Work and Energy - and the great conservation laws

Physics has as among its most powerful tools the conservation laws. Energy, momentum (both linear and angular), charge and other quantities are conserved. Basically, this means that in an interaction the value of these quantities before and after the interaction remain unchanged. A familiar example of a conservation law in action is the conservation of angular momentum exhibited by a figure skater during a spin. The law states that that a system's (the skater’s) angular momentum, defined as the product of the skater's mass M, rotation speed V and radius R (the average distance to the various body parts, measured from the rotation axis) remain unchanged during the spin. As an equation this becomes (MVR) _initial = (MVR) _final. The skater begins with arms and legs extended as much as possible (a large value of R). Pushing to gain maximum spin velocity (V), the skater obtains a fixed value of angular momentum - the initial value in the equation. Once spinning, the arms and legs are drawn in, decreasing the effective value of R. Since the skater's mass remains constant, the decrease in R is matched with the increase in V that makes the spin so spectacular. On a more universal scale, the same conservation law keeps the planets in stable orbits about the Sun.

Closely associated with energy is the concept of work. It is, in fact, the means by which we define energy. These topics are discussed in what follows.

Conservation of Momentum

Newton spoke of mass as the quantity of matter, and momentum as the quantity of motion. Both these quantities lead to important results. Look first at momentum:

Typically the letter p is used to represent the momentum of an object. We write p = m v.  (The momentum of an object is, as defined by Newton, its mass times its velocity). Starting from Newton's second law we can write

F = ma = m Δv/Δt, since acceleration a is just a measure of how velocity changes with time. As was implied in the page on Newton's laws, this form of the second law assumes the mass remains constant. The more general statement of the second law moves the mass back along side the velocity, providing F = Δ (m v)/Δt, or F = Δp/Δt. This form reduces to the familiar form F = m a in the case of constant mass, but allows for a time varying mass if the occasion demands (as, for example, in the case of the launch of the space shuttle, where the fuel greatly outweighs the payload).

Suppose two objects (we like to call them particles) collide head on, one coming in from the left, the other from the right. According to Newton's third law, the force that the first particle exerts on the second is equal and opposite to the force the second exerts on the first. As an equation, we write F_1-2 = - F_2-1 where the subscript 1-2 means the force Particle #1 exerts on #2, and 2-1 the force #2 exerts on #1. In other words, the forces are equal and opposite. Using Newton's second law we can rewrite this as Δp₂/Δt = - Δp₁/Δt. This simply states that when the force of #1 is felt by #2 the result is a change in #2's momentum, and likewise in the other direction. If the right-hand term is moved to the left we have Δp₂/Δt + Δp₁Δt = 0, which says that the total change in momentum of both particles is zero. How can the total change be zero? Only if the total momentum (p₁ + p₂) remains constant. Note that either one or both can change, but the total remains constant. This is the law of conservation of momentum. It is the first of the great conservation laws that provide us with powerful concepts for our study of physical phenomenon. Here's an example of how it works:

In the game of billiards the cue ball strikes an (at rest) object ball, and then stops. How fast does the object ball move after the collision? Answer: Just as fast as the cue ball was moving. Why? Conservation of momentum!

Conservation of Energy

What is energy? Everyone knows what energy is - we use it all the time, but how is it defined? Write down your definition before you read further. Chances are you had a difficult time defining energy. Here's a definition from Webster's New World Dictionary, Third College Edition: "the capacity for doing work" (along with many other, non-scientific definitions). OK, then what is work? "the means by which energy is transferred" (again, along with many other definitions). Here's how we choose to define (one kind) of energy:

We start with a definition of work: Work is defined as the product of the force applied to an object times the displacement of the object due to the force. We are careful to count only the force which is applied in the direction of the motion. Here's an example: You push a box along the floor, pushing exactly in the direction the box moves. If your force is 10 Newtons and the box moves 3 meters you have done 30 Newton-meters of work. (A Newton in the SI system of units is equal to about 0.225 pounds; a Newton-meter is more usually called a Joule). Now suppose you push the same box the same distance, but this time you push downward at an angle of, say 60⁰ from the floor.  Now some of your effort is trying to push the box into the floor. To get the same 10 Newtons of force in the direction the box moves you have to exert a total force of 20 Newtons. You do the same work - 10N times 3 m, but you may be a bit more tired if you move the box this way. (Actually, it's a bit more involved, because the downward part of your applied force has the effect of adding to the weight of the box, but the point's the same: Only the portion of the force in the direction of the motion counts towards the work done). 

