A piece of string and a small weight are all it takes to build a simple pendulum. It looks trivial, yet it hides a wonderfully strange rule: the time for one complete back-and-forth swing (the period) depends only on the pendulum's length — the mass of the bob and the size of the swing make almost no difference.

This guide has students measure the period with their own hands first, then use a draggable interactive simulation to verify quickly that a longer pendulum swings more slowly.

Safety first (read this before you start)

  • Hang the pendulum from a solid stand or door frame, and make sure it cannot come loose and hit anyone or anything.
  • Clear breakables out of the swing path. Keep the swing small (no more than about 15°) — it matches the physics better and is safer too.

Materials and equipment

  • A length of strong thin string (cotton string or kite line);
  • A reasonably heavy small object for the bob — a nut, a key or a small sandbag;
  • A fixed support (lab stand, door frame, or a clamp on the table edge);
  • A phone stopwatch, plus a ruler or tape measure.

Step-by-step procedure

  1. Tie the bob to one end of the string, fix the other end to the support, then measure and record the pendulum length — the distance from the pivot to the center of the bob.
  2. Pull the bob to one side through a small angle and let go. Be careful not to give it a push.
  3. To reduce error, time 10 complete swings in a row with the stopwatch, then divide by 10 to get one period.
  4. Change the length (halve it, for example), measure again, and compare the periods.
  5. Keep the length fixed and swap in a heavier bob, or release from a different height, and time the period once more.
Why time 10 swings: starting and stopping a stopwatch each carry reaction-time error. Timing 10 swings and averaging shrinks that error to one tenth — the classic "measure many times and average" technique of middle school labs.
Interactive simulation: this experiment has a drag-and-play simulator in our Interactive Lab. The lab interface is currently Chinese-only — the controls are simple sliders and switches, so it is still easy to explore.

Recording what you see

Log each trial's length, the total time for 10 swings, and the calculated period. A typical record (a coin on thin string, indoors with no draft) looks like this:

  • Length 25 cm: 10 swings in about 10.0 s → period about 1.00 s;
  • Length 50 cm: 10 swings in about 14.2 s → period about 1.42 s;
  • Length 100 cm: 10 swings in about 20.1 s → period about 2.01 s.

Stare at those three rows for a moment: the length went from 25 to 100 — 4 times longer — yet the period only went from 1.00 to 2.01, 2 times longer. Quadruple the length, double the period. That is the most direct evidence that the period grows with the square root of the length, and it's the one number from this experiment worth remembering.

The other two comparisons matter just as much:

  • Swap the bob. Keep the length at 50 cm, replace the coin with a heavier nut, and measure again — the period stays right around 1.4 s. Mass has no effect on the period.
  • Change the amplitude. Still at 50 cm, release once from about 5° and once from about 15°. The two periods usually differ by less than 0.05 s — within stopwatch error. That is isochronism. But pull the pendulum out to something like 60° and the period gets noticeably longer; the simple rule no longer applies.

The science: what decides the period

As a pendulum swings, the component of gravity along its arc always pulls the bob back toward the lowest point, so it oscillates back and forth around it. For small swings, the period T of a simple pendulum obeys:

T = 2π√(L ⁄ g)

where L is the pendulum length and g is the acceleration due to gravity. Three conclusions every middle schooler should take from this — all visible in the simulation above:

  • The longer the pendulum, the longer the period (the slower the swing); length is the only factor that changes the period noticeably;
  • The period is independent of the bob's mass: hang a heavier bob and the period doesn't budge;
  • For small swings, the period is nearly independent of amplitude — that's isochronism.

Isochronism is exactly why pendulums were used to build pendulum clocks: fix the length, and every swing takes the same reliable time. As a teenager, Galileo noticed that a chandelier in the cathedral seemed to take the same time for every swing — and timed it with his own pulse. Doing this experiment with a phone is considerably easier than what he had to work with.

Pendulums in everyday life

  • Grandfather clocks. The little nut under the pendulum bob that screws up and down is a length fine-tuner: if the clock runs fast, lower the bob a touch (longer pendulum, longer period); if it runs slow, raise it. Isochronism, applied directly.
  • Playground swings. Swinging while seated and while standing feel different: standing raises your center of gravity, shortening the effective pendulum length, so you swing faster. It's also why you stand up near the lowest point and crouch at the highest to pump a swing higher and higher.
  • Metronomes. A metronome is an upside-down pendulum: slide the weight up and the center of gravity rises — it ticks slower; slide it down and it ticks faster. Tempo control by changing the effective length.
  • Skyscraper dampers. A steel sphere weighing hundreds of tons hangs near the top of Taipei 101 as a pendulum. Its period is tuned close to the building's own sway period, canceling out the rocking caused by wind and earthquakes.

Four mistakes that ruin the measurement

  • Measuring the length wrong. Pendulum length runs from the pivot to the bob's center of gravity — not just to the top of the bob. With a nut as the bob, forgetting that extra half-nut of height can throw the result off by several percent. This is the most common source of error in the whole experiment.
  • Timing only one period. Human reaction time on a stopwatch is about 0.2 s, and a period is barely over 1 s — that's up to 20% error. Time 10 swings and divide by 10, and the error drops below 2% instantly.
  • Miscounting the swings. The moment of release counts as "zero"; only when the bob returns to its starting point do you count "one." Many students count "one" when the bob first reaches the far side — and get half the true period. The more reliable method is to count from the lowest point, where the bob moves fastest and its position is easiest to judge.
  • Swinging too wide. "Period doesn't depend on amplitude" holds only for small angles (5°–15° is the usual range). Release from beyond 45° and the period visibly lengthens — and students wrongly conclude the rule is false.
One more, beyond the mistakes: "A heavier bob feels more gravity, so it should swing faster." Gravity does increase — but so does the mass that must be moved, and the two exactly cancel. It's the same fact as "heavy and light objects hit the ground together": the acceleration due to gravity, g, does not depend on mass.

Tips for teachers and parents

  • Timing and groups: 30 minutes, in groups of three — one releases, one times, one records; rotate roles.
  • Say it before hands touch equipment: clear the area under the bob, and keep swings under 15°.

The step most people skip. Give the "swap in a heavier bob" trial to the student most convinced it will change the result. Measuring an unchanged period with their own hands beats hearing it ten times. And have everyone count swings from the lowest point, consistently.

How to tell they really understand. If a student can state "the period is decided by length alone" and explain why quadrupling the length only doubles the period, they've met the standard.

Take it further

  • Make the pendulum 4 times longer and guess: how many times longer is the period? (Hint: T is proportional to √L, so 4× the length means 2× the period.) Check it in the simulator or with the real thing.
  • Build a "one-second pendulum": tune the length so each half-swing takes exactly 1 second, and try using it as a metronome. Working backward from T = 2π√(L/g), a 2-second period needs a length of about 99 cm — calculate first, then measure, and see whether students can nail it on the first try.
  • Time the same pendulum on the ground floor and on a high floor. In theory g is slightly smaller up high, so the period is slightly longer — but the difference is far smaller than stopwatch error. Think about it: how much would the experiment need to improve to detect it?
  • Replace the string with a rubber band and try again. It's no longer a proper simple pendulum: the rubber band stretches, so the length keeps changing throughout the swing.

When a student can explain why a pendulum clock keeps time with its pendulum's length rather than its weight, this experiment has been understood all the way through.