What if a pendulum swung very high?

A pendulum is a weight that swings back and forth.

If you pull it back a little, it swings a little. What if you pull it way, way up high?

A pendulum is a weight hanging from a rod or string that swings back and forth. A playground swing is one, and so is the long swinging weight inside an old grandfather clock.

Famous fact: small swings take the same time, whether you pull the weight back a little or a bit more. But what if you pulled it back really far, almost upside down? Does the swing still take the same time?

For small swings, the period of a simple pendulum depends only on its length and on gravity. OpenStax notes that the period is nearly independent of amplitude, especially below about 15°, which is what made pendulums useful for timekeeping.

That word 'nearly' hides the question for this lab: what happens to the period at large amplitudes, like 90° or 150°, and why did it matter so much to clockmakers?

Olly's pendulum is 1 meter long. He pulls it back a little and lets go.

One swing over and back takes about 2 seconds.

In the model, the pendulum is 1 meter long on Earth. Pull it back 10° and let go. One full swing, there and back, takes 2.010 seconds. That's about 30 swings a minute.

The small-swing formula, which assumes tiny swings, predicts 2.006 seconds: almost the same. Try 5° or 15° and the time barely changes. That steadiness is what makes a pendulum a good timekeeper.

With L = 1 m, g = 9.81 m/s² and no damping, the small-angle formula T₀ = 2π√(L/g) gives 2.006 s. Released from 10°, the model's exact period is 2.010 s, only about 0.2% longer, and the bob passes the bottom at 0.55 m/s.

This near-independence from amplitude is called isochronism. It is an approximation: it comes from replacing sin θ with θ, which is very good for small angles.

Now Olly pulls the pendulum way up high, almost upside down.

It swings fast at the bottom. But one big swing takes much longer than a small one!

Now pull the same pendulum back to 150°, almost upside down. Our model uses a stiff rod, so it can swing that high; a string would go floppy.

One swing now takes 3.535 seconds instead of 2.010. That's about 76 percent longer, and only about 17 swings a minute. The bob also whooshes through the bottom at about 6 meters per second, but it creeps very slowly near the top of each swing, and that slow part adds lots of time.

At θ₀ = 150° the model's exact period is 3.535 s, 1.76 times the 10° value, and the bob passes the bottom at 6.05 m/s. Intermediate values: 2.086 s at 45°, 2.368 s at 90°, 2.754 s at 120°.

The model uses a rigid, massless rod, so swings above 90° are allowed; a string would go slack there. The period grows without limit as θ₀ approaches 180°, where the bob would balance upside down: at 170° the model already gives 4.894 s.

Near the top of a big swing, the pendulum is almost sideways or upside down.

There, gravity hardly pulls it back down. So it moves very slowly up there, and that takes time.

Gravity pulls the pendulum back toward the bottom. For small swings, pulling it twice as far makes the pull back about twice as strong, so a bigger swing just goes faster and the timing stays the same.

For big swings that rule breaks. When the rod points sideways or higher, gravity mostly pulls along the rod instead of swinging the bob back down. The pull back grows less than the distance, so the bob dawdles near the top, and every swing takes longer.

The restoring term is −(g/L) sin θ, not −(g/L) θ. Since sin θ < θ, large-amplitude swings get proportionally less restoring acceleration and spend longer near their turning points. The exact undamped period is T = T₀ / AGM(1, cos(θ₀/2)), equivalent to (2/π) T₀ K(sin(θ₀/2)), with the series T ≈ T₀(1 + θ₀²/16 + …) for moderate angles.

Model limits: a point bob on a rigid, weightless rod in one plane, uniform g, and simple linear damping when switched on. Real pendulums are physical pendulums with rod mass and bob size, which shift the period slightly.

Old clocks have a pendulum inside. It swings back and forth to keep time.

Clockmakers made it swing only a little bit. Small swings keep the clock on time!

In 1656, the Dutch scientist Christiaan Huygens built the first pendulum clock, NIST explains, and it was off by less than a minute a day, far better than earlier clocks. As OpenStax's physics textbook notes, for small swings a pendulum's timing depends almost only on its length and on gravity.

But early clock mechanisms made pendulums swing wide, sometimes more than 50° each way, according to engineers at the University of Belgrade. Wide swings meant timing errors. Around 1670 a new part, the anchor escapement, cut the swing to about 5° to 10°, and later designs to about 2° to 3°.

NIST's history of timekeeping credits Huygens with the first pendulum clock in 1656 (error under a minute a day) and George Graham with reaching about a second a day in 1721 by compensating for temperature changes in the pendulum's length. The anchor (recoil) escapement, designed by William Clement and Robert Hooke, dates from about 1670; NIST says Clement was building clocks with it in London from 1671.

A University of Belgrade review of escapements explains why: verge escapements drove amplitudes above ±50°, increasing the circular error. Recoil anchor escapements cut this to ±5–10°, and deadbeat escapements, refined by Graham, to about ±2–3°, where the effect of circular error is slight.