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Free online simple pendulum calculator. Calculate pendulum period and frequency based on length and gravity, or reverse-calculate length and acceleration of gravity.
Select the quantity to calculate; the other two will be used as inputs.
Formula (Small Angle Approximation)
T = 2π × √(L / g)
L = (T / 2π)² × g
g = 4π² × L / T²
This formula applies to an ideal simple pendulum with small oscillation angles (typically < 15°), neglecting air resistance and string mass.
Overview
Understand what the tool solves, how it works, and the boundaries of its data.
For an ideal simple pendulum with a small swing angle, the period is T = 2π√(L/g), where L is pendulum length in meters and g is gravitational acceleration in meters per second squared. Period T is seconds per complete back-and-forth cycle. Frequency is f = 1/T in hertz, and angular frequency is ω = 2π/T in radians per second. OpenStax University Physics derives this model using the small-angle approximation (checked 2026-09-29).
The calculator also rearranges the same relationship to solve for length, L = g(T/2π)², or gravity, g = 4π²L/T². It requires positive values and reports the solved quantity alongside length, gravity, period, frequency, and angular frequency. The displayed calculation uses SI inputs; NIST's Guide to the SI covers consistent unit expression (checked 2026-09-29).
The starting values are length 1 m and standard gravity 9.80665 m/s². Substitution gives T = 2π√(1/9.80665) ≈ 2.0064 s and f ≈ 0.4984 Hz. Presets are also provided for the equator, pole, Moon, Mars, Jupiter, and Venus; these are selectable reference values, not measurements for a specific location. You can enter a custom acceleration when a problem or experiment supplies one.
Guide
Follow the workflow and verify inputs and outputs with practical examples.
Choose period, length, or gravity. The form shows the other two required quantities.
Use meters for length, seconds for period, and m/s² for gravity. Convert centimeters or milliseconds before entering them; the calculator does not convert units.
Check the solved value and units, then compare the accompanying frequency and angular frequency with the situation you expect. Copy the solved value when needed.
Use cases
See how the tool fits into real work and everyday tasks.
Hold gravity fixed and vary length to see the square-root relationship: doubling length does not double the period. Use the output to check a worked physics problem.
For a target period and a chosen gravity value, solve for length to get an ideal-model starting estimate, then compare it with a physical setup.
Enter an observed period and measured length to solve for an implied gravity. Compare only after checking how length and the period were measured.
Q&A
Find concise answers to common questions and confusing cases.
Mass is not part of the ideal simple-pendulum formula, so it does not affect this calculated value. Real setups may differ because of friction, string mass, or other departures from the ideal model.
The formula uses the small-angle approximation. At larger amplitudes, the period depends on the amplitude and this simple formula becomes less suitable; OpenStax discusses the approximation and its scope.
No. Convert the length to meters first. Likewise, enter period in seconds and acceleration in m/s².
No. Presets are convenient reference inputs. Use a location-specific value from a suitable measurement or source when the calculation requires local precision.
Notes
Review scope, result limitations, and important precautions before use.
This is an idealized calculation, not an experimental measurement. It assumes a simple pendulum and the small-angle model; it does not account for amplitude correction, friction, air resistance, string mass, or measurement uncertainty. For lab work, measure multiple cycles consistently, define length from pivot to bob center, and report the assumptions and uncertainty. The number of decimal places shown should not be mistaken for measurement accuracy.
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