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Pendulum lab: period T = 2π√(L/g), large-swing error, time it with a stopwatch and find g from a T² against L graph

Swing a pendulum of any length on Earth, the Moon, Mars or Jupiter and see its period worked out. At Class 11 see how much longer big swings take, then time a pendulum with the stopwatch and fit a straight line of T² against L to measure g.

Keep it one tap away for everyday use.

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  1. Set length
  2. Time swings
  3. T² vs L
  4. Find g
Nothing is uploadedWorks offline once open

Swing a pendulum

Time it yourself

Time the pendulum above, or a real one (a nut on a thread works well). Start when the bob is at one end, count the swings back to that end, and stop at the last one. Timing many swings makes your reaction time matter less.

0.00 s

The readings to start with are made up, for practice (10 swings each). Replace them with your own.

Try this

  • Make the string four times longer: the period doubles. Twice as long? The period grows by √2, about 1.41.
  • Change the release angle from 5° to 20°: the period hardly changes. That is why pendulum clocks work.
  • At Class 11, let go from 90° or 150°: now the period is clearly longer than 2π√(L/g).
  • Take it to the Moon: the same pendulum swings about 2.5 times more slowly.
  • Find the length that ticks once a second each way (a period of 2 s) on Earth.

Check yourself

What it does

A pendulum is the simplest clock there is: a weight on a string, swinging back and forth. This page swings one for you. Change the length, the angle you let it go from, and the planet, and it shows the period (the time for one full swing there and back) with the formula T = 2π√(L/g) filled in with your numbers. The motion on screen is the true motion, worked out step by step, so a big swing really does take longer.

Then do the experiment. Use the stopwatch to time 10 swings of the pendulum on screen, or of a real one at home, and add each reading to the table. The page draws T² against L, fits the best straight line and works out g from its slope, with an uncertainty.

It serves Class 7 Science (motion and time: the simple pendulum and its time period), Class 9 (motion and gravity, measuring time) and Class 11 Physics, Oscillations (the simple pendulum as simple harmonic motion) and the Class 11 practical on finding g with a pendulum. See SHM circle for why a pendulum moves in simple harmonic motion, and pendulum lab analyser for a fuller analysis of your lab readings.

How to use it

  1. Level: Class 7–9 gives the period and swings per minute, with release angles up to 45°. Class 11 adds the exact period for big swings (up to 170°) and a graph of how far off the formula is.
  2. String length L: from 10 cm to 2.5 m. The picture is drawn to fit the box.
  3. Let go from: the release angle. Let go again starts the swing over; the counter shows swings and time.
  4. Gravity g: Earth, Moon, Mars or Jupiter.
  5. Time it yourself: press Start when the bob is at one end, count 10 returns to that end, press Stop. Check Swings timed and Length used, then Add to my readings. You can also type or paste readings, one “length, time” pair per line.

Worked example

As the page opens, a 1 m pendulum on Earth: T = 2π√(1 / 9.8) = 2.007 s, so 10 swings take about 20.07 s and it makes about 29.9 swings a minute. At Class 11, letting go from 60° gives an exact period of 2.154 s, 7.32 % longer; from 10° it is only 0.19 % longer.

The practice readings (lengths 0.2 to 1.0 m, times for 10 swings 9.0 to 20.1 s) give the best line T² = 4.044 × L − 0.00509, so g = 4π² / 4.044 = 9.763 m/s² ± 0.049 m/s², about 0.38 % below 9.8.

Check yourself

Five fresh questions: a period from a length, the length for a given period, g from a timing, what a longer string does, and what changes the period. Answers are checked on the page and the working is shown. Nothing is saved.

Tips

  • Time 10 or 20 swings, not one. Your reaction time (about 0.2 s) is then shared out over all of them.
  • Start and stop at the end of a swing, where the bob is slowest and easiest to see; or at the middle with a marker, where it passes quickly and the moment is sharp.
  • Measure L to the middle of the bob, not just the string.
  • Keep swings small (under about 10°) for the formula to hold.

FAQ

Why plot T² and not T? T² = (4π²/g) L is a straight line through the origin, so its slope gives g directly. A line also shows at once if one reading is off.

Why is there a small intercept? If you measure the length a little short (to the top of the bob, say), the line shifts. The slope, and so g, is not affected, which is the beauty of the graph method.

Does the mass of the bob matter? No: heavier bobs feel more pull but need more pull to speed up, and the two cancel.

Where the numbers come from

  • Simple pendulum: T₀ = 2π√(L/g) for small swings; the exact period T = T₀ / AGM(1, cos(θ₀/2)) (complete elliptic integral of the first kind written with the arithmetic–geometric mean): standard mechanics.
  • The motion shown is the full equation θ″ = −(g/L) sin θ solved step by step (Runge–Kutta).
  • g = 9.8 m/s² (school value); Moon about 1.62, Mars about 3.71 and Jupiter about 24.8 m/s² are rounded general values.
  • The starting readings are made up for practice.
  • Checked against an independent Python calculation (numpy polyfit for the fit, a fine-step integration and the textbook series for the large-swing period).

Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.

Last updated 2026-11-09.

Change log for Pendulum lab

  1. Add #1351 Pendulum lab Details on GitHub

Every change to fibub

Related tools

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