The repackaging tool: Changing physics equations to math (and back)

Prerequisites

When you're given a messy equation, you might be tempted to try to simplify it by putting in numbers. Don't do it! When you put in specific numbers, you lose your understanding of how the various parameters affect your result. A much better technique is to "put the parameters into a box" and change the physics equation so it looks like a math one. You can change it back when you're done with the math without having lost anything.

Our equations in physics often look like a mess compared to what you might be used to seeing in a math class. This is because equations in math focus on mathematical relationships, there may be a only a small number of symbols standing for the variables in the problem, with everything else as numbers. 

Equations in physics typically have many symbols. Often these are parameters representing how the equation depends on a variety of measurements in the physical situation. 

Also, in math, they like to use letters at the end of the alphabet for variables (x, y, z) and those at the beginning of the alphabet for parameters (a, b, c). We can't get away with that in physics. We have so many symbols, we need to use symbols that remind us of what the symbol represents physically. But when we have an equation, we can temporarily change it to math form, solve it as a math problem, and then change it back to interpret the result in physics.

An example

Here's an example. Note that you don't need to have studied the physics in the problem to solve it!

For very small objects in a fluid, the viscous resistive force, $F_{visc} = 6\pi \mu Rv$, dominates, where $\mu$ is the viscosity of the fluid, $R$ is the radius (size) of the object, and $v$ is its speed through the fluid.

For large objects, the drag resistive force, $F_{drag} = C\rho R^2 v^2$,  dominates, where $C$ is a dimensionless constant, $\rho$ is the density of the fluid, $R$ is the radius (size) of the object, and $v$ is its speed through the fluid.

If we model the resistive force on an intermediate size object as feeling both these forces, is there ever a speed where the magnitude of these two forces is equal?  

These forces have very complicated forms. How do we deal with this?

The first thing to do in making sense of an equation, is to identify, for the problem at what, what do we consider as changing (are variables) and what are we keeping fixed (are constants). Here, we're asked to find a speed, so $v$ is the variable. Everything else are constants since we have a fixed fluid (so the parameters of the fluids, $\mu$ and $\rho$, don't change. We're talking about a single fixed object, so $R$ doesn't change. 

We're asked if the forces are ever equal so let's write the equation that they are equal:

$$F_{visc} = F_{drag}$$ 

 $$6\pi \mu Rv = C\rho R^2v^2$$

We are interested in finding  $v$. For this problem, everything else is a constant. Let's group all constants that multiply each other into a single constant and give them a name like in math:

 $$(6\pi \mu R)v = (C\rho R^2)v^2$$

Let call the combinations of constants, $6\pi \mu R = A$  and  $C\rho R^2 = B$. With these definitions, and keeping $v$ for our variable 

$$Av = Bv^2$$

(We don't want to use $x$ so as not to confuse it with position here! Since the letter $v$ is from the end of the alphabet it still looks like a math variable.)

Now the math is easy! We can divide both sides of the equation by $v$. We then get a simple equation with only one $v$ in it. We can solve for $v$ giving $v = A/B$.

Now all we have to do is substitute our definitions of $A$ and $B$ to get our physics equations back:

$$v =(6\pi \mu R)/(C\rho R^2) = 6\pi \mu /C\rho R$$ 

We've not only answered the question with a resounding "yes", we've found for what speed that happens. We've changed from physics to math and back. Of course, to evaluate whether this answer makes any biological sense, we have to consider whether this organism can actually attain this speed!

This "change the look of the problem" trick can really help you to disentangle equations that contain a pile of unfamiliar symbols. Make it a standard element of your toolbelt!

Joe Redish 7/11/17

Article 294
Last Modified: May 30, 2019