The Physics Gallery.

Principles of
Unworkable Devices.

1. The principle of unlimited possibility. Anything is possible in nature.
2. The "heavier on one side" seduction.
3. The "more weight on one side" distraction.
4. The "unbalanced torque" deception.
5. The cyclic disappointment.
6. The elastic/inelastic dilemma.
7. Failure to isolate the system.
8. The static/dynamic trap.
9. The apples/oranges equation.
10. The artistic illusion.
11. Experimental "gotchas".
12. The "dog chasing its tail" principle.
13. Reinventing the square wheel.

Anything is possible

Those who think this way are much like the child who puts two blocks in a box, closes the box, shakes it, then opens it, hoping to find three blocks inside. The child hasn't yet had sufficient experience with how nature works to realize that nature doesn't work that way. Perpetual motion machine seekers hope that by using some new combination of well-known materials and well-known mechanisms, they just might make a machine that outputs more work than is required to run it.

The "heavier on one side" seduction.

We can test its stability by turning the wheel to any position whatever. No matter what position it happens to be in at rest, it has no tendency to turn in either direction, and will not initiate motion by itself.

The earliest unbalanced wheel of Bhaskara underwent countless modifications and embellishments over centuries. But the seductive principle that motivated the inventors was that "If you can keep the wheel continuously heavier on one side as it rotates, then it will continue rotating with constant speed forever." Their overbalanced wheels had weights that moved, and in the static picture at least, there were always more weights on one side of the wheel axle. But, when the system is analyzed, one finds that the torques are balanced (add to zero). A wheel will not initiate motion unless the torques are greater in one direction than the other. If the net torque about an axis are zero, the system will not start turning by itself.

The situation is worse than that. Newton's third law ensures that all internal forces in a system sum to zero, and all internal torques do also. If the system is to move, some external source must supply force or torque. A system with unbalanced torque from external sources moves toward a position where the torques will become zero. That's a position where the system will be in static equilibrium. Typically the system will oscillate around that position, slowing down gradually to rest as its initial stored energy dissipates and it comes to rest.

Critical readers will note that the above comments assumed a uniform gravitational field. Actually the gravitational field decreases with height, and the field direction diverges from the center of the earth. These effects are negligible for a machine that would fit in a laboratory. But they would cause a slight preference for one position of this wheel (which one?). Still, they won't provide any force that would cause a consistently preferred direction of motion through many cycles.

The "more weight on one side" distraction.

The inventor may even test his assumptions by removing the two balls resting on the right ramp and weighing them on a balance. He does the same with the four balls on the left. The weights are not the same. He says, "I have tested this experimentally, and the weight of balls on the left of the vertex is twice as great as the weight of the balls on the right."

But this isn't a valid test. By removing balls for weighing, he's removing them from the parts of the system they interact with, and thereby changing the geometric and physical relation they had with each other and with the ramps.

"And if you still are not convinced," the inventor says, "I'll prove my assertion with another independent experimental test. I will put an electronic scale under the left corner of the ramp, and another under the right corner, to see which weighs more. As you can plainly see, the scales indicate different weights, proving that one side is heavier."

These last two tests are irrelevant to the inventor's purposes, for he is weighing the balls and the ramp. The mass of the ramp and the distribution of mass in the ramp is irrelevant to the operation of the machine, even the sort of operation that the inventor assumes.

Inventors have found many ingenious ways to make a wheel have more weight on one side of the axle, and to maintain that condition continually as the wheel is rotated. But as soon as it is left to its own devices the wheel finds an equilibrium position and just sits there unmoving. This happens when reaction forces reach just the right sizes to produce a net force of zero and a net torque of zero on the wheel.

But, during all these tests on the actual system, the balls sit there unmoving. The inventor rationalizes: "If I could only get rid of that pesky friction, I'd have this thing running. Just a little more work tweaking the design, and it will sustain its motion forever."

