Most physics teachers emphasize units in problem solving. Far fewer use the concept of dimensions and dimensional analysis. The author finds this regrettable, especially in view of the contemporary emphasis on conceptual physics. Dimensional analysis gives beginning students near miraculous power to predict physics behavior. Few first year students can master enough theory to do as well. I have been developing this topic for many years and find high school students can be very successful with it. However like any profound idea it must be taught over a long period of time and applied again and again.

The pamphlets accompanying this introduction are the students' text in dimensional analysis and the labs are ideal practical examples. This introduction shows the author's strategies for introducing the topic.

The concept of dimension can be introduced early in motion studies. The four basic quantities of physics are distance d, time t, mass M, and electric charge q. To distinguish them from less basic quantities we call them by the name dimensions. Any conceivable physics quantity is some combination of these dimensions. * Beginning lessons emphasize the dimensions of compound quantities like velocity (d/t = dt^-1), acceleration (d/t² = dt^-2), etc. Some students are surprised that gravity acceleration is also dt^-2 . (Students often resist writing dimensions with negative exponents; for example they prefer a = d/t² to
a = dt^-2. For purely practical reasons, this should be overcome.)

Of special significance are the inverse quantities of period (t) and frequency (1/t = t^-1). Illustrating period as seconds per cycle and frequency as cycles per sec may help. In any case master this before proceeding further!

Most students (especially those taking chemistry the previous year) will have trouble with density. The standard definition of density = mass/volume = m/d³ = md^-3 is too restrictive. If we have thin sheet materials like paper or stretched membranes, the appropriate definition is density = mass/area = m/d² = md^-2. For thin strings or wires, density = mass/length = m/d = md^-1. The better student notes that these definitions are moot with thick sheets or fat wires. True enough. The definitions are idealized. This issue is similar to "point masses" or "small pendulum swings" etc. It will soon be important.

If students master the preceding it is sufficient for the time being. Their algebra skills need the honing of motion physics. In particular, most students need to review exponent manipulations like x^y * x² = x^y+2 etc.

A good first example of dimensional analysis is to predict the behavior of falling bodies. We expect that the distance an object falls from rest is related to the acceleration of gravity and how long it falls. We can express this as:

(1) d = f (a,t)                  This is shorthand for "distance has something to do with acceleration and time".

(2) d = c a^x t^y                 c is a constant, x and y are the exponents of a and t.

(3) d = c (dt^{- 2}) ^x t^y         This plugs in dimensions of acceleration.

(4) d = c d^x t^{- 2x} t^y

(5) d = c d^x t ^y-2x

Equations must have the same dimensions on both sides. The left side has no time variable. We can remedy this by multiplying the left side by t^o. Since t^o = 1, this is acceptable.

(6) t^o d = c d^x t ^y-2x

Now solve for exponents of like dimensions.

X = 1, y - 2x = 0 thus y = 2

Knowing the value of the exponents is not the finish! Now take these results back to expression 2 on the previous page to get:

d = cat²

This tells us that the distance covered by falling bodies varies as the first power of the acceleration and with time². In other words, if an object falls from rest it covers quadruple the distance in twice the time. Of course dimensional analysis cannot give us the constant but knowing the square proportion is the big point. Indeed, not knowing the constant has its advantages! Students require lots of practice to appreciate what equations mean as opposed to their purely computational role. For example, a diver normally makes one flip from a 2 meter high diving board. How high should the board be to make 3 flips? D µ t² thus the board must be 3² higher = 9 x 2 meters = 18 meters high. Not knowing the constant does not impede us and indeed, inspires proportion thinking!

Ideally the students would previously do a pendulum lab in which they discover that the period µ Ö L and that pendulum mass and width of swing from side to side ( if not too great ) are irrelevant. This involves a lot of work in measurement and graphing. If done honestly the labor will make the student appreciate the value of dimensional analysis. For we can get the same result in seconds that the students took hours to do empirically!

(1) t = f (l,g,m)                This says the pendulum period seems related to pendulum length 1, the acceleration of gravity g
                                       and possibly mass m.
(2) t = c l^x g^y m^z              recall that g is dt^-2

(3) t = c d^x(dt^-2)^y m^z       Note that l has the dimension of d.

