Pendulum period formula and simple pendulum simulator • The Science 360

By The Science 360 · Updated October 2026 · 9 min read

What is the formula for the period of a pendulum?

For small swings, the period of a simple pendulum is T = 2π√(L ÷ g): it depends only on the string length L and the strength of gravity g, not on the mass of the bob or (to a good approximation) on how far you pull it. A pendulum 0.994 m long swings once to and fro in 2.00 s on Earth. A pendulum 10 m long takes 6.34 s. The simulator above lets you change the length, amplitude and world and compares this formula with a numerical solution of the real equation.

The "small swing" part matters. The formula assumes the restoring force is proportional to the angle, which is only true when sin θ ≈ θ. At 20° the true period is 0.8% longer than the formula says, and at 90° it is 18% longer. The simulation shows both numbers, plus the energy swapping between kinetic and potential and a phase-plane plot of speed against angle. If you would rather build one, a bob on a string and a stopwatch is the classic of the 5 amazing physics experiments for your kids type.

Key results from the simulation

Period formula: T = 2π√(L ÷ g). At L = 0.994 m on Earth that is 2.00 s, so each one-way swing takes about a second.
Length, not mass: quadrupling the length doubles the period. A bob ten times heavier changes nothing.
Gravity: the same pendulum takes 4.92 s on the Moon, 2.46 times longer, because g is 6.1 times smaller.
Big swings: at 90° the exact period is 2.36 s against 2.00 s from the formula, 18% longer.

Six pendulums to try

Each section gives the numbers the simulation shows for that setting and has a button that loads it, including the amplitude, the world and the friction switch. The ruler on the left of the picture is drawn to the string length, so the scene zooms as you change it.

A grandfather clock: 0.994 m for a 2-second period

The seconds pendulum

A grandfather clock keeps time with a pendulum of about 0.994 m. The formula gives T = 2π√(0.994 ÷ 9.81) = 2.00 s. One swing to the right and back is 2 s, so each one-way swing, which is one tick or one tock, lasts 1 s. That is why it is called a seconds pendulum.

Clocks keep the swing small, here 4°, so the formula is excellent: the exact period is 2.00 s, only 0.03% longer. A clock gains or loses time if the length changes, so the bob sits on a screw that moves it up or down by tiny amounts. The top speed at the bottom of the swing is only 0.22 m/s.

Small-angle T 2.00 s True period 2.00 s Frequency 0.500 Hz Top speed 0.22 m/s

A one-second pendulum is about 25 cm long

A desk-top experiment

Solving T = 2π√(L ÷ g) for L gives L = g T² ÷ 4π². For T = 1 s that is 24.8 cm, so a string of about a quarter of a metre with a small weight swings once per second. In the simulation, 0.248 m gives 1.00 s, and with a 10° swing the true period is 1.00 s.

This is an easy home experiment: time ten swings with a phone stopwatch and divide by ten. A bigger swing than about 15° starts to show up in the third decimal, so keep it small and compare the length with the period. Quadruple the string and the period doubles.

Length 0.248 m Small-angle T 1.00 s True period 1.00 s Frequency 0.999 Hz

Pull it out to 90°

Where the formula starts to fail

Release the clock pendulum from horizontal and the formula still says 2.00 s, but the real period is 2.36 s. The extra 18% comes from the restoring force being proportional to sin θ: near the ends it is weaker than the straight-line approximation, so the bob lingers there. The top speed is 4.42 m/s.

Watch the phase plane. For a small swing the solid curve and the dotted ellipse lie on top of each other, which is simple harmonic motion. At 90° the true orbit is visibly squarer than the ellipse. The series T ≈ T₀(1 + θ₀²/16 + 11θ₀⁴/3072) gives 1.176 × T₀ here against the exact 1.180, because at 90° more terms are needed.

Small-angle T 2.00 s True period 2.36 s Difference +18% Top speed 4.42 m/s

The same clock on the Moon

Gravity 1.62 m/s²

With g = 1.62 m/s² the 0.994 m pendulum has T = 4.92 s (4.93 s for a 10° swing), which is 2.46 times as long as on Earth. A grandfather clock taken to the Moon would run slow by that factor.

Period scales as 1 ÷ √g. On Mars, with g = 3.71 m/s², the factor is 1.63. On Jupiter, at the level where the pressure is 1 bar, g = 24.79 m/s² and the factor is 0.63. Pendulums were used for centuries to measure the local value of g, because timing a swing is easy to do accurately.

Small-angle T 4.92 s Moon vs Earth ×2.46 Mars vs Earth ×1.63 Jupiter vs Earth ×0.63

A 10 m pendulum and the Foucault note

Slow and stately

At the top of the length slider, 10 m, the period is 6.34 s from the formula (6.37 s for a 15° swing) and the bob reaches 2.59 m/s at the bottom. The simulation slows nothing down here, but it speeds the playback up a little so the swing fits in a few seconds, and the badge shows by how much.

Léon Foucault hung a much longer one in 1851: a heavy bob on a wire about 67 m long in the Panthéon in Paris. By the formula its period is 16.4 s. Its plane of swing slowly turns because the Earth rotates underneath it: at the latitude of Paris (about 48.85°) a full turn takes about 32 hours, and at a pole exactly one day. A Foucault pendulum is the simplest visible proof that the Earth spins. This simulator has no rotation, so it swings in a fixed plane.

