\(\def \u#1{\,\mathrm{#1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\)

Energy Conservation

Energy is one of the most fundamental quantities in physics, although it can be a little hard to define for all that. There are many different forms that energy can take: motion, warmth, sound, light, electrical, the energy in a stretched spring. Even mass itself is a sort of compressed energy (as given by Einstein's famous equation, $E=mc^2$.)

The Law of Conservation of Energy is one of the cornerstones of physics. When something is conserved, that means it cannot suddenly appear or disappear: if an object suddenly has more or less energy than it did before, that means energy flowed into or out of it from somewhere else. Energy conservation is so important to physicists that when it appears to be violated, physicists will invent new types of energy to account for it! For example, when a block sliding across a table comes to a stop, it loses its motion energy but the table warms up slightly: this led physicists to realize that heat was a type of energy flow, rather than a separate phenomenon as they’d originally assumed. Mass energy is required to explain how a uranium sample sitting quietly on a shelf can suddenly release a very energetic alpha particle (two protons and two neutrons): during nuclear decay, the total mass of the nucleus and the ejected particle is slightly smaller than the mass of the original nucleus. Mathematically, we express energy conservation with the equation

where $E_i$ is the initial energy of an object, $E_f$ is its final energy, and $\Delta E$ is the total flow of energy into (when positive) or out of (when negative) of the object. This equation will be the framework for everything we do with energy.