(4) t = c d^x d^y t^{- 2y} m^z

(5) t = c d^x+y t^{- 2y} m^z

Since we have d and m on the right side we need them on the left side as well. This can be done by multiplying the left side
          by d^o and m^o

(6) d^o m^o t = d^x t ^{x- y} m^z

Now solve for the exponents and find l = 1/2, y = -1/2, z = 0. Put these values back into equation (2) above to get:

T = c l ^1/2 g ^-1/2           The mass is irrelevant

The proportion is probably more familiar if rewritten as:
           ____
T = c Ö l/g

Students are often astounded to find the hours of lab time needed to discover the pendulum variables empirically can be equaled by a few moments of algebra. Yet that is the power of dimensional analysis! The question can be raised " if dimensional analysis is so powerful, why do labs at all? Of what use is experimenting? ". of course this can be debated in various ways. Dimensional analysis is very powerful, yet basic experience is necessary to decide which variables to include in your analysis. After all, if in place of the pendulum length you substitute the distance between the students eyes you still get the same proportion but it makes an incorrect prediction!

Doing the dimensional analysis of the pendulum after the lab gives us an unfair advantage. If we did not know the width of the swing was irrelevant, by all means we should have included it. If we do, the analysis proceeds as follows:

(1) t = f (l, g, m, swing width)

(2) t = c (l ^x g ^y m ^z w ^A) we express width of swing as w and its exponent as A.

(3) t = c d^x (dt^{- 2})^y m ^z d ^A width w is just d

(4) t = c d^x d^y t^{- 2y} m ^z d ^A

(5) t = c d ^x+y+A t^{- 2y} m ^z

We have m and D on the right side but not on the left. Thus:

(6) d^o m^o t = c d ^x+y+A t^{- 2y} m ^z

You can quickly find that z = 0 and y = -1/2. When you try to solve for the d exponents you have 1 = x + y + A and cannot succeed. There are too many variables. What to do?

The way out of our dilemma is to see that there are two kinds of distances in our variables. For small swings the cord length is measured vertically but width of swing is measured from side to side - horizontally. The acceleration of gravity works vertically. Thus we can differentiate vertical and horizontal distances as d _v and d _h. Rewriting equation (3) above we get:

t = c d _v x (d _v t- 2 )y d _h A m z

t = c d _v ^x+y d _h ^A t^{- 2y} m ^z

d _v ^o d _h ^o t = c d _v^x+y t^{- 2y} d _h ^A m ^z
                                                                                                                                            ___
now x + y = 0, y = -1/2, z = 0, A = 0 so the mass and swing width are irrelevant and T = c Ö l/g as before.

The alert student will object that for large swings the pendulum arc will move in both the horizontal and vertical directions so of what use is our analysis? Fair enough! In fact our dimensional analysis is not at fault. Rather, there is no exact equation for the period of the pendulum for large swings. Instead we must make do with an infinite series expression - see p. 8 of chapter 117 on advanced differential equations.

This is a good place to point out that most physics situations have no analytic solutions. Neither do most mathematical equations. Of course idealized equations have been selected for use in textbooks. The down side of this is most students (and many teachers!) fail to appreciate that most real life situations elude exact analytical solutions.* This is the nature of reality, not limitations of the physicist. This realization is an important goal of science literacy! (Surely it ranks with Godel's principle in significance.)

Mastering F = ma greatly expands the reach of dimensional analysis. Students readily see that force is a compound quantity and its units are F = ma = mdt^-2. A frequent problem is failing to see that weight is just another force thus the dimensions of weight are also mdt^-2. Tension is another force which students often fail to recognize as such. Appropriate applications abound. Now is the time to review the difference between period and frequency and also the three types of density. All this is discussed in the pamphlet "Further Dimensional Analysis".

Harking back to the pendulum there is the occasional student who wants to include the weight of the object rather than its mass. This objection can be met in two ways. Of course you can point out that W = mg thus including m and g in fact does mean weight is considered. If this doesn't convince them, then by all means include W as a separate variable instead of mass as well as 1 and g. After all, the period of a given pendulum does vary as you move it to locales where the weight is different. If this is done the weight exponent proves to be zero so weight itself is irrelevant.

In friction we define the coefficient of friction as u = F_fric/N. Since both F_fric and N are forces it is often said that u has no dimensions. Wrong! It has no units but it does have dimensions. This is clear if you look at our apparatus below. The man tries to drag the block to the left so the friction is to the right (horizontal) - The normal force is vertical. Thus u = F_friction/F_normal = md_ht^-2/ md_vt^-2 = d_hd_v^-l. So u does have dimensions! With this surprising discovery the whole realm of friction physics is open to dimensional analysis. This is covered in the pamphlet "Dimensional Analysis With Too many Variables".