What's the error here? When analyzing a system or a part of a system, one must consider all of the forces and torques acting on that part of the system. Yes, the torques due to gravity acting on the balls are unbalanced about the upper vertex. But other forces also exert torques on the balls. The reaction forces of the ramps exert forces on the balls. The tension in the links joining the balls is a consideration, too. When all these things are considered, we find to our dismay that the net torque on any part of this system is zero. More to the point, the net forces and net torques on the ball-chain are zero. So the chain doesn't move by itself.

By the way, if you think that trick of balancing the system with a fulcrum, or with the two balance-scales, is absurdly blatant, let me tell you that a perpetual motion machine inventor very recently (in 2002) tried to pull that very trick on me. But he honestly and seriously thought it provided conclusive evidence in favor of his hypothesis. I provide another example below in the Experimental "gotchas" section.

The "unbalanced torque" deception.

• Continually move mass from one side of the axle to the other just fast enough to replace mass lost elsewhere by the turning of the wheel, but accomplish those transfers without doing work on the mass from any external energy source.
• Keep some part of the mass continually on one side of the axis, even as the wheel turns, so the wheel remains unbalanced.
• Introduce some force that is always greater on one side of the axle than the other, which provides torque that is continually greater in one direction of motion.
• Shield some portion of the machine from external forces.

Attempts to transfer mass run up against a cruel fact of nature. Even in the absence of friction the work required to take a mass from point A to point B and back again is exactly zero, and all wheel machines move masses around closed loops. Also, it takes the same amount of energy to transfer mass from A to B in a gravitational field no matter what path you choose between those points.

Consider the simple wheel shown. If it rotates clockwise, moving the light blue sector at the top to the right of the axis, the same size sector at the bottom moves to the left of the axis. The net result is that the mass distribution of the wheel remains the same as before.

The sector at the top has greater gravitational potential energy than the one at the bottom, so the inventor is encouraged. But then he discovers that the top sector loses exactly that amount of potential energy as the wheel rotates it to the bottom, while the sector at the bottom gains the same amount, for a net energy change of zero. Drat!

The nature of the gravitational field is at fault. Nature insists that the net amount of work done in carrying a mass around a closed loop in a gravitational field be exactly zero. (The same is true for electric fields.) Such fields are called "conservative" meaning "energy-conserving". There are advanced mathematical ways of saying this, using vector calculus, but that goes beyond our present needs or purposes.

Machines are cyclic. Wheels turn around an axle. Any process intended to produce power must move its components cyclically.

Many of the previously noted delusions about nature have the common feature that they are based on true observations about how something behaves over a finite time and distance (such as a ball rolling down an inclined plane) then assuming that such a process can also be worked into a wheel or other cyclic device. The trouble is, that with a cyclic device:

• The weight that gains energy falling on one side of a wheel must lose the same amount of energy when being lifted up the other side.
• The steel ball attracted to a magnet gains energy, but to return it to its original position and repeat the process, at least that much energy must be given to it.
• A buoyant object rising in water gains energy, but it takes at least that much work to return it back to its starting point at the bottom in order to repeat the cycle.
• A charged body in an electric field can be set in motion, gaining energy. But to return it to its starting point requires at least that much work.

So, how about a non-cyclic machine? Can a ball roll down an incline forever? You'd need an infinite length incline, and a gravitational field of infinite spatial extent. All fields known in the universe have two features that make this impossible: (1) they have "sources" at a particular location in space, and (2) their strength decreases with distance from the source, becoming zero at infinity.

To make a gravitational field nearly uniform over a large region of space would require a distribution of mass over a volume of space as large as that region. To assemble and "nail down" that much mass requires a huge expenditure of energy. Finally, the energy available to any machine operating in that field is no greater than the energy you expended in assembling that mass to produce the field.

To increase the size of that uniform region to infinite extent requires an infinite amount of mass, and where do you put that mass? We are forced to make do with cyclic processes occurring in a relatively small finite region of space.

