Class Content I >Modeling with mathematics > Building your mathematical toolbelt
2.3.6
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Prerequisites:
Although we don't always think of a "box" as a tool, it can be, if we want to organize, move things, or hide a mess. |
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Our equations in physics often look like a mess compared to what you might be used to seeing in a math class. Since equations in math are focusing on relationships -- typically between only two or perhaps three mathematical objects -- that may have few symbols with everything else being numbers. Equations in physics typically have multiple symbols. Often these are parameters representing how the equation depends on a variety of measurements in the physical situation.
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.
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 you of what the symbol represent 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. Here's an example. Note that you don't need to have studied the physics in the problem to solve it!
Example:
For very small objects in a fluid, the viscous resistive force, Fvisc = 6πμRv, dominates, where μ is the viscosity of the fluid, R is the radius (size) of the object, and v its speed through the fluid. For large objects, the drag resistive force, Fdrag = CρR2v2 dominates, where C is a dimensionless constant, ρ is the density of the fluid, and R is the radius (size) of the object, and v 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 is to identify, for this problem, what are variables and what are constants? 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, and we're talking about a single 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:
Fvisc = Fdrag
6πμRv = CρR2v2
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:
(6πμR)v = (CρR2)v2
6πμR = A
CρR2 = B
With these definitions, and keeping "v" for our variable (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.)
Av = Bv2
Now the math is easy! One power cancels and 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 = (6πμR)/(CρR2) = 6πμ/CρR
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" trip 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