Perpetual Motion Machines Math21aABSTRACT. Is it possible to produce devices which produce energy? Such a machine is called a perpetualmotion machine. It is also called with its Latin name perpetuum mobile.LINE INTEGRALS. If F (x, y, z) is a vector field and γ : t 7→ r(t) is a curve, then W =RbaF (r(t)) · r′(t) dt iscalled a line integral along the curve. The short-hand notationRγF · ds is also us e d. If F is a force field,then W is work.CLOSED CURVES. The fundamental theorem of line integrals assures that the line integral a long a clo sedcurve is zero if the vector field is a gradient field. The work done along a closed path is zero. In a physicalcontext, this can be understood as e nergy conservation.ENERGY CONSERVATION. All physical experiments so far confirm that static force fields in our universeare of the form F = −∇U, where U is a function called the potential e nergy. If a body moves in this forcefield, its acceleration satisfies by Newton’s laws m¨x = F (x). The function H(x(t)) = ˙x2/2 + U(x) is conserved:d/dtH(x(t)) = ¨x ˙x + ∇U (x) ˙x = (¨x + ∇U(x)) ˙x = 0. The law of energy conservation is also called the first lawof thermodynamics.PERPETUUM MOBILES.A machine which implements a force field whichis not a gradient field is called a perpetuummobile or perpetual motion machine. Math-ematically, it implements a force field in whichsome closed loops have a non-vanishing lineintegral. By possibly changing the direction ofmotion, the energy change can always be madepositive.The first law of thermodynamics forbids theexistence of such a machine. It is informative tostudy some of the ideas people have come up inhistory (and today!) and to see why they don’twork. Also the drawings of Escher suggest, usinggraphical tricks, that perpetual motion should ex-ist.WHY ARE THERE NO PERPETUUM MOBILES?As an ”explanation” one can invoke the antropic principle:In a universe with a perpetuum mobile, an arbitrary amount of energy could be produced out of nothing andnothing would prevent this to happen spontaneously. This energy pr oduction co uld lea d to energy fluctuations.These energy bursts would destroy an observer.This argument is not a proof for the nonexistence of perp e tua l motion machine but it provides a plausiblereason, why the first law of thermodynamics should hold. A world without this law would likely be toohostile for life to exist.WHEEL SUGGESTIONS (1235).The oldest known attempt for a pe rpetuum motion machineis from the year 1245. There are seven hammers attachedto a wheel. On the top, the weights flip on the other sidein order that on the left hand side, there will always be 4weights and on the right hand side three. A similar ideawas designed by Leonardo da Vinci around 1500. You mightthink that such inventions from the past. The web is how-ever full with suggestions See for example http://www.free-energy.co.uk.WATE R MACHINE (162 0).Water flowing down turns an pump which transpor ts thewater back up. There ar e many variants of this idea: aelectric motor which turns a generator whose electricity isused to power the motor. In all of these ideas, there isa twist like some kind of gear or change of voltage whichconfuses a naive observer.BALL MACHINE (1997).The energy needed to lift the balls up is claimed to besmaller than the energy gained from pushing the otherballs. The website http://jnaudin.free.fr/html/rsmot.htmon which the idea was posted says: ”This devic e is NOTTESTED today”U TUBE SUGGESTION.The first suggestion for a perpetual motionmachine is a O shaped tub e which is filled onone side with water. There are two valves eachof which when closed prevent the water fromfalling. Opening and closing the walls doesnot use energy. A wooden ball falls in the airand is dragged up inside the water.MAGNETIC MACHINES on http ://www.geocities.com/k pullo/P M 3.htmm, there isan attempt for a perpetual motion machine based onmagnets. The author seems serious in claiming that themagnetic field from the moving mag net will change themagnetic field from the sta tionary magnets in such a waythat the magnet will move.A REC ENT INVENTION.The web is full of suggestions. An example of a perpetuum mo-bile which had actually been built is a machine by Reidar Finsrudhttp://www.theverylastpageoftheinternet.com/magneticDev/finsrud/finsrud.htm2A historical remark: Benjamin Peirce writes in his book ”A system of analytic mechanics” of 1855 :”Such a series of motions would rece ive the technical name of a ”perpetual motion” by which is to be understood,that of a system which would consta ntly return to the same pos itio n, with an incre ase of power, unless a po rtionof the power were drawn off in some way and appropriated, if it were desir ed, to some species of work. Aconstitution of the fixed forces, such as that here supposed and in which a perpetual motion would bpossible,may not, per haps, be incompatible with the unbo unded power of the C reator; but, if it had been introducedinto nature, it would have proved destructive to human belief, in the spiritual origin of force, and the necassityof a First Cause superior to matter, and would have subjected the grand plans of Divine benevolence to thewill and caprice of
View Full Document
Unlocking...