A Glossary of Frequently Misused or Misunderstood Physics Terms and Concepts.

By Donald E. Simanek, Lock Haven University.

Technical terms of science have very specific meanings. Standard dictionaries are not always the best source of useful and correct definitions of them.

This glossary is not intended to be complete. It focuses on those terms that give students particular difficulties. Some words have subtle and intricate meanings that cannot be encapsulated in a short definition. That's why textbooks exist. A good glossary for elementary physics may be found in Appendix G-1 of Kirkpatrick & Wheeler, Physics, A World View, Saunders, 1992.

This document is continually under development and may never be finished. Input and suggestions for additional troublesome terms, or for clearer statements about these, are invited. Use the address at the right.

Links are built into this document. Feel free to use your browser’s search and find tools to extend its usefulness.

A related document, Jack Holden's Illustrated Dictionary of Physics has additional cartoon definitions, not included in this document.

Accurate. Conforming closely to some standard. Having very small error of any kind. See: Uncertainty. Compare: precise.

Absolute uncertainty. The uncertainty in a measured quantity is due to inherent variations in the measurement process itself. The uncertainty in a result is due to the combined and accumulated effects of these measurement uncertainties that were used in the calculation of that result. When an uncertainty is expressed in the same units as the quantity itself it are called an absolute uncertainty. Uncertainty values are usually attached to the quoted value of an experimental measurement or result, one common format being: (quantity) ± (absolute uncertainty in that quantity). Compare: relative uncertainty.

Action. This technical term is an historic relic of the 17th century, before energy and momentum were understood. In modern terminology, action has the dimensions of energy×time. Planck's constant has those dimensions, and is therefore sometimes called Planck's quantum of action. Pairs of measurable quantities whose product has dimensions of energy×time are called conjugate quantities in quantum mechanics, and have a special relation to each other, expressed in Heisenberg's uncertainty principle. Unfortunately the word action persists in textbooks in meaningless statements of Newton's third law: "Action equals reaction." This statement is useless to the modern student, who hasn't the foggiest idea what action is. See: Newton's 3rd law for a useful definition. Also see Heisenberg's uncertainty principle.

Avogadro's constant. Avogadro's constant has the unit mole-1. It is not merely a number, and should not be called Avogadro's number. It is correct to say that the number of particles in a gram-mole is 6.02 × 1023. Some older books call this value Avogadro's number, and when that is done, no units are attached to it. This can be confusing and misleading to students who are conscientiously trying to learn how to balance units in equations.

One must specify whether the value of Avogadro's constant is expressed for a gram-mole or a kilogram-mole. A few books prefer a kilogram-mole. The unit name for a gram-mole is simply mol. The unit name for a kilogram-mole is kmol. When the kilogram-mole is used, Avogadro's constant should be written: 6.02252 × 1026 kmol-1. The fact that Avogadro's constant has units further convinces us that it is not "merely a number."

Though it seems inconsistent, the SI base unit is the gram-mole. As Mario Iona reminds me, SI is not an MKS system. Some textbooks still prefer to use the kilogram-mole, or worse, use it and the gram-mole. This affects their quoted values for the universal gas constant and the Faraday Constant.

Is Avogadro's constant just a number? What about those textbooks that say "You could have a mole of stars, grains of sand, or people." In science we do use entities that are just numbers, such as π, e, 3, 100, etc. Though these are used in science, their definitions are independent of science. No experiment of science can ever determine their value, except approximately. Avogadro’s constant, however, must be determined experimentally, for example by counting the number of atoms in a crystal. The value of Avogadro's number found in handbooks is an experimentally determined number. You won't discover its value experimentally by counting stars, grains of sand, or people. You find it only by counting atoms or molecules in something of known relative molecular mass. And you won't find it playing any role in any law or theory about stars, sand, or people.

The reciprocal of Avogadro's constant is numerically equal to the unified atomic mass unit, u, that is, 1/12 the mass of the carbon 12 atom.

1 u = 1.66043 × 10-27 kg = 1/6.02252 × 1023 mole-1.

Because. Here's a word best avoided in physics. Whenever it appears one can be almost certain that it's a filler word in a sentence that says nothing worth saying, or a word used when one can't think of a good or specific reason. While the use of the word because as a link in a chain of logical steps is benign, one should still replace it with words more specifically indicative of the type of link that is meant. See: Why?

Illustrative fable: The seeker after truth sought wisdom from a Guru who lived as a hermit on top of a Himalayan mountain. After a long and arduous climb to the mountain-top the seeker was granted an audience. Sitting at the feet of the great Guru, the seeker humbly said: "Please, answer for me the eternal question: Why?" The Guru raised his eyes to the sky, meditated for a bit, then looked the seeker straight in the eye and answered, with an air of sagacious profundity, "Because!"
Capacitance. The capacitance of a physical capacitor is measured by this procedure: Put equal and opposite charges on the capacitor's plates and then measure the potential between the plates. Then C = |Q/V|, where Q is the charge on one of the plates.

Capacitors for use in circuits consist of two conducting bodies (plates). We speak of a capacitor as "charged" when it has charge Q on one plate, and -Q on the other. Of course the net charge of the entire object is zero; that is, the charged capacitor hasn't had net charge added to it, but has undergone an internal separation of charge. Unfortunately this process is usually called charging the capacitor, which is misleading because it suggests adding charge to the capacitor. In fact, this process usually consists of transfering charge from one plate to the other. The capacitance of a single object, say an isolated sphere, is determined by considering the other plate to be an infinite sphere surrounding it. The object is given charge, by moving charge from the infinite sphere, which acts as an infinite charge reservoir ("ground"). The potential of the object is the potential between the object and the infinite sphere.

Capacitance depends only on the geometry of the capacitor's physical structure and the dielectric constant of the material medium in which the capacitor's electric field exists. The size of the capacitor's capacitance is the same whatever the charge and potential (assuming the dielectric constant doesn't change). This is true even if the charge on both plates is reduced to zero, and therefore the capacitor's potential is zero. If a capacitor with charge on its plates has a capacitance of, say, 2 microfarad, then its capacitance is also 2 microfarad when the plates have no charge. This should remind us that C = |Q/V| is not by itself the definition of capacitance, but merely a formula that allows us to relate the capacitance to the charge and potential when the capacitor plates have equal and opposite charge on them.

A common misunderstanding about electrical capacitance is to assume that capacitance represents the maximum amount of charge a capacitor can store. That is misleading because capacitors don't store charge (their total charge being zero). They "separate charge" so that their plates have equal and opposite charge. It is also wrong because the maximum charge one may put on a capacitor plate is determined by the potential at which dielectric breakdown occurs. Compare: capacity.

We probably should avoid the phrases "charged capacitor", "charging a capacitor" and "store charge". Some have suggested the alternative expression "energizing a capacitor" because the process is one of giving the capacitor electrical potential energy by rearranging charges on it (or within it).

Some who agree with most everything I have said on this topic still defend "stored charge". They say that the capacitor circuit separates charge and then stores equal and opposite charges on the capacitor plates presumably for release by discharge through a circuit (rather than by discharge within the capacitor). That's a correct description for it puts the capacitor in the context of the circuit to which it is attached. But the abbreviated phrase "The capacitor stores charge" is still misleading and should be avoided unless it is explained as I have done here. And it's still more to the point to say the capacitor stores electrical potential energy.

