Following the basic pattern established at the start of our seasonal countdown to Newton's birthday, today's equation defines a piece that was left hanging in yesterday's post:
This is the technical definition of "work" in physics terms. It's also probably the scariest-looking equation to this point, as it explicitly involves vector calculus-- there's an integral sign, and a dot product. The basic concept is simple enough, though: you look at the force F exerted on an object, multiply it by the distance dr that the object moves under the influence of that force, and then add up the Fdr values for every step along the path from your initial position to your final position.
So, a simple idea expressed with scary math. Why is this important?
Well, if you recall yesterday's equation, the Energy Principle, you'll remember that work is how you change the energy of a system. If you do work on something, you can increase or decrease the energy available to it-- if I take a heavy book and lift it up to a high shelf, I have done work on the book (exerting the force required to lift it through the distance that it is raised). In the process, I have added energy to the system consisting of the book and the Earth-- if the book is tipped off the shelf, it will fall due to the gravitational interaction with the Earth, and acquire kinetic energy.
Work makes the essential connection between the more concrete world of Newton's laws-- where you deal with directly measurable quantities like force, mass, and velocity-- and the slightly higher-level abstraction that is energy. In the language of Newton's laws, forces act to change momentum, while in the language of energy, forces are responsible for the flow of energy within and between systems, through work.
If you're comfortable with calculus, you can also turn this around, and learn something useful. Energy accumulates due to work, which is force integrated over position. We can reverse this process by taking the derivative of energy with respect to position, which, for the right kind of interactions, gets us back to force.
This is why physicists interested in the motion of some system of objects will often talk about it in terms of a potential energy curve. If you know the potential energy as a function of position, you can figure out exactly what happens to the system at any point, just by taking derivatives (which are much easier than integrals). This is an extremely important result, because it carries over into quantum mechanics, where force and position and velocity are not well-defined quantities (unless you're a Bohmian. You can still think about a quantum system in terms of the energy, though, so energy takes on a central role when talking about the quantum world.
So, take a moment to appreciate the importance of hard (mathematically, anyway) work to physics. And tomorrow, we'll have another equation for the season.
One take: conservation of energy must apply in conservative fields - sounds tautological, but given radial field under some power law, that equation maintains no net energy change over any given closed path. If we have something unusual, like circulating lines of E near a solenoid with changing current, then subtle interactive connections must be taken into account - the field itself is not conservative.
I look at this equation and I see roller coasters!