1. Find period from length
For L = 1.000 m and g = 9.80665 m/s² at 10°, T = 4√(1/9.80665)K(sin 5°) ≈ 2.0102 s.
Simple Pendulum Calculator – Period, Frequency, Length & Gravity
Solve an ideal simple pendulum for period, frequency, length, or gravitational acceleration. For small angles, T = 2π√(L/g); when an amplitude is supplied, the calculator automatically applies the exact large-angle correction. Bob mass does not affect the ideal pendulum period.
Calculator inputs
Choose the unknown; only the needed known quantities remain active.
Choose a preset or enter local gravity.
Advanced, copy & sharing options
Use 0–10 digits after the decimal point.
Results
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How to use the pendulum calculator
- Choose whether to solve for period, frequency, length, or gravity.
- Enter the known quantities and choose their units. For length, measure from the pivot to the bob's center of mass.
- Enter the peak release amplitude if known. The calculator chooses the exact large-angle model automatically; clear amplitude to use only the small-angle approximation.
- Read the dominant answer, interpretation, conversions, and substituted calculation steps. Results update as inputs change.
Pendulum formulas and variables
Small-angle period: T0 = 2π√(L/g). Rearranging gives L = g(T/2π)² and g = 4π²L/T². Frequency is f = 1/T, and small-angle angular frequency is ω = 2πf = √(g/L).
Amplitude-corrected period: T = 4√(L/g)K(k), where k = sin(θ0/2) and K is the complete elliptic integral of the first kind. This page evaluates K using the arithmetic-geometric mean, then uses the same factor to rearrange for L or g.
- T
- Period: time for one complete back-and-forth oscillation.
- f
- Frequency: complete oscillations per second (Hz).
- L
- Effective length from pivot to bob center of mass.
- g
- Local gravitational acceleration.
- θ0
- Maximum release angle measured from vertical.
When does the small-angle approximation work?
At small release angles, sin θ is close to θ (in radians), so T0 is highly accurate. The correction is small near 5–10°, noticeable in careful measurements by roughly 15–30°, and important at large angles. The live comparison table in the results uses your current solved length and gravity, so it is more useful than a fixed rule of thumb.
The exact conservative-model period grows without bound as the release approaches 180°. That mathematical result assumes an ideal release from rest and does not make the model exact for a real apparatus.
How to measure effective pendulum length
Measure from the pivot's axis of rotation to the bob's center of mass while the pendulum hangs vertically. For a uniform spherical bob, the center is one radius below its top. Do not use string length alone unless it ends at the bob's center. A thick hook, knot, moving support, or extended bob can make the effective pivot or center uncertain.
For an experiment, time many complete cycles and divide by the cycle count. Starting and stopping the timer over 10–20 cycles reduces reaction-time error. Keep the release gentle so you do not push the bob or introduce sideways motion.
Assumptions and model boundaries
This calculator describes an ideal simple pendulum: a point-like bob on a massless, rigid, inextensible string or rod; a fixed, frictionless pivot; constant uniform gravity; planar motion; release from rest; and no air drag or damping.
The large-angle result is exact for that ideal conservative model, not for every real pendulum. A compound or physical pendulum needs its moment of inertia; an elastic string changes length; and air resistance, pivot friction, string mass, finite bob size, a moving support, or a release extremely close to 180° can require a different model. In laboratories, length uncertainty, reaction time, local gravity, non-planar motion, and falling amplitude commonly explain differences from the prediction.
Worked pendulum examples
2. Length of a seconds pendulum
A seconds pendulum takes 1 s each way, so T = 2.000 s. At g = 9.80665 m/s² and negligible amplitude, L = g(T/2π)² ≈ 0.99362 m.
3. Estimate gravity
For L = 0.750 m and measured T = 1.7357 s at negligible amplitude, g = 4π²L/T² ≈ 9.8281 m/s².
4. Compare Earth and Moon
A 1.000 m, small-angle pendulum has T ≈ 2.0064 s on Earth but T ≈ 4.9365 s on the Moon (g = 1.62 m/s²).
5. Compare small and large angles
For L = 1.000 m, g = 9.80665 m/s², and θ₀ = 60°, T₀ ≈ 2.0064 s while the exact result is ≈ 2.1532 s, a difference of about 7.32% relative to T₀.
Frequently asked questions
Does the bob's mass affect a simple pendulum's period?
No. Bob mass cancels from the ideal simple-pendulum equation, so length, gravity, and—outside the small-angle approximation—release angle determine the period.
What is the difference between period and frequency?
Period T is the time for one complete back-and-forth cycle. Frequency f is the number of complete cycles per second, measured in hertz, and f = 1/T.
What counts as one pendulum oscillation?
One oscillation is a complete cycle from one extreme to the other and back to the starting extreme. A one-way swing takes half a period.
How should pendulum length be measured?
Measure the effective length from the pivot axis to the bob's center of mass, not to the top or bottom of the bob.
Why does amplitude matter?
The familiar formula T = 2π√(L/g) assumes a small release angle. At larger angles the exact period is longer, so this calculator applies an elliptic-integral correction whenever amplitude is supplied.
Which gravity value should I use?
Use local gravitational acceleration when known. The Earth preset is standard gravity, 9.80665 m/s²; actual Earth gravity varies slightly with latitude and elevation.
Does this formula apply to a physical or compound pendulum?
Not generally. An extended rigid body's period depends on its moment of inertia and pivot-to-center-of-mass distance, so it needs the physical-pendulum formula rather than a simple length substitution.
Why does my laboratory period differ from the calculator?
Timing reaction, length measurement, air resistance, pivot friction, string mass or stretch, bob size, changing amplitude, and local gravity can all shift a measured result.
Which formulas does this calculator use?
It uses T₀ = 2π√(L/g) for the small-angle period and T = 4√(L/g)K(sin(θ₀/2)) for the amplitude-corrected period, where K is the complete elliptic integral of the first kind. Frequency is f = 1/T.
What units can I use?
Length can be entered in millimetres, centimetres, metres, inches, or feet; period in milliseconds or seconds; frequency in hertz or cycles per minute; and angle in degrees or radians. Calculations convert internally to SI units.
When is the small-angle approximation accurate?
It is highly accurate at about 10° or less. The period difference grows with release angle, so use the calculator's exact comparison when accuracy matters.
Methodology, review and sources
Author and reviewer: Starlight Tools science editorial team · Reviewed: 14 July 2026.
The calculation converts inputs to SI units, evaluates the small-angle formula, and—when amplitude is present—computes the complete elliptic integral by arithmetic-geometric mean iteration. Inverse answers algebraically rearrange the same selected model. Display rounding is applied only after the full-precision calculation.
Physics reference: OpenStax University Physics, §15.4 Pendulums. Constant reference: NIST SP 811, standard acceleration of free fall, defining standard gravity as exactly 9.80665 m/s².