Ultimately the common feature underlying all of these natural limitations, and underlying all of physics, for that matter, is geometry of space and time.

The elastic/inelastic dilemma

All real materials deform to some extent under application of force or pressure. In fact, one can argue that perfectly rigid bodies are a physical impossibility because then impacts between such bodies would give rise to infinite reaction forces for infinitesimal time durations.

In many materials, such as steel and glass, collisions can be over 95% elastic, that means that less than 5% of the impact energy is dissipated irreversibly. Can we assume elastic materials in our initial analysis, confident that this will set an upper limit on a machines efficiency?

If we do so, we encounter a serious dilemma. Consider any machine that has rolling balls, or shifting weights that suddenly change velocity due to impacts. Many classic perpetual motion machine designs have swinging hammers suddenly hitting pegs or "stops", as in Villard's wheel. Others have balls rolling down inclines and then stopped suddenly. Whenever there's such an impact, the hammer or ball will bounce or rebound. Invariably the inventor has not taken this fact into account, and it always decreases the assumed performance of the machine.

So can we redesign the machine for "softer" impacts? Perhaps we can put cushioning material at the impact point, or some sort of a catch or latch to prevent rebound. Alas, this is worse, for any such cushion or latch makes the impact inelastic, and dissipates energy as heat or sound. [In fact, chatter, clatter, whirring, humming, ticking or banging sounds from a machine are a sure sign of irreversible energy dissipation—meaning the machine is losing energy. The perfect perpetual motion machine would necessarily be perfectly silent.

A collision between two nearly elastic objects is not necessarily an elastic collision.

It is not generally appreciated, even by some physicists and engineers, that a collision or impact can involve very nearly elastic materials and still be a significantly inelastic (energy-dissipating) impact. An elastic material is one that can be slowly deformed, then relax to its original state without energy loss due to the deformation. But during a rapid deformation due to collision or impact, energy is temporarily stored (as elastic or vibrational energy) within each body, and when the bodies separate, one or both of them may still retain some of that stored energy, which is subsequently slowly dissipated as sound or heat as the body vibrates. Think of a bell being hit by a clapper. The bells are made of elastic metal, but they release energy as sound long after the imapact is over.

These phenomena are yet another "gotcha" nature springs on the over-optimistic perpetual motion machine inventor. Whenever the mechanism of a machine involves changing accelerations (and what mechanism doesn't?), mechanical energy will be dissipated irreversibly. But, as with friction, if we idealize these for analysis and the machine still won't work as we hope, then it will perform even more poorly with real materials and machinery. If we idealize everything and the machine's efficiency is still less than one, then we haven't achieved perpetual motion.

Generally we avoid bringing up classical thermodynamics, but here we can't resist mentioning the implications of what we've said above. One statement of the second law of thermodynamics tells us that the efficiency of a cyclic engine can never exceed that of a Carnot engine operating between the same two temperatures. A Carnot engine is an hypothetical "ideal" and "perfect" engine, assumed for comparison purposes. No one could ever build one. The efficiency of a Carnot engine depends on its input and exhaust temperatures, according to the formula e = 1 - (T_c/T_h), where T_c is the kelvin temperature of the cold reservoir and T_h is the kelvin temperature of the hot reservoir. (The kelvin temperature scale begins at absolute zero.) For real engines this efficiency is much less than one, only approaching a value of one when the temperature difference between output and input becomes infinite.

For example, the Carnot engine cycle is shown here for an ideal gas, on a pressure-volume-temperature graph. Everything about the Carnot engine is idealized, to properly represent the upper limit of what's attainable. The gas is an ideal gas, in which all gas particle collisions are perfectly elastic. The cycle has two portions during which the gas expands and contracts along isothermal (constant temperature) paths on the graph. It has two portions where, thanks to perfect insulation, the gas expands and contracts along two adiabatic lines (no thermal energy loss or gain). (T₁ and T₂ are the temperatures of the cold and hot reservoirs, while A₁ and A₂ are adiabats.) The changes occur infinitesimally slowly, to avoid inertial effects.