Small-angle T 6.34 s True period 6.37 s Top speed 2.59 m/s 67 m wire 16.4 s

Switch friction on

Energy leaks away

With friction on, the equation gains a term −c θ′ with c = 0.25 per second. The amplitude of a small swing then halves about every 5.5 s (2 ln 2 ÷ c). The energy bars shrink with it and the phase-plane loop spirals in to a point. The first full swing from 40° takes 2.04 s in the simulation, against 2.06 s without friction: friction trims the amplitude as it goes, and smaller swings are quicker.

Without friction the motion is the same run backwards as forwards. With it, a film of the pendulum running in reverse would show it gaining energy from nothing, which never happens, and that one-way direction is the everyday arrow of time. See The Arrow of Time Paradox for more.

Friction c 0.25 /s Amplitude halves 5.5 s Simulated T 2.04 s No friction 2.06 s

How much longer big swings take

Amplitude	True period	Small-angle formula	Longer by	Series estimate
5°	2.00 s	2.00 s	0.05%	0.05%
10°	2.00 s	2.00 s	0.2%	0.2%
20°	2.02 s	2.00 s	0.8%	0.8%
30°	2.04 s	2.00 s	1.7%	1.7%
45°	2.08 s	2.00 s	4.0%	4.0%
60°	2.15 s	2.00 s	7.3%	7.3%
90°	2.36 s	2.00 s	18%	18%

All rows are for a 0.994 m pendulum on Earth with no friction. The true period is exact (it uses the complete elliptic integral, evaluated with the arithmetic-geometric mean) and agrees with the simulation to many digits. The series estimate keeps the first three terms and drifts at large angles.

How the simulation works

A bob on a massless string of length L obeys θ″ = −(g ÷ L) sin θ, where θ is the angle from the vertical. For small angles sin θ ≈ θ and this becomes θ″ = −(g ÷ L) θ, the equation of simple harmonic motion, whose period is T = 2π√(L ÷ g). The simulation does not use that approximation: it steps the full sin θ equation with fourth-order Runge-Kutta in steps of about 4 ms or less, and the bob you see is the result. With friction on a term −c θ′ is added, with c = 0.25 per second.

The true period comes from the complete elliptic integral of the first kind, T = 4√(L ÷ g) K(sin(θ₀ ÷ 2)), which equals T₀ ÷ AGM(1, cos(θ₀ ÷ 2)) where AGM is the arithmetic-geometric mean. The "Simulated T" readout is found separately by integrating the equation and timing two successive passes through the bottom of the swing, so the two columns agree without friction and the simulated one differs honestly with it.

Energy per unit mass is KE = ½ (L θ′)² and PE = g L (1 − cos θ). The bars show them as a share of the energy at release, and the dashed line is their sum, which stays level without friction. In the phase plane the horizontal axis is θ and the vertical axis is θ′: the solid curve is the exact frictionless orbit, and the dotted ellipse is what simple harmonic motion would give. The playback shows each period in between 1 and 6 seconds, so very short pendulums are slowed down and very long ones sped up; the badge says by how much.

What the model leaves out

Real bobs are not points: a physical pendulum has a bob with size and a rod or string with mass, so its period depends on how that mass is distributed. The simulation is the ideal simple pendulum.
Friction is a simple model: the linear term c = 0.25 per second lumps air resistance and pivot loss together, and is the same for every length and world. Real air drag grows with speed squared.
Constant gravity and a fixed pivot: g does not change with height, and the pivot does not move or flex.
No rotation of the Earth: the plane of swing stays fixed. A real long pendulum precesses, as Foucault showed.
Temperature: a metal rod expands when warm, which lengthens a clock pendulum and slows the clock. Clockmakers use compensated pendulums for this.

Common misconceptions

“A heavier bob swings faster.” Mass cancels from the equation: a heavier bob feels a proportionally bigger pull from gravity and also has proportionally more inertia. Only length and g set the period.

“The period never depends on the swing size.” It almost does not for small swings, which is the useful result Galileo noticed, but it is not exact. Look at the 90° row in the table.

“A pendulum is a perfect clock.” Friction slowly shrinks the swing, and temperature changes the length. A clock needs a mechanism that tops up the energy and a pendulum that is compensated for heat.

Frequently asked questions

What is the formula for the period of a pendulum?

T = 2π√(L ÷ g), where L is the length of the string in metres and g is the acceleration of gravity (9.81 m/s² on Earth). It holds for small swings, up to roughly 10 to 15°. For larger swings the true period is longer, by 0.8% at 20° and 18% at 90°.

Does the mass of the bob affect the period?

No. In the ideal pendulum the mass cancels: the pull of gravity along the arc and the inertia are both proportional to the mass. In a real pendulum the bob must be heavy compared with the string and small enough that air resistance matters little, but the mass itself is not in the formula.

How long is a pendulum with a period of one second?

About 24.8 cm, from L = g T² ÷ 4π². The famous seconds pendulum, with 1 s per swing in one direction, has a full period of 2 s and a length of about 0.994 m.

Why does the period change with the swing angle?

The restoring force on the bob is proportional to sin θ, not θ. For a big swing sin θ is smaller than θ, so the pull at the ends is weaker than the simple formula assumes and the swing takes longer. The exact result uses an elliptic integral, which the simulator evaluates.

How does gravity change a pendulum’s period?

The period scales as 1 ÷ √g. The same pendulum swings 2.46 times slower on the Moon, 1.63 times slower on Mars and 1.59 times as fast on Jupiter's 1-bar level. That is how pendulums have been used to measure local gravity.

What is a Foucault pendulum?

A long, heavy pendulum free to swing in any direction. Foucault showed in 1851 that its plane of swing turns slowly relative to the floor because the Earth rotates beneath it. At the latitude of Paris the turn takes about 32 hours. The simulator does not include this, but the 10 m scenario is a good place to see a slow, long swing.