# 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.

- Calculator id: `pendulum` · Category: Physics (`physics`) · Tool name: `calculate_pendulum_period`
- Canonical page: https://tttkmbb.com/physics/pendulum · This document: https://tttkmbb.com/physics/pendulum.md · JSON definition: https://tttkmbb.com/physics/pendulum.json

## Purpose

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.

**Use when:** 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).

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `length_m` | number | m | required | Distance from the pivot to the centre of mass of the bob, in metres. (> 0) |
| `gravity_m_s2` | number | m/s² | optional, 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). (> 0, max 10000) |
| `amplitude_degrees` | number | ° | optional | Optional maximum swing angle from vertical; enables the exact (large-amplitude) period. (min 0, max 170) |

## Output

| Field | 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). |

## 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.

## Data Sources

- HyperPhysics – Simple pendulum — http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html (reference, retrieved 2026-09-23)
- Wikipedia – Pendulum (mechanics) — https://en.wikipedia.org/wiki/Pendulum_(mechanics) (reference, retrieved 2026-09-23)
- NIST CODATA – standard acceleration of gravity g_n = 9.80665 m/s² — https://physics.nist.gov/cgi-bin/cuu/Value?gn (standard, retrieved 2026-09-23)

Data freshness: `static`. Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output.

## API

- `GET https://tttkmbb.com/api/v1/calculate/pendulum?length_m=…`
- `POST https://tttkmbb.com/api/v1/calculate/pendulum` with JSON body `{"inputs": {…}}`
- Response: unified envelope (`success`, `request`, `result.values`, `result.units`, `sources`, `freshness`, `timestamp`, `next_actions`, `links`); see https://tttkmbb.com/docs/response-format.md
- Schema: https://tttkmbb.com/api/v1/calculators/pendulum · OpenAPI operationId `calculate_pendulum_period` in https://tttkmbb.com/openapi.json
- Authentication: none. Rate limit: fair use, see https://tttkmbb.com/docs/rate-limits.md.

## MCP

- Server: `https://tttkmbb.com/mcp` (Streamable HTTP, JSON-RPC 2.0, no auth)
- Tool:  `run_calculator` with `{"calculator_id": "pendulum", "inputs": {…}}`

## Example

- 1 m pendulum, 30° amplitude: inputs `{"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: inputs `{"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
```

## Limitations

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). 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. All values are computed from the formula above; no measurement or live data is involved.

## 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

- [Hooke's Law Calculator](https://tttkmbb.com/physics/hookes-law.md) — Spring–mass oscillator period.
- [Free Fall Calculator](https://tttkmbb.com/physics/free-fall.md) — Another way to measure g.
- [Wavelength & Frequency Calculator](https://tttkmbb.com/physics/wavelength-frequency.md) — Frequency and period of waves.