Real engines, if they are to be any use, must complete a cycle much faster, so inertial effects are unavoidable. Therefore the path (on this graph) of such a real engine will have "rounded corners" and will lie completely within the path of the Carnot engine. By the definition of work, it can be shown that the work done by engines is proportional to the area within their closed path on this graph. Therefore the real engine will do less work than the Carnot engine operating between the same temperatures, since its area is within that of the Carnot.

The Carnot engine's efficiency is proportional to the area within this boundary (the sum of the light blue and dark blue areas). Compare the diagram for a real heat engine (the smooth closed curve filled with light blue) operating between the same two temperatures, and you see that those sharp corners of the graph are rounded off, and the area within this curve is smaller. The efficiency of this engine is therefore smaller. While the ideal engine for other sets of variables will be different, the same general conclusions apply.

The Carnot engine has greater efficiency than any other engine operating between the same temperatures. If we idealize everything in a device and its efficiency still isn't greater than that of a Carnot engine operating between the same temperatures, then the second law of thermodynamics is still secure.

Textbooks always stress the point that the Carnot engine operates with strictly reversible processes. But sometimes they don't sufficiently emphasize that the operation of the Carnot engine requires infinitesimally slow operation, to avoid infinite accelerations of its material components. One method of analyzing the entropy changes in a real engine is to decompose its operating cycle into an infinite number of infinitessimal Carnot cycles. This is a valid method, but doesn't eliminate this problem—it only disguises it.

Even in the realm of smaller materials, atoms, molecules, and electrons, similar "gotchas" arise to thwart inventors' aspirations.

Failure to isolate the system

This simple machine has been constructed to illustrate such errors with a minimum of complicating details. In fact, this picture is the essence of the idea behind hundreds of perpetual motion machine proposals of the 17th and 18th centuries, which were often fantastically Baroque in their embellishment of mechanical details.

A wheel is free to turn on axle A. A heavy mass M is fastened to an L-shaped member MXY, hinged at pin X near the edge of the wheel. Point Y is linked to a member YZ that has the same effective length as AX.

The inventor points out that there's obviously more mass on the left side. Clearly if the machine were in this position and then released, the wheel would turn counter-clockwise as the mass M moves downward along the dotted line. The only path this mass can take is the dotted circle, which is entirely on the left of the axle A. So the mass always provides a counterclockwise torque anywhere on this circle. Also, wherever the mass is along this circle, the mass is greater on the left of the axle.

The inventor thinks he has solved the problem of keeping the mass and the torques unbalanced as the wheel turns.

This drawing is unlikely to deceive anyone, for the error is right out there for anyone to see. But the same fundamental deception is at the heart of many machines seen in the perpetual motion machine literature, but in those machines the deception is buried in a mechanic's nightmare of wheels, belts, linkages, springs, and other distractions.

The give-away here is the fact that the mass M must rise as high as it falls, so the net work done on it by gravity is zero around one cycle. But if the inventor focuses attention on the torques, some people may be seduced into thinking that the torques are unbalanced at any position of the wheel. How can that be?

John Haywood (Br. mechanic), 1790, British patent No. 1750. Dircks (1861) p. 66.

It can't. The loose language above failed to specify two things: (1) the body on which the torques act, and (2) the axis we choose to express those torques. We must choose a portion of the machine as our system, then carefully look at all external torques acting on that portion. It's the important principle that you must isolate the system before applying F = ma or τ = Iα.

