Pendulum Period Calculator

Background.

This pendulum period calculator computes T = 2π√(L/g) for a simple pendulum — a point mass swinging on a massless string or rod — and reports frequency, angular frequency, and something most pendulum calculators leave out entirely: an honest estimate of how wrong that famous formula is at your chosen swing amplitude. Enter a length of 2 meters at standard gravity and the calculator returns a period of 2.8375 seconds, a frequency of 0.3524 Hz, and an angular frequency of 2.2143 rad/s. That much any pendulum calculator can do. What this one adds is the caveat every introductory physics course states and every online tool quietly omits: T = 2π√(L/g) is a small-angle approximation, exact only in the limit as the swing amplitude θ₀ approaches zero.

The approximation comes from replacing sin(θ) with θ in the pendulum's equation of motion, a substitution that is excellent for small angles and progressively worse as the swing widens. At a 10° amplitude — a fairly typical clock or classroom swing — the true period is about 0.19% longer than the small-angle formula predicts. That sounds negligible, and for most purposes it is. But the error does not grow linearly; it grows roughly with the square of the amplitude. Double the swing to 20° and the error nearly quadruples to about 0.77%. Push to 45°, a swing you'd see in a playground or a wrecking-ball demonstration, and the true period is running nearly 4% longer than the textbook formula claims. This calculator computes that error directly from the amplitude you enter, using the next terms in the exact series expansion of the pendulum period, so you can see exactly when the small-angle shortcut is safe to trust and when it starts to matter.

The formula's most celebrated property is that the period depends on length and gravity alone — not on mass, and, to the small-angle approximation, not on amplitude either. Galileo is credited with first noticing this near-independence from amplitude while watching a swinging lamp in the Pisa cathedral, timing it against his own pulse. That near-independence (not exact, as this calculator's error output demonstrates) is what made pendulum clocks the most accurate timekeepers in the world for nearly three centuries, from Christiaan Huygens' 1656 pendulum clock design until quartz and atomic clocks took over in the twentieth century. A clockmaker exploits the fact that a slightly shorter or longer swing barely changes the period — which is also exactly why escapement mechanisms are engineered to keep the amplitude small and consistent, since any residual amplitude-dependence directly becomes a timekeeping error.

Gravitational acceleration g defaults to 9.80665 m/s², the standard gravity value fixed by international agreement at the 3rd General Conference on Weights and Measures in 1901 — not a measurement, but an exact reference constant. Because real gravity varies slightly with latitude, altitude, and local geology (from about 9.78 m/s² near the equator to about 9.83 m/s² near the poles), the calculator lets you override g to match your actual location, or to explore other worlds entirely: a pendulum that takes 2.84 seconds to swing on Earth takes about 4.93 seconds on the Moon, where g is only 1.625 m/s². The length and gravity you enter, plus the amplitude you're actually swinging through, together tell you not just the textbook period but how much you should trust it.

What is pendulum period calculator?

A simple pendulum is an idealized mechanical system: a point mass (the bob) suspended from a fixed pivot by a massless, inextensible string or rod, free to swing under gravity in a single vertical plane. When displaced from its resting (vertical) position and released, the restoring component of gravity pulls it back, and it oscillates back and forth. For small swing angles, this motion is very close to simple harmonic motion, and the time for one complete back-and-forth swing — the period T — is given by T = 2π√(L/g), where L is the pendulum's length and g is the local gravitational acceleration.

The formula is a small-angle approximation. It comes from the pendulum's exact equation of motion, d²θ/dt² = −(g/L)sin(θ), which has no closed elementary-function solution in general. Substituting sin(θ) ≈ θ — valid to within about 1% for angles under roughly 15°, per OpenStax's University Physics treatment — linearizes the equation into the same differential equation that describes a mass on an ideal spring, and its solution is the familiar sinusoidal small-angle period. The exact period, expressible as a complete elliptic integral of the first kind, is always somewhat longer than this small-angle value, and by how much depends only on the swing amplitude θ₀, not on L or g individually.

This calculator returns the small-angle period, frequency (f = 1/T), and angular frequency (ω = 2π/T = √(g/L)), plus an estimate of the amplitude-dependent error so you know when the approximation is trustworthy and when it isn't.

How to use this calculator.

1. 01Enter the pendulum length L — the distance from the pivot to the bob's center of mass — in meters.
2. 02Enter the local gravitational acceleration g in m/s², or keep the default standard value of 9.80665 m/s².
3. 03Enter the swing amplitude θ₀ in degrees — the maximum angle the pendulum reaches from vertical on each side.
4. 04Read the period T, the primary result, alongside frequency and angular frequency.
5. 05Check the small-angle approximation error. Below about 10° it is under 0.2% and safe to ignore for most purposes; above 20-30° it grows quickly and the small-angle period should be treated as an estimate, not an exact answer.

The formula.

The period of a simple pendulum in the small-angle limit is T = 2π√(L/g). L is the length from pivot to bob in meters, g is the local gravitational acceleration in m/s², and T comes out in seconds. Frequency is simply the reciprocal, f = 1/T, and angular frequency is ω = 2π/T, which simplifies algebraically to ω = √(g/L) — the same angular frequency formula as a mass on a spring with an 'effective spring constant' of g/L.

The formula assumes the small-angle approximation sin(θ) ≈ θ, which is only exact as the swing amplitude θ₀ approaches zero. The true period is always slightly longer, and the standard way to quantify how much longer is a series expansion in powers of θ₀ (in radians): T_exact ≈ T₀ × [1 + (1/16)θ₀² + (11/3072)θ₀⁴ + ...], where T₀ = 2π√(L/g) is the small-angle period this calculator returns as its headline result. Subtracting 1 and multiplying by 100 gives the percentage error this calculator reports as smallAngleErrorPercent. At θ₀ = 10°, that error works out to about 0.19%; at θ₀ = 20° it's about 0.77%; at θ₀ = 45° it has grown to nearly 4%. Because the leading correction term scales with θ₀², doubling the amplitude roughly quadruples the error — which is why the error stays negligible for small clock-like swings but becomes impossible to ignore for wide swings.

The formula also reveals what does not affect the period: mass. A heavier bob and a lighter bob of the same length, released from the same angle, swing with the identical period, because the mass cancels out of the pendulum's equation of motion just as it does in free fall — both are consequences of gravitational and inertial mass being equivalent. What does matter is length (period grows with the square root of L, so quadrupling the length only doubles the period) and gravitational acceleration (period shrinks with the square root of g, so a pendulum swings faster on a planet with stronger surface gravity).

A worked example.

Frequently asked questions.

References& sources.

1. [1]Bureau International des Poids et Mesures (BIPM). 3rd CGPM (1901), Resolution 2 — definition of standard gravity, g_n = 980.665 cm/s² = 9.80665 m/s² exactly.
2. [2]National Institute of Standards and Technology (NIST). CODATA Value: standard acceleration of gravity, 9.806 65 m/s² (exact, by definition).
3. [3]OpenStax. University Physics Volume 1, Section 15.4 'Pendulums' — derivation of T = 2π√(L/g) and the small-angle approximation's accuracy limits.
4. [4]Dourmashkin, P. (MIT 8.01 Classical Mechanics). 'Appendix 24A: Higher-Order Corrections to the Period for Larger Amplitudes of a Simple Pendulum,' hosted on Physics LibreTexts — the amplitude-correction series used for the error estimate.