The pendulum and simple harmonic motion: where the formula stops working

The pendulum is the first thing most physics courses model properly, and it is chosen for a good reason: it is simple enough to solve and interesting enough to be worth solving. It is also, quietly, a lie. The formula everyone learns is an approximation that is excellent for small swings and increasingly wrong for large ones — and understanding exactly where it fails teaches more than the formula itself.

What makes motion simple harmonic

Simple harmonic motion has one defining property: the restoring force is proportional to the displacement. Pull it twice as far and it pulls back twice as hard. A spring does this almost perfectly, which is why springs are the textbook example.

The consequence is remarkable. When that proportionality holds, the time for one full cycle does not depend on how far you displaced it. A spring pulled 1 cm and one pulled 5 cm take exactly the same time to complete a cycle — the larger swing simply moves faster to cover the extra ground. That property is called isochronism, and it is the reason oscillators can keep time at all.

The pendulum only approximately qualifies

For a pendulum, the restoring force is proportional not to the displacement angle but to its sine. That is a problem, because sine is not proportional to anything — except that for small angles it very nearly is:

At 5° the two agree to four decimal places. At 60° sine has fallen 17% below the angle. The famous pendulum formula comes from simply replacing sin θ with θ and solving what is left:

Notice what is not in that expression. There is no mass — a heavy bob and a light one on the same string keep identical time, because gravity pulls harder on the heavier bob in exactly the proportion that its extra inertia resists. And there is no amplitude, which is precisely the assumption that will fail.

How wrong is it?

The exact period can be expanded as a series, and the first correction term is θ²/16. Running it out gives the error at each release angle:

Below about 10° the approximation is better than you can measure with a stopwatch. At 30° it is off by roughly a second every minute. At 90° it is hopeless. This is why every textbook problem quietly specifies a small angle, and why a real pendulum clock swings through only a few degrees.

The direction of the error is worth remembering: the real pendulum is always slower than the formula predicts, never faster. Wide swings spend proportionally more time near the extremes where the restoring force has gone soft.

Length is everything

Since period goes as the square root of length, quadrupling the length doubles the period. A 1 m pendulum takes 2.01 s per full swing; a 4 m one takes 4.01 s.

Run that backwards and you get the seconds pendulum — the length that takes exactly one second to swing each way, so two seconds for a full cycle. It works out at 0.994 m, just under a metre, and it is why longcase clocks are the height they are. The case exists to house the pendulum, not the other way round.

Weighing the Earth with a piece of string

Rearrange for g and you have a genuine measuring instrument:

Time fifty swings rather than one, divide by fifty, and a school pendulum gives g to within a percent or so. For two centuries this was the best method available, and careful pendulum work mapped real variations in gravity across the Earth's surface — g is measurably weaker at the equator and on mountaintops.

The catch is the one this whole article is about: the method only works if the swing is small, because a wide swing reads slow and hands you a value of g that is too low. Our pendulum simulator shows the exact and approximate periods side by side as you change the release angle, so you can watch the gap open. Set it to 60° and the 7% error stops being an abstraction.