Energy (science)

Energy is a word with several connotations. The word goes back to the natural philosophy of Aristotle, ενέργεια (energeia), where it means roughly "efficacy". In the early nineteenth century the word was incorporated into science by Thomas Young. In science, the concept energy has a clear meaning, which, however, is not easy to explain because of the many forms in which energy manifests itself. The word energy is also very commonly used outside science, where it means physical or mental power to achieve something. "Negative energy" is a mental power that is in the way of achieving things.

In this article we will restrict attention to the scientific meaning of energy. In science, energy is a measurable property of a physical or chemical system, i.e., the energy of a system may be given by a single real number. Roughly speaking, the energy of a system is a measure of the amount of work that the system is able to perform on its environment. As stated, energy has many manifestations, be it chemical energy of a certain amount of gasoline, the kinetic energy of a moving cannon ball, the heat stored in a steam boiler, the potential energy of water in a reservoir, the fusion energy contained in a hydrogen bomb, the electricity in a battery. All these manifestations obey the same very important law: energy is conserved in conversion from one form of energy to the other. This law of conservation of energy is known as the first law of thermodynamics. This law pervades all of science, and is probably science's most important principle.

Let us consider an example. Assume we use a pump, running on gasoline, to pump water up to a reservoir, and when the reservoir is filled, we let the water flow down to drive an electrical generator. Doing this, we convert the chemical energy of the gasoline to (i) the mechanical energy of the pump to (ii) the potential energy of the water in the reservoir to (iii) the kinetic energy of the falling water, and finally to (iv) the electric energy generated by the generator. If we use the generated electric current for lighting, then the light bulbs convert the electric current to yet another form of energy, namely (v) light (electromagnetic radiation). During these energy conversion processes, the law of conservation of energy assures us that no energy is lost. To non-scientists the contrary may seem the case sometimes, because heat is generated (especially in burning the gasoline to drive the pump), and the heat will escape to the environment without any useful, or directly noticeable, effect. However, since heat is also a form of energy, it must be included in the energy balance of the first law.

Energy in classical mechanics

To keep the discussion simple we will consider a point particle of mass m in one-dimensional space. That is the position of m at time t is given by x(t). For more details and extension to the three-dimensional case, see classical mechanics. Let us assume that a force F(x) is acting on the particle. As an example we may think here of a mass in the gravitional field of the earth. The one dimensional space in this example is a line perpendicular to the surface of the earth. Actually, we will consider the slightly more complicated case of F being a function of x, because remember that the gravitational force F does not depend on x near the surface of the earth: F = mg. (Here g is the gravitational acceleration, which is constant and approximately 9.8 m/s².)

So, the physical system that we are considering is the particle of mass m in a force field F(x). Earlier we defined energy as the work that a system can deliver to its environment. If work is done by the system its energy decreases. If work is done on the system its energy increases. Think now, for example, of Galileo Galilei (GG), carrying mass up the stairs of the leaning tower of Pisa. Doing this GG has the overcome the gravitational force, which works downward. The work ΔW performed by GG is proportional to the gain in height Δx and the absolute value |F| of the force. Since the work done by GG is positive and F is directed downward (F = − |F|), we have

$\Delta W=-F\Delta x\,,$

for the work done by GG on the system while carrying the mass up over a distance Δx. The gain in potential energy dU of the system is the work done on it by GG,

$dU=-Fdx\Longrightarrow U(x)=-\int _{x_{0}}^{x}\,F(x')\,dx',$

where we made the choice of zero of potential energy: $\scriptstyle U(x_{0})=0\,$. By the fundamental theorem of integral calculus, we have the important expression that relates force F(x) and potential energy U(x),

$F(x)=-{\frac {dU(x)}{dx}}.$

Besides potential energy, classical mechanics knows another form of energy: kinetic energy. Suppose GG drops the mass to the bottom of the tower after arriving at its top. The mass will pick up speed, (we will neglect air resistance, which we will put some brake on the falling mass and generate some heat) and get the kinetic energy

$T\equiv {\tfrac {1}{2}}mv^{2}\quad {\hbox{with}}\quad v\equiv {\frac {dx}{dt}},$

where the speed of the particle is the absolute value of its velocity v.

This dropping of mass off the tower of Pisa is a good example of conversion of energy: potential energy is converted into kinetic energy. We will prove that energy is conserved, that is, the sum of kinetic and potential energy is constant in time.

${\frac {d}{dt}}(T+U)=mv{\frac {dv}{dt}}+{\frac {dU}{dx}}{\frac {dx}{dt}}=mva-Fv,\quad {\hbox{with}}\quad a\equiv {\frac {dv}{dt}}.$

Invoke Newton's second law (see classical mechanics):

$F=ma\,,$

and we have proven that the time derivative vanishes of the total energy E ≡ T + U of the physical system under consideration.