Copyright © 2008 jsd
There is a right way and a wrong way to express the units associated with a given quantity. Good examples include:
To a good approximation you can make a yardstick by gluing three one-foot rulers together end-to-end, as shown in figure 1. This gives us a concrete, physical basis for saying that 1 yard “equals” three feet.
This physical interpretation is captured in the algebra, if we do things right. For example, if we multiply two quantities together, the numerical magnitudes get multiplied and the units get multiplied. For example, if the length L is 30 feet and the width W is 2 feet, the area A is given by:
It would be quite improper to say «the area was 60», or that «the area was 60 feet». The area is 60 square feet.
Let us now consider an example of what not to do. This is taken from page 13 of reference 1.
In this document, we put equations and tables on a red background to warn you about things that are not recommended. Similarly we use «⋯» as scare-quotes around improper expressions.
This equation uses the unit of measurement (C or F) as a stand-in for the thing being measured. This is a Bad Idea. It is diametrically inconsistent with modern notions of unit-analysis.
There is a very important difference between the thing being measured and the unit used to measure it, as discussed in section 2.2.
Here’s another example: Recently I was looking at the data plate on an electric motor. It said:
The colons on the data plate were meant to be interpreted as equal signs; that is:
Using this approach, it is possible to write equations such as
If you find this confusing, good for you. This usage is very old-fashioned and not recommended, as indicated by the red background. This usage is exactly backwards relative to modern usage. However, you still see it from time to time, so it is worth learning to recognize it, and to appreciate just how different it is from modern usage.
Modern usage would say “input frequency = 60 Hz” instead of «HZ = 60», and would say “rotation rate = 1725 RPM” instead of «RPM = 1725», et cetera. Instead of equation 3 we would write the conversion factor in the form:
There is quite a stark difference between these two equations. In equation 4, Hz appears in the denominator of the RHS, whereas in equation 3 it appears in the numerator. These two equations are using the same symbols with essentially opposite meanings. Since equation 4 is consistent with our policy of attributing physical reality to units (as depicted in figure 1) and equation 3 is not, we accept equation 4 and reject equation 3.
If you are going to use the modern system, you must strictly avoid mixing it with the other system. This leaves us with the question, what to do if you encounter something expressed in the bad old system, such as the motor data plate mentioned above.
The answer is to consider the RPMs mentioned on the data plate, and in equation 3, as being essentially the inverse of real RPMs ... and similarly for the other abused units on the data plate. The rationale for this is as follows:
which we see is the same as equation 4, as it should be.
Here’s a trickier example,from page 190 of reference 1. For a steam engine:
I wish to make two different points about this. Please let’s not confuse the two points.
1) First, I call attention to the LHS, which is “indicated horsepower”. This is another an example of using the unit of measurement as a stand-in for the thing being measured. This is a Bad Idea, as discussed in section 2.2.
The thing that is really interesting here is that the way units are used on the LHS of equation 6 is inconsistent with the way units are used on the RHS, and in particular with the way the quantities P, L, A, and N are defined on the RHS of equation 7.
This is relevant because if the people who use backwards units would use them consistently backwards, we might chalk it up to a difference of taste, not worth arguing about. But when they are inconsistent about it, it’s more than a question of taste; it’s an out-and-out blunder.
The best way to remove the inconsistency is to standardize on the modern approach.
2) Secondly, let us turn to the RHS. The book says:
which is interesting because in some sense those four statements are dimensionally correct, if we consider P, L, A, and N to be “pure numbers” (unitless and dimensionless) standing in front of explicit units.
This is, however, still not recommended. Even though it is not wrong in any deep sense, it is not recommended. The problem is that the numerical value of (say) A is tied to the units in use.
In contrast, modern practice is to do algebra using variables that have not had units factored out, for example
At this point you may be wondering about an interesting issue, namely that when doing a practical calculation it has always been necessary (with rare, recent exceptions) to calculate using pure numbers.
(Nowadays you can find software that will keep track of the units while doing algebra, but this was not the case in 1948, and is still quite rare today, and in particular not widely available in the physics classroom.)
So you could argue that a formula involving pure numbers has value for practical computations.
That is a valid argument, but there is a stronger counterargument. The need for pure numbers can be accommodated within the formalism of modern unit analysis. One could write:
|indicated power/HP =|
(where spm means power strokes per minute).
which has the advantage of being absolutely true no matter what units the variables are measured in. If L is measured in some units other than ft, then the problem is obvious and more importantly the solution is obvious if you follow the factor-label method.
Units are not the same as dimensions. A foot and a yard have the same dimensions, namely dimensions of length. But a foot and a yard are very different units; one is smaller than the other by a factor of three. Length and width have the same dimensions. Sometimes length and width are measured in the same units, and sometimes in different units. The distinction between units and dimensions comes into sharp focus when we consider dimensionless units, as in reference 2.
See reference 3 for an introduction to dimensional analysis. All notions of units and dimensions rest on deeper notions of scaling, as discussed in reference 4.
There is a profound difference between the unit of measurement and the thing being measured. For example, if you are measuring a piece of fabric with a yardstick, the fabric is not to be confused with the yardstick. The fabric is the thing being measured, and the yardstick embodies the unit of measurement.
Another example can be found in equation 2. Temperature is the thing being measured; degrees C or degrees F are the units of measurement.
Yet another example is the frequency in equation 5. The frequency is the thing being measured, and the hertz (denoted Hz) is the unit of measurement. Just as the rulers in figure 1 have physical significance, the Hz has physical significance. If the frequency changes from 50 Hz to 60 Hz, the Hz is not what changes; it is the frequency that changes.
Sometimes, the thing being measured is named after a unit. Examples include
It must be emphasized that the physical quantity is not required to be measured in the named units.
Note that mileage (as a measure of fuel economy) is exceptional because it measures distance per unit volume, even though the name would suggest plain distance.
Copyright © 2008 jsd