The Hidden Flaw

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The Work Energy Theorem

Before illustrating a hidden but oddly obvious flaw, the Work Energy Theorem is first briefly stated mathematically.  It is the idea that the work (Fd) done to an object is exactly equal to the change in its kinetic energy mv2).

Fd cos θ =   mv2)final - mv2)initial


F is the amount of force applied, a vector

d is the displacement, a vector

cos is short for cosine (a trig term)

θ is the angle between the applied force and the resulting motion of the object

m is the mass of the object

v is the object's velocity


Fd is the formula for work

½ mv2 is the formula for kinetic energy

The Flaw No One Sees

In short, physicists understand that a form of mechanical energy (work) is defined as a force acting through a displacement.  However, when electrical energy is converted into mechanical,Solenoid the definition / mathematics of "work" actually changes; it becomes a force acting over time

To convert electrical energy into mechanical energy is rather easy.  It can be done using  only four things a 9 volt battery, some magnet wire, a neodymium magnet, and a paper tube just slightly bigger than the magnet.  Wind the wire around the tube with 50 turns of wire, and place the magnet inside the tube (see illustration).   Once that is done, connect the two ends of the magnet wire to the battery and the magnet will shoot out of one of the tube's ends.   Electrical energy has just been converted into mechanical.

A quantity of electrical energy is defined as a function of voltage, current and time.   Electrical utilities know and use this fact; they typically charge for electricity by the kilowatt hour (essentially voltage x current x time).   This is rather well known amongst electricians, engineers and physicists.   Nothing new in other words but when you convert electrical energy into mechanical, a fact unknown to all turns up.

When electrical energy is fed into the solenoid, a mechanical force is the result.   This mechanical force occurs because the solenoid produces a magnetic field that interacts with the magnets inside the solenoid.  The magnetic force is the direct result of an applied voltage which in turn causes an electrical current to flow.   In other words, it takes both voltage and an electrical current to produce magnetic force that manifests as a mechanical force.   So, when you convert electrical energy into mechanical (the acceleration of the neodymium magnets resulting in a change in kinetic energy), the amount of mechanical energy is a function of force (voltage x current) and time.  

Mechanically, work is defined as a force acting through a distance (displacement in formal terms).  Electrically, "work" is defined as a force (voltage x current) acting over time.   This is an anomaly or a paradox, if you will, that physics students are never told about because their teachers and professors were never shown this when they were students. 

Mathematical Example 

Imagine converting a known quantity of electrical energy into mechanical.   And to make this very easy to follow, we will use the simplest case possible — accelerating an object from rest.   Electrical energy could be configured to provide an average mechanical force of 10 newtons that acts for 1 second.  We will use this exact amount of electrical energy twice; first to accelerate a 1 kg object and then then on a 2 kg object.  

The equations that apply are F=ma, v=ta, and ke=½mv2.  These are, of course, standard physics formulas; see any standard textbook.

The one-kilogram object

It will accelerate at: a = F ÷ m = 10 ÷ 1= 10 m/s2

Its final velocity will be: v = ta = 1 x 10 = 10 m/s

It experiences a change in Kinetic Energy of: ½mv2 =  ½ x 1 x 10 x 10 = 50 joules

The two-kilogram object

It will accelerate at: a= F ÷ m = 10 ÷ 2 = 5 m/s2

Its final velocity will be:  v = ta = 1 x 5 = 5 m/s

It experiences a change in Kinetic Energy of:  ½mv2 =  ½ x 2 x 5 x 5 = 25 joules

The final values of mechanical energy (50 joules and 25 joules) do not match despite the fact that the exact same amount of electrical energy (10 newtons of force acting for 1 second from voltage x current x time) was used for each occurence.   In terms of the Law of Conservation of Energy, we have either lost or gained 25 joules of energy depending on your particular viewpoint.

Taking the Mathematics Further

On seeing the above mathematical example, some might get the idea that Newton's Third Law might play a role; it might explain why an apparent problem exists but not an actual one.  Instead of assuming this is the case, the proper thing to do is to run the numbers and know.  Say that the objects in the mathematical example above are magnets and the mechanism used to accelerate them is a solenoid connected to a battery.  And let us further state that the mass of the solenoid and battery is 2 kilograms.

When this setup fires the 1 kg magnet, it goes in one direction and the solenoid in the other.   The amount of electrical energy used equates to 10 newtons of force acting for 1 second (voltage x current x time).   So in this first situation, the solenoid, having a mass of 2 kg, changes its kinetic energy by 25 joules (as calculated above); the 1 kg magnet changes its kinetic energy by 50 joules.  In both cases, the solenoid and magnets begin with no kinetic energy since both are initially at rest. The total amount of mechanical energy after the electrical energy is converted to mechanical is the sum of both — 75 joules (25+50).

When this same setup is used to accelerate the 2 kg magnet, that object's kinetic energy changes by 25 joules.  The solenoid, also having a mass of 2 kg, changes its kinetic energy by the same amount 25 joules.  The total amount of mechanical energy converted from electrical is the sum of both — 50 joules (25+25).

Once again, the final values of electrical energy converted in mechanical do not match.  This is essentially a simplified version of one of the rocketry situations mentioned on the previous page; the acceleration of two payloads of unequal mass.  In other words, if you use identical quantities of chemical or electrical energy and convert that amount of energy into mechanical, the Law of Conservation of Energy does not hold true.   Depending on your viewpoint, 25 joules of energy has simply vanished or mysteriously appeared.

Experimental Proof

There may be some who will come up with some means of explaining that there is no problem.  And to those, I say they should do an actual experiment.  Rig up a system to provide a fixed amount of electrical energy to a solenoid that is repeatable.  Then have that solenoid fire magnets having differing amounts of mass each time but always using the same amount of electrical energy.  Measure the mass, the velocity of the magnets as they leave the soleniod, and then calculate the kinetic energy.   If the kinetic energy varies, your explanation has no value.  And just for fun, see if the momentum of the magnets is the same but do not draw any conclusions just yet; there is more to know that you were never taught.

Moving Forward

At this point, you should at least see that there just might possibly be an unseen issue with the Work Energy Theorem.   And if this is the case, the best way to move forward is by going backwards.  In other words, it is time to examine the history of Momentum and the Work Energy Theorem.  Here we will find how and why the physics community accepted both physics principles — momentum and kinetic energy.   And in doing that, we will uncover something astonishing providing we observe closely enough.

Click on this link, "History", to see the answers to questions no physics students ask.


NOTE:  This website is under construction.  There will be more information in the next few days and weeks, all of which can be verified independently and add to the woes of the Work Energy Theorem.

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