The other day, I read yet again about the use of a pendulum for timekeeping, based on the fact (supposedly first observed by Galileo) that the period of the pendulum was independent of its arc, and was only a function of pendulum's length.
Well, that's true. . . . and it's not true. If you look at the differential equation of motion for a simple pendulum, it's a short but complex equation that cannot be solved analytically, see the analysis here. Luckily, the core of the stumbling block can be overcome by making the modest assumption that sin x ≈ x (x in radians, of course) for “small” angles. This is a very legitimate approximation, and it is the one I think is we most frequently use when modeling systems, to reduce complex equations to something more solvable or to an analytical form.
At the same time, though, we have to keep in mind that it is only an approximation, and correlate our required accuracy with the error we incur to this assumption. I have seen two problems when using realistic, valid assumptions such as this one:
- Depending on the situation and application, as well as cumulative buildup of errors through the equation chain, the final error maybe much larger or more significant than the approximation itself. Just because sin 0.1 (5.73°)= 0.0998, a “mere” 0.2% difference, does not mean your final analysis will be good to 0.2%.
- Second, it's too easy to get so wrapped up in the analysis, that you forget that you made an approximation way back there at the beginning, and somehow your final equation comes to represent reality in your mind; the specifics of the “approximation” you used and its range of validity is lost in history, so to speak.
So while approximations are useful and very necessary, always state the ones you make clearly, and keep them clearly in front of you, so you don't gloss over their long-term impact and implications. And it doesn't hurt to review their implications when you are done with your analysis, either. ♦