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Simple Pendulum Calculator

Find the period, length or local gravity of a simple pendulum from T = 2π√(L/g), with frequency, angular frequency and a large-amplitude correction.

2.00641 s
Period T
0.4984 Hz
Frequency
3.13156 rad/s
Angular frequency ω
2.00736 s
Corrected period

T = 2π√(L ÷ g)

  • T = 2π × √(1 m ÷ 9.80665 m/s²) = 2.006409 s
  • Frequency f = 1 ÷ T = 0.498403 Hz — that is 29.904 full swings per minute.
  • Angular frequency ω = √(g ÷ L) = 3.131557 rad/s (equal to 2π ÷ T).
  • Half-period (one swing from side to side) = 1.003205 s.
  • At an amplitude of 5° the first-order correction gives T ≈ T₀(1 + θ₀²/16) = 2.007364 s — 0.0476% slower than the small-angle answer.
  • Adding the next term, T ≈ T₀(1 + θ₀²/16 + 11θ₀⁴/3072) = 2.007365 s.
  • The period does not depend on the bob's mass — only on length and gravity.

T = 2π√(L/g) is the small-angle approximation for an ideal simple pendulum: a point mass on a massless, inextensible string, swinging without air resistance. A one-metre pendulum where g = 9.81 m/s² has a period of 2.006 seconds, which is why a "seconds pendulum" beating once per second is about 0.994 m long. The correction terms above come from the exact elliptic-integral solution, and real clock pendulums also need allowances for buoyancy, bob shape and temperature expansion of the rod.

What is the Simple Pendulum Calculator?

A simple pendulum swings with period T = 2π√(L ÷ g), where L is the length from pivot to bob centre and g is local gravity. A one-metre pendulum where g = 9.81 m/s² has a period of 2.006 seconds.

  • Solves T = 2π√(L/g) for period, length or local gravity
  • Frequency, angular frequency and half-period reported with every answer
  • First and second order large-amplitude corrections from the exact solution
  • Shows the percentage error the small-angle assumption introduces
  • Length units from millimetres to feet, converted to SI before solving
  • Fully client-side — nothing you enter leaves the browser

How to use the Simple Pendulum Calculator

  1. 1

    Choose whether to solve for the period, the length or local gravity.

  2. 2

    Enter the two known values, picking a length unit from millimetres up to metres or feet.

  3. 3

    Set the swing amplitude θ₀ in degrees to control the large-amplitude correction.

  4. 4

    Read the period, frequency, angular frequency and corrected period in the four tiles.

  5. 5

    Open the working panel for the full substitution and copy it if you need to show your steps.

About the Simple Pendulum Calculator

The ByteTools Pendulum Calculator rearranges T = 2π√(L ÷ g) three ways: it gives the swing period from a length, the length needed for a chosen period, or the local value of gravity from a measured length and period. That last option is the classic school experiment for measuring g, and this tool does the algebra and the unit conversion for you.

Each answer comes with the frequency f = 1 ÷ T, the angular frequency ω = √(g ÷ L) and the half-period for a single side-to-side swing. Because T = 2π√(L/g) is a small-angle approximation, the tool also applies the first-order correction T₀(1 + θ₀²/16) and the next term for the amplitude you enter, so you can see exactly how much a wide swing slows the pendulum down.

Everything is worked out in your browser in JavaScript, with no uploads, no tracking and no stored data. Numbers refresh as you type, which suits a classroom demonstration where you change the length and watch the period follow, and the page carries on working offline once it has been loaded.

Frequently asked questions

What is the formula for the period of a pendulum?

The period of a simple pendulum is T = 2π√(L ÷ g), where L is the length from the pivot to the centre of mass of the bob. It depends only on length and gravity, so the period does not change when you swap the bob for a heavier one.

How long is a pendulum with a period of two seconds?

Rearranging gives L = g·T² ÷ 4π², which for T = 2 s and g = 9.81 m/s² is about 0.994 m. That length is known as the seconds pendulum, because it takes one second for each single swing.

Does the mass of the bob change the period?

No. Mass cancels out of the equation of motion, so a heavy bob and a light one on the same string swing at the same rate. Only the length, local gravity and, to a small extent, the amplitude affect the period.

How accurate is the small-angle approximation?

T = 2π√(L/g) assumes sin θ is close to θ, which is good to within about 0.2% below 10° of amplitude. At 30° the true period is roughly 1.7% longer, which is why this tool shows the corrected value alongside the simple one.

Can I measure gravity with a pendulum?

Yes — time a number of swings, work out the period, and use g = 4π²L ÷ T². Timing many swings and dividing reduces the error, and keeping the amplitude small keeps the small-angle formula valid.

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