Dimensions and Units

Numbers are meaningless for the physicist without the correct use of units. It makes no sense to say "the distance from my house to school is two", unless we follow that statement with "miles" or "kilometers", or whichever unit makes the statement true. We will distinguish between dimension, the abstract quality of a measurement without scale (ie., length), and unit, the quality of a number which specifies a previously agreed upon scale (ie., meters). While dimensional arguments are of primary importance in understanding the qualitative behavior of systems, the use of units are necessary for the predictions we seek.

The four fundamental dimensions are length, time, mass and electric charge. These qualities of numbers are fundamentally different, and make up the building blocks for all of our studies. We perceive three spatial dimensions of length, upon which we build dimensions of area and volume (notice the different use of the word dimension). Area and volume have "composite dimension", by which we mean that more than one factor of a fundamental dimension appears (area = length squared, volume = length cubed). Time, likewise, will often appear in higher powers in a composite dimension. Consider the rate of change of speed: "my car can accelerate at 6 miles per hour per second":

6 miles / hour / sec = 6 miles / (hour sec).

Whenever we work with a rate of change, we get another power of time in the denominator of the composite dimension.

Mass and electric charge only occur as multiple factors in a composite dimension when the quantity in question involves the interaction of multiple masses or charges. The gravitational force between the Earth and its moon is proportional to the product of their masses. We will later define mass and charge in terms of force. For the moment, we associate those dimensions with the quantity of matter, literally the number of atoms, molecules, ions, etc., in an object. It is important to distinguish between mass and volume, although for objects of a relatively constant density, they can be considered to be proportional. In those cases, we say that mass "scales with" volume.

In order to make quantitative measurements, we need to agree on units: one of the basic tenets of science is the premise that any experiment should be independently verified in order to be accepted. This validation process requires consensus on the units of measurement. There are two commonly used systems of measurement: SI, or Systeme Internationale, and cgs. The basic units of SI are the meter ("m", with dimensions of length), the second ("s", with dimensions of time), the kilogram ("kg", with dimensions of mass) and the Coulomb ("C", with dimensions of electric charge). The units of the cgs system are the centimeter ("cm"), the second, the gram ("g") and the Coulomb. So we see that the only difference between SI and the cgs system of units is a metric one: cgs units are used whenever centimeters and grams are more appropriate units for the scale of the system under investigation. We will freely use both systems as necessary.

We will be dealing with what sometimes seems to be a plethora of composite units: Newtons, dynes, Joules, ergs, Poise, Watts, Pascals, mmHg, etc. These units are all expressible in terms of the fundamental units in either system. Many differ from each other only by factors of ten, but some units are related by odd factors whose value derive from historical accident. Conversion of units will be a frequent necessity for the student of physics.

In addition to composite units, we will encounter dimensionless quantities. These will be ratios of quantities with identical units: in the common parlance, the units "cancel". These dimensionless or unitless quantities will often serve as characteristic values which signal the change from one qualitative behavior, or "regime", to another. For instance, the Reynolds number for fluid flowing through a pipe signals the onset of turbulence when its value exceeds 2000. Finally, you should always remember that the arguments of trigonometric, logarithmic and exponential functions must be dimensionless. While we speak of angles being measured in "degrees" or "radians", these pseudo-units are really the ratio of arc length to radius of an arc of a circle (p / 2 radians is the ratio of the arc length of one quarter of a circle to its radius), and are therefore dimensionless.

We will use many different equations in our predictive computations. If you have a thorough knowledge of the units of the quantities relevant to a phenomenon, and some empirical knowledge about how those quantities are related, you can construct an equation relating them. This technique is useful both for finding new relationships, and for determining which equations to use to solve a given problem. You can use dimension, or any system of units to do this; we will most often use units which are natural for the system, so that once we have "derived" the equation, we will know in what units to make our measurements and state our answers.

For example, suppose we wish to be able to compute the power output of the heart. From the everyday experience of buying a light bulb, we know that power is measured in Watts. We will learn later that a Watt is equivalent in SI units to a kilogram - meter squared per second cubed. We suspect that the power output of the heart will be related to the pressure it develops at the outlet, the stroke duration and the stroke volume. That these quantities are the only necessary ones is perhaps not obvious at this time, but the reasoning we will use will tell us if we are missing something. The SI units for pressure are kilograms per meter - second squared, and for volume are (of course) meters cubed. What then is the relationship between power, pressure, volume and time?

We seek an equation with power on the left hand side (LHS) and the other quantities on the right hand side (RHS). It will not be an equation unless all of the terms have the same units. Since no two of our RHS quantities have the same units, we must multiply or divide them in some combination to produce units of Watts. We seek the simplest equation possible: students find physics hard enough without unnecessarily complicating things! To come up with units of seconds cubed in the denominator, the simplest thing to do is to divide pressure by time:

P / t ~ (kg / m s 2) / s = kg / m s 3.

However, this leaves us with meters in the denominator, and we need meters squared in the numerator. The obvious choice is to multiply by the volume:

P V / t ~ (kg / m s 3) m 3 = kg m 2 / s 3 = W.

And so we have an equation with units of Watts on both sides which expresses the average power output of the heart as the product of the output pressure and the stroke volume, divided by the stroke duration:

Power = P V / t.

This same result might have been derived from purely qualitative considerations: the power should increase if the pressure is increased, or if the volume is increased, but should be smaller if the heart beats more leisurely. This more physical "derivation" can either be the initial reasoning or the check on our work, but it must never be omitted from our process.

For the chemists among us, this process of dimensional reasoning can be likened to stoichiometry. Instead of "balancing" atoms on both sides of a reaction, here we are balancing each of the fundamental units on each side of an equation. The main difference is that you modify the coefficients in stoichiometry, where in dimensional analysis you essentially modify the powers of the various quantities you have to work with.

We will have many occasions to do similar dimensional or unit analyses in the future. In each case, we perform the following steps:

  1. Determine the units of the LHS.
  2. Choose one particular unit to match on both sides.
  3. Multiply or divide quantities with differing units to get that unit to match.
  4. Repeat this process until all of the units match.
  5. Verify that the behavior of the equation is physically appropriate by imagining changes in the RHS quantities and deciding how the LHS will change. Quantities which change in proportion with the LHS will be in the numerator of the RHS, those which change inversely will be in the denominator.

Throughout the history of science, this procedure has been a useful theoretical tool: determine the independent variables for a given dependent variable, and construct a relation based on units. Then perform an experiment to check the relationship: if the measured quantities behave as expected, the equation is consistent with the data within the accuracy of the experiment. If not, we have obviously overlooked some independent variables, and we start over.

Problems

The next section is about coordinate systems.



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1996, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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