You can derive the pendulum formula without knowing any physics. Almost.
Length divided by gravity, square-rooted, is 0.319 seconds. The pendulum takes 2.006. The gap between those two numbers is 2π, and no amount of algebra with units will ever produce it.
Suppose you have forgotten the pendulum formula and want to reconstruct it without opening a textbook.
List what the period could possibly depend on: the length L, in metres. The gravitational field g, in metres per second squared. The mass of the bob m, in kilograms. You want an answer in seconds.
Now look at the mass. Kilograms appear in exactly one place on that list and nowhere in the answer, so no combination of these quantities can produce seconds while including m. The mass is out. Not because you know anything about pendulums, but because there is nothing for the kilograms to cancel against.
That leaves L and g. Metres divided by metres-per-second-squared gives seconds squared, so the only combination that produces a time is
T ∝ √(L/g)
You now have the entire shape of the law. The period grows as the square root of length, so quadrupling the string doubles the swing. It falls as the square root of gravity, so the same clock runs slower on the Moon. And it does not depend on the mass at all: Galileo's result, recovered from bookkeeping.
Where it stops
Put numbers in. For L = 1 m and g = 9.81 m/s², √(L/g) = 0.319 s. The pendulum tool's small-angle formula gives 2.006 s.
The ratio is 6.28. Dimensional analysis got everything except that, and it could never have got it, because 2π has no units. Any dimensionless number — 2π, ½, 137, one — is invisible to the method. Units constrain the shape of a law completely and its magnitude not at all.
This is the honest boundary, and it is what makes the technique trustworthy rather than magical: you always know precisely which part of the answer you have not earned. In this case the missing factor happens to be large. A physicist who reconstructed T ≈ √(L/g) and built a clock on it would be out by a factor of six.
The same trick, twice more
Once you have noticed the pattern, it is everywhere on this site.
Orbits. A period T, an orbit size a, and the gravitational parameter GM, whose units are m³/s². The only combination giving seconds is √(a³/GM): Kepler's third law, exponent and all, from the units. The orbital period tool's scaling table shows the consequence directly: double the axis and the period goes up by 2.83, which is 23/2. Dimensional analysis predicts that exponent exactly. It does not predict the 2π, which is again what stands between the shape and the answer.
Escape. A speed from G, M and R: √(GM/R) is the only option. The true escape velocity is √(2GM/R), so the method delivers everything but a √2 — and it delivers, for free, the fact that escaping is only 41% faster than orbiting, since the missing factors differ by exactly that.
Three laws, three correct shapes, three missing pure numbers. The pattern is the method's signature.
Why it is worth doing anyway
Buckingham formalised this in 1914 as the π theorem: a relation among n quantities built from k independent units can always be rewritten as a relation among n − k dimensionless groups. The pendulum has three quantities and two units, so one group survives, which is why the answer was forced rather than merely suggested. That is the whole content of the technique, and it is why the pendulum argument could not have come out any other way.
Three practical uses follow.
It checks your algebra. If you derive a period and it has metres left in it, you have made a mistake, and you know that before you check anything against reality.
It tells you what to vary. The pendulum argument says mass cannot matter, so an experiment that carefully measures the period against bob mass is measuring nothing, and its result was known before the apparatus was built.
And it makes models scale. A ship model in a tank is useless unless the dimensionless groups match the full-size ship, which is why naval testing is organised around Froude and Reynolds numbers rather than around lengths. The groups are what transfer between the model and the real thing; the units are what tell you which groups exist.
The method gives you the skeleton of a law for the cost of listing what it depends on. Just never forget that the number in front — the 2π, the √2, the ½ — is not in the skeleton, and has to come from the physics.
References (1)
- the π theorem, and the paper that made the method a method Buckingham (1914). On Physically Similar Systems; Illustrations of the Use of Dimensional Equations. Physical Review 4(4).