Physics with Calculus/Mechanics/Energy and Conservation of Energy

One of the most fundamental concepts in physics is energy. It's difficult to define what energy actually is, but one useful definition might be "a measure of the amount of change taking place within a system, or the potential for change to take place within the system".

Roughly speaking, energy can be divided into two forms, kinetic and potential. Kinetic energy is the energy of movement or change. Potential energy is the energy a system has as a result of being able to undergo some change. To provide a specific example, a falling book has kinetic energy, because its position in space is changing (it is moving downwards). A book resting on a shelf has no potential energy relative to the shelf since it has a height of zero meters relative to the shelf. However, if the book is elevated to some height above the shelf, then it has potential energy proportional to the height at which it resides above the shelf.

An object can have both kinetic and potential energy at the same time. For example, an object which is falling, but has not yet reached the ground has kinetic energy because it is moving downwards, and potential energy because it is able to move downwards even further than it already has. The sum of an object's potential and kinetic energies is called the object's mechanical energy.

As an object falls its potential energy decreases, while its kinetic energy increases. The decrease in potential energy is exactly equal to the increase in kinetic energy.

Another important concept is work. Similarly to the way we defined energy, we may define work as "a measure of the amount of change brought about in a system, by the application of energy". For instance, you might do work on a book by picking it up off the floor and putting it on a shelf. In doing so, you have increased the potential energy of the book (by increasing its potential to fall down to the floor). The amount of potential energy you have "given" to the book is exactly equal to the amount of work you do by lifting it onto the shelf.

Mathematically, however, energy is very easy to define. Kinetic energy is 1/2 m v^2. Potential energy is a little bit trickier. Say we have a force which can be written as the gradient (a three-dimensional derivative. If you don't know what it is, pretend it is a normal derivative and you should be able to understand things in one dimension.) of some function, $$\phi$$ times the mass of the particle. That is $$ \vec{F} = m \vec{\nabla} \phi $$. Then potential energy is just $$m \phi + C$$ where C is an arbitrary constant. What arbitrary definitions, you might say. At first, you might think so, but it turns out, the work done by the force is the change in kinetic energy (see ../Work and Energy/). They are actually very closely related. In fact, the potential energy plus the kinetic energy due to the force is constant! Aha, so this "arbitrary" potential energy decreases at exactly the same rate this "arbitrary" kinetic energy increases. They must be the same thing in different forms! It is not so arbitrary after all. This is the conservation of energy. In fact, since the particles are moving at finite velocities, this is the much stronger local conservation of energy for mechanical systems. Another amazing fact is that it appears that all forces are conservative (this changes in electrodynamics, but energy is still conserved)! Even friction appears to be conservative on a molecular level. The slightly more mathematical treatment is available in ../Work and Energy/.

We may concisely state the following principle, which applies to closed systems (i.e. when there are no interactions with things outside the system):
 * In all physical processes taking place in closed systems, the amount of change in kinetic energy is equal to the amount of change in potential energy. If the kinetic energy increases, the potential energy decreases, and vice-versa.

When we consider open systems (i.e. when there are interactions with things outside the system), it is possible for energy to be added to the system (by doing work on it) or taken from the system (by having the system do work). In this case the following rule applies:
 * The total energy of a system (kinetic plus potential) increases by the amount of work done on the system, and decreases by the amount of work the system does.

This leads us to consider the conservation of energy and other quantities. In many cases, "you get out what you put in".

If you put 3 pairs of socks into an empty dryer, you don't need to analyze the exact configuration of the dryer, the temperature profile, or other things to figure out how many socks will come out of the dryer. You'll get 3 pairs of socks out[*].

A conservation law, in its most general form, simply states that the total amount of some quantity within a closed system doesn't change. In the example above, the conserved quantity would be socks, the system would be the dryer, and the system is closed as long as nobody puts socks into or takes socks out of the dryer. If the system is not closed, we can always regard a larger system which is closed and which encompasses the system we were initially considering (e.g. the house in which the dryer is located), even though, in extreme cases, this may lead us to consider the number of socks (or whatever) in the entire Universe!

