Energy and work

Energy

The energy is an important property of physical systems. It is a fundamental property about which we learn by studying examples, similarly as in the case of force or mass. The basics of how energy is defined in mechanics are presented in the section Energy in mechanics below. operational definition of energy as a start:

Energy is the capacity of a system to do work.

This is not completely satisfactory, as we have to know what "work" is. The concept of work is discussed in a separate document Work.

Internal energy U

A particle may have two kinds of energy - kinetic energy, related to the velocity of motion and potential energy related to the position. The sum of all kinetic and potential energies of all the particles contained in the system is the total system energy. Of this total energy, we can define a part that resides in the system - called the internal energy U. This is the part that is of interest to us in thermodynamics. We can speak of the internal energy of a candy bar, a tank of gasoline or a nuclear bomb.

The remaining part of the total energy, sometimes called the external energy, will not be considered in this course. This energy is related to the motion of the system as a whole and to the position of the system in relation to external fields, such as the Earth's gravitational field. The discussion below tries to illustrate the difference between internal and external energy. Imagine that you have three identical candy bars. You eat one sitting on the sofa in your room, the second on the top of a tall mountain and the third in an airplane traveling at 600 mph. According to our definition, the internal energy content of the candy bar, which enters your body, should be the same in the three situations. You can check this by designing some experiments, e.g. by checking your weight, etc. However, the total energy of each candy bar would be slightly different - we would have to add a term equal to mgh in the second situation and $\frac{1}{2}mv^2 + mgH$ in the third (m - mass of candy, g - gravitational acceleration, h - height of mountain above sofa level, H - elevation of plane).

Energy conservation

In physical science a quantity is conserved when it's value does not change in some process. The principle of energy conservation has been observed in many areas of science, from classical mechanics and thermodynamics to electromagnetism, theory of relativity and quantum mechanics. As far as we know, energy is conserved exactly in all processes from collisions of elementary particles to motions of galaxies. Why is energy conserved? The real answer is that we do not know. A sort of answer is presented in the section below. The summary of that section may be given by saying "if particles obey Newton's Law's of motion, then energy is conserved". But why do particles obey Newton's Laws? At some point we have to stop and just say "that is what we observe in nature".

Energy in mechanics

Consider a particle of mass m moving in one dimension. The particle position is x and its velocity is v. Newton's second law may be written

$\begin{displaymath}\frac{dp}{dt} = F where p = mv and v = \frac{dx}{dt} \end{displaymath}$

where p is the momentum, F is the force and t is time. Multiplying both sides of Newton's Law by v we obtain:

$\begin{displaymath}v\frac{d(mv)}{dt} = Fv \end{displaymath}$

Case of free particle: F = 0.

Transform the left-hand-side of equation above:

$\begin{displaymath}v\frac{d(mv)}{dt} = mv\frac{dv}{dt} = m\frac{d(\frac{1}{2}v^2)}{dt} = \frac{d(\frac{1}{2}mv^2)}{dt} \end{displaymath}$

in which we used the fact that the particle mass m is a constant. Setting this new left-hand-side equal to the right-hand-side (Fv = 0 when F = 0):

$\begin{displaymath}\frac{d(\frac{1}{2}mv^2)}{dt} = 0 \end{displaymath}$

This says in mathematical language : "the quantity $\frac{1}{2}mv^2$ does not change with time for a free particle. In other words, this quantity remains constant. This interesting and important quantity is called the kinetic energy of the particle

$\begin{displaymath}E_k = \frac{1}{2}mv^2 (definition) \end{displaymath}$

What we have found is that when F=0 the kinetic energy is conserved

$\begin{displaymath}E_k = \frac{1}{2}mv^2 = const (free particle) \end{displaymath}$

Case of particle in external potential.

Suppose the particle is moving in external potential V(x). In that case the force acting on the particle may be calculated as

$\begin{displaymath}F = -\frac{dV}{dx} \end{displaymath}$

Simple examples of potentials

Harmonic potential.
For example if our particle is attached to a spring with a spring constant f and x measures the displacement of the particle from the spring equilibrium position

$\begin{displaymath}V = \frac{1}{2}fx^2 F = -\frac{dV}{dx} = -fx \end{displaymath}$

i.e. the force will be proportional to the displacement from equilibrium and in the opposite direction (Hooke's Law).

Gravitational potential.
If there is a gravitational field with constant acceleration g directed in the -x direction then

$\begin{displaymath}V = mgx F = -\frac{dV}{dx} = -mg \end{displaymath}$

i.e. the force will be constant, proportional to particle mass and in direction of the acceleration.

Transform the Fv term using the chain rule of derivatives:

$\begin{displaymath}Fv = -\frac{dV(x)}{dx} \frac{dx}{dt} = - \frac{dV}{dt} \end{displaymath}$

Bring together left and right-hand sides:

$\begin{displaymath}\frac{d(\frac{1}{2}mv^2)}{dt} = - \frac{dV}{dt} \end{displaymath}$

$\begin{displaymath}\frac{d}{dt}(E_k + V ) = 0 \end{displaymath}$

This tells us the quantity does change in time. This is called the total energy E.

$\begin{displaymath}E = E_k + V (definition) \end{displaymath}$

Our general result is that the total energy is constant for a particle moving in an external potential V:

$\begin{displaymath}E = E_k + V = const (particle in potential V) \end{displaymath}$

In plain English, we can say that for a particle moving according to Newton's laws of motion, we can introduce a quantity called energy which is conserved, i.e. does not change in time. The total energy has two parts - kinetic energy, which depends on velocity, and potential energy, which depends on position.

The introduction of work

So far we have seen no trace of the work w in the theoretical treatment. Let's look for it by coming back to our basic equation

$\begin{displaymath}\frac{d(\frac{1}{2}mv^2)}{dt} = Fv \end{displaymath}$

We will integrate both sides over time

$\begin{displaymath}\int_{t_1}^{t_2} \frac{d(\frac{1}{2}mv^2)}{dt} dt = \int_{t_1}^{t_2} Fv dt \end{displaymath}$

which leads to

$\begin{displaymath}\frac{1}{2}mv^2(t_2) - \frac{1}{2}mv^2(t_1) = \int_{x_1}^{x_2} Fdx \end{displaymath}$

where we have used the theorem about the integral of a derivative on the left-hand-side and a change of variables from t to x on the right-hand-side. Of course, is the particle position at time and the position at time . Translating the above equation into English, we can say that as the particle moves along its trajectory the change in it's kinetic energy $\Delta E_k$ is equal to the work of external forces w: