Simple Pendulum Period Calculator
Calculate the period T and frequency f of a simple pendulum instantly. Enter length (L) and gravity (g) — the formula T = 2π√(L/g) gives the exact answer. Includes a reference table for common lengths.
- Data verified · June 2026
- Edited by Martín Rodríguez
- Private — runs on your device
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When to use this calculator
- Designing a grandfather clock escapement: a 0.994 m pendulum (the 'seconds pendulum') gives exactly T = 2.000 s, one tick per second.
- Physics lab experiment: measuring local gravitational acceleration g by timing 50 oscillations of a known-length pendulum and solving g = 4π²L/T².
- Seismic station calibration: long-period pendulums (L = 10–100 m equivalent) are used as reference oscillators in gravimeters to detect g variations across geological formations.
- Amusement park ride engineering: swing rides and pendulum-style attractions are modeled as simple pendulums to set safe oscillation periods and predict maximum angular velocity at the bottom of the arc.
Pendulum Period & Frequency by Length (g = 9.80665 m/s²)
| Length L (m) | Period T (s) | Frequency f (Hz) | Real-World Example |
|---|---|---|---|
| 0.248 | 1.000 | 1.000 | 1-second frequency standard |
| 0.994 | 2.000 | 0.500 | Seconds pendulum (grandfather clock) |
| 1.000 | 2.006 | 0.499 | Classic 1 m lab pendulum |
| 2.000 | 2.837 | 0.352 | Slow decorative wall clock |
| 4.000 | 4.013 | 0.249 | Large hall pendulum |
| 9.810 | 6.283 | 0.159 | Period = exactly 2π seconds |
| 25.000 | 10.037 | 0.100 | Long structural/architectural pendulum |
| 100.000 | 20.074 | 0.050 | Geophysical / seismic reference scale |
Fuente: NIST standard gravity g = 9.80665 m/s² (physics.nist.gov); fórmula T = 2π × √(L/g), válida para ángulos < 15°.
How it works
How It's Calculated
The period of a simple pendulum under the small-angle approximation is:
T = 2π × √(L / g)Where:
Frequency and period are reciprocals:
f = 1 / T = (1 / 2π) × √(g / L)Angular frequency (rad/s):
ω = 2π / T = √(g / L)Validity condition: This formula holds when the maximum angle θ₀ < ~15°. For larger angles, the true period is:
T_exact = 2π × √(L/g) × [1 + (1/16)θ₀² + (11/3072)θ₀⁴ + ...]At θ₀ = 15°, the error from using the small-angle formula is only ~0.5%.
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Period Reference Table
All values use g = 9.80665 m/s² (standard gravity, NIST definition)
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Deriving Pendulum Length from a Target Period
To find what length gives a desired period T:
L = g × T² / (4π²)Example — seconds pendulum (T = 2.000 s):
L = 9.81 × 4.000 / 39.478 = 0.9940 m---
Typical Cases
Case 1 — The Classic 1-Meter Lab Pendulum
Case 2 — The Seconds Pendulum (Clockmaking)
Case 3 — Measuring g in the Field
Case 4 — Pendulum on the Moon
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Common Errors to Avoid
1. Using diameter instead of effective length: L must be measured from the pivot point to the center of mass of the bob, not to its bottom edge. For a 3 cm steel ball, this error introduces ~1.5 cm offset, adding ~7 ms to T at L = 1 m.
2. Applying the formula at large angles: At θ₀ = 30°, the true period is ~1.7% longer than the small-angle prediction. At θ₀ = 90°, the error exceeds 18%. Always keep swings below 15° for formula accuracy within 0.5%.
3. Assuming g = 9.81 m/s² everywhere: Standard gravity (9.80665 m/s²) is a defined NIST constant, but real g ranges from 9.764 m/s² (equatorial mountain tops) to 9.834 m/s² (polar sea level). Using the wrong g introduces proportional T error via √g.
4. Ignoring air resistance and pivot friction: In a real pendulum, damping gradually reduces amplitude. The formula gives the undamped natural period; real clocks use an escapement mechanism to compensate for this energy loss and maintain constant amplitude.
5. Confusing period with half-period: A "tick" on a grandfather clock is a half-period (one directional swing). T = 2.00 s means 1 second per tick, not 2 seconds. Misreading this doubles or halves all timing calculations.
Worked Example: 1-meter lab pendulum
Frequently asked questions
What is the period of a simple pendulum formula?
What is the period of a 1-meter simple pendulum?
What length pendulum gives exactly a 2-second period (seconds pendulum)?
Does the mass of the pendulum bob affect the period?
What value of g should I use for my calculation?
At what angle does the small-angle approximation break down?
How can I measure local gravity g with a pendulum?
How does the pendulum period change on the Moon vs. Earth?
Why does doubling the pendulum length not double the period?
Sources & references
Methodology & trust
ciencia calculator reviewed by the Hacé Cuentas editorial team, checked against NIST — Standard Acceleration of Gravity (g = 9.80665 m/s²), following our editorial policy and methodology.
Last reviewed: June 2026. Parameters are verified periodically against the cited sources.
Calculations run 100% in your browser. We do not store or transmit your data.
Indicative results. For critical decisions, consult a professional.
📌 How to cite this calculator
Rodríguez, M. (2026). Simple Pendulum Period Calculator. Hacé Cuentas. https://hacecuentas.com/en/simple-pendulum-period
@misc{hacecuentas_simple_pendulum_period_2026,
author = {Rodríguez, Martín},
title = {{Simple Pendulum Period Calculator}},
year = {2026},
howpublished = {\url{https://hacecuentas.com/en/simple-pendulum-period}},
note = {Hacé Cuentas}
} Content licensed under CC-BY 4.0 — reuse it citing the source with a link to Hacé Cuentas.