Capacity. This word is properly used in names of quantities that express the relative amount of some quantity with respect to a another quantity upon which it depends. For example, heat capacity is dU/dT, where U is the internal energy and T is the temperature. Electrical capacity, usually called capacitance is another example: C = |dQ/dV|, where Q is the magnitude of charge on each capacitor plate and V is the potential difference between the plates.

Consistent use of the word "capacitance" for C avoids this conceptual error. But the same misconceptions can occur with the others, and we don't have other names for them that might help avoid this. Heat capacity isn't the maximum amount of heat something can have. That would also incorrectly suggest that heat is a "substance", which it isn't.

Centrifugal force. When a non-inertial rotating coordinate system is used to analyze motion, Newton's law F = ma is not correct unless one adds to the real forces a fictitious force called the centrifugal force. The centrifugal force required in the non-inertial system is equal and opposite to the centripetal force calculated in the inertial system. Since the centrifugal and centripetal forces are concepts used in two different formulations of the problem, they can not in any sense be considered a pair of reaction forces. Also, they act on the same body, not different bodies. See: centripetal force, action, and inertial systems.

Centripetal force. The centripetal force is the radial component of the net force acting on a body when the problem is analyzed in an inertial system. The force is inward toward the instantaneous center of curvature of the path of the body. The size of the force is mv2/r, where r is the instantaneous radius of curvature. See: centrifugal force.

cgs. The system of units based upon the fundamental metric units: centimeter, gram and second.

Classical physics. The physics developed before about 1900, before we knew about relativity and quantum mechanics. See: modern physics.

Closed system. A physical system on which no outside influences act; closed so that nothing gets in or out of the system and nothing from outside can influence the system's observable behavior or properties.

Obviously we could never make measurements on a closed system unless we were in it, for no information about it could get out of it! In practice we loosen up the condition a bit, and only insist that there be no interactions with the outside world that would affect those properties of the system that are being studied.

† Besides, when the experimenter is a part of the system, all sorts of other problems arise. This is a dilemma physicists must deal with: the fact that if we take measurements, we are a part of the system, and must be very certain that we carry out experiments so that fact doesn't distort or prejudice the results.
Conserved. A quantity is said to be conserved if under specified conditions it's value does not change with time.

Example: In a closed system, the charge, mass, total energy, linear momentum and angular momentum of the system are conserved. Philosophers debate whether mass and energy are fundamentally the same thing, and whether we should have a conservation of mass-energy law. If you want to learn more about this, see The Equivalence of Mass and Energy.
Current. The time rate at which charge passes through a circuit element or through a fixed place in a conducting wire, I = dq/dt.

Misuse alert. A very common mistake found in textbooks is to speak of "flow of current". Current itself is a flow of charge; what, then, could "flow of current" mean? It is either redundant, misleading, or wrong. This expression should be purged from our vocabulary. Compare a similar mistake: "The velocity moves West." Sounds absurd, doesn't it?
Data. The word data is the plural of datum. Examples of correct usage:

"The data are reasonable, considering the…"
"The data were taken over a period of three days."
"How well do the data confirm the theory?"

Dependent variable. See variable.

Derive. To derive a result or conclusion is to show, using logic and mathematics, how a conclusion follows logically from certain assumed facts and principles.

Dimensions. The fundamental measurables of a unit system in physics—those that are defined through operational definitions. All other measurable quantities in physics are defined through mathematical relations to the fundamental quantities. Therefore any physical measurable may be expressed as a mathematical combination of the dimensions. See: operational definitions.

Example: In the MKSA (meter-kilogram-second-ampere) system of units, length, mass, time and current are the fundamental measurables, symbolically represented by L, M, T, and I. Therefore we say that velocity has the dimensions LT-1. Energy has the dimensions ML2T-2.

Discrepancy. (1) Any deviation or departure from the expected. (2) A difference between two measurements or results. (3) A difference between an experimental determination of a quantity and its standard or accepted value, usually called the experimental discrepancy.

Empirical law. A law strictly based on experimental data, describing the relations in that data. A law generally describes a very specific and limited phenomenon, and does not have the broader scope of a theory.

Electricity. This word names a branch or subdivision of physics, just as other subdivisions are named ‘mechanics’, ‘thermodynamics’, ‘optics’, etc.

Misuse alert: Sometimes the word electricity is colloquially misused as if it named a physical quantity, such as "The capacitor stores electricity," or "Electricity in a resistor produces heat." Such usage should be avoided! In all such cases there's available a more specific or precise word, such as "The capacitor stores electrical energy," "The resistor is heated by the electric current," and "The utility company charges me for the electric energy I use." (I am not being charged based on the power, so these companies shouldn't call themselves Power companies. Some already have changed their names to something like "... Energy")
Energy. Energy is a property associated with a material body. Energy is not a material substance. When bodies interact, the energy of one may increase at the expense of the other, and this is sometimes called a transfer of energy. This does not mean that we could intercept this energy in transit and bottle some of it. After the transfer one of the bodies may have higher energy than before, and we speak of it as having "stored energy". But that doesn't mean that the energy is "contained in it" in the same sense as water in a bucket.

Misuse example: "The earth's auroras—the northern and southern lights—illustrate how energy from the sun travels to our planet." —Science News, 149, June 1, 1996. This sentence blurs understanding of the process by which energetic charged particles from the sun interact with the earth's magnetic field and our atmosphere, causing the light seen in auroras.

Whenever one hears people speaking of "energy fields", "psychic energy", and other expressions treating energy as a "thing" or "substance", you know they aren't talking physics, they are talking moonshine.

In certain quack theories of oriental medicine, such as qi gong (pronounced chee gung) something called qi is believed to circulate through the body on specific, mappable pathways called meridians. This idea pervades the contrived explanations/rationalizations of acupuncture and other quack medical theories. The word qi is generally translated into English as energy. No one has ever found this so-called "energy", nor confirmed the uniqueness of the meridian pathways, nor verified, through proper double-blind tests, that any therapy or treatment based on the theory actually works any better than placebos. The proponents of qi can't say whether it is a fluid, gas, charge, current, or something else, and their theory requires that it doesn't obey any of the physics of known carriers of energy. But, as soon as we hear someone talking about it as if it were a thing we know they are not talking science, but quackery.

Note: Some medical researchers claim there is evidence that in some limited applications, sticking thin needles into a person can help alleviate chronic pain, backache, etc. And it doesn't matter whether the needles are inserted at the "correct" points predicted by "chi and meridian" theory, but can be inserted anywhere on the body. This does not in any way support or confirm the theory that chi flows along meridians, nor that the theory determines the best location for needling. You will notice that in the previous paragraphs I spoke of "quack medical theories" and I'll stand by that characterization of acupuncture.

The statement "Energy is a property of a body" needs clarification. As with many things in physics, the size of the energy depends on the coordinate system. A body moving with speed V in one coordinate system has kinetic energy ½mV2. The same body has zero kinetic energy in a coordinate system moving along with it at speed V. Since no inertial coordinate system can be considered "special" or "absolute", we shouldn't say "The kinetic energy of the body is ..." but should say "The kinetic energy of the body moving in this reference frame is ..."

