The pendulum

A pendulum 99.4 cm long takes 2 seconds for a full swing, one second each way.

Loading the 3D model

Drag to turn. Scroll or pinch to zoom.

Press Release. The lab times every full swing like a stopwatch and compares the average with 2π√(L/g). Raise the amplitude to 60° and the difference grows to about 7%.

A weight on a string

A simple pendulum is a small heavy bob on a light string. Pull it aside and let go and it swings back and forth. The time for one full swing, there and back, is called the period. Two forces act on the bob: its weight pulling down and the tension of the string pulling towards the pivot. Tension acts along the string and has no component tangent to the arc. The tangential part of the weight, m g sin θ, pulls the bob back towards the bottom. That restoring force is what keeps the pendulum swinging.

Forces on a pendulum bob: weight mg and string tension T. Length L, angle θ.

The period formula

For small swings OpenStax derives the period as T = 2π√(L ÷ g). Only two things appear in it: the length L and the gravity g. The mass of the bob is not there, so a heavy bob and a light one of the same length swing in step. The size of the swing is not there either, as long as it stays below about 15°. A 1.00 m pendulum on Earth has T = 2π√(1 ÷ 9.81) ≈ 2.006 s. Because of the square root, making the pendulum four times longer only doubles the period: 4 m gives about 4.01 s.

Turn the formula round and you get the length for any period: L = g T² ÷ 4π². For a period of exactly 2 s, L = 9.81 × 4 ÷ 4π² ≈ 0.994 m. This is the seconds pendulum, the long pendulum of a tall case clock that ticks once each way. On the Moon the same 0.994 m pendulum would take about 4.9 s per swing.

Where the formula stops working

The formula uses the approximation sin θ ≈ θ, which is good for small angles only. The lab does not use the formula to move the bob. It solves the full equation of motion step by step and times the swings like a stopwatch. At 10° the measured period is about 0.2% longer than the formula. At 30° it is 1.7% longer, at 60° about 7.3% longer and at 80° almost 14% longer. Big swings are slower, and that is why precise pendulum clocks keep their swing small.

BackgroundAround 1602 Galileo noticed that a pendulum keeps nearly the same period whether it swings widely or narrowly, a property called isochronism. Half a century later the Dutch scientist Christiaan Huygens used it to regulate a clock. He had a working pendulum clock by the end of 1656, had local clockmaker Salomon Coster build them, and patented the design on 16 June 1657. The American Physical Society notes that his clocks drifted about 15 seconds a day, against about 15 minutes for the spring-driven clocks before them. Pendulum clocks remained the most precise timekeepers until the 1930s.

Measuring g with a pendulum

If you know the length and measure the period, you can work out g = 4π² L ÷ T². OpenStax gives an example: a 75.000 cm pendulum with a period of 1.7357 s gives g ≈ 9.828 m/s². You can try it at home with a key on a 1 m string: start it at less than 10°, time 10 full swings and divide by 10. If you get 20.1 s for 10 swings, T = 2.01 s and g ≈ 9.77 m/s². Timing many swings shrinks the error from starting and stopping the stopwatch.

Key points

T = 2π√(L ÷ g) for small swings.
The period does not depend on mass, and barely on amplitude below about 15°.
Four times the length, twice the period.
Big swings take longer than the formula says: about 7% at 60°.

Period of a 1.00 m pendulum on Earth (g = 9.81), in seconds? (three decimals)
Period of a 0.25 m pendulum on Earth, in seconds? (two decimals)
You make a pendulum four times longer. The period becomes…
- twice as long
- four times as long
- half as long
You swap the bob for one twice as heavy, same length. The period…
- doubles
- stays the same
- halves
Period of a 1.00 m pendulum on the Moon (g = 1.62), in seconds? (two decimals)
What length, in metres, gives a 2 s period on Earth? (three decimals)
At 60° amplitude, the real period compared with 2π√(L ÷ g) is…
- about 7% longer
- about 7% shorter
- the same
A key on a 1 m string makes 10 swings in 20.1 s. Estimate g in m/s². (two decimals)
Who built the first pendulum clock, in 1656?
- Galileo
- Christiaan Huygens
- Isaac Newton
Which force pulls the bob back towards the bottom?
- The tension alone
- The part of the weight along the arc, mg sin θ
- Air resistance

Sources for this course (10)

OpenStax College Physics 2e, chapter 4 (Newton's laws)
5.1 (Friction, Table 5.1 coefficients)
5.2 (Drag)
7 (Work and energy)
16.4 (The simple pendulum, worked example 75.000 cm and 1.7357 s) (openstax.org/books/college-physics-2e)
NIST CODATA standard acceleration of gravity 9.80665 m/s² and Newtonian constant of gravitation (physics.nist.gov/cuu/Constants)
NASA Moon and Mars Fact Sheets (surface gravity 1.62 m/s²; Mars 3.69 to 3.73 m/s²; Earth GM 0.39860 × 10⁶ km³/s², mean radius 6371 km; Earth surface acceleration 9.78 equator, 9.83 pole) (nssdc.gsfc.nasa.gov/planetary/factsheet/)
Newton, Philosophiae Naturalis Principia Mathematica (1687)
Galileo, Two New Sciences (1638)
American Physical Society, "Christiaan Huygens Patents the First Pendulum Clock" (aps.org/apsnews/2017/06/huygens-patents-first-pendulum-clock).

Checked: 2026-10-17