When the student masters energy and power concepts the realm of dimensional analysis expands still more. It is essential to see that the dimensions of work and energy are the same. Spring physics is a natural for dimensional analysis. Students are often surprised that the "constant" in the spring PE equation PE = 1/2kd² has units and dimensions. This is all discussed in the pamphlet "Still More Dimensional Analysis".

Classic examples of dimensional analysis are found in simple harmonic motion. It works so well that the accompanying lab is titled " Proof of God ". This may seem excessive but in fact is a version of the argument where a watch is found on a desert island and is considered proof of higher intelligence. Then by analogy, the presence of order in the universe is proof of an even higher being. Surely when students do the very detailed experimental study of SHM over several classes then find that dimensional analysis achieves the same result in a minute or so, they may indeed gain faith that there is some underlying order and theme in the universe. Of course we are close to the limits of science here!

If the student has made it this far he is ready to work more independently. However dimensional analysis is no substitute for conceptual insight. The following hints are in order.

Gravity - Students must first be able to find the dimensions of G. This is done by solving the universal gravity law for G then proceed as normal.:

F_g = Gm_lm₂ so G = F d ² /m ² = m^{- 1} d ³ t^{- 2}
           d²

The difficulty is to know this is the proper way to proceed. If the students have had practice in finding dimensions of things like the k in F = kD and PE = 1/2kD², they should be able to succeed.

One other issue needs our attention. The gravity field of a planet or star depends on the mass of the body but not the density. If the sun were to vary its radius markedly, the motion of the planets would not alter at all. After all, the d in the universal gravity law is the distance between the centers of the objects, not their surfaces. Clearly then we need not consider the radius of the earth when using dimensional analysis to find an expression for velocity or period of an earth satellite. Instead it is the radius of the satellite orbit that matters. Without this insight the student can get sidetracked. See "Gravitation and Dimensional Analysis" for details.

Another classic exercise is the lab on Kepler's 3rd Law where students first use extensive log paper plotting to find the proportion then by dimensional analysis get the result in a minute or two. Finally they can read Kepler's statement of joy after making the same discovery after 20 years of labor. Somehow it seems unfair!

Heat and Temperature

The nature of temperature rapidly yields to dimensional analysis. See the pamphlet "The Kinetic Molecular Theory". The big task is to show that the number of molecules/unit volume = N/V is dimensionally l/V since the number of molecules N is dimensionless. This bothers some students. A little previous experience such as " what are the dimensions of the number of ants on a square foot of sidewalk " etc. may help. When the students actually do the dimensional analysis, it is too late to make this point without spoiling the impact of the exercise. Do it the day before!

Electricity

In electricity we finally get to include electric charge in our analysis. Of course this increases the algebraic complexity. It seems best to begin by writing the dimensions of current, voltage, resistance, the electric field E etc. then showing that formulas like V = Ed, work = QV, P=I ²R etc. are dimensionally correct. Finally you can give exercises like "write a proportion for electric power in terms of the voltage applied and the resistance" etc. Good students are then able to tackle full size problems like the following:

What is the particle range R in terms of voltage V,                          What is the particle velocity in terms of
velocity v, particle charge Q and mass m,                                        voltage, particle mass and charge.
height H and plate separation D. Quite tricky!

There seems to be no limit to the types of proportions to which dimensional analysis applies. The pamphlet "Dimensional Analysis and Exponentials" also includes trig functions. The best students notice that, strictly speaking, dimensional analysis is meant for monomials. Of course binomial equations like D = V_ot + 1at² are dimensionally identical term by term, but discovering this equation with dimensional analysis might be difficult. However D = V_ot + 1at² is not a law of nature while
D = 1/2at² is. The great laws of physics are monomials and the better students will look through their texts to see if this is true.

This about wraps up the introduction to the teachers guide. Of course the real meat of the topic is the various problems and labs that give practice in dimensional analysis. As the students proceed through the year they get more and more confidence in the method. Many look back after graduation and find dimensional analysis is the most memorable topic. Very insightful students have a more reserved judgement. Clearly the use of dimensional analysis requires proper selection of variables to try. Without a solid conceptual foundation dimensional analysis fails. But then, so does the whole physics course!