Energy (take two). Elementary textbooks often say "there are many forms of energy, kinetic, potential, thermal, nuclear, etc. They can be converted from one form to another." Let's try to put more structure to this. There are really only two functional categories of energy. The energy associated with particles or systems can be said to be either kinetic energy or potential energy.

  • The kinetic energy of a particle of mass m and speed v is ½mv2. The kinetic energy of a system of particles is ½MV2 where M is the system mass and V is the speed of its center of mass. One part of a system's energy may be thermal energy, the kinetic energy due to disordered motions and vibrations of particles, on the microscopic scale of molecules, atoms, and even smaller particles.
  • The potential energy of a system is always due to some other system exchanging energy with it by forces moving the system or parts of the system. Potential energy is a way of accounting for the work done by or on another system interacting with the system of interest. Gravitational potential energy is the work we must do against the force due to gravity to move an object to a new position. Once we have accounted for the effect of other systems we can treat our system as if it were "isolated", which is often convenient.

Systems may exchange energy in two ways, through work or heat. Work and heat are never in a body or system, they measure the energy transfered during interactions between systems. Work always requires motion of a system or parts of it, moving the system's center of mass. Heating does not require macroscopic motion of either system. It involves exchanges of energy between systems on the microscopic level, and does not move the center of mass of either system.

Equal. [Not all "equals" are equal.] The word equal and the symbol "=" have many different uses. The dictionary warns that equal things are "alike or in agreement in a specified sense with respect to specified properties." So we must be careful about the specified sense and specified properties.

The meaning of the mathematical symbol, "=" depends upon what stands on either side of it. When it stands between vectors it symbolizes that the vectors are equal in both size and direction.

In algebra the equal sign stands between two algebraic expressions and indicates that two expressions are related by a reflexive, symmetric and transitive relation. The mathematical expressions on either side of the "=" sign are mathematically identical and interchangeable in equations.

When the equal sign stands between two mathematical expressions with physical meaning, it means something quite different than when standing between two numbers. In physics we may correctly write 12 inches = 1 foot, but to write 12 = 1 is simply wrong. In the first case, the equation tells us about physically equivalent measurements. It has physical meaning, and the units are an indispensable part of the quantity.

When we write a = dv/dt, we are defining the acceleration in terms of the time rate of change of velocity. One does not verify a definition by experiment. Experiment can, however, show that in certain cases (such as a freely falling body) the acceleration of the body is constant.

The three-lined equal sign, ≡, is often used to mean "defined equal to".

When we write F = ma, we are expressing a relation between measurable quantities, one that holds under specified conditions, qualifications and limitations. There's more to it than the equation. One must, for example, specify that all measurements are made in an inertial frame, for if they aren't, this relation isn't correct as it stands, and must be modified. Many physical laws, including this one, also include definitions. This equation may be considered a definition of force, if m and a are previously defined. But if F was previously defined, this may be taken as a definition of mass. But the fact that this relation can be experimentally tested, and possibly be shown to be false (under certain conditions) demonstrates that it is more than a mere definition.

Additional discussion of these points may be found in Arnold Arons' book A Guide to Introductory Physics Teaching, section 3.23, listed in the references at the end of this document.

Usage note: When reading equations aloud we often say, "F equals m a". This, of course, says that the two things are mathematically equal in equations, and that one may replace the other. It is not saying that F is physically the same thing as ma. Perhaps equations were not meant to be read aloud, for the spoken word does not have the subtleties of meaning necessary for the task. At least we should realize that spoken equations are at best a shorthand approximation to the meaning; a verbal description of the symbols. If we were to try to speak the physical meaning, it would be something like: "Newton's law tells us that the net vector force acting on a body of mass m is mathematically equal to the product of its mass and its vector acceleration." In a textbook, words like that would appear in the text near the equation, at least on the first appearance of the equation.
Error. In colloquial usage, "a mistake". In technical usage error is a synonym for the experimental uncertainty in a measurement or result. See: uncertainty.

Error analysis. The mathematical analysis (calculations) done to show quantitatively how uncertainties in data produce uncertainty in calculated results, and to find the sizes of the uncertainty in the results. [In mathematics the word analysis is synonymous with calculus, or "a method for mathematical calculation." Calculus courses used to be named Analysis.]

See: uncertainty

Extensive property. A measurable property of a thermodynamic system is extensive if, when two identical systems are combined into one, the value of that property of the combined system is double its original value in each system. Examples: mass, volume, number of moles. See: intensive variable and specific.

Experimental error. The uncertainty in the value of a quantity. This may be found from (1) statistical analysis of the scatter of data, or (2) mathematical analysis showing how data uncertainties affect the uncertainty of calculated results.

Misuse alert: In elementary lab manuals one often sees: experimental error = |your value - book value| /book value. This should be called the experimental discrepancy. See: discrepancy.
Factor. One of several things multiplied together.

Misuse alert: Be careful that the reader does not confuse this with the colloquial usage: "One factor in the success of this experiment was…"

Fictitious force. See: inertial frames.

Focal point. The focal point of a lens is defined by considering a narrow beam of light incident upon the lens, parallel to the optic (symmetry) axis of the lens and centered on that axis. The focal point is that point to which the rays converge or from which they diverge after passing through the lens. The convergent case defines a converging (positive) lens. The second case defines a diverging (negative) lens. It’s easy to tell which kind of lens you have, for converging lenses are thicker at their center than at the edges, and diverging lenses are thinner at the center than at the edges.

FPS. The system of units based on the fundamental units of the ‘English system’: foot, pound and second.

Function. A relation between the elements of one set, X (the domain), and the elements of another set, Y (the range), such that for each element in the domain X there's only one corresponding element in the range Y. When a function is written in the form of an equation relating values of variables, y = y(x), y must be single-valued, that is each value of x corresponds to only one value of y. While y = x2 is a function, x = y1/2 is not. Both equations express relations, however. Experimental science deals with mathematical relations between measurements. Physical laws express these relations. Physical theories often include entities that are defined to be functions of other quantities. Scientists often use the word function colloquially in the sense of "depends on" as in "Pressure is a function of volume and temperature", when they really mean just "Pressure depends on volume and temperature."

Heat. Heat, like work, is a measure of the amount of energy transferred from one body to another because of the temperature difference between those bodies. Heat is not energy possessed by a body. We should not speak of the "heat in a body." The energy a body possesses due to its temperature is a different thing, called internal thermal energy. The misuse of this word probably dates back to the 18th century when it was still thought that bodies undergoing thermal processes exchanged a substance, called caloric or phlogiston, a substance later called heat. We now know that heat is not a substance. Reference: Zemansky, Mark W. The Use and Misuse of the Word "Heat" in Physics Teaching" The Physics Teacher, 8, 6 (Sept 1970) p. 295-300. See: work.

Heisenberg's Uncertainty Principle. Pairs of measurable quantities whose product has dimensions of energy×time are called conjugate quantities in quantum mechanics, and have a special relation to each other, expressed in Heisenberg's uncertainty principle. It says that the product of the uncertainties of the two quantities is no smaller than h/2π. So if you improve the measurement precision of one quantity the precision of the other gets worse.