The torque due to gravity on the mass M, about the axle A, is always counter-clockwise in this machine. But that's not the only torque acting on the rigid combination of mass M and the member MXY. At point Y there's a force due to tension in member YZ, and it is clockwise around the axle A. There's also a force that the wheel exerts upward at joint X, and that also gives a clockwise torque around A. Now when the mass M is rising after it reaches its lowest point, the sense of these torques due to forces at X and Y changes sign. The net torque on MXY during the fall is counter-clockwise, while the net torque during the rise is clockwise. The torque analysis gets a bit messy, but the bottom line is that the net work done by all of these torques acting on member MXY is zero during each cycle.

The machine shown on the right was invented by John Haywood, a draftsman and mechanic. He obtained a British patent for it. The small rollers at the end of each spoke are supposed to be quite heavy, to provide the "overbalance". The error of physics that led him (and the patent examiners) to think it might work is the same as in our simpler machine described above.

The static/dynamic trap

The classic overbalanced wheel is such a case. As shown here it's easy to calculate the torques due to gravity acting on each of the balls. It's a correct static picture of one of the resting positions of the wheel. But is it a correct snapshot of the wheel in motion?

We should always be suspicious of computer animations, for while they look impressive, they may very well be constructed on mathematical models the programmer assumes, but which may not be the ones that nature uses. This animation of the overbalanced wheel is unphysical in several respects. Can you identify its errors?

Answers left as an exercise for the reader.

The apples/oranges conversion

Here's an example. A system consists of three pulleys, R1, R2, and R3 and a flexible belt. Pulleys R1 and R2 are very heavy, while R3 can be lightweight. The whole system is hung from support S. The function of R2 is supposed by the inventor to provide continuous unbalance of the system. The inventor thinks that this makes the system "heavier on the right side" and therefore the belt will turn clockwise.

The left drawing deceives us. It shows the belt with two strictly vertical portions. But in reality, any belt flexible enough to move around the pulleys is also flexible enough to respond to the force of the pulley R2 upon it. The system, when assembled, actually assumes a configuration something like the drawing on the right. In so doing, the center of mass of the system moves until it is located directly under the axle of the upper pulley. The system won't budge thereafter.

Experimental "gotchas".

But again the diagram deceives us. The version on the left doesn't accurately depict the experimental situation; the one on the right is more like what happens. Our picture is exaggerated for clarity. The reconfiguration of the system may not be noticed by the experimenter. But why should this cause different spring scale readings and unbalance of the bar?

We are now dealing with the real world of real materials, not the idealized world we sometimes assume for initial analysis of perpetual motion machine puzzles. As the offside weight R2 is allowed to rest against the belt, the belt bends there until its reaction force is sufficient to bring the system to equilibrium. The belt, however, isn't perfectly flexible, nor is it frictionless. Friction and elastic deformations of the belt may supply forces that prevent the pulleys from assuming their smallest potential energy. Good lab practice requires fiddling and jiggling the system to find the range of positions the system can assume and still be unmoving. However, if the initial measurement confirms what the inventor expects, he may have no motivation to do that.

The last figure shows how mere shift of the center of mass CM with respect to the support points P1 and P2 can cause the tensions T1 and T2 in the supporting wires to be different. The system will come to rest when the center of mass is directly below the intersection point S of lines extended along the two support wires. If the angle between these wires is small, the ever-optimistic perpetual motion machine inventor may not even notice.

Advocates of political correctness may object to my reference to perpetual motion believers with the masculine pronoun. It seems appropriate to me, since to my knowledge there has never been a woman who was afflicted by this delusion to the point of inventing perpetual motion devices.

The "dog chasing its tail" principle.

Oroborus. A sacred symbol of alchemy. It could also be an appropriate symbol for the futility of certain pursuits.

This principle is sometimes called the "Baron Munchausen" principle after the 18th century teller of tall tales. He said he got of a desperate situation by grabbing his own pigtail and lifting himself up from a quagmire. (Some derivative tall tales have one lifting himself by his bootstraps, the origin of the term "bootstrap princicple".) Use of this trick is one example of an inventor's ignorance of (or disdain for) well-known valid physical principles.