# History of the Work Equation

1. May 27, 2010

### Jacksilver

Hey everyone!

I want to know how the equation W=Fd was formulated?
That is to say: why is Work determined as the total force TIMES the distance traveled in the direction of the force.
It seems almost arbitrary though perfectly logical (more force or more distance means more work).

Was this equation the result of some sort of experiment (empirical equation)
or it's possible to mathematically derive it from other equations? If so: how?

Thanks

2. May 27, 2010

### Jerbearrrrrr

Oh interesting question.
Well, I'm sure Newton or Hooke did the first major "play around" with work and energy using calculus. Whatever arguments they used would be nowhere near the standard of rigour required today, but the answers were still right (for a couple of centuries).
If you haven't calculused before, for Newton, calculus was basically a way of approximating a curve (difficult thing that changes) as a sequence of straight lines (easy things, but many of them), and so was able to calculate the "product" of two changing things (W=Fd - but what if the force you apply goes up and down? calculus methods let you get W in this general case).

Using calculus, you can show a number of things that point towards W=Fd. (F*parallel d, that is)

For example, you can show that the integral of force with respect to distance (by this, I mean F*d's total for varying forces) is how much you increase something's kinetic energy. You might like to even define that as the work done by a force (how much energy you give it).
[Even for a constant force, it's best to use calculus here because applying a force accelerates the body (in general) and so you need to take into account of changes in the system.]

He might also have played around with gravitational potentials, where if you equate various expressions for energy, you can pull out W=Fd analogues, where F is weight.

3. May 27, 2010

### Jacksilver

Wow! very quick answer and even though I don't know anything about calculus - I got it! :)
So the root of the equation is, as I suspected, mathematical (or "calculustical"..) if I understand correctly?

Thank you!

4. May 27, 2010

### dulrich

I like to think of it the other way around. To define kinetic energy as the energy generated by doing work on a force-free particle. In this way one is able to show why there is the one-half factor in the formula. It also helps explain why the formula for kinetic energy is different in relativity.

If I remember my history correctly, the idea of virtual work predates even Newton. The simplest example is the lever. In order to raise a 2 newton weight 1 cm, you can push down on a lever with 1 newton of force -- but you have to push 2 cm in order to make it work. In other words, in any simple machine (ignoring friction, etc) the product of force and distance is the same in both the input and output. This conservation motivates giving it a name.

Last edited: May 27, 2010