Misuse alert: Folks who don't pay attention to details of science, are heard to say "Heisenberg showed that you can't be certain about anything." We also hear some folk justifying belief in esp or psychic phenomena by appeal to the Heisenberg principle. This is wrong on several counts. (1) The precision of any measurement is never perfectly certain, and we knew that before Heisenberg. (2) The Heisenberg uncertainty principle tells us we can measure anything with arbitrarily small precision, but in the process some other measurement gets worse. (3) The uncertainties involved here affect only microscopic (atomic and molecular level phenomena) and have no applicability to the macroscopic phenomena of everyday life.

Hypothesis. An untested statement about nature; a scientific conjecture, or educated guess. Elementary textbooks often declare that a hypothesis is made prior to doing the experiments designed to test it. However, we must recognize that experiments sometimes reveal unexpected and puzzling things, motivating one to then explore various hypotheses that might serve to explain the experiments. Further testing of the hypotheses under other conditions is then in order, as always. Compare: law and theory.

Ideal-lens equation. 1/p + 1/q = 1/f, where p is the distance from object to lens, q is the distance from lens to image, and f is the focal length of the lens. This equation has important limitations, being only valid for thin lenses, and for paraxial rays. Thin lenses have thickness small compared to p, q, and f. Paraxial rays are those that make angles small enough with the optic axis that the approximation (angle in radian measure) = sin(angle) may be used. See: optical sign conventions, and image.

Independent variable. See variable.

Inertia A descriptive term for that property of a body that resists change in its motion. Two kinds of changes of motion are recognized: changes in translational motion, and changes in rotational motion.

In modern usage, the measure of translational inertia is mass. Newton's first law of motion is sometimes called the "Law of Inertia", a label that adds nothing to the meaning of the first law. Newton's first and second laws together are required for a full description of the consequences of a body's inertia.

The measure of a body's resistance to rotation is its Moment of Inertia.

See: moment of inertia,

Misuse alert: One sometimes sees "A force arises because of inertia." This misleads one into supposing that the inertia is a cause of the force. It is not hard to discuss all of the physics of force, mass and acceleration without ever using the word "inertia". Unfortunately we are stuck with it in the widely used name "moment of inertia".

To be provocative, one might ask "Does a mass at rest have inertia?" Since inertia is a synonym for mass, then the answer must be "Yes."

Inertial frame. A non-accelerating coordinate system. One in which F = ma holds, where F is the sum of all real forces acting on a body of mass m whose acceleration is a. In classical mechanics, the real forces on a body are those that are due to the influence of another body. [Or, forces on a part of a body due to other parts of that body.] Contact forces, gravitational, electric, and magnetic forces are real. Fictitious forces are those that arise solely from formulating a problem in a non-inertial system, in which ma = F + (fictitious force terms)

Infinity. (Symbol ∞) is a mathematical shorthand for an uncountable set of things. Sometimes we used the symbols lim A→∞ to represent a process of taking A "to its limit, that is to values so large that further increase in its size changes nothing else. Perhaps the most important lesson we can offer is this: Infinity is NOT a number. If it were a number, then ∞+2 would be a still larger number. Therefore algebraic expressions combining ∞ with numbers or with letters representing numbers, should be regarded as potentially misleaing.

In optics one sometimes reads "the real image is located at infinity". This translates to "The light rays forming the image do not converge to or diverge from any finite distance." The rays forming the image are, in fact, parallel. In the lens equation 1/p + 1/q = 1/f one can "get away" with treating the expression 1/∞ as zero, which may be the reason students come to accept all such algebraic expressions with infinity as valid. In other physial situations this can lead to trouble. In fact, even in optics, one can, with this phony algebra, "logically" conclude that when a real image is at infinity, there's also a virtual image at minus infinity [Proof: 1/∞ and 1/-∞ are both equal to zero, since -0 = 0]. So "the" image formed by a lens system can be in two places at once, "infinitely" far apart!

Intensive variable. A measurable property of a thermodynamic system is intensive if when two identical systems are combined into one, the variable of the combined system is the same as the original value in each system. Examples: temperature, pressure. See: extensive variable, and specific.

Image. (Optics) A surprising number of physics glossaries omit a definition of this! No wonder. It's difficult to put in a few words, and still be comprehensive in scope. Try this.

Image: A point mapping of luminous points of an object located in one region of space to points in another region of space, formed by refraction or reflection of light in a manner that causes light from each point of the object to converge to or diverge from a point somewhere else (on the image).

The images that are useful generally have the character that adjacent points of the object map to adjacent points of the image without discontinuity, and is a recognizable (though perhaps distorted) mapping of the object. This qualification allows for anamorphic images, that are stretched or compressed in one direction, as well as the sort of distorted (but recognizable) images you see in a fun-house mirror. See: real image and virtual image.

Law. A statement, usually mathematical, that describes the relation between variables in some particular physical phenomena. Compare: hypothesis and theory.

Lens. A transparent object with two refracting surfaces. Usually the surfaces are flat or spherical. Sometimes, to improve image quality, lenses are deliberately made with surfaces that depart slightly from spherical (aspheric lenses).

Kinetic energy. The energy a body has by virtue of its motion. The kinetic energy is the work done by an external force to bring the body from rest to a particular state of motion. See: work.

Common misconception: Many students think that kinetic energy is defined by ½mv2. It is not. That happens to be approximately the kinetic energy of objects moving slowly, at small fractions of the speed of light. If the body is moving at relativistic speeds, its kinetic energy is γ mc2, which can be expressed as ½ mv2 + an infinite series of terms. γ2 = 1/(1-(v/c)2), where c is the speed of light in a vacuum.

Macro-. A prefix meaning ‘large’. See: micro-

Macroscopic. A physical entity or process of large scale, the scale of ordinary human experience. Specifically, any phenomena in which the individual molecules and atoms are neither measured, nor explicitly considered in the description of the phenomena. See: microscopic.


Two kinds of magnification are useful to describe optical systems and they must not be confused, since they aren't synonymous. Any optical system that produces a real image from a real object is described by its linear magnification. Any system that one looks through to view a virtual image is described by its angular magnification. These have different definitions, and are based on fundamentally different concepts.

Linear Magnification is the ratio of the size of the object to the size of the image.

Angular Magnification is the ratio of the angular size of the object as seen through the instrument to the angular size of the object as seen with the 'naked eye' under the best viewing conditions. The 'naked eye' view is without use of the optical instrument, but under optimal viewing conditions.

Certain 'gotchas' lurk here. What are 'optimal' conditions? Usually this means the conditions in which the object's details can be seen most clearly. For a small object held in the hand, this would be when the object is brought as close as possible and still seen clearly, that it, to the near point of the eye, about 25 cm for normal eyesight. For a distant mountain, one can't bring it close, so when determining the magnification of a telescope, we assume the object is very distant, said to be "at infinity". [But heed the cautions about the use of the word "infinity" in this informal manner. See: macro]

And what is the 'optimal' position of the image? For the simple magnifier, in which the magnification depends strongly on the image position, the image is best seen at the near point of the eye, 25 cm. For the telescope, the image size doesn't change much as you fiddle with the focus, so you likely will put the image at infinite distance for relaxed viewing. The microscope is an intermediate case. Always striving for greater resolution, the user may pull the image close, to the near point, even though that doesn't increase its size very much. But usually, users will place the image farther away, at the distance of a meter or two, or even "at infinity". But, because the object is very near the focal point, the magnification is only weakly dependent on image position.

