted by the non-scientist reader.

Can these wheels be analyzed using simple, conceptual physics, and simple concepts such as force, mass, velocity, acceleration and work, with appeal to examples from everyday experience that most people understand and accept? I shall try, and I'll also try to keep this discussion at the level of a freshman college physics course.

Overbalanced wheels are of little interest to the physicist, for physicists understand, from many well-established laws and many lines of argument that even if there were such a thing as perpetual motion, this is certainly not the way to achieve it.

All of these wheels use geometric design or mechanical devices to shift mass to a larger radius from the wheel's axle. We can only speculate what the inventors assumed to be the operating principle. Here are three possible reasons for motion, expressed in modern terminology:

1. All of these designs have more mass continually on one side of the axle, presumably unbalancing the wheel to bias rotation in one direction.
2. The masses on one side of the axle continually have larger lever arm, supposedly unbalancing the wheel to bias rotation in one direction.
3. As each weight shifts to larger radius, an impulse is given to the wheel, sustaining rotation.

While these "reasons" for the wheel's supposed operation are invalid, they are still very seductive to someone with only casual experience with mechanisms. Such minds will then suppose that the wheel's operation causes the output to be somehow greater than the input. Greater what? Greater force? Greater work? Greater power? In the days before force, work, and power were understood, this question would not have been asked, for inventors were probably thinking only of what we now call "work", the ability to raise heavy weights (as in pumping water), or run a gristmill or sawmill. Such a machine would supposedly perform these operations with smaller input effort or fuel consumption. In later centuries, when concepts of force, work and power were better understood, PMM inventors still thought that perhaps such a machine would multiply force, as well as produce more work and power.

In any of these cases, the motive principle might be supposed to (a) cause the wheel to accelerate, or (b) simply deliver greater power at constant speed. The second possibility is seldom mentioned in the perpetual motion literature. But some inventors showed a brake in their designs, to prevent the velocity to increase to dangerous values. If the machine were driving a grinding wheel or lift pump, presumably the load on the output would be considered sufficient to maintain constant speed of the PMM wheel.

But all of these wonderful speculative fantasies mean nothing if the wheel simply refuses to turn, and that's exactly what happens with any of these wheel designs. Such failure could always be rationalized by saying "The friction prevents the motion. Reduce the friction and it will surely work." But friction isn't the problem. The machines still refuse to turn even when all friction and other energy-dissipative processes are totally eliminated.

The inventors overlook some important basic physics. The earlier inventors can be excused, since those physics principles were not yet well understood. Inventors today who recycle modifications of the same designs have no excuse.

Imagine a wheel with masses on spokes. The masses are located at the middle of each spoke. Let the wheel axis be vertical, so the wheel is rotating in a horizontal plane. Suppose one of the masses is suddenly released so it moves out to the end of its spoke, where it is stopped by some restraint or catch. Will this cause the wheel to turn faster due to the greater torque on this mass? The inventor assumes that this sudden process occurs without doing any work.

To illustrate what really happens, consider an experiment that anyone can perform. Attach a weight on the end of a rope and swing the weight around a horizontal circle. Anyone who has done this knows that if the rope is released to allow the weight to suddenly move to a larger radius, the stone's angular speed (revolutions/time) decreases. Also, its speed decreases, though that fact is not readily apparent to the eye. Its kinetic energy of motion has gotten smaller. That's not encouraging. This happens because the rope does negative work on the stone during the change of radius. If the rope is "let out" gradually, or if the rope only exerts force to stop the stone at the new radius, the force the rope exerts on the stone is opposite to the stone's displacement. Therefore the rope does negative work on the stone. This is equivalent to saying that the stone does positive work on the rope, and ultimately work is done on whatever the rope is attached to at the center of rotation. [For physicists reading this, we note that conservation of angular momentum applies, and the rope tension provides nearly zero torque, so the product of moment of inertia and angular velocity remains nearly constant.]

Perpetual motion wheels are invariably cyclic, that is, all motions of the wheel and its parts are repeated exactly during each complete revolution. So if a weight moves to a larger radius once per cycle, it must also be pulled back to the original radius later during the cycle. The work done in changing the radius by a certain amount from large to small is equal and opposite to the work done in changing the radius by the same amount from small to large. We gain no net energy per cycle.

Now imagine such a rotation in a vertical plane, so that gravity can play a role. Since the wheel's motion is cyclic, and the motion of the mass is cyclic, the work done on the mass by gravity as the mass moves down is of equal size to the work it does against gravity moving back up. There's no gain in net energy per cycle.

One of the most fundamental and well-tested facts of physics is that forces combine by vector addition. [This is sometimes called the "superposition principle" of force.] When two forces act on a body simultaneously, the net effect on the body is simply the vector sum of the forces. From this follows the result that when two forces do work on a body, the net work done on that body is the same as the work done by the vector sum of those forces.

We have seen that the radially shifting mass alone does no net work on the wheel during each cycle. We have also seen that gravitational force does no net work on the wheel or the mass during each cycle. So the two processes acting together will not do net work on the wheel.

This is an outline of the reasons why physicists understand that all overbalanced wheels, no matter how ingeniously constructed, can never provide more energy than they were given initially. In fact, the situation is even worse, for the more ingenious and complex is the mechanism for maintaining the overbalance, the more poorly the wheel will perform due to mechanical inefficiency.

Since our theme is puzzles, consider the pretty design shown here. The idea for this as a perpetual motion puzzle was sent to me by Slawek Liszewski. It is a wheel with three equal compartments, constructed using arcs of circles. Put equal amounts of liquid (red) half-filling each compartment. (1) In what position(s) will the wheel be in static equilibrium? Or will the wheel be in static equilibrium in every position? (2) Will these answers be the same for any chosen amount of liquid (but with the same amount in each compartment)? Show and justify all mathematical steps in your solution.

This puzzle illustrates a general question that is at the heart of all shifting-weight overbalanced wheels. I do not know whether it yields to a simple conceptual/physical/mathematical argument. It might require calculus. Remember that we are not asking for a complete analysis. We only ask whether there are positions of stable equilibrium, and where they are.

Re: Answers left as exercise for the student. Send your answers to Donald Simanek at the address shown to the right. The earliest good one(s) that arrive may be posted here, with credit to author. I will post (at my discretion) answers that are in simple to explain, clear, correct, perceptive, and which stimulate thinking and further discussion. Posted answers, whether written by me or by others, do not always represent the final word on a given proposal. On several occasions perceptive readers have noticed things we missed, or suggested simpler ways to explain something. So don't hesitate to skeptically rethink given "answers".

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