An Anomalous Perpetual-Motion Machine

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    Properly speaking, what I describe in this essay does not constitute a true perpetual-motion machine: it won’t create energy ex nihilo. It has more in common with the perpetual-motion machine of the second kind. But instead of violating the law of entropy, it violates the law of conservation of angular momentum. Call it a perpetual-motion machine of the third kind.

    For this analysis we want something with a lot of rotational energy that we can tap, something readily available. The obvious candidate is Earth itself. The planet contains approximately 2.57x1029 joules of kinetic energy in its rotation. If we were to tap that energy at a rate of one billion kilowatts, roughly order-of-magnitude equal to the electric generating capacity of the United States, that energy would last over eight billion years. So how do we tap it? Could we actually tap any of Earth’s rotational energy through some mechanism?

    Imagine mounting a gyroscope on Earth’s equator with its axis oriented vertically. We permit that axis to move only in the equatorial plane. Conservation of angular momentum keeps that axis pointing in one direction in space and obliges us to exert a torque upon the system to change its orientation. Earth rotates from west to east, so we would see the axis of that gyroscope appearing to turn from east to west, making one complete revolution in a day.

    Using analytic geometry, we represent angular momentum with a vector (actually an axial vector or pseudovector, but that’s a technicality that we can ignore) that we describe through the right-hand rule. If the fingers of your right hand curl in the direction of an object’s rotation, then we represent the object’s angular momentum by an arrow whose direction is indicated by your right thumb and whose magnitude is indicated by the length of the arrow. That arrow represents the quantity and direction subject to the conservation law. For example, Earth’s angular momentum would appear as a giant arrow jutting up through the North Pole. Our imaginary gyroscope’s angular momentum appears as an arrow pointing along the gyroscope’s axis, which lies in Earth’s equatorial plane and points at some distant galaxy or quasar.

    If we stand on the south side of the gyroscope and face north, the gyroscope will look to us as if it were rotating counterclockwise about its axle, which is oriented north-south in the foundation frame that holds the gyroscope up. If we attach that axle, through appropriate gearing, to a shaft, we can make that shaft spin rapidly enough to turn the rotor of an electric generator. Of course, this would not be a practical device, but in concept it should work as I describe it. As the generator produces electricity, the electromagnetic interaction will produce equal and opposite torques on the rotor and the stator.

    Seen from our south outlook, the stator exerts a torque that acts to twist the foundation frame counterclockwise. That torque presses the west footing of the foundation down and pulls the east footing of the foundation up (so that part of the foundation must be anchored with weight and/or by being firmly attached to bedrock). As a consequence, Earth’s rotation slows at a minuscule rate as the device takes rotational energy and transforms it into electricity.

    The rotor of our generator exerts a torque clockwise upon the gyroscope. That causes the gyroscope’s angular momentum vector to turn in Earth’s equatorial plane. Over a minuscule elapse of time that turning produces a minuscule angular momentum vector lying in Earth’s equatorial plane and perpendicular to the gyroscope’s angular momentum vector. That minuscule change represents a reaction torque that acts to change the orientation of Earth’s angular momentum vector. But as the gyroscope makes a complete rotation about its axle those changes come full circle and cancel each other out. Thus there will be no change in the orientation of Earth’s angular momentum.

    So now we have conceived a system that someone could use to convert a planet’s kinetic energy of rotation into electricity. It’s perfect, except for one minor detail – there’s a conservation law.

    Nothing exists outside the Universe and that fact determines what exists inside the Universe. It necessitates that the Universe as a whole can have no properties. For relevant example, the Universe can have no rotary motion. But rotary motion exists within the Universe. We are obliged to infer, then, that all of the rotary motions within the Universe must always and forever add up to a net zero. That’s the conservation law pertaining to angular momentum, the measure of rotary motion. If the angular momentum of some body changes, then at the same place and the same time the angular momentum of another body must change by the same amount in the opposite direction.

    By that reasoning we understand that our imaginary device cannot diminish Earth’s angular momentum without producing an equal and oppositely directed angular momentum in some other body. But it doesn’t do that, so what have we failed to take into our account?

    Look again at the reaction torque acting to twist the gyroscope’s axis out of Earth’s equatorial plane. It doesn’t succeed in that action because the gyroscope’s axle is firmly mounted, through bearings, on the foundation frame. Thus, instead of turning the gyroscope, the reaction torque gets transferred to the footings of the foundation and absorbed by Earth. The result is a non-cumulative, sub-minuscule, undetectable, and unmeasurable tilt of Earth’s axis. The gyroscope continues to point in the equatorial plane.

    I have tacitly assumed that, because of its angular momentum, the gyroscope will resist any effort to change the orientation of its axis. The experience of turning a spinning gyroscope and feeling it bucking seems to support that supposition. But that’s an illusion: the bucking occurs perpendicular to the turning of the gyroscope’s axis. There is actually no resistance to the gyroscope being turned over. The gyroscope, constrained as we have it, exerts no torque upon the generator and, through it, upon Earth to slow Earth’s rotation.

    Like all perpetual-motion machines, this one looks good on first impression, but further analysis reveals that it just won’t work.

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