Some texts express angular magnification as the ratio of the angles, some express it as the ratio of the tangents of the angles. If all of the angles are small, there's negligible difference between these two definitions. However, if you examine the derivation of the formula these books give for the magnification of a telescope fo/fe, you realize that they must have been using the tangents. The tangent form of the definition is the traditionally correct one, the one used in science and industry, for nearly all optical instruments that are designed to produce images that preserve the linear geometry of the object.

Micro-. A prefix meaning ‘small’, as in ‘microscope’, ‘micrometer’, ‘micrograph’. Also, a metric prefix meaning 10-6. See: macro-

Microscopic. A physical entity or process of small scale, too small to directly experience with our senses. Specifically, any phenomena on the molecular and atomic scale, or smaller. See: macroscopic.

MKSA. The system of physical units based on the fundamental metric units: meter kilogram, second and ampere.

Modern physics. The physics developed since about 1900, which includes relativity and quantum mechanics.

The transition from classical physics to modern physics was gradual, over about 30 years. Classical physics is still a part of physics, and the demarcation between classical and modern physics relates to the size and character of the systems studied. Classical physics applies to bodies of sizes larger than atoms and moleculels, moving at speeds much slower than the speed of light. Quantum mechanics applies at size scales of atoms or smaller. Relativity is necessary at speeds near the speed of light.

See: classical physics.

Mole. The term mole is short for the name gram-molar-weight; it is not a shortened form of the word molecule. (However, the word molecule does also derive from the word molar.) See: Avogadro’s constant.

Misuse alert: Many books emphasize that the mole is "just a number," a measure of the number of particles in a collection. They say that one can have a mole of any kind of particles, baseballs, atoms, stars, grains of sand, etc. It doesn't have to be molecules. This is misleading.

To say that the mole is "just a number" is simply wrong, from physical, pedagogical, philosophical and historical points of view. There's no physical significance to a mole of stars or a mole of grains of sand, or a mole of people. The physical significance of the mole as a measure of quantity arises only when dealing with physical laws about matter on the molecular scale. The only physical and chemical laws that use the mole are those dealing with gases, or systems behaving like gases.

Molecular mass. The molecular mass of something is the mass of one mole of it (in cgs units), or one kilomole of it (in MKS units). The units of molecular mass are gram and kilogram, respectively. The cgs and MKS values of molecular mass are numerically equal. The molecular mass is not the mass of one molecule. Some books still call this the molecular weight.

One dictionary definition of molar is "Pertaining to a body of matter as a whole: contrasted with molecular and atomic." The mole is a measure appropriate for a macroscopic amount of material, as contrasted with a microscopic amount (a few atoms or molecules). See: mole, Avogadro's constant, microscopic, macroscopic.

Moment of Inertia. A property of a body that relates its angular velocity about a particular axis to the net torque on the body about that axis. τ = Iω. The moment of inertia is very much dependent on the chosen axis, for it may have a different value for different axes. In fact, the moment of inertia is best expressed as a three dimensional array (matrix) of values measured with respect to a three dimensional coordinate system. There is always one particular coordinate system in which this matrix is diagonal, having only three distinct values along its diagonal, and zeros elsewhere. These axes are called the principal axes of the body, and the three values are the principal moments of the body.

This may be thought of as analogous to Newton's second law F = m a, where m (mass) is a measure of translational inertia, and I (moment of inertia) is rotational inertia. But always be suspicious of analogies, except as memory clues.

The moments of inertia of an extended body can be calculated directly by volume integrals taken over the volume of the body. The formula is I = ∫ r2 dm where r is the perpendicular distance from the mass element dm to the chosen axis.

Newton's first and second laws of motion. F = d(mv)/dt.

F is the net (total) force acting on the body of mass m. The individual forces acting on m must be summed vectorially. In the special case where the mass is constant, this becomes F = ma.

Newton's third law of motion. When body A exerts a force on body B, then B exerts and equal and opposite force on A. The two forces related by this law act on different bodies. The forces in Newton's third need not be net forces, but because forces sum vectorially, Newton's third is also true for net forces on a body.

Ohm's law. V = IR, where V is the potential across a circuit element, I is the current through it, and R is its resistance. This is not a generally applicable definition of resistance. It is only applicable to ohmic resistors, those whose resistance R is constant over the range of interest and V obeys a strictly linear relation to I.

Materials are said to be ohmic when V depends linearly on R. Metals are ohmic so long as one holds their temperature constant. But changing the temperature of a metal changes R slightly. (More than slightly if it melts!) When the current changes rapidly, as when turning on a lamp, or when using AC sources, non-linear and non-ohmic behavior can be observed.

For non-ohmic resistors, R is current-dependent and the definition R = dV/dI is far more useful. This is sometimes called the dynamic resistance. Solid state devices such as thermistors are non-ohmic and non-linear. A thermistor's resistance decreases as it warms up, so its dynamic resistance is negative. Tunnel diodes and some electrochemical processes have a complicated I-V curve with a negative resistance region of operation.

The dependence of resistance on current is partly due to the change in the device's temperature with increasing current, but other subtle processes also contribute to change in resistance in solid state devices.

Operational definition. A definition that describes an experimental procedure by which a numeric value of the quantity may be determined. See dimensions.

Example: Length is operationally defined by specifying a procedure for subdividing a standard of length into smaller units to make a measuring stick, then laying that stick on the object to be measured, etc....

Very few quantities in physics need to be operationally defined. They are the fundamental quantities, which include length, mass and time. Other quantities are defined from these through mathematical relations.

Optical sign conventions. In introductory (freshman) courses in physics a sign convention is used for objects and images in which the lens equation must be written 1/p + 1/q = 1/f. Often the rules for this sign convention are presented in a convoluted manner. A simple and easy to remember rule is this: p is the object-to-lens distance. q is the lens to image distance. The coordinate axis along the optic axis is in the direction of passage of light through the lens, this defining the positive direction. Example: If the axis and the light direction is left-to-right (as is usually done) and the object is to the left of the lens, the object-to-lens distance is positive. If the object is to the right of the lens (virtual object), the object-to-lens distance is negative. It works the same for images.

For refractive surfaces, define the surface radius to be the directed distance from a surface to its center of curvature. Thus a surface convex to the incident light is positive, one concave to the incident light is negative. The surface equation is then n/s + n'/s' = (n'-n)/R where s and s' are the object and image distances, and n and n' the refractive index of the incident and emergent media, respectively.

For mirrors, the equation is usually written 1/s + 1/s' = 2/R = 1/f. A diverging mirror is convex to the incoming light, with negative f. From this fact we conclude that R is also negative. This form of the equation is consistent with that of the lens equation, and the interpretation of sign of focal length is the same also. But violence is done to the definition of R we used above, for refraction. One can say that the mirror folds the length axis at the mirror, so that emergent rays to a real image at the left represent a positive value of s'. We are forced also to declare that the mirror also flips the sign of the surface radius. For reflective surfaces, the radius of curvature is defined to be the directed distance from a surface to its center of curvature, measured with respect to the axis used for the emergent light. With this qualification the convention for the signs of s' and R is the same for mirrors as for refractive surfaces.

