Conservation of energy

Notes from "Chapter 4: Conservation of Energy", from The Feynman Lectures on Physics: Volume I.

4.1 What is energy?

The law known as the conservation of energy is a mathematical principle that says there is a numerical quantity called “energy” that does not change when something in nature happens.

Energy has a large number of different forms, and there is a formula for each one. These are: gravitational energy, kinetic energy, heat energy, elastic energy, electrical energy, chemical energy, radiant energy, nuclear energy, mass energy. If we total up the formulas for each of these contributions, the total amount of energy in a system will not change except for energy going in and out.

In physics today, we have no knowledge of what energy actually is. However, there are formulas for calculating some numerical quantity, and when we add it all together, it stays consistent when we account for all things in a system. It is an abstract thing in that it does not tell us the mechanism or the reasons for the various formulas.

4.2 Gravitational potential energy

Every reversible machine, no matter how it operates, which drops one pound one foot and lifts a three-pound weight always lifts it the same distance, X. It is impossible to build a machine that will lift a weight any higher than it will be lifted by a reversible machine.

In general, if one pound falls a certain distance in operating a reversible machine; then the machine can lift p pounds this distance divided by p. Put another way:

weight~lifted~\times height~lifted \\ = weight~lowered~\times distance~lowered

3lbs \times \frac{1}{3}ft = 1lb \times 1ft

Generalized, we can say that

\begin{pmatrix} \text{gravitational}\\ \text{potential energy}\\ \text{for one object} \end{pmatrix}= (\text{weight})\times(\text{height})

The general name of energy which has to do with location relative to something else is called potential energy. In this particular case, of course, we call it gravitational potential energy.

In high school we learned a lot of laws about pulleys and levers used in different ways. We can now see that these “laws” are all the same thing, and that we did not have to memorize 75 rules to figure it out.

For example, how heavy must W be to balance 1 lb on this inclined plane? To keep things in balance, the gravitational potential energy of the left block must equal the gravitational potential energy of the right block.

Since we can define GPE as weight * height near the surface of the Earth, we can represent the balance of these two blocks as

WH_1 = W_2H_2 \\ WH_1 = 1lb \cdot H_2 \\ W = \frac{H_2}{H_1} \\ W = \frac{3}{5}lb

The height of the left block, $H_1$ , is considered 5 ft because it can be lowered 5 ft by the right block. The height of the right block, $H_2$ , is 3 ft, because it is 3 ft at its maximum height. This problem can also be solved using force components with Newton’s second law, but here we’ve used conservation of energy.

Similarly, suppose in this scenario, when the weight of W is lowered by 4”, suppose the center of the beam is raised by 2”, and the right quarter is raised by 1”. What is the weight of W?

4~in \cdot W~lb = 2~in \cdot 60~lb + 1~in \cdot 100~lb \\ 4W = 220~lb \\ W = 55~lb

4.3 Kinetic energy

Consider the motion of a pendulum as it swings back and forth. Since gravitational potential energy is $weight \times height$ (near the surface of the Earth), the ball of the pendulum obviously loses GPE as it loses height at the center of the pendulum. But naturally, it swings upward again. The type of energy that allows that to happen is kinetic energy.

In the motion at the bottom must be a quantity of energy which permits the ball to rise a certain height, which has nothing to do with the machinery by which it comes up or the path by which it comes up.

The kinetic energy at the bottom equals the weight times the height that it could go, corresponding to its velocity: $KE = W\cdot H$ . So, how do we find the height that it could reach? In terms of velocity, this can be defined as

KE = \frac{WV^2}{2g}

The fact that motion has energy has nothing to do with the fact that we are in a gravitational field. It makes no difference where the motion came from. But both this formula for kinetic energy and there prior formula for gravitational potential energy are approximate. The first because it is incorrect when the heights are great, i.e., when the heights are so high that gravity is weakening; the second, because of the relativistic correction at high speeds.

4.4 Other forms of energy

We can continue in this way to illustrate the existence of energy in other forms. If we pull down on a spring, we must do some work, for when we have it down, we can lift weights with it. Elastic energy is the formula for a spring when it is stretched.

So far, we’ve cheated by saying that machines are reversible, or that they go on forever, but in reality, everything stops eventually. Where is the energy when the spring has finished moving up and down? This brings in another form of energy: heat energy. Heat energy is really just kinetic energy, but of another form–the internal motion of the atoms of an object.

There is electrical energy, which has to do with pushing and pulling by electric charges. There is radiant energy, the energy of light, which we know is a form of electrical energy because light can be represented as wigglings in the electromagnetic field. There is chemical energy, the energy which is released in chemical reactions.

Chemical energy is composed of two parts: kinetic energy of the electrons inside atoms, electrical energy in the interaction between electrons and protons.

Next we have nuclear energy, which is involved with the arrangement of particles inside the nucleus. We have formulas for that, but don’t have fundamental laws yet. We know that it is not electrical, gravitational, and not purely kinetic, but we do not know what it is. It seems to be an additional form of energy.

Finally, associated with the relativity theory, there is a modification of the laws of kinetic energy, so that kinetic energy is combined with another thing called mass energy. An object has energy from its sheer existence. If a positron and an electron collide, radiant energy is released in a fixed amount that is wholly dependent on its mass. This is the famous $E = mc^2$ equation.

The law of conservation of energy is very useful, because we can use energy formulas to solve problems that would otherwise take much more effort (for example, we solved the pulley weight problem by balancing gravitational potential energy instead of needing force diagrams).

There are a few other conservation laws which are analogous to the conservation of energy:

the conservation of linear and angular momentum (it doesn’t matter where you do experiments or what orientation the apparatus is at to achieve consistent results)
the conservation of charge (no net new positive or negative charges are created)
the conservation of baryons and leptons (the number of baryons, like neutrons and protons, and leptons, like electrons, muons, and neutrinos, does not change in a reaction)

You may have heard the the energy of a photon is simply Planck’s constant times the frequency. While this is true, the frequency of light can be anything—there is not law that says a unit of energy needs to be certain fixed amount.

So we do not understand this energy as counting something at the moment, but just as a mathematical quantity, which is an abstract and rather peculiar circumstance.

The laws which govern how much energy is available are called the laws of thermodynamics and involve a concept called entropy for irreversible thermodynamic processes.