In addition to forces and accelerations, energy is an alternative tool we can use to understand how systems behave. We will spend a lot of time on energy, and the closely related idea of electric potential, in EMP-Cubed. This page contains brief reminders of the key ideas about energy from your mechanics course; for more details refer to the readings from Physics 183.

It's actually not easy to give a precise definition of energy that covers all of the situations in which it occurs. For our course, a reasonable way to think about it is that energy is either motion, or the ability to produce motion. (However, some forms of this motion, like random molecular motion associated with thermal energy produced by friction, may be difficult to see.)

There are two important types of energy we talk about: kinetic energy, which is the energy associated with objects currently moving in a particular direction, and potential energy, which is energy stored somehow in a system that could cause things to move in the future (like a compressed spring that could push a block and make it move).

A really nice feature of energy is that energy is a scalar, not a vector. Energy is just a number without any direction associated with it, which means there are no x and y components and no trigonometry to worry about.

Several symbols are commonly used to represent energy. $E$ is the most common symbol representing any kind of energy. If we want to remind ourselves that the energy is kinetic energy, we often use $K$, and $U$ is generally used to represent potential energy (sometimes with a subscript to specify what kind of potential energy, like $U_g$ for gravitational potential energy). Regardless of the type of energy, the SI unit of energy is the joule (J), which is equivalent to 1 kg m²/s². Using the definition of a newton, you can also write this as 1 J = 1 N m.

An alternative (non-SI) unit of energy that may be more convenient in this course is the electron-volt, eV, which has a value of 1 eV = $1.602 \times 10^{-19}$ J.

Energy is important because energy is a conserved quantity – energy is never created or destroyed, it only changes form, for example from $K \rightarrow U$. This means that for any isolated system, if you compare two snapshots of the system at different times, the total amount of energy in the system has to be the same. Mathematically, we express this as

$$E_f = E_i,$$ and then write down expressions that include all the types of energy that are present, such as $E_f = K_f + mgh_f$ for an object with kinetic and gravitational potential energy.

Often, this means that you can figure out the state of the system at some final $t_f$ based on what you know about some initial time $t_i$, without having to calculate anything about what happened in between: no tracking forces and accelerations, no worrying about vectors and trig components, and so on.

The idea of a system is important for using energy correctly. When we set up a problem, we select the objects whose behavior we want to keep track of: these are our system. If all of the interactions/forces that the objects experience come from other objects in the system, then we say that this is an isolated system, and the total energy of the system cannot change: $E_f = E_i$.

But if the objects in our system experience forces or interactions coming from their surrounding environment – external forces – then the system isn't isolated and you need to keep track the energy flowing into or out of the system:

$$E_f = E_i + \Delta E,$$ where $\Delta E$ is the energy exchanged between the system and its environment. (Be careful of the sign! As written here, $\Delta E$ should be positive is energy is flowing in, and negative if energy is flowing out of the system.)

For example, if two blocks slide together and collide on a table, we might say that our system consists of the two blocks, and define the table to be outside the system, part of the environment. Then if we don't include any forces from the table in our model – maybe because it's an air table so that friction is negligible, or because we're only interested in comparing the moment just before the collision to the moment right afterward – we would say the blocks are isolated and their total energy is conserved: $E_{1,f} + E_{2,f} = E_{1,i} + E_{2,i}.$ But if we do include friction from the table in our model, then that friction is an external force that changes the energy of the system: $E_{1,f} + E_{2,f} = E_{1,i} + E_{2,i} - \Delta E_{fric}$.

It's a very good idea to make note of what you are defining as your system when setting up your model of the problem, such as with a circle on your free-body diagram or interaction diagram.

If an external force acts on our system, the energy it transfers in or out is the work done by the force on an object in the system. The work is calculated as

$$W = \Delta E = \int \vec{F}_{ext} \cdot d\vec{x},$$ where $\Delta \vec{x}$ is the displacement the object experiences while the force acts on it. If the magnitude of the force is constant, this integral reduces to simply $W = \Delta E = \vec{F}_{ext} \cdot \Delta \vec{x}$.

A few things to note about this expression: if the object doesn't move, no work is done and so no energy is transferred to the system. Forces perpendicular to the object's velocity also do no work. (The force produces an acceleration changing the direction of the velocity, but the object's speed doesn't change.)

If the force is exactly parallel to the velocity, then work is just $W = F \, \Delta x$, but in general you need to take the dot product of the vectors and there is will a trig term: $W = F \, \Delta x \cos \theta$, where $\theta$ is the angle between the force and the velocity. If $\theta = 180^\circ$, meaning that the force is pushing in the direction opposite to the object's velocity, the dot product is negative because the force is reducing the object's energy (slowing it down).

We say that energy is a conserved quantity, meaning that it is never created or destroyed (although it may flow in our out of your system). What energy can do is change form, such as from potential to kinetic energy, or vice versa. You've seen several forms of energy in your mechanics class, and we'll learn about some important new forms in EMP-Cubed.

Kinetic energy, abbreviated $K$, is the energy of a macroscopic object in motion. (In a microscopic picture, it is the energy associated with particles moving in a single direction: coherent linear motion.) The kinetic energy of an object depends on its mass and its speed,

$$K = \tfrac{1}{2} mv^2.$$ Since the object's speed $v$ is squared in this formula, the kinetic energy is always a positive number.

Rotational energy

Potential energy, generically denoted $U$, is the energy associated with forces between objects in our system. Gravitational potential energy $U_g$ is the potential energy associated with the force of gravity. There are two expressions we sometimes use for $U_g$:

The most general expression is the Newtonian form,

$$U_g = - \, \frac{G \, m_1 \, m_2}{r}.$$ This form depends on the masses $m_1$ and $m_2$ of the two interacting objects, the distance $r$ between them, and Newton's constant $G = 6.674 \times 10^{-11} \;{\rm N} \; {\rm m}^2/{\rm kg}^2$. This expression can always be used, but in practice we normally only use it when thinking about masses on orbital or planetary scales. Note that this expression is always negative, and goes to zero as $r$ becomes very large. This convention means that the gravitational potential energy is the binding energy holding the two objects together; i.e., the amount of energy that must be added to the system to let the objects escape from each other.

For masses close to the Earth's surface, we use a simpler approximation that combines the values of $G$, $m_{\rm Earth}$ and $r_{\rm Earth}$ into a single constant $g = 9.8 \; {\rm m}/{\rm s}^2$, leading to the expression

$$U_g = mgy.$$ When using this approximation, we often set $y=0$ to the lowest position we're interested in, but that choice is completely arbitrary and negative values of $U_g$ are perfectly valid.