In advanced optics courses, a cartesian sign convention is used in which all things to the left of the lens are negative, all those to the right are positive. When this is used, the lens equation must be written 1/p + 1/f = 1/q. (The sign of the 1/p term is opposite that in the other sign convention). This is a particularly meaningful version, for 1/p is the measure of vergence (convergence or divergence) of the rays as they enter the lens, 1/f is the amount the lens changes the vergence, and 1/q is the vergence of the emergent rays.

Particle. This word, lifted from colloquial usage, means different things in science, depending on the context. To the Greek philosophers it meant a "little piece" of matter, and Democritus taught that these pieces, that he called "atoms" had different geometric shapes that governed how they could combine and link together. This idea was speculative, and not supported by any specific experiments or evidence. The "atomic theory" didn't arise until the 19th century, motivated primarily by the emerging science of chemistry, though at first some scientists rejected the reality of atoms, considering atoms to be no more than a "useful fiction" since they weren't directly observable. In the early 20th century the Bohr theory gave a detailed picture of atoms as something like "miniature solar systems" of electrons orbiting an incredibly small and dense nucleaus. This "classical" picture proved to be misleadingly simplistic, though it is still the "picture" in most people's minds when they think of atoms. Since then experimentalists have identified a whole "zoo" of particles that arise in nuclear reactions. But are these "little pieces" of matter as the Greeks thought? Or are they a convenient fiction to describe what we measure with increasingly sophisticated "particle detectors". Perhaps what we are measuring is nothing more than "events" resulting from complex interactions of wave functions. Did you really expect a definite and final answer to this here?

Pascal's Principle of Hydrostatics. Pascal actually has three separate principles of hydrostatics. When a textbook refers to Pascal's Principle it should specify which is meant.

Pascal 1: The pressure at any point in a liquid exerts force equally in all directions. This shorthand slogan means that an infinitesimal surface area placed at that point will experience the same force due to pressure no matter what its orientation.

Pascal 2: When pressure is changed (increased or decreased) at any point in a homogenous, incompressible fluid, all other points experience the same change of pressure.

Except for minor edits and insertion of the words 'homogenous' and 'incompressible', this is the statement of the principle given in John A. Eldridge's textbook College Physics (McGraw-Hill, 1937). Yet over half of the textbooks I've checked, including recent ones, omit the important word 'changed'. Some textbooks add the qualification 'enclosed fluid'. This gives the false impression that the fluid must be in a closed container, which isn't a necessary condition of Pascal's principle at all.

Some of these textbooks do indicate that Pascal's principle applies only to changes in pressure, but do so in the surrounding text, not in the bold, highlighted, and boxed statement of the principle. Students, of course, read the emphasized statement of the principle and not the surrounding text. Few books give any examples of the principle applied to anything other than enclosed liquids. The usual example is the hydraulic press. Too few show that Pascal's principle is derivable in one step from Bernoulli's equation. Therefore students have the false impression that these are independent laws.

Pascal 3. The hydraulic lever. The hydraulic jack is a problem in fluid equilibrium, just as a pulley system is a problem in mechanical equilibrium (no accelerations involved). It's the static situation in which a small force on a small piston balances a large force on a large piston. No change of pressure need be involved here. A constant force on one piston slowly lifts a different piston with a constant force on it. At all times during this process the fluid is in near-equilibrium. This "principle" is no more than an application of the definition of pressure as F/A, the quotient of net force to the area over which the force acts. However, it also uses the principle that pressure in a fluid is uniform throughout the fluid at all points of the same height.

This hydraulic jack lifting process is done at constant speed. If the two pistons are at different levels, as they usually are in real jacks used for lifting, there's a pressure difference between the two pistons due to height difference ρgh where ρ is the density of the liquid. In textbook examples this is generally considered small enough to neglect and may not even be mentioned.

Pascal's own discussion of the principle is not concisely stated and can be misleading if hastily read. See his On the Equilibrium of Liquids, 1663. He introduces the principle with the example of a piston as part of an enclosed vessel and considers what happens if a force is applied to that piston. He concludes that each portion of the vessel is pressed in proportion to its area. He does mention parenthetically that he is "excluding the weight of the water..., for I am speaking only of the piston's effect."

Percentage. Older dictionaries suggested that percentage be used when a non-quantitative statement is being made: "The percentage growth of the economy was encouraging." But use percent when specifying a numerical value: "The gross national product increased by 2 percent last year."

One other use of "percentage" is proper, however. When comparing a percent measure which changes, it's common to express that change in "percentage points." For example, if the unemployment rate is 5% one month, and 6% the next, we say "Unemployment increased by one percentage point". The absolute change in unemployment was, however, an increase of 20 percent. The average person hearing such figures seldom stops to think what the words mean, and many people think that "percent" and "percentage point" are synonyms. They are not. This is one more reason to avoid using the word "percentage" when expressing percent measures. The term "percentage point" is almost never used in the sciences. (Unless you consider economics a science.)

Students in the sciences, unaware of this distinction will say "The experimental percentage uncertainty in our result was 9%." Perhaps they are trying to "sound profound". In view of the above discussion, this isn't what the student meant. The student should have simply said: "The experimental uncertainty in our result was 9%."

Related note: Students have the strange idea that results are better when expressed as percents. Some experimental uncertainties must not be expressed as percents. Examples: (1) temperature in Celsius or Fahrenheit measure, (2) index of refraction, (3) dielectric constants. These measurables have arbitrarily chosen ‘fixed points’. Consider a 1 degree uncertainty in a temperature of 99 degrees C. Is the uncertainty 1%? Consider the same error in a measurement of 5 degrees. Is the uncertainty now 20%? Consider how much smaller the percent would be if the temperature were expressed in degrees Kelvin. This shows that percent uncertainty of Celsius and Fahrenheit temperature measurements is meaningless. However, the absolute (Kelvin) temperature scale has a physically meaningful fixed point (absolute zero), rather than an arbitrarily chosen one, and in some situations a percent uncertainty of an absolute temperature is meaningful.

Per unit. In my opinion this expression is a barbarism best avoided. When a student is told that electric field is force per unit charge and in the MKS system one unit of charge is a coulomb (a huge amount) must we obtain that much charge to measure the field? Certainly not. In fact, one must take the limit of F/q as q goes to zero. Simply say: "Force divided by charge" or "F over q" or even "force per charge". Unfortunately there is no graceful way to say these things, other than simply writing the equation.

Per is one of those frustrating words in English. The American Heritage Dictionary definition is: "To, for, or by each; for every." Example: "40 cents per gallon." We must put the blame for per unit squarely on the scientists and engineers.

Precise. Sharply or clearly defined. Having small experimental uncertainty. A precise measurement may still be inaccurate, if there were an unrecognized determinate error in the measurement (for example, a miscalibrated instrument). Compare: accurate.

Proof. A term from logic and mathematics describing an argument from premise to conclusion using strictly logical principles. In mathematics, theorems or propositions are established by logical arguments from a set of axioms, the process of establishing a theorem being called a proof.

The colloquial meaning of ‘proof’ causes many problems in physics discussions and is best avoided. Since mathematics is such an important part of physics, the mathematician’s meaning of proof should be the only one we use. Also, we often ask students in upper level courses to do proofs of certain theorems of mathematical physics, and we are not asking for experimental demonstration!

