Re: How can the equation for work (W=Fx) be proven?

Hi Joel,

That's a very interesting question, indeed. It does seem mighty strange when someone says you haven't done any work while you're sweating from holding a 30lb bag for the last 1/2 hour. But, as I hope you'll soon see, the scientific definition makes the popular definition of work a little more precise. I'm not sure if this will convince you, as there are a multitude of approaches to this question. But, this approach satisfied me when I was wrestling with this same question. So, here goes!

Let's start out by "deriving" the formula for work. One way to get to the formula for work is to go from Force considerations to Energy considerations. All moving objects have kinetic energy and what we want to do here is to relate the kinetic energy to the forces that are acting on that object. Let's start off by building on one of the better-known kinematic equations (to simplify things, the following discussion is restricted to the one-dimensional case):

1) v² - v₀² = 2ax

where v is the velocity, v₀ is the initial velocity, and a is the acceleration. And, since we know that:

2) F_net = ma

we can get:

3) v² - v₀² = 2(F_net/m)x

Here, x is the change in the object's position (i.e., the displacement from the intial position to the final position). Now if we divide by 2 and multiply out by m, we end up with something a little more familiar on the left side:

4) ¹/₂ mv² - ¹/₂ mv₀² = F_netx

Please note three things about this: first, both the F and a are constant; and second, there is no time dependence anywhere. If the object is subjected to a constant net force, all we need to see how the speed changes with distance is (4). Finally, the third thing to notice is that the term on the left is just our regular KE; i.e., the change in Kinetic Energy! So what we've managed to stumble across is that the change in energy can be found by seeing what F_netx is. This term, F_netx, is what we call work and, from the above "derivation", we can see that work is really just a measure of the change in energy of an object. This is why work is defined as Fx. I guess another way of looking at it, in regards to your question of Ft, is that F is also p/t. Thus, if you multiply F by t, you get p, the change in momentum. Getting back to the physical definition of work in terms of energy, we have:

5) Work = F_netx = ¹/₂ mv² - ¹/₂ mv₀² = KE

This has very important consequences for objects subjected to conservative forces (the fundamental forces, as it turns out, are all conservative). The work that these conservative forces do to make an object gain or lose energy can be regained! That work is "stored" and and can also be thought of as a form of energy. This energy is termed potential energy and, appropriately enough, this is a function of the position. And, as the famous conservation of energy states, it is this combination of potential energy and kinetic energy that is a constant; thus, energy is conserved for conservative forces. Nonconservative forces, on the other hand, dissipate energy... e.g., friction converts energy to heat, abrasion, noise, etc. However, at the microscopic level, everything can be broken down so that only the fundamental foces apply, and, since all fundamental forces are conservative, this is a very useful concept, as you can see.

All right, this is all well and good, but it might not be completely satisfactory so I'll try to address the specific questions you raise. I hope I've shown why work is defined as Fx and why Ft is different, but now let's look at what's going on when someone says you're not doing any work while you're sweating holding that 30lb weight suspended in space. The work that you do in "fighting" gravity is actually chemical work. It's at odds with the physical definition of work because that's being applied, in this case, to the macroscopic definition, so to speak.

Energy, of course, is also conserved at the microscopic level. What we usually do is differntiate between chemical work and physical work but, in the end, everything can be broken down to a physical description. To demonstrate this, consider what's actually going on at the cellular level in your body when you lift a weight (without going into too many details). Your muscles are made up of muscle fibers, which come in two varieties. They are the fast twitch (these are the red muscle fibres because they're so rich in myoglobin) and the slow twitch fibres (this is the white meat you usually eat in a chicken :). Fast twitch fibres come into play when you need a lot of strength for a short duration and slow twitch fibres come into play when you need longer exertion, though not at maximal intensity.

