Dimensional analysis is a simple, physically transparent and intuitive method for obtaining approximate solutions to physics problems, especially in mechanics. It may—indeed sometimes should—precede or even supplant mathematical analysis. And yet dimensional analysis usually is given short shrift in physics textbooks, presented mostly as a diagnostic tool for finding errors in solutions rather than in finding solutions in the first place. Dimensional analysis is especially well suited to estimating the magnitude of errors associated with the inevitable simplifying assumptions in physics problems. For example, dimensional arguments quickly yield estimates for the errors in the simple expression 2h/g for the descent time of a body dropped from a height h on a spherical, rotating planet with an atmosphere as a consequence of ignoring the variation of the acceleration due to gravity g with height, rotation, relativity, and atmospheric drag.

Topics
Dimensional analysis, Educational aids
1.
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7.
It is common to find in textbooks a linear drag law of the form kv, where k is independent of speed. This yields an equation of motion that is readily solved at the expense of the solution being largely irrelevant. Such a law is valid only for Reynolds numbers less than 1, which except for exceedingly short time intervals is not satisfied by objects with the dimensions of tennis balls (or even lead shot). The ratio μ/ρ (called the kinematic viscosity) for air at 15 °C is about 0.15 cm2/s. A ball or other object of comparable size dropped from rest reaches 1 cm/s in about 10−3 s. For d=10 cm and v=1 cm/s, the Reynolds number is of order 102, so the linear drag law is invalid for such objects during all but a tiny fraction of their trajectories. This law is valid, however, for cloud droplets, which have diameters of order 10 μm. A rule of thumb is that if you can readily see a falling body, the linear drag law is not applicable to it.
8.
For the consequences of a variable density with height to the motion of a body dropped at rest in Earth’s atmosphere see
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The solution here for v and z is essentially the same as that given by
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© 2004 American Association of Physics Teachers.
2004
American Association of Physics Teachers
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