One of the most valuable tools that physicists use to understand the universe is to analyze what is happening to the energy of an object. Let’s start by looking at an example.
If you set a coffee mug down your desk, you don’t expect it to spontaneously move. In fact, if you see it begin to move, your first assumption probably be that something is causing the desk to move: perhaps there is an earthquake (in which case you would notice other objects begin to move as well); or someone is hiding under the desk and lifting one side to tilt it. We know from experience* that an object at rest will not start moving all by itself. We also know from experience that in order to make something move, you need to directly interact with it. For small objects like a mug of coffee this is not a problem, but if you want to move a heavy object, you will become tired and your muscles will feel weak—you will have lost some energy.
We could say that a moving coffee mug has some amount of energy, but a stationary one in the same location does not. (The location is important in ways we’ll see when we discuss the different kinds of energy.) Another way of saying that an object will not start moving all by itself would be to say that an object does not spontaneously gain energy. And we can say that for one object to gain energy, another one will lose energy.
At the most fundamental level, we can define energy as follows:
Energy is a property of an object which can be transferred to other objects or converted into different forms, but cannot be created or destroyed.
The SI unit for energy is the joule (J). It is named after James Prescott Joule, a Scottish brewer and scientist who extensively studied heat and energy in the 1800s. One joule is roughly the energy required to lift a small apple up by a distance of a meter.
6.1 Conservation
In colloquial, everyday language, when we say we are “conserving” something, we mean that we are minimizing how much of it we use. This is not the same as the scientific meaning of conservation. In physics, a quantity that is conserved is one that does not change: a conserved quantity can neither be created nor destroyed.
For example, mass is a conserved quantity. You can take a block of wood, and break it up into many little pieces, and scatter them around, but the total amount of mass has not changed. Or, you could burn the block; you would end up with some ash and many gasses released as vapor (for example, any water in the cells of the wood would boil and be released into the air), but the total amount of matter would not change. There is the same amount of mass in the universe today as there was fourteen billion years ago. It may change forms, but the total amount is constant.
The same is true of energy—there is the same amount of energy in the universe now as there was at the big bang, and as there will be far into the future. Everything we do involves the energy changing forms, but the total amount is constant. The fact that energy is conserved is precisely why it is such a useful tool for doing physics.
6.1.1 Budgeting
A good analogy is to think of energy as a form of currency. Say you have $15, and you want to buy a drink that costs $5. You would give $5 to the bartender, and you would be left with $10. You now have less money available than before to buy more drinks, but the total amount of money in circulation has not changed.
Now, say you want to lift that drink up to your mouth so that you can drink it. The glass is at rest, so you need to transfer some energy from you to the glass. You have stored energy in your muscles in the form of chemical energy, derived from the food you eat (and that energy ultimately comes to us from nuclear fusion in the core of the sun). After setting the glass in motion, you have less energy, but the glass has more than before—the total amount of energy has not changed.
A handy way of visualizing this is with a bar graph, as shown in the interactive below. We have bars for the kinetic energy (energy related to motion) and potential energy (energy related to position) as well as the total energy. You can imagine this like a person skiing down a mountain. At the very top, they at a high elevation but not moving very quickly (large potential energy, small kinetic energy). As they go down the mountainside, they gain speed, increasing their kinetic energy, as the potential energy decreases. However, the total energy remains constant.
A mathematical statement of conservation of energy is
\[
E_i = E_f \tag{6.1}
\]
This equation says that whatever total amount of energy we have before some event takes place (call this the initial energy), we have the the same amount of energy after the event takes place (the final energy).
6.1.2 Mass-energy equivalence
The total amount of mass, and the total amount of energy, in the universe has remained constant throughout all time. In the early 1900s, Albert Einstein revealed that mass and energy are actually two different ways of quantifying the same thing. This seems like an extraordinary claim, but it has been borne out by a glut of evidence from experiments of subatomic physics. (In fact, mass-energy equivalence plays a fundamental role in the production of energy from nuclear fusion—the process going on at the core of stars. If it were not for mass-energy equivalence, the sun would not produce energy and life on Earth could not exist!) Einstein’s famous equation relating the two is
\[
E = mc^2 \tag{6.2}
\]
Where \(c\) is the speed of light, which has a value of \(2.99 \times 10^8\) m/s in SI units.
Example 6.1
This example is adapted from a problem in Conceptual Physics by Benjamin Crowell, licensed under CC-BY-SA 3.0. The book can be found at lightandmatter.com/cp; the license can be found at creativecommons.org/licenses/by-sa/3.0.
A free neutron (not bound to the nucleus of an atom) is unstable, and decays into a proton, an electron, and an antineutrino (another kind of subatomic particle). The three particles fly off in different directions. A neutron has a mass of \(1.67495 \times 10^{-27}\) kg, a proton has a mass of \(1.67265 \times 10^{-27}\) kg, an electron has a mass of \(0.000910938 \times 10^{-27}\) kg, and the mass of an antineutrino is so much smaller that it can be ignored. (The mass of an antineutrino is \(5.705 \times 10^{-37}\) kg—this is roughly 10 billion times smaller than masses of the other particles we’re dealing with.) We want to know how much energy is released when the free neutron decays.
First find the total mass of the proton and electron:
\[
m_p + m_e = 1.67265 \times 10^{-27}\ \textrm{kg} + 0.000910938\times 10^{-27}\ \textrm{kg} = 1.67356 \times 10^{-27}\ \textrm{kg}
\]
This is less mass than the neutron had to begin with, but mass is a conserved quantity; because of mass-energy equivalence, we know that the difference in mass was released as energy. Use mass-energy equivalence to find how much energy is released from the “missing” mass:
\[
\begin{align*}
E &= (m_n – (m_p + m_e))c^2 \\
&= \left(1.67495\times 10^{-27} – 1.67356 \times 10^{-27}\right)\left(2.99 \times 10^8\right)^2 \\
&= 1.25 \times 10^{-13}\ \textrm{J}
\end{align*}
\]
At the mass and energy scales that we work with in day-to-day life, mass-energy equivalence is inconsequential and does not factor into our analysis.
*This is actually a risky train of thought. Relying on simple everyday experience can often lead us to the wrong conclusions. In this case, everyday experience is in agreement with rigorous controlled experiments.