The Overbalanced Wheel
In deducing the law pertaining to conservation of energy I noted that the law is not perfect, like the one pertaining to linear momentum, but admits of at least one exception (the origin of matter in the Universe). Where we find one exception we may find others, so I want to revisit the issue of conservation of energy to see whether we have found any such exceptions and to clarify our knowledge of the conservation law.
Long before physicists clarified the concept of energy and inferred the conservation law pertaining to it, inventors had come up with devices that allegedly created endless work out of nothing. Such devices fall into the category physicists call Perpetual Motion Machines of the First Kind (perpetual motion machines that violate the law of conservation of energy: those of the second kind violate the second law of thermodynamics).
There is almost a tradition among various believers in perpetual motion regarding what physicists call the overbalanced wheel. It's a favorite device in all of its forms because it seems so easy to understand. For centuries people had seen waterwheels grinding grain or driving other important manufacturing processes and understood that these wheels were doing work based only on the weight of the water flowing over them. Some of those people speculated that they might be able to drive a wheel with a weight that some mechanism would raise back to the top of the wheel to be reused, as the water in the rivers that drove waterwheels appeared to be reused. Without a proper, mathematical understanding of how weight relates to work, those people could not have known that they would have to put as much work into raising the weight as they would derive from the weight's descent. Their successors don't have that excuse.
Nonetheless, the past few centuries have yielded up numerous designs for overbalanced wheels. Drawings of these things depict virtually every variation we can imagine on the theme of a modified waterwheel. If one version turns out not to work, then another version springs into being to take its place, like the mythical Hydra. Not wanting to play Heracles, the patent office does not accept, much less consider, applications for patents on perpetual motion machines. And yet they continue to proliferate.
So now the time has come to create our own overbalanced wheel. In order to understand why these things fail to create energy, as their inventors have hoped, we will create the simplest version that we can subject to mathematical analysis. To facilitate that analysis we may want to draw a diagram of the wheel in our imaginations. We can make that diagram as fancy or as plain as we want, though I recommend making it plain enough that we don't obscure the principle's of the wheel's operation.
Also we must be sure to draw our wheel with a brake. We must be able to restrain the wheel in its motion, lest its creation of energy accelerate to a rate that tears the wheel apart. Now imagine gasping in astonishment at the thought that the wheel might be capable of such a thing. This perpetual motion must be very potent indeed!
Imagine a large wheel of wide radius, say 100 meters: in your imagination you should see one huge Ferris wheel. Imagine that on the outside of its rim we have mounted U-shaped brackets strong enough to hold the cylinders that we will place in them. At the top of this wheel we place concrete-filled steel cylinders in the brackets and we let the wheel carry them down and set them onto a ramp at the bottom of the wheel's descent. That ramp has a very shallow slope down to our left, so the cylinders accumulate to the left of the wheel.
In this situation the wheel turns as we load cylinders onto it and allow them to descend. Indeed, the wheel will even produce energy: we can connect its axle to an electric generator and produce electricity. In this way our wheel differs little from an overshot waterwheel. It certainly doesn't seem that it would violate any laws of physics. And, in fact, as I have described it so far it does not violate any laws of physics.
But now let's think about all of those heavy cylinders accumulating at the base of the wheel. We would like to reuse them. So let's imagine setting up a vertical conveyor with brackets that scoop up the cylinders as they come down the ramp and carry them up to a point just above level with the top of the wheel. There the conveyor deposits the cylinders onto a shallow ramp that takes them back to the top of the wheel for another cycle. To drive the conveyor we pass it over a wheel near the top of our large wheel. We make the smaller wheel with a radius of, say, 10 meters and so position it that its top comes just a little above level with the top of the larger wheel.
