# (p.252) Appendix C Energy, Mass, Momentum, Angular Momentum

# (p.252) Appendix C Energy, Mass, Momentum, Angular Momentum

Energy is both a mathematical concept and, in certain circumstances, a measurable quantity or one whose value can be deduced. Yet it is one of the most elusive entities in science. To paraphrase the outstanding twentieth-century physicist and educator Richard Feynman, Nobody really knows what energy is. Nonetheless, one of the greatest laws of physics states that energy is conserved in all physical processes occurring in closed systems (i.e., those arranged to be isolated from external influences). This means that the amount of energy in the closed system is the same before, during, and after any process that takes place in it.

Energy takes many forms, e.g., kinetic (motion), gravitational, heat, elastic, electrical, chemical, nuclear, or mass (via the famous formula $E=M{c}^{2}$, where *E* = energy, *M* = mass, and *c* = speed of light). Some forms of them can be converted to others.

As an example of energy conservation and conversion in a familiar setting, let a ball be tossed up in the air in an ideal situation where there is no air friction. During its flight up to its highest point and then back down, conservation of energy means that its total energy *E* is an unchanging constant. In the assumed absence of air friction, the total energy is equal to the sum of the ball’s kinetic energy *KE* and its gravitational energy *GE*: *E* = *KE* + *GE*. Both *KE* and *GE* will vary during the ball’s journey up and down. By a standard convention, the ball’s initial gravitational energy, say $G{E}_{\text{i}}$, is chosen to be zero. Hence, it starts up having only its initial kinetic energy, say $K{E}_{\text{i}}$. So at the beginning $E=K{E}_{\text{i}}$. At its highest elevation *e*, its speed is zero, as is its kinetic energy $K{E}_{\text{e}}$: $K{E}_{\mathit{\text{e}}}=0$. This means that at *e*, its total energy is all gravitational: $E=G{E}_{\text{e}}$. What’s happened is that at *e* the initial kinetic energy has been converted entirely to gravitational energy: $K{E}_{\text{i}}=G{E}_{\text{e}}$.

In the real macroscopic world, air friction cannot be totally ignored. As a result, some of the ball’s kinetic energy during the upward flight is lost to friction and the ball will not reach the elevation *e*. The lost kinetic energy is converted to heat—i.e., to speeding up some of the air molecules. Denoting the heat energy by *HE*, the energy conservation relation now reads $E=KE+GE+HE$. Fortunately, neither friction nor heat plays any role in the microscopic phenomena considered in this book.

(p.253)
While Feynman’s comment about energy holds true,^{1} that has not prevented physicists from writing formulas for it, for example, for kinetic energy. If the mass of an object is *M* and its speed *v*, then its kinetic energy is given by the formula $KE=M{v}^{2}/2$, where *Mv*^{2} means the product of *M* and *v*^{2}: as is usual in this book (and textbooks in general) the times symbol “×” is normally omitted in expressions involving products of two or more symbols (but not, of course, when indicating a product of two numbers!).

You may be wondering what is meant by mass. It is the quantity of matter in a body, measured in units of kilograms, abbreviated kg.^{2} Mass is related to weight: the relation between a body’s mass *M* and its weight *W* near the surface of the earth is $W=Mg$, where *g* is the acceleration of gravity at the earth’s surface. Mass is also a measure of a body’s inertia, which is its resistance to a change in its motion, in that the larger the mass, the greater the resistance to a change in motion. Consider a child’s wagon and a passenger car, each stationary. As measured by their weights, the mass of the wagon is obviously much less than that of the car. Now imagine pushing on each to get it to move. A far greater effort will be needed for the car than for the wagon, and the reason is that the effort required in each case is proportional to the mass involved.

Mass also enters the definition of momentum, whose symbol is *p*. The momentum of a body whose mass is non-zero is the product of the body’s mass *M* times its speed *v*: $p=Mv$. You will have noted the phrase “non-zero” in the preceding sentence. It suggests that there may be zero-mass entities. There are! One is the photon, the massless particle of light and all electromagnetic radiation, whose existence as a light quantum was first postulated by Albert Einstein, following the initial but restricted proposal of Max Planck (Chapter 4). Photons, which move with the speed of light *c*, have both energy and momentum, despite their mass being zero. Because they are particles of electromagnetic radiation, photons have a frequency *f* and a wavelength $\mathrm{\lambda}$. These attributes, along with Planck’s constant *h* (introduced in Chapter 4) enter into the definitions of a photon’s energy and momentum. A photon’s energy ${E}_{\text{ph}}$ is the product of Planck’s constant times its frequency: ${E}_{\text{ph}}=hf$. Correspondingly, its momentum ${p}_{\text{ph}}$ is the ratio of Planck’s constant divided by its wavelength: ${p}_{\text{ph}}=h/\mathrm{\lambda}$. The first of these two formulas is discussed in Chapter 4, the second in Chapter 6. (Note that all the symbols introduced so far, plus many others, are listed in Appendix A.)