A muscle is usually composed of both these kinds of fibres. So what happens when you hold a weight for a long time is mainly the slow twitch fibres fire. And this mechanism is very interesting. The arrival of a signal (Ca²⁺) causes the muscle fibre to contract. But, it needs some kind of energy input in order to relax and elongate. The universal energy carrier in biology is the all-purpose ATP. Here, ATP attaches to the fibre, then hydrolyzes to ADP, causing it to relax (myosin head releases actin filament) and also releasing energy. And the fibre is ready to start the process again. This process continues as long as your brain continues to signal the muscle to keep contracting (as long as Ca²⁺ is present). The hydrolysis of ATP to ADP releases approximately 54kJ of energy for each mole of ATP. This energy, if the reaction had proceeded in vitro, would just be wasted as heat to surroundings. But in a system like the body, much of this energy is utilized to do chemical work (e.g., in our case, helping the muscle fibre enter the relaxed state). But this process is still not 100% efficient. Some of the energy is used to power other processes (this is called coupling between reactions and is the essence of biophysics, especially in applying thermodynamics at the cellular/molecular level) and some is again lost to the environment as heat.

This increase in temperature of the surroundings is "carried off" by the blood in order to prevent the core temperature from getting too high. The hot blood is sent to the skin and subcutaneous areas where most of the heat is eliminated via evaporation (through sweating) and conduction (some is also radiated away, of course). The sweat that shows you you're doing "work" is actually an indication of the chemical work your body did and not "physical" work that you're doing against that object. Of course, what's "actually" going on is much more complex than this simple summary (you might want to refer to one of my previous answers here). The main thing to remember is that all energy is conserved. It certainly changes form, from mechanical to chemical, but it is conserved. The body is a wonderfully complex and, in it's way, efficient machine but this also shows that when you consider the body as other than an automaton, you have to bring in many more concepts. Most of science is looking at simplified cases to gain some measure of insight because what's "actually" going on is usually too intricate to study thoroughly and simply.

The next example, that of earth's gravity adding more and more energy to a falling object, is not exactly true. As you know from the previous discussion, an object has a certain potential energy by virtue of it's position. Thus, the total energy, the potential plus the kinetic, is a constant. An object has a certain amount of total energy and, when it is subjected to a gravity field, converts some energy from one form to the other (in this case, from potential to kinetic). You can think of a falling object as converting some of the potential energy it had at it's initial position (the top) to kinetic energy as it falls to it's final position (the bottom). Thus, the total energy remains the same... i.e., constant. And, as you also know from the previous discussion, this potential energy reflects the amount of work it can do.

I hope that helped. That was probably a lot more info than you wanted but it seemed worth the risk to go off on tangents to make it, I hope, a little clearer. Still, I know a lot of these concepts can seem obscure and strange but if you stick to it, and look at explanations from many different perspectives, I'm sure you'll also see that they do make a certain sense. I can't emphasize enough looking at things from different perspectives. Usually, if you read more than just one explanation of some phenomenon, it helps develop a better understanding of the principles. To this end, I can't recommend anything more highly than Prof. R. P. Feynman's excellent series, The Feynman Lectures. The first volume is especially pertinent to our discussion. If you're still feeling adventurous after reading the relevant sections of that volume, you can plow right into any good introductory physics text. The ones I'd recommend are by Giancoli, Halliday & Resnick, or Fishbane, et. al. This can seem like quite an intimidating reading assignment but remember that you don't have to read it all... just skim the relevant sections when you have time. If you really give this variety a try, you'll find that one explanation strikes a chord and will suddenly make everything completely clear and lucid. That moment of insight is definitely worth the work you might have to put into it! And, of course, you can always email me (at science@zentropy.com) if you have any further questions.

Good hunting!

Rick.

P.S., if you want to explore the biological/chemical side of work a bit more, I highly recommend reading the "bible" of MCB: Molecular Biology of the Cell by Alberts, Lewis, et. al. (I've got the 2nd edition but I know there's a newer edition out with a lot more pictures :) Although it is intended for the advanced undergraduate/graduate level, you'll find it surprisingly readable. Give it a look!