So, in a laboratory report, we should not say "We proved Newton's law." Rather say, "Today we demonstrated (or verified) the validity of Newton's law in the particular case of…"

Science doesn't prove, but it can disprove. See: Why?

Radioactive material. A material whose nuclei spontaneously give off nuclear radiation. Naturally radioactive materials (found in the earth's crust) give off alpha, beta, or gamma particles. Alpha particles are Helium nuclei, beta particles are electrons, and gamma particles are high energy photons.

Radioactive. A word distinguishing radioactive materials from those which aren't. Usage: "U-235 is radioactive; He-4 is not."

Note: Radioactive is least misleading when used as an adjective, not as a noun. It is sometimes used in the noun form as an shortened stand-in for radioactive material, as in the example above.
Radioactivity. The process of emitting particles from the nucleus. Usage: "Certain materials found in nature demonstrate radioactivity."
Misuse alert: Radioactivity is a process, not a thing, and not a substance. It is just as incorrect to say "U-235 emits radioactivity" as it is to say "current flows." A malfunctioning nuclear reactor does not release radioactivity, though it may release radioactive materials into the surrounding environment. A patient being treated by radiation therapy does not absorb radioactivity, but does absorb some of the radiation (alpha, beta, gamma) given off by the radioactive materials being used.

This misuse of the word radioactivity causes many people to incorrectly think of radioactivity as something one can get by being near radioactive materials. There is only one process which behaves anything like that, and it is called artificially induced radioactivity, a process mainly carried out in research laboratories. When some materials are bombarded with protons, neutrons, or other nuclear particles of appropriate energy, their nuclei may be transmuted, creating unstable isotopes which are radioactive.

Rate. A quantity of one thing compared to a quantity of another. [Dictionary definition]

In physics the comparison is generally made by taking a quotient. Thus speed is defined to be the dx/dt, the ‘time rate of change of position’.

Common misuse: We often hear non-scientists say such things as "The car was going at a high rate of speed." This is redundant at best, since it merely means "The car was moving at high speed." It is the sort of mistake made by people who don't think while they talk.

Ratio. The quotient of two similar quantities. In physics, the two quantities must have the same units to be ‘similar’. Therefore we may properly speak of the ratio of two lengths. But to say "the ratio of charge to mass of the electron" is improper. The latter is properly called "the specific charge of the electron." See: specific.

Reaction. Reaction forces are those equal and opposite forces of Newton's Third Law. Though they are sometimes called an action and reaction pair, one never sees a single force referred to as an action force. See: Newton’s Third Law.

Real force. See: inertial frame.

Real image. The point(s) to which light rays converge as they emerge from a lens or mirror. See: virtual image.

Real object. The point(s) from which light rays diverge as they enter a lens or mirror. See: virtual object.

Reality. We say that science studies the "real" world of perception and measurement. If we can apprehend something with our senses, or measure it, we treat it as "real". We have learned not to completely trust our unaided senses, for we know that we can be fooled by illusions, so we rely more on specially designed measuring instruments. Yet much of the language of science has entities that are not directly observable by our senses, such as "energy", and "momentum". These are, however, directly related to observables and defined through exact equations. Philosophers may argue whether the "real" world exists, but so long as our sense impressions and measurements of this real world are shared by independent observers and are precisely repeatable, we can do physics without philosophical concerns.

Relation A rule of correspondence between the set of values of one quantity to the values of another quantity, often (but not always) expressible as an equation. See function.

Relative. Colloquially "compared to". In the theory of relativity observations of moving observers are quantitatively compared. These observers obtain different values when measuring the same quantities, and these quantities are said to be relative. The theory, however, shows us how the differing measured values are precisely related to the relative velocity of the two observers. Some quantities are found to be the same for all observers, and are called invariant. One postulate of relativity theory is that the speed of light is an invariant quantity. When the theory is expressed in four dimensional form, with the appropriate choice of quantities, new invariant quantities emerge: the world-displacement (x + y + z + ict), the energy-momentum four-vector, and the electric and magnetic potentials may be combined into an invariant four-vector. Thus relativity theory might properly be called invariance theory.

Misuse alert: One hears some folks with superficial minds say "Einstein showed that everything is relative." In fact, special relativity shows that only certain measurable things are relative, but in a precisely and mathematically specific way, and other things are, not relative, for all observers agree on them.

Relative uncertainty. The uncertainty in a quantity compared to the quantity itself, expressed as a ratio of the absolute uncertainty to the size of the quantity. It may also be expressed as a percent uncertainty. The relative uncertainty is dimensionless and unitless. See: absolute uncertainty.

Rigid body. Classical mechanics textbooks have a chapter on the mechanics of perfectly rigid bodies, but may fail to define what they are. If one thinks about it, one must conclude that there's no such thing as a perfectly rigid body. All bodies are compressible because of the inherent atomic structure of materials. Even if you look at purely classical phenomena, such as the collision of two billiard balls, the observed physics couldn't happen if the bodies were perfectly rigid. (The forces at impact would have to be infinite.) The "rigid body" assumption is a mathematical convenience that is useful and gives correct results for many important phenomena, much as it is sometimes useful to analyze systems by assuming that friction is negligible.

Scale-limited. A measuring instrument is said to be scale-limited if the experimental uncertainty in that instrument is smaller than the smallest division readable on its scale. Therefore the experimental uncertainty is taken to be half the smallest readable increment on the scale.

Specific. In physics and chemistry the word specific in the name of a quantity usually means divided by an extensive measure, that is, "divided by a quantity representing an amount of material". Specific volume means volume divided by mass, which is the reciprocal of the density. Specific heat capacity is the heat capacity divided by the mass. See: extensive, and capacity.

Tele-. A prefix meaning at a distance, as in telescope, telemetry, television.

Term. One of several quantities that are added together.

Confusion can arise with another use of the word, as when one is asked to “Express the result in terms of mass and time.” This means that the result is “dependent on mass and time”. Obviously it doesn’t mean that mass and time are to be added as terms.

Truth. This is a word best avoided entirely in physics except when placed in quotes, or with careful qualification. Its colloquial use has so many shades of meaning from "it seems to be correct" to the absolute truths claimed by religion, that it’s use causes nothing but misunderstanding. Someone once said "Science seeks proximate (approximate) truths". Others speak of provisional or tentative truths. Certainly science claims no final or absolute truths.

Theoretical. Describing an idea that is part of a theory, or a consequence derived from theory.

Misuse alert: Do not call an authoritative or ‘book’ value of a physical quantity a theoretical value, as in: "We compared our experimentally determined value of index of refraction with the theoretical value and found they differed by 0.07". The value obtained from index of refraction tables comes not from theory, but from experiment, and therefore should not be called theoretical. The word theoretically suffers the same abuse. Only when a numeric value is a prediction from theory, can one properly refer to it as a "theoretical value".

Theory. A well-tested, usually mathematical, model of some part of science. In physics a theory usually takes the form of an equation or a group of equations, along with explanatory rules for their application. Theories are said to be successful if (1) they synthesize and unify a significant range of phenomena; (2) they have predictive power, either predicting new phenomena, or suggesting a direction for further research and testing. Compare: hypothesis, and law.