(p.254)
The kinetic energy formula $K=M{v}^{2}/2$ can be rewritten in terms of the momentum, turning it into the form used in quantum theory (e.g., Chapter 7). The numerator of the preceding formula would contain the momentum if $(Mv{)}^{2}={p}^{2}$ replaced *Mv*^{2}. This is achieved by multiplying the kinetic energy formula by the ratio *M*/*M*, which then turns it into *KE* = $(Mv{)}^{2}/2M={p}^{2}/2M$, the desired result.

Momentum may be thought of as a measure of an object’s straight-line motion, if only for the moment before it might change direction. Correspondingly, *angular momentum* is a measure of a body’s circular motion (hence the presence of “angular”), again if only for a moment before it might stop moving in a circle. The symbol for angular momentum is *L*, and since it relates to circular motion, the radius of the circle enters into its definition: it is the product of the body’s momentum *p* times the distance *r* of *p* from an arbitrary center about which the body is moving, at least instantaneously: $L=rp=rMv$. There is also an extended definition that applies to rotating bodies such as an ice skater or a spinning top.

Angular momentum can be conserved (recall conservation of energy), either exactly in ideal (mostly textbook) situations or approximately, as in the case of an ice skater. The latter situation means that the value of *L* doesn’t change much between the beginning and the selected end of the event, whereas in the ideal case it remains the same throughout. Suppose an ice skater starts spinning with her arms spread out. Since there is very little friction between the skates and the ice, her rotational speed will remain constant for some time. In the extended definition of *L*, her outstretched arms act like the radius *r* in the relation *L* = *rMv*, so if she pulls her arms in to her body she will speed up, exactly as occurs in real situations. Why does she speed up? Because if *r* becomes smaller and *M* remains the same, the only way *L* = *rMv* can remain constant (at least temporarily) is if *v* increases. Angular momentum is a key quantity in Niels Bohr’s ad hoc theory of the hydrogen atom, discussed in Chapter 4.

Like energy, momentum and angular momentum have their quantum counterparts, but unlike energy, each is actually a *vector*, which is a quantity that has magnitude as well as direction. The vectorial aspect is crucial for quantum theory, which is why I mention it here and describe it in Appendix D.

Just as speed and mass have units (m/sec and kg, respectively), so does energy. There is no single convention for specifying the unit of energy, and many exist. A standard one used in physics is the Joule, abbreviated J. It is probably unfamiliar to you, but enters the definition of a watt (watts are units of power, which is energy emitted per unit of time): one watt is one Joule of energy radiated per second. Thus, a 100-watt bulb radiates 100 Joules per second. Unfortunately, the Joule turns out to be an inappropriate unit in the microscopic world. A much more convenient one for atomic and molecular (p.255) phenomena is the electron volt (eV) and its higher multiples (e.g., kilovolts (keV) = 1000 volts).

One electron volt is the energy a single electron or proton will gain if it moves in the electric field generated by a one-volt battery. Electron volts are the units appropriate for this book because it deals mainly with atomic and molecular phenomena, the energies for which are at most a few tens of eV. An example is the energy with which an electron is bound to the proton in the hydrogen atom, namely, 13.6 eV.

To see why electron volts are so much more convenient than Joules for energies in the microscopic world, consider the relations between the two units:

and

Expressed in Joules, the binding energy of the electron in hydrogen is $2.2\times {10}^{-18}$ J! This example should make clear the inconvenience of using Joules, since it would always mean writing 10 to large negative powers for the energies of atoms or molecules. An analog would be expressing one cent in units of thousand dollar bills: doable but inconvenient.

Systems such as atoms are made up of electrons and nuclei bound together by the electrical force between the positively charged protons in the nucleus and the negatively charged electrons. Hence, energy must be *added* to an atom to remove one or more of its electrons. Because of this, the convention is that the energies of bound systems such as atoms or molecules are *negative*. An example is the value of the lowest, or ground state, energy of hydrogen, namely, − 13.6 eV (which is why it’s *binding energy* is 13.6 eV). Thus, to ionize hydrogen when in its normal ground state, that is, to free its electron from captivity, requires a photon or other projectile whose energy is at least 13.6 eV.

## Notes:

(^{1})
Energy has sometimes been defined as “the capacity to do work.” This statement refers to the physics definition of “work,” a concept almost without meaning for microscopic phenomena.

(^{2})
Although not intended to confuse Americans who visit Europe, in that continent the kilogram (or “kilo”) is a unit of weight! One kilo is a little more than a weight of two pounds.