# Organizing the idea of energy

#### Prerequisites

The idea of energy is a critical organizing principle for all of science, not just physics. You've already read lots about many different kinds of energy and various principles about it — conservation (or not) of mechanical energy, the first law of thermodynamics, etc., etc. — and you have likely seen discussions of energy in all of your previous science courses. But it's more complicated and subtle than it may seem at first. Different disciplines have different ways of talking about energy and may even use different language. Before we move on to try to study some of the more complex ways of talking about energy (in particular, when is energy "usable"? — whatever that means), here is an overview to help us organize our thinking about energy.

## Energy, motion, and forces

The idea of energy arises out of trying to understand how forces lead to a change in speed. The speed of an object does not depend on its direction of motion. To change an object's speed, we needed a force either along or against the direction of motion. Forces perpendicular to the direction of motion only change the object's direction, not its speed.

The result is the emergence of the kinetic energy — $½mv^2$. Since it depends on the magnitude of the velocity (squared), it is not a directed measure of motion; that is, it doesn't tell you anything about what direction the object was going in, just how fast.

From taking Newton's 2nd law and multiplying it (really, taking the dot product) with the displacement in the direction of the force, we obtain the work-energy theorem:

$$\Delta \big({1 \over 2} \; m_A v_A^2 \big) = \sum_{j = 1}^N \int \overrightarrow{F}_{j \rightarrow A} \cdot \overrightarrow{dr}_A$$

This tells us that it is the work done by forces acting on an object along the direction of motion that cause the object to change its kinetic energy (KE).

For some forces (gravity, electric, springs), we are then able to define the negative of the work done by a force as a kind of energy of place — a potential energy. With these forces, the kinetic energy added or removed from an object by these forces can be reversed by going back to the starting point. The energy change can be associated with a change in position.

This didn't work for other forces (resistive ones, like friction, drag, and viscosity) which seem to "steal" kinetic energy from systems of objects. (Though recall, that these forces can speed up individual objects if they are slowing down others.) We are left at the macro level with a description of energy that is only conserved if certain forces are absent.

### Going micro — hidden energies, thermal and chemical

When we bring to bear our understanding of the structure of matter, we learn that energy that looks "lost" to us at the macro level is not actually lost, but just hidden.

Macroscopic matter is made up of atoms and molecules and our concepts of kinetic and potential energies, developed at the macro level, also apply at the micro level. The "lost" energy just goes into other forms of kinetic and potential energy — either the motion and interaction of the atoms and molecules of which matter is made, or at an even finer level, into kinetic and potential energy in the internal structures of the atoms and molecules themselves and their bindings to each other.

### Forms of energy — Choosing a system and transduction

Although the result of our developing an understanding of the structure of matter is the emergence of the idea that the total energy of the universe is actually conserved, that's not good enough. The fact that we convert energy stored in plants into our own life and motion has a different interest to us than the conversion of the energy stored in us into the life and motion of worms and saprophytic bacteria after we are dead. The fact that the total energy of the universe is conserved is not really the point. Where the energy is and what happens to it matters!

In order to use energy in a way that is useful and informs us about biological processes, we need to understand not just that energy is conserved, but to answer the questions:

• In what forms does energy reside? and
• What determines what happens to energy?

In order to answer the first, we have to choose a system decide what objects we are interested in, look at the energies of those objects, and understand how energy is transformed including moving into the system of interest, transforming forms within the system of interest, and moving out of the system of interest. The transformation of energy from one form to another is called transduction. The tool that we use to keep track of our system is the system schema.

Even though all energy is in some sense "the same thing", it is crucial to specify where and how it's localized. For convenience in describing transformations between macro and micro scales, we choose to define four forms of energy. (There are more — such as the energy of matter and the energy of light — that we will consider later and in other contexts.)

1. Kinetic energy — This is the energy of coherent motion, $½mv^2$. We use this term when we are talking about the motion of a single object and that object also has a (directed vector) momentum associated with it, $m\overrightarrow{v}$. We use this term both when we are talking about a macroscopic object and when we are talking about the motion of an individual atom or molecule.
2. Potential energy — This is what we have called the energy of place, associated with (conservative) forces between objects such as electricity or gravity. Again, we use this term when we are talking about specific objects that interact with forces we are identifying. We use this term whether we are talking about a ball moving in response to the gravitational force of the earth or about an electron moving within a molecule in response to the forces from the other charges (electrons and nuclei) in the molecule.
3. Thermal energy — This is just kinetic and potential energies of the atoms and molecules of a macroscopic object, but it is associated with the random chaotic motion that those atoms and molecules have as a result of their temperature. Since we are adding up the energies of many particles moving every which way, there is no (directed vector) momentum associated with it. It is useful to talk about the energies of atomic and molecular motion in this way when we are not focusing on individual molecules but on objects containing a large number of them.
4. Chemical energy — This is again just kinetic and potential energies, this time of the internal structures of atoms and molecules, particularly the kinetic energy of the electrons and their potential energies from their interactions with other electrons and with nuclei.

Although all these different energies are in some sense "the same thing", we find it useful to use these terms to differentiate where the energy is in the system are talking about.

If we want to talk about macroscopic objects, then the energy of the motion of the atoms and molecules in the object are "hidden", and we will just refer to "thermal energy" without specifying the velocities of individual atoms and molecules. (We can measure the temperature of the object and get a measure of this hidden motion, but we don't view it directly.) If we want to talk about either macroscopic objects or even about atoms and molecules, then we might choose to leave the internal structures of the atoms and molecules "hidden" and just talk about "chemical energy" without specifying the motion of electrons and nuclei explicitly.

Of course as we develop our sense of the mechanism of various biological processes, we might have to "open up" the  boxes of thermal or chemical energy in order to better understand how energy is organized in a living system.

This view of choosing a system and looking at the distribution of a conserved energy is our analysis of how the 1st law of thermodynamics works. Where we have to go next is to ask: When energy moves through a system, how does it tend to distribute itself spontaneously? This leads to concepts associated with the 2nd law of thermodynamics and the subtle ideas of entropy and free energy.

Joe Redish 1/27/12 and Wolfgang Losert 1/25/2013

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