Uncertainty. Synonym: error. A measure of the inherent variability of repeated measurements of a quantity. A prediction of the probable variability of a result, based on the inherent uncertainties in the data, found from a mathematical calculation of how the data uncertainties would, in combination, lead to uncertainty in the result. This calculation or process by which one predicts the size of the uncertainty in results from the uncertainties in data and procedure is called error analysis.

See: absolute uncertainty and relative uncertainty. Uncertainties are always present. The experimenter’s job is to keep them as small as required for a useful result. We recognize two kinds of uncertainties: indeterminate and determinate. Indeterminate uncertainties are those whose size and sign are unknown, and are sometimes (misleadingly) called random. Determinate uncertainties are those of definite sign, often referring to uncertainties due to instrument miscalibration, bias in reading scales, or some unknown influence on the measurement.

"Uncertainty" and "error" have colloqual meanings as well. Examples: "I have some uncertainty how to proceed." "The answer isn't reasonable; I must have made an error (mistake or blunder)."

Units. Labels that distinguish one type of measurable quantity from other types. Length, mass and time are distinctly different physical quantities, and therefore have different unit names, meters, kilograms and seconds. We use several systems of units, including the metric (SI) units, the English (or U.S. customary units), and a number of others of mainly historical interest.

Note: Some dimensionless quantities are assigned unit names, some are not. Specific gravity has no unit name, but density does. Angles are dimensionless, but have unit names: degree, radian, and grad. Some quantities which are physically different, and have different unit names, may have the same dimensions, for example, torque and work. Compare: dimensions.

Variable. A symbol representative of a set of numbers, points, values, etc. In science, variables represent values of measurements of quantities.

Much confusion exists about the meanings of dependent and independent variables. In one sense this distinction hinges on how you write the relation between variables.

(1) If you write a function or relation in the form y = f(x), y is considered dependent on x and x is said to be the independent variable.

(2) If one variable (say x) in a relation is experimentally set, fixed, or held to particular values while measuring corresponding values of y, we call x the independent variable. We could just as well (in some cases) set values of y and then determine corresponding values of x. In that case y would be the independent variable.

(3) If the experimental uncertainties of one variable are smaller than the other, the one with the smallest uncertainty is often called the independent variable.

(4) As a general rule, independent variables are plotted on the horizontal axis of a graph, but this is not required if there's a good reason to do it otherwise.

y ≠ ƒ(x)

This x is a very independent variable.

In many cases these four different practical definitions do not conflict with each other and one may choose language, form of equation, and method of plotting graphs so that all definitions are satisfied. But not always. Let common sense and the need for clear communication decide how to deal with situations where there seems to be conflict.

Some common statistical packages for computers can only deal with situations where one variable is assumed error-free, and all the experimental error is in the other one. They cavalierly refer to the error-free variable as the independent variable. But in real science, there's always some experimental error in all values, including those we "set" in advance to particular values.

Oh, yes. There's an exception. If one variable can take only integer values it can often be assumed error-free. One cannot have 2±0.4 billiard balls. It is assumed we can accurately count small numbers of discrete objects.

Virtual image. The point(s) from which light rays converge as they emerge from a lens or mirror. The rays do not actually pass through each image point. [One and only one ray, the one which passes through the center of the lens, does pass through the image point.] See: real image.

Virtual object. The point(s) to which light rays converge as they enter a lens. The rays do not actually pass through each object point. [One and only one ray, the one which passes through the center of the lens, does pass through the object point.] See: real object.

Weight. The size of the external force required to keep a body at rest in its frame of reference.

Elementary textbooks almost universally define weight to be "the size of the gravitational force on a body." This would be fine if they would only consistently stick to that definition. But, no, they later speak of weightless astronauts, loss of weight of a body immersed in a liquid, etc. The student who is really thinking about this is confused. Some books then tie themselves in verbal knots trying to explain (and defend) why they use the word inconsistently. Our definition has the virtue of being consistent with all of these uses of the word.

In the special case of a body supported near the earth's surface, where the acceleration due to gravity is g, the weight happens to have size mg. So this definition gives the same size for the weight as the more common definition.

This definition is consistent with the statement: "The astronauts in the orbiting spacecraft were in a weightless condition." This is because they and their spacecraft have the same acceleration, and in their frame of reference (the spacecraft) no force is needed to keep them at the same position relative to their spacecraft. They and their spacecraft are both falling at the same rate. The gravitational force on the astronauts is still mg (though g is about 12% smaller at an altitude of 400 km than it is at the surface of the earth. It is not zero).

This definition is consistent with statements about the "loss of weight" of a body immersed in a liquid (due to the buoyant force). The "weight" meant here is the external force (not counting the buoyant force) required to support the body in equilibrium in the liquid.

Why? Students often ask questions with the word why in them. "Why is the sky blue?" "Why do objects fall to earth?" "Why are there no bodies with negative mass?" "Why is the universe lawful?" What sort of answers does one desire to such a question? What sort of answers can science give? If you want some mystical, ultimate or absolute answer, you won't get it from science. Philosophers of science point out that science doesn't answer why questions, it only answers how questions. Science doesn't explain; science describes. Science postulates models to describe how some part of nature behaves, then tests and refines that model till it works as well as we can measure (as evidenced by repeated, skeptical testing). Science doesn’t provide ultimate or absolute answers, but only proximate (good enough) answers. Science can't find absolute truth, but it can expose errors and identify things which aren't so, thereby narrowing the region in which truth may reside. In the process, science has produced more reliable knowledge than any other branch of human thought.

Work. The amount of energy transferred to or from a body or system as a result of forces acting upon the body, causing displacement of the body or parts of it. More specifically the work done by a particular force is the product of the displacement of the body and the component of the force in the direction of the displacement. A force acting perpendicular to the body's displacement does no work on the body. A force acting upon a body which undergoes no displacement does no work on that body. Also, it follows that if there's no motion of a body or any part of the body, nothing did work on the body. See: kinetic energy.

Zeroth law of thermodynamics. If body A is in thermal equilibrium with body B, and B is also in thermal equilibrium with C, then A is necessarily in thermal equilibrium with C.

This is equivalent to saying that thermal equilibrium obeys a transitive mathematical relation. Since we define equality of temperature as the condition of thermal equilibrium, then this law is necessary for the complete definition of temperature. It ensures that if a thermometer (body B) indicates that body A and C give the same thermometer reading, then they bodies A and C are at the same temperature.


Arons, Arnold B. A Guide to Introductory Physics Teaching. Wiley, 1990.

Arons, Arnold B. Teaching Introductory Physics. Wiley, 1997.

Iona, Mario. The Physics Teacher. Regular column, titled "Would You believe?", which documents and discusses errors and misleading statements in physics textbooks.

Swartz, Clifford and Thomas Miner. Teaching Introductory Physics, A Sourcebook. American Institute of Physics, 1997.

Symbols, Units, Nomenclature and Fundamental Constants in Physics. From Document U.I.P 11 (S.U.N. 65-3) International Union of Pure and Applied Physics. Contained in the Handbook of Chemistry and Physics, The Chemical Rubber Company. Online PDF.

Warren, J. W. The Teaching of Physics. Butterworth's, 1965, 1969.

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Revised 1997, 2004, 2014.