Conservation laws help us solve problems quickly because we know that we will have the same amount of the conserved quantity at the end of some process as we did at the start. The fundamental laws of conservation are;
 * conservation of mass
 * conservation of energy
 * conservation of momentum
 * conservation of angular momentum
 * conservation of charge

Returning to our example above, the 'conservation of socks' is, in fact, a consequence of the law of conservation of mass. It should be noted that in the context of nuclear reactions, energy can be converted to mass and vice-versa. In such reactions, the total amount of mass plus energy doesn't change. Therefore the first two of these conservation laws are often treated as a single law of conservation of mass-energy

Combining these laws with Newton's laws gives other derived conserved quantities such as
 * conservation of angular momentum

Within a closed system, the total amount of energy is always conserved. This translates as the sum of the n changes in energy totaling to 0.


 * $$\sum_{k=1}^n \Delta \mathbf{E}_k = 0$$

An example of such a change in energy is dropping a ball from a distance above the ground. The energy of the ball changes from potential energy to kinetic energy as it falls.



\begin{matrix} U_g &=& m\mathbf{g}h \\ K &=& {1 \over 2} m\mathbf{v}^2 \end{matrix} $$

Because this is the only change in energy within our system, we will take a simple physical problem and model it in order to demonstrate.

An object of mass 10kg is dropped from a height of 3m. What is its velocity when it is 1m above the ground?

We start by evaluating the Potential Energy when the object is at its initial state.



\begin{matrix} U_g &=& m\mathbf{g}h \\ U_g &=& 30\mathbf{g} \\ \mathbf{g} &=& 9.807ms^{-2} \\ U_g &=& 30 \cdot 9.807 \\ U_{g3} &=& 294.21J \end{matrix} $$

The Potential Energy of the object at a height of 1m above the ground is given in a similar fashion.



\begin{matrix} U_g &=& m\mathbf{g}h \\ U_g &=& 10\mathbf{g} \\ U_g &=& 10 \cdot 9.807 \\ U_{g1} &=& 98.07J \end{matrix} $$

Hence the change in Potential Energy is given



\begin{matrix} \Delta U_g &=& 294.21 - 98.07 = 196.14J \end{matrix} $$

By definition, the change in Potential Energy is equivalent to the change in Kinetic Energy. The initial KE of the object is 0, because it is at rest. Hence the final Kinetic Energy is equal to the change in KE.



\begin{matrix} \Delta U_g &=& \Delta K \\ 196.14J &=& {1 \over 2} m\mathbf{v}^2 \\ 196.14 &=& 5\mathbf{v}^2 \end{matrix} $$

Rearranging for v



\begin{matrix} {196.14 \over 5} &=& \mathbf{v}^2 \\ \sqrt &=& \mathbf{v} \\ \mathbf{v} &\approx& 6.263 ms^{-1} \end{matrix} $$

We can check our work using the following kinematic equation.


 * $$\begin{matrix}

\mathbf{v}^2 &=& \mathbf{u}^2 + 2\mathbf{as} \\ \mathbf{v}^2 &=& 0^2 + 2\mathbf{gs} \\ \mathbf{v} &=& \sqrt{2\mathbf{gs}} \\ \mathbf{v} &=& \sqrt{2\cdot 9.807 \cdot 2} \\ \mathbf{v} &\approx& 6.263 ms^{-1} \end{matrix} $$

This follows because we can actually use the equations for energy to generate the above kinematic equation.


 * $$\begin{matrix}

\Delta U_g &=& \Delta K \\ m\mathbf{g}h &=& {1 \over 2}m\mathbf{v}^2 \\ m\mathbf{g}\Delta h &=& {1 \over 2}m{\Delta (\mathbf{v}}^2) \\ \mathbf{s} &=& \Delta h \\ \mathbf{gs} &=& {1 \over 2} \Delta (\mathbf{v}^2) \\ 2\mathbf{gs} &=& \mathbf{v}^2 - {\mathbf{v}_0}^2 \\ 2\mathbf{as} &=& \mathbf{v}^2 - \mathbf{u}^2 \\ \mathbf{v}^2 &=& \mathbf{u}^2 + 2\mathbf{as} \end{matrix} $$