Calculate the period of a simple pendulum using T = 2π√(L/g). Includes large-angle elliptic integral correction for accurate results.
The **Pendulum Period Calculator** computes the oscillation period of a simple pendulum — the time for one complete back-and-forth swing. The classic formula T = 2π√(L/g) works well for small amplitudes, but real pendulums swung at larger angles have longer periods that require elliptic integral corrections.
This calculator provides both the small-angle approximation and the corrected period using a high-order series expansion of the complete elliptic integral of the first kind. You can immediately see how much the small-angle formula underestimates the true period at any amplitude. For a 30° swing, the error is about 1.7%, but at 60° it grows to 7.3% and at 90° it reaches 18%.
Beyond period calculation, the tool can solve for the required pendulum length to achieve a target period (useful for clock design) or the gravitational acceleration implied by observed period and length (used in gravimetry). The angle correction table with visual error bars makes it easy to determine when the simple formula is good enough and when you need the full correction.
Whether you are designing a pendulum clock, calibrating a physics experiment, or studying oscillatory motion, accurate period calculation is essential. The small-angle formula is taught in every introductory physics course, but real pendulums rarely swing at tiny angles. This calculator bridges the gap between textbook simplicity and real-world accuracy.
Clock designers need precise periods — a 1 m pendulum ticking at exactly 2 seconds was historically the basis for the meter definition. Geophysicists use pendulum period measurements to determine local gravitational acceleration, where even small period errors translate to significant gravity measurement errors.
Small-angle period: T₀ = 2π√(L/g) Large-angle correction: T = T₀ × [1 + (1/4)sin²(θ/2) + (9/64)sin⁴(θ/2) + (25/256)sin⁶(θ/2) + ...] Frequency: f = 1/T, Angular frequency: ω = 2πf Required length: L = g(T/2π)² Variables: L = pendulum length, g = gravitational acceleration, θ = amplitude angle
Result: 2.0174 s (corrected period)
A 1 m pendulum on Earth has small-angle period T₀ = 2π√(1/9.807) = 2.0064 s. At 30° amplitude, the elliptic correction factor is 1.0055, giving corrected period T = 2.0174 s — about 0.55% longer than the small-angle approximation.
The period of a simple pendulum — a point mass on a massless, inextensible string — is one of the most important formulas in classical mechanics. For small oscillations, Newton's second law applied to the tangential direction gives the familiar result T = 2π√(L/g). This formula is remarkable in its simplicity: the period depends only on the length and local gravity, not on the mass of the bob or the amplitude of the swing.
However, "small oscillation" is a crucial qualifier. The derivation assumes sin θ ≈ θ, which is the first term in the Taylor expansion. For amplitudes beyond about 15°, the nonlinearity of the sine function causes measurable deviations. The exact solution involves the complete elliptic integral of the first kind, K(k), where k = sin(θ₀/2).
The exact period is T = T₀ × (2/π) × K(sin(θ₀/2)), where K is the complete elliptic integral. Since K has no closed-form expression, we use its series expansion: T = T₀ × [1 + (1/4)k² + (9/64)k⁴ + (25/256)k⁶ + (1225/16384)k⁸ + ...]. Each additional term improves accuracy for larger angles. This calculator uses terms through k⁸, which provides excellent accuracy up to 90°.
Pendulums have been central to physics for centuries. Galileo first observed that the period is independent of amplitude (approximately) around 1602. Huygens built the first pendulum clock in 1656. Since then, pendulum-based measurements have been used to determine the shape of the Earth (which affects local g), prospect for mineral deposits underground, and calibrate precision instruments. Understanding the period formula — including its large-angle corrections — remains a cornerstone of physics education.
At larger angles, the pendulum travels a longer arc, and the restoring force is not proportional to displacement (the sin θ ≈ θ approximation breaks down). The result is a longer period, quantified by the elliptic integral correction factor.
The calculator uses a 4th-order series expansion that is accurate to better than 0.001% for angles up to 90°. For angles beyond 90° (approaching inverted), higher-order terms or numerical integration would be needed.
No. In the ideal simple pendulum model, mass cancels out completely. The period depends only on length and gravitational acceleration. This is true for both small and large angles.
A seconds pendulum has a period of exactly 2 seconds (1 second per half-swing). On Earth at sea level, this requires a length of about 0.994 m. It was historically important for timekeeping and was once proposed as the basis for the meter.
Yes! By measuring the period T and knowing the length L precisely, you can calculate g = 4π²L/T². This is the principle behind the reversible (Kater's) pendulum used in gravimetry.
Air resistance slightly increases the period compared to the ideal case and causes the amplitude to decrease over time (damping). For typical laboratory pendulums, the effect on period is less than 0.01% and can be neglected for most purposes.