We assume a uniform gravitational field of 9.81 newtons per kilogram. Let's also assume into our premises the statement that we have put the cylinders, the weights, so close together that we can accurately model them with a continuous belt that ponders dM/dl kilograms per meter of length. We begin by calculating the amount of torque that this belt exerts upon the larger wheel. We imagine the belt riding the right side of the wheel from θ=0 at the top of the wheel to θ=π at the bottom of the wheel, where the wheel deposits it onto the ramp that takes it to the conveyor that takes it back to the top of the wheel. Each element of mass dM (with length dl=Rdθ, in which R represents the radius of the wheel) exerts an element of torque equal to its weight multiplied by its lever arm, the distance between it and the wheel's axle measured in the direction perpendicular to the direction of the gravitational force, that distance being equal to Rsinθ. Thus we have
That result should tell us right away that this device will never create energy. We can see the truth of that statement if we imagine letting the belt come loose from the wheel at θ=π/2 and hang free as it descends to the ramp at the bottom of the wheel. The part of the belt that lies on the wheel in this example generates half the torque calculated above. The part of the belt that hangs free has length R and applies its weight (gRdM/dl) to a lever arm of length R, thereby generating a torque equal to half the torque in Equation 1. Thus the net torque acting on the wheel has the same value, whether the belt clings to the wheel as it descends from the highest point to the lowest or whether the belt comes away from the wheel at θ=π/2 and hangs freely. That fact tells us that, however we contrive to hang the belt off the wheel, the belt's weight will generate the same amount of energy for every radian the wheel turns under the torque produced by that weight.
Once the belt has descended onto the ramp at the bottom of the large wheel, it goes to the point at which it begins to ascend back to the top of the large wheel as a consequence of the turning of the small wheel lifting it. The amount of work that the small wheel must do in order to lift the belt equals the product of the torque that the belt exerts upon the wheel and the angle through which the wheel turns. If we let r represent the radius of the small wheel, then the belt going over the wheel exerts the torque
But that only accounts for the amount of the belt that extends from a point level with the bottom of the wheel to the point at the top of the wheel. We also have to account for the weight of the portion of the belt that hangs free between the point level with the bottom of the small wheel and the point level with the bottom of the large wheel, a portion of length 2R-2r. That portion exerts a torque of
If we now add the torques of Equations 2 and 3, we find that the net torque acting on the small wheel equals
That looks promising; that looks like something that we can manipulate to apply different torques to our two wheels in a way that creates energy. However we have one more necessary criterion to fulfill before we can generate energy with this thing: we must ensure that for every meter of belt that goes over the large wheel one meter of belt goes over the small wheel. I don't have to be a master geometer to know that I can only fulfill that criterion by so building my machine that for every radian the large wheel turns the small wheel turns R/r radians. And that fact is the wooden stake pounded through the heart of this perpetual motion machine.
Each wheel generates or absorbs energy equal to the torque acting on the wheel multiplied by the angle through which the wheel turns. Thus, in turning through an angle ΔθL the large wheel generates energy
Because we use the torque on the large wheel to drive the small wheel, the small wheel will absorb energy. For the small wheel we have
But we know that we must have built into our machine the relation
so we can rewrite Equation 6 as
Finally, we calculate the net energy that the machine creates by subtracting Equation 8 from Equation 5.
Regardless of how we build this thing, regardless of how we operate it, that subtraction must always yield a zero, which means the this machine absolutely will not create or destroy energy. However impressive it may look in the drawing, however many wheels and idlers we build into the device, it will not do anything impressive. It also will not do anything useful beyond illustrating the law of conservation of energy.
In the 1970's a margarine advertisement proclaimed "It's not nice to fool Mother Nature!" In fact, it's not possible to fool Mother Nature. Inventors may devise clever machines that send their weights on journeys that make the Minoan Labyrinth look as simple as a checkerboard by comparison. They may devise machines whose self-manipulations display an elaboration that would leave Rube Goldberg speechless in astonished awe. Yet none has ever succeeded in fooling Mother Nature.
We need only look at the definition of work and the fact that we can represent any path followed by the weights as a chain of minuscule straight horizontal and vertical elements. A weight does no work and requires no work be done upon it when it follows a horizontal element. On the vertical elements, for every upward increment of motion that obliges us to do work on the weight we find a downward increment of motion that gives back the same amount of work, regardless of how many twists and turns we put in the weight's path. When we carry out the integration along the path we get the result that tells us that the amount of work that we must put into a weight or that we can get out of it depends only upon the relative altitude between the weight's starting point and its ending point.
In adding unnecessary complexity to a system we succeed only in deceiving ourselves. Complexity tempts us to take shortcuts, to make guesstimates that may lead us astray from the path to the truth of the matter (and I have so strayed myself in another matter). Thus we see the wisdom of Occam's Razor (entities are not to be multiplied beyond necessity): the more we can simplify our description of a situation, the less likely we will go wrong.
We also see the wisdom of carrying out the full
mathematical calculations pertaining to the situation, however tedious and
boring they may be. Having been stung myself, I can say that there is a very
thin, barely visible line between simplifying assumption and mathematical booby
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