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Pendulum Calculator

Find how long a pendulum takes to swing from its length, or the length you need for a given swing time, with an optional wide-swing correction.

Level: Grades 9–10

Printable worksheet

Pendulum Calculator

Know the answer you need? Pick an input to work backward.

Inputs
Find

From the pivot to the center of the bob.

The largest angle from vertical. Leave blank for small swings (under about 10°); fill it in to correct for wider swings.

Result

Period (small swings)

2.0064 s

A 1 m pendulum has a period of 2.0064 s (0.4984 Hz).

Frequency
0.4984 Hz
Full swings per minute
29.9042
Angular frequency (ω = 2π ÷ T)
3.1316 rad/s
Gravity used
9.8067 m/s²

See the step-by-step solution ↓

Step-by-step solution

  1. Use the surface gravity of Earth (standard gravity)

    g=9.80665 m/s²

  2. Find the small-swing period T₀ from the length L and gravity g: 2π × the square root of L ÷ g

    T₀=2×π×19.80665=2.0064092926 s

  3. Frequency f is the number of full swings per second, so it's 1 ÷ the period T

    f=12.0064092926=0.4984027953 Hz

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Find out how long a pendulum takes to swing, or how long to make one so it swings at the pace you want. A simple pendulum is a weight (the bob) on a light string or rod. Its rhythm is set almost entirely by two things: its length and the strength of gravity.

The weight of the bob doesn't matter. For small swings, neither does how far it swings. That steady beat is why pendulums kept time in clocks for nearly three centuries.

The calculator gives the period, meaning the time for one full swing out and back. It also gives the frequency and the swings per minute for a pendulum of any length. Or it works backward from a period to the length you need. Gravity can be Earth's standard value, the Moon's, Mars's or your own. You can also enter how wide it swings, and the calculator applies the exact correction for wide swings.

Where it's used

Why this math matters outside the classroom:

  • Physics labs. Students time 10 swings of a 0.5 m pendulum, find about 1.42 s each, and compare with 2π√(0.5 ÷ 9.81).
  • Clock repair. A clockmaker shortens a pendulum slightly to make a clock that runs slow keep better time.
  • Measuring gravity. Rearranging the formula, a student finds g = 4π²L ÷ T² from a measured length and period.
  • Playground swings. A child on a swing with 2.5 m chains goes back and forth in about 3.2 s, whoever is sitting on it.
  • Space comparisons. A class finds a 1 m pendulum would take about 4.9 s per swing on the Moon versus 2.0 s on Earth.

Grade level and standards: Grades 9–10; Common Core State Standards for Mathematics HSA.CED.A.4. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Choose whether to find the period from a length, or the length for a period.
  2. Enter the length from the pivot to the center of the bob, or the period in seconds.
  3. Choose the gravity, or enter your own value, such as your textbook's 9.8 m/s².
  4. For swings wider than about 10°, enter the amplitude to include the wide-swing correction.

What the inputs mean

Pendulum length (L)
Measure to the bob's center of mass (its balance point), not to its bottom. If the bob isn't small compared with the string, the simple formula becomes less accurate.
Period (T)
The time for one full swing out and back. A clock's "seconds pendulum" has a 2-second period.
Custom gravitational acceleration
For example 9.81 or 9.8 m/s² to match a textbook, or 32.174 ft/s².
Swing amplitude (optional)
The amplitude is the largest angle from straight down, on either side. Leave it blank for the usual small-swing formula.

Formula and method

T₀ = 2π √(L ÷ g)
f = 1 ÷ T
L = g × (T ÷ 2π)²
T = T₀ ÷ AGM(1, cos(θ₀ ÷ 2))

T is the period in seconds, and T₀ is the period for small swings. L is the length and g is the strength of gravity. f is the frequency, the number of full swings per second. θ₀ (theta zero) is the amplitude, the widest angle of the swing. π (pi) is about 3.14159, and √ is the square root.

For small swings, the pull back toward the middle grows in step with how far the bob is pulled aside. That gives a smooth, regular back-and-forth motion (called simple harmonic motion) with a period of 2π√(L/g). Making the pendulum four times longer doubles the period. Stronger gravity shortens it.

Wide swings take a little longer. The exact period needs a harder kind of math (an elliptic integral). The calculator works it out with the arithmetic–geometric mean, or AGM, a quick repeated averaging method. (cos, the cosine, is a standard function of an angle.) At 10° the swing is about 0.19% slower than the simple formula. At 90° it's about 18% slower.

Worked example

How long does one full swing of a 1 m pendulum take on Earth?

  1. Use standard gravity, g = 9.80665 m/s².
  2. T = 2π × √(1 ÷ 9.80665) = 2.0064092926 s.
  3. f = 1 ÷ 2.0064092926 = 0.4984 Hz, or about 29.9 swings per minute.

Result: A 1 m pendulum swings out and back in about 2.006 s. So each one-way swing takes very nearly one second.

Understanding the result

The period counts a complete cycle. A clock pendulum that ticks once per second ticks on each one-way swing. So it has a 2-second period and is about 0.994 m long.

The period depends on gravity. So carefully timing a pendulum of known length was once a standard way to measure g. It also shows why a pendulum clock runs slow high up a mountain.

Assumptions and limitations

  • A tiny bob on a weightless, rigid string, swinging back and forth in one plane. There's no air resistance or friction at the pivot.
  • Gravity is the same all through the swing.
  • Real pendulums with heavy rods or large bobs are called physical pendulums. They swing at different rates.
  • Air and friction slowly shrink the swing but barely change the period. That isn't modelled.
  • Gravity on Earth varies by about 0.5% between the equator and the poles. That changes the period by about 0.25%.

Frequently asked questions

Does the mass of the bob affect the period?

No. A heavier bob needs more force to get moving, but gravity pulls it harder by the same amount, so the two cancel. Only length and gravity matter for a simple pendulum.

How long is a pendulum that beats seconds?

A seconds pendulum has a 2-second period (one second each way). With standard gravity its length is 9.80665 ÷ π² = 0.9936 m, just under a meter.

Why does the swing angle matter?

The 2π√(L/g) formula assumes small angles, where sin θ ≈ θ (the sine of a small angle is almost equal to the angle itself). At larger angles, the pull back grows more slowly than that, so each swing takes longer. Below about 10° the difference is under 0.2%.

How would a pendulum swing on the Moon?

More slowly. The Moon's gravity is about 1.62 m/s², roughly a sixth of Earth's. So the period is about 2.46 times longer: a 1 m pendulum takes about 4.9 s per swing.

How this calculator is checked

Method: Simple pendulum T₀ = 2π√(L/g); exact large-amplitude period T = T₀ / AGM(1, cos(θ₀/2)) (complete elliptic integral of the first kind); standard gravity 9.80665 m/s². The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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