Work and Energy
Overview
By considering the work and energy of a system, we can deduce behavior without working too hard.
Newton's Third Law
Newton's Third Law is that any force acting on a body has a counter-force acting on the actor, in equal but opposite magnitude.
In other words, when you sit in your chair, you're pushing your chair down as hard as it is pushing you up.
This law has some rather remarkable consequences.
First, you can't get something for nothing! There are no ex nihilo forces. Everything comes in pairs. If you want 5 pounds of force, you have to have -5 pounds of force appear somewhere else. (Accountants should be familiar with this concept.)
Second, this leads to several conservation laws.
Conservation Laws
Physics is littered with conservation laws. These laws simply state that some quantity will never change over time.
Most come with caveats. For instance, mass is conserved in the macro, non-relativistic world we live in. But in the extremes of quantum or relativity, mass can be created or destroyed.
Conservation of Momentum
Because every change in momentum (= Force) is marked by a change in momentum of equal but opposite magnitude on the acting particle, the momentum of a system of particles remains constant over time, provided that there are no particles from outside the system acting on particles inside.
This is the Conservation of Momentum. Of all the physical conservation laws, this one seems to be the most absolute, and extends in all directions and all extremes.
Conservation of Energy
Energy is also conserved, provided no external forces are causing any work to be done in a system of particles.
What is Energy?
First, let's start with a precise definition of "work". This is very different from the word "work" you are familiar with, such as, "I'm going to go to work now", or "I've got lots of work to do for my physics assignment."
Digression on Terminology
This is an unfortunate pattern in physics. Physicists borrow words from every day language and then change the meaning ever so slightly. This confuses the layman to no end. It also means that physicists sometimes confuse themselves in casual conversation.
The other option was to invent entirely new words for things. "Force" could have been "Foo", and Work could have been "Woo". The problem with this is that now it sounds like physicists speak a gibberish code that has nothing to do with reality. (The names of the quarks are a good example of this: up, down, top, bottom, charm, strange. There were more fantastic names considered but rejected.)
This is a problem in any field. Medicine is littered with Greek and Latin terminology that is absolute gibberish to the common non-medical person. Psychology seems to speak their own language about the human soul. Mathematicians talk about fields and spaces and folds and layers but they are always in their own world anyway.
If you want to be a physicist, you have to master the terminology used. You have to know the precise meaning of the words we use, and you have to use them in the same way. If you are confused by a new term, then don't allow it to slide.
Work
Work is simply this: a force is applied, and the object is moving. Work is the force times the distance. In vector multiplication, it is <math>W = \vec F \cdot \vec d\,</math>. Note that because we are doing a dot-product, we are only interested in the distance traveled by the target in the direction (or anti-direction) of the force applied. The dot product is simply <math>\vec a \cdot \vec b = |a| |b| \cos \theta\,</math>, where <math>\theta\,</math> is the angle between the two vectors.
Work is a scalar, a one-dimensional quantity that doesn't have a direction or components. Work can be positive (the object is moving in the same direction as the force) or negative (they are opposed to each other.) When a force exists but work is zero, that means the particle is traveling perfectly perpendicular to the force. Such a situation arises when you have orbits of one object circling another, with a force that pulls the object inward.
What does work mean? Work is basically a measure of how much the system has changed due to a force. If a large force is applied, and an object moves along that force a large distance, then a large amount of work has been done. Small forces and small distances make smaller amounts of work done.
Energy
Work has a time component to it. Each moment, a certain quantity of work is done. But rarely do we think about the work done at an instant. We are usually concerned with the amount of work done over a period of time. This gives us a sense of how much has changed between the two points in time.
This sum of all work done on a particle over a period of time is the Energy of the particle. Think of energy as a battery. Energy is stored energy. The work one particle does on another ends up stored in a battery that the other particle has. It can later, possibly, put that energy to work on another particle.
Energy comes in many forms.
Kinetic Energy
Kinetic Energy is the energy stored in the motion of the particle. It is simply <math>K = {1 \over 2} m v^2\,</math>. (I won't bother showing how to derive this: it's simply the integral of the momentum.) <math>v^2 = \vec v \cdot \vec v = |v|^2\,</math>, in case it's not clear.
Note that Kinetic Energy is stored energy that can later be released without any loss in motion. Other forms of energy are not so kind. (We'll study thermodynamics later.)
Note also that a particle traveling at twice the speed has 4 times as much Kinetic Energy, while a particle twice as massive only has 2 times the Kinetic Energy. This is why auto crashes become devastating at 60 MPH while they seem to be survivable at 30 MPH. Cars traveling at 60 MPHs have to dissipate 4 times as much energy to come to a full stop. This energy ends up in the structure of the car and in the bodies of the passengers. If you were to travel at the dangerous speed of 120 MPH, your car would have 16 times as much energy as it did at 30 MPH. To put this in human perspective, if a crash at 30 MPH had enough energy to kill a single person, then a crash at 60 MPH could kill 4 and a crash at 120 MPH could kill 16.
Potential Energy
Consider the way gravity works near the surface of the earth. Stuff near the surface accelerates downwards due to the gravitational force at the speed of <math>g = 9.8 m/s^2\,</math> which means you have a force of gravity <math>\vec F_g = -mg\hat k\,</math>, where <math>\hat k\,</math> is the unit vector pointing up. It takes an opposite force simply to hold a particle still.
Suppose you wanted to move the particle up. You'd have to apply a greater force to get the particle moving up, and then a lesser force to have the particle stop where you wanted it to. The greater and lesser forces contribute, 100%, to the Kinetic Energy. But throughout the process, you had to apply a force to oppose gravity along the distance the particle traveled. You had to do additional work above and beyond the work for Kinetic Energy.
The potential energy due to gravity in this scenario is simply <math>P = mgh\,</math>. (<math>h\,</math> is the height of the particle.) You get to choose where your zero point is, either at ground level or some arbitrary height. No matter where the particle is, if it moves down, work is being done on the objects around it, while if it is moving up, work is being done on it.
There are other potentials energies we will discuss in the future. For now, this will suffice.
Simple Analysis: Particle Sliding Down a Hill
With these two energies fully analyzed, we can correctly determine the behavior of certain problems without considering forces at all.
Consider a particle of mass <math>m\,</math> at the top of a hill inclined <math>45^\circ</math>. After it has fallen from rest a certain distance vertically, how fast is it going? Well, <math>mgh\,</math> gives us the amount of potential energy that is transferred into Kinetic Energy. So:
<math> \begin{align} mgh &= {1 \over 2} mv^2 \\ gh &= {1 \over 2} v^2 \\ v^2 &= 2gh \\ v &= \sqrt{2gh} \\ \end{align} </math>
We don't consider the negative velocity because this is the magnitude of a vector. Note that it doesn't matter what angle the hill was at. That only affects the direction, not the magnitude of the velocity.
Energy Lost Due to Friction and Drag
Some forces act in opposition to motion, and end up simply decreasing the amount of useful energy available. Friction and drag are such forces. Since these forces do not behave like batteries, we simply write off the energy lost, never to be recovered.
Plastic deformations, such as the mangling of a car in a wreck or the modeling of clay, is also energy lost that cannot be recovered.
These energies tend to gum up all of our beautiful equations, so we pretend to ignore them, or we just take into account what it would take to counteract them. For instance, a plane flying experiences drag. The thrust is in opposition to this, just to keep the system moving. So we can calculate, rather easily, the amount of fuel burned just to maintain speed at different velocities. We don't even really need to understand how the forces relate to velocity; simply measuring the amount of fuel consumed to maintain constant speed is enough.
