Another Stupid Physics Thing

One of my sophomore physics instructors was kind of a goofball, and he told a snarky anecdote which I will now paraphrase.

First, the physics. There's this thing called gravitational potential energy--and when an object can decrease its gravitational potential energy, it will. This is almost an overly complicated way of saying "things fall towards the earth," but in fact it is a quantitative rule one an use to calculate not only that an object will fall, but how hard it will splat when it hits. This is accomplised by writing the gravitational potential energy as a mathematical expression:

potential energy = mass × gravitational acceleration × height

or for the symbolophiles among us,

U = mgh

As an object falls, its gravitational potential energy gets converted into kinetic energy (motion), or we say, "The gravitational field does work on the object, to increase its speed in a downward direction." Potential energy stores work, so when that initial mgh of potential energy decreases by a certain amount, the falling object's kinetic energy gets that certain amount back because of the work done. As the kinetic energy increases in this way, so does the falling object's speed. In fact, the total energy--gravitational potential energy plus kinetic energy--is a constant throughout (until the object hits the ground and, with any luck, goes splat).

This is sort of why we say "energy is conserved"--energy is neither created or destroyed for a falling object. In fact, a force field like that of the earth's gravitational field is called a conservative force field.

Similarly, if we increase the potential energy of an object during some process, then we have to have "done work" some time in the process. We have to supply energy by lifting with our muscles or burning fuel to run an engine (which is actually what our muscles are anyway) or make some other object decrease in potential energy (which is really what the engine is doing, too) in order to increase the object's potential energy.

Another property of a conservative force field, related to the total (kinetic + potential) energy being a constant, can be seen from the expression U=mgh for the gravitational field. This forumla makes reference to an object of mass m at a height h above some reference point (such as the ground). It makes no reference to how the object of mass m got to height h. Those details don't matter: all that matters is the height h.

So the physics slapstick goes like this. A weightlifter is bench-pressing a barbell. After his workout, he replaces the barbell back onto its rack, where he found it. At this point, a fruity physicist walks by and remarks, unkindly, "Since the barbell is at its original height h, its potential energy is mgh, exactly the same as before you started your workout, when its energy was also mgh. So therefore the energy change of the barbell is zero. This means that you did no work on the barbell! If you had done work on the barbell, its height would have changed during the process (your workout)."

Now, I don't know that "you did no work on the barbell" is exactly what my instructor said. If he had it would be totally wrong, so wrong, in fact, as to smell.

It is true that the potential energy of the barbell is the same before and after the workout. But certainly the weightlifter feels like he has done work! This is because the weightlifter has done work. He has expended chemical energy pushing on the barbell in order to make it go up and down at a controlled rate. In order to push the barbell up, let's say he has expended an amount of energy Δe.

Energy, however, cannot be created or destroyed, so that energy Δe must have gone somewhere. It can't have ended up in the barbell--remember, its energy is the same before and after the workout, since its height is the same before and after and all of its energy is potential energy, which is proportional to the height.

Where did the energy Δe go, then? We can figure this out by breaking down the process of pushing a barbell up from its original height h then letting it fall back down to the same height h.

1. As the barbell goes up, there are two forces acting on it: the force of gravity, which pulls it down, and the weightlifter, which pushes it up. The weightlifter expends energy, and, since the barbell increases in height, this energy goes into increasing the barbell's potential energy.

2. As the barbell goes down, the same two forces are acting. Gravity is pulling down, and, since the weightlifter is probably not letting the barbell fall freely but is controlling its speed, he is still pushing up (just not as hard as gravity is pulling down). In any case, gravity is now doing work on the barbell to pull it back down to its original height.

So two things happened during this stroke (and as it is repeated). The weightlifter did work on the barbell. Then, gravity did work on the barbell. And since the energy of the barbell is the same before and after, since it starts and ends at the same height h, the work that the weightlifter did and the work that gravity did are exactly the same magnitude and opposite in sign.

That is to say, all of the following statements are correct:

During the workout, the weightlifter did work Δe.
During the workout, gravity did work -Δe.
The net work done on the barbell is the work done by the weightlifter plus the work done by gravity.
*The net work done on the barbell is zero, as can be seen since its energy is the same before and after the workout.*

But the statement marked with the asterisks (*) is not the same as the following completely untrue statement

The work done on the barbell is zero.

It's like so: if the taco truck guy gave me $5, then I gave him $5, then the net change in my wealth is $0, but that doesn't mean that no money was exchanged. Don't trust physicists who are trying to snark at weightlifters!

Update: this morning, I got a call from a telemarketer offering me a free one-month trial of life insurance (so I get to try it out!). Except that it wasn't free, per se. I would be charged about $60, then if I decided the life insurance didn't fit or was the wrong color, I could return it for a full refund. They'd give me my $60 back. Except that this isn't a free offer. My net change in dollars would be zero, but that doesn't mean there was no exchange of money, just like zero net work doesn't mean that nothing happened ...