So: Work = Force x distance (remembering that it’s the force in the direction of movement that counts). Suppose a force F causes a movement Δx. The work done is FΔx. Since F = m a, it follows that F Δx = m a Δx. Remember Galileo's equations of motion? I reproduce them here for convenience (I've changed y to x and x to x - x₀ to make things a bit more general):

  
A little algebra should convince you that, if you eliminate time from these two equations you arrive at an expression relating velocity, position and acceleration. It turns out to be (you should verify the result for yourself)

.

Now x - x₀ is just Δx and, if I solve this equation for a and put the result into the equation for work above (F Δx = maΔx), the result is

.

F Δx is the work done on the mass m. The expression on the right is defined to be the kinetic energy of the mass m. (Actually the right hand side represents the change in kinetic energy - its initial kinetic energy was 1/2 m v₀² and its final kinetic energy was 1/2 m v²). 

So, energy is a concept. It is an idea useful in describing what happens to something when work is done on it. The change in kinetic energy of an object is a measure of the work done on it. We call kinetic energy "energy of motion" (for obvious reasons). 

Remember when we multiplied F by Δt and eventually arrived at conservation of momentum? It turns out that multiplying F by Δx provides another conservation law, conservation of energy. In its most general form it states that the total energy in the universe is conserved. We may change its form from one kind of energy to another, but the total remains constant. Perhaps the most remarkable thing about this law is that it deals with our invented concept of energy. It is actually a convenient bookkeeping scheme upon which the very foundations of physics are built.

I speak of conservation of energy, but it’s easy to find a problem with what I have said so far: Think of two identical pucks on an air table coming toward each other at some speed v. They each have kinetic energy 1/2 m v². After they collide they are each moving away from each other and it's easy to show that they will each have kinetic energy 1/2 m v² as before - energy is conserved. But for a brief time both pucks have stopped, at the instant they collide. Where was the energy then? Well, as they collide the two pucks distort somewhat and then regain their original shape. Then the pucks push away from each other, thereby getting their kinetic energy back. We say that there is energy stored in the distorted pucks (as if the pucks were made of tiny springs that compress and then release). This form of stored energy is named "potential energy". Here's an example of potential energy in a different form: Suppose you have a weight (give it mass m) that you lift a height h above the floor. How do you do his? You apply a force that overcomes Earth's gravity and exert that force over a distance h - you do work! The work done on the weight results in it’s upwards movement. Once at the height h you can hold it there (or put it on a shelf) and, whenever you want, let it fall to the ground. As it falls, it converts that stored potential energy into energy of motion - kinetic energy. So it is the combined energy that is conserved - kinetic plus potential (in this case gravitational potential energy). Carry the process one step further: On the floor under the weight place a coiled spring. The weight falls, lands on and compresses the spring and then rebounds up again, gaining (gravitational) potential energy, reaches the height h, falls, rebounds, on and on forever. This is great! You do work raising a ball once and, since energy is conserved, it bounces forever as it converts gravitational energy into kinetic energy into spring energy (another form of potential energy), back to kinetic energy, etc. This is a perpetual motion machine! Try it! It won't work. Why? We forgot to consider friction. Eventually the weight will stop bouncing. Well, there goes conservation of energy. Not quite. Let's just include the frictional heat as part of the energy. Run another thought experiment: Give a weight a push across the floor: You push (apply a force) over a distance to get the weight going (and so do work equal to FΔx on the weight). This changes its kinetic energy from zero (the weight was at rest to begin with) to 1/2 m v². After a while the weight stops, as friction opposes its motion and brings it to a stop. What has happened to the energy? The work done was converted to the heat due to friction. As was argued by Julius Robert von Meyer(5), (with some paraphrasing by me) "If energy, once in existence, cannot be annihilated but only change its form, what other forms is it capable of assuming?" He then argues "If work can be converted to heat, then heat must be a form of energy, and must naturally be equivalent to kinetic and potential energy." The direct measurement of the "mechanical equivalent of heat" was done by James Prescott Joule in his famous Paddle Wheel experiment. This continually expanding role of energy was finally set into comprehensive form by Hermann Helmholtz who concluded that energy was the sum total of kinetic, potential, heat, electrical and "other forms to be discovered or invented". 

So we have it: Energy has many forms (but only one form of kinetic energy, 1/2 m v²). If in our searching we find a process that seems to violate the conservation of energy, we simply invent another form to take up the slack. Then, of course, begins the task of defining its properties and experimentally verifying its existence.

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