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Pendulum Period Calculator

Computes the period, frequency and angular frequency of a simple pendulum from its length with the small-angle formula T = 2π√(L/g); when an amplitude is given it also returns the exact period from the arithmetic–geometric mean.

When to use

You need how long a pendulum of known length takes to swing on Earth or another body, or how much a large swing angle lengthens the period.

Do not use when: The bob's size or the rod's mass is not negligible (physical pendulum, T = 2π√(I / (m·g·d))), or the oscillator is a mass on a spring (use hookes-law).

Formula

T = 2π × √(length_m / gravity_m_s2); f = 1 / T; ω = √(gravity_m_s2 / length_m); exact period for amplitude θ0: T_exact = T / agm(1, cos(θ0 / 2)) (arithmetic–geometric mean)

The small-angle formula is within 0.1 % up to about 7° amplitude, 1.7 % low at 30° and 18 % low at 90°; the AGM result is exact for a point mass on a massless rod without friction.

Inputs

Parameter	Type	Unit	Required	Description
`length_m`	number	m	yes	Distance from the pivot to the centre of mass of the bob, in metres. Range: > 0
`gravity_m_s2`	number	m/s²	default 9.80665	Local gravitational acceleration. Default is standard gravity g_n = 9.80665 m/s² (Moon ≈ 1.62, Mars ≈ 3.72, Jupiter ≈ 24.8). Range: > 0, ≤ 10000
`amplitude_degrees`	number	°	no	Optional maximum swing angle from vertical; enables the exact (large-amplitude) period. Range: ≥ 0, ≤ 170

Outputs

Output	Type	Unit	Description
`period_s`	number	s	T = 2π√(L/g), one full back-and-forth swing.
`frequency_hz`	number	Hz	f = 1 / T.
`angular_frequency_rad_s`	number	rad/s	ω = √(g / L).
`period_at_amplitude_s`	number	s	Exact period T / agm(1, cos(θ0/2)) (only when amplitude_degrees is given).
`amplitude_correction_percent`	number	%	How much longer the exact period is than the small-angle value (only when amplitude_degrees is given).

Example

1 m pendulum, 30° amplitude: {"length_m":1,"amplitude_degrees":30} → {"period_s":2.0064,"frequency_hz":0.4984,"angular_frequency_rad_s":3.1316,"period_at_amplitude_s":2.0413,"amplitude_correction_percent":1.7409}

0.25 m pendulum: {"length_m":0.25} → {"period_s":1.0032,"frequency_hz":0.9968}

GET https://tttkmbb.com/api/v1/calculate/pendulum?length_m=1&amplitude_degrees=30

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/pendulum (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/pendulum · Markdown: https://tttkmbb.com/physics/pendulum.md · JSON definition: https://tttkmbb.com/physics/pendulum.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="pendulum"
OpenAPI operationId: calculate_pendulum_period
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

HyperPhysics – Simple pendulum (reference)
Wikipedia – Pendulum (mechanics) (reference)
NIST CODATA – standard acceleration of gravity g_n = 9.80665 m/s² (standard)

FAQ

Does the mass of the bob change the period?

No. For a simple pendulum the period depends only on length and gravity; mass cancels out.

What length gives a period of exactly 2 s?

About 0.994 m at g = 9.80665 m/s² (L = g·T² / 4π²), the classic 'seconds pendulum' with a one-second half swing.

Related calculators

Hooke's Law Calculator — Spring–mass oscillator period.
Free Fall Calculator — Another way to measure g.
Wavelength & Frequency Calculator — Frequency and period of waves.