Class Content I  >Modeling with mathematics  > Building your mathematical toolbelt

2.3.1

Prerequisites: Measurement Dimensions and units

Since we are mapping physical measurements into math, most of the quantities we use in physics are NOT NUMBERS. They are MEASUREMENTS.This means they depend on an arbitrary scale we have chosen.

In order that the equations we write keep their validity (the equation still holds) when we change our arbitrary scale, dimensions must match on both sides of the equation.

Dimensions are arbitrary and depend on what choices we choose to think about changing. (e.g., moles, angles).

This means that equations in physics (and science in general) tend to be different from what you have seen in math. Figuring out what kinds of measurement were used in getting a value is called dimensional analysis.

In our class we will typically use five different kinds of measurement:

A measurement made with a ruler (a length)
A measurement made with a clock (a time)
A measurement made with a scale (a mass)
A measurement of warmth (a temperature)
A measurement of electric strength (a charge)

When we have an equation in physics, it will typically contain a lot of symbols. To make sense of the equation we have to ask each symbol: "What measurements are you made of and how?" For now, we will indicate the question by using double square brackets and the answer by using the symbols given above.

1. A displacement (change in position) is found using a ruler (making a length measurement - L)   2. A time interval is found using a clock (making a time measurement - T)   3. A mass is found using a scale (making a mass measurement - M)   4. A temperature is found using a thermometer (making a temperature measurement - Θ)   5. A charge is found using an ammeter (making a current-time measurement - Q)

(Note that the variable for temperature is typically, T, but the symbol for the dimension of temperature is a capital theta (Θ). There is always the possibility of confusion when we use the same symbol for different things. Keep track by context.)

When we combine measurements, we express it by showing how these measurements are combined.

When we have correct equations for symbols that we know it can tell us what measurements were combined to create that symbol:

We've been very careful in the above discussion to use markers that are very different from what we do in standard algebra and writing equations. Double brackets are rarely used in doing algebra and our funny little icons, while being a very good reminder of what we are doing (specifying the kind of measurement, NOT a value), they are not commonly used (and are not available on typical keyboards).

As a result, instead of using a little ruler, we will write "L", instead of using a little clock, we will write "T", instead of using a little scale, we will write "M", and instead of using a little ammeter, we will write "Q". 

Worse than that, instead of writing double brackets, from laziness, we will use single brackets! So our "what measurements were used to get your value?" question will look like this for velocity and force:

[v] = L/T

[F ] = ML/T²

This can be a real pain! In particular, we might sometimes see the equation v = L/T, where we mean L and T to be values rather than icons. You just have to be careful and pay attention to the context. What question are you asking? Are you doing a dimensional analysis (asking what measurements are made to get the value) or doing a calculation (using the values of the measurements)? The symbols mean very different things in the two cases. It might help you when doing dimensional analysis to think of the symbols M, L, T, and Q as our little icons.

Here's an example that shows the difference.

Example 1:

A traveler needs to transfer to get to his final destination. If he flies a distance d₁ on his first flight and a distance d₂ on his second, write an equation to express the total distance he has flown, d, in terms of d₁ and d₂. Perform a dimensional analysis of your equation.

While this problem is pretty trivial, the tough part is thinking about the last part of the question. The equation is straightforward. The total distance he travels is the sum of the two distances:

d = d₁ + d₂. 

But how we write the dimension? Each is a length so our dimensional analysis equations look like this:

[d] = L     [d₁] = L     [d₂] = L

[d] =  [d₁] + [d₂]  

so  L = L + L
 

This seems decidedly weird! Shouldn't L+L = 2L? Not here! Since L doesn't stand for a value here but for a statement "this was measured with a ruler" the sum of two things measured with a ruler was also measured with a ruler so "L = L+L" is correct. It makes more sense if you think in terms of the little icons!

Here's an example that shows the power of dimensional analysis.

Example 2:

Which equation correctly represents the surface area of a sphere?

You might have memorized this, but, as we know, memory is not always reliable. Can you figure this out without remembering the answer? A good first step is to check the dimensions. The radius of a sphere is a length ([R ] = L) and an area is the product of two lengths ([area] = L²). Therefore, the answer must have two factors of R. We can therefore rule out answers 1 and 3 right away without using any memory of the equations at all. We now only have to choose between 2 and 4. If we remember that 4 is the area of a circle and we realize that this is the circle that cuts through the center of the sphere, it's clear that the area of the surface of the sphere is bigger than that — and since the surfaces of both the top and bottom hemispheres are also bigger than that circle, the answer has to be bigger than 2πR². This tells us the answer must be 2. (Correct)

We'll also find throughout the course that dimensional analysis lets us generate new equations as well as check old ones! This is an extremely powerful tool that we will be using extensively throughout the class.

Joe Redish 7/3/17