Section 7.1 What is a Mathematical Model?
Engineers use mathematical models all the time to represent physical phenomena. For example, in the last chapter we learned about Hooke’s Law, \(F=kx\) for springs, which is a mathematical model. You may have also heard about Ohm’s Law in physics, \(V=IR\) for circuits, which is also a mathematical model. There is also Pascal’s Law for fluid pressure, \(\Delta P =\rho g \Delta H\text{,}\) Stoke’s Law for drag force, \(F_d = 6\pi \mu R v,\) and the list goes on and on.
The point is that these mathematical models tell us something about the way things actually occur in the real world. Again, returning to Hooke’s Law from the previous chapter, we saw that as we increase the force stretching a spring, the spring will stretch linearly proportional to the force applied. That means if we double the force, the distance the spring will stretch will double. We can intuitively understand this because we have all played with springs, and we can also see this reflected in the mathematical equation that describes the phenomena.
Almost all mathematical models will have the following characteristics:
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They will take the form of an equation \(y=\) something with an \(x\text{,}\) \(m\text{,}\) and \(b\text{.}\)
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\(y\) and \(x\) are variables of interest.
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\(b\) is a constant (keep in mind sometimes it is 0 or 1 meaning it might be invisible)
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\(m\) is the “slope”. This will intuitively make sense to you for linear models but you need to keep in mind that the interpretation of slope with the other models will be slightly different.
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