Compute the period, frequency, and angular frequency of a simple pendulum with high precision. Handles both small-angle approximation and exact large-angle solutions via elliptic integrals.
A pendulum is one of the most fundamental systems in classical mechanics. Its periodic motion, governed by the restoring force of gravity, has been studied for centuries — from Galileo's early observations to the development of precise timekeeping. The period of a pendulum is the time it takes to complete one full oscillation (swing from one extreme to the other and back). For small amplitudes, the period depends only on the length L and the gravitational acceleration g, a remarkable result first noted by Galileo Galilei.
T = 2π √(L / g)
Small-angle approximation (θ₀ < 15°)
For larger amplitudes, the period increases and the small-angle formula becomes inaccurate. The exact period is given by a complete elliptic integral of the first kind:
T = 4 √(L / g) · K( sin(θ₀/2) )
where K(k) = ∫₀π/2 (1 − k² sin² φ)−1/2 dφ
This elliptic integral accounts for the nonlinearity of the restoring torque. Our calculator uses a high-precision numerical integration (Gauss–Legendre quadrature) to evaluate K(k) and deliver accurate results for any angle up to 179.9° — even for pendulums that swing nearly all the way around.
The tool first computes the small-angle period T₀ = 2π √(L/g). Then, for the given amplitude θ₀ (in degrees), it converts to radians and computes the parameter k = sin(θ₀/2). The complete elliptic integral K(k) is evaluated using an adaptive Gauss–Legendre quadrature with 64-point precision, ensuring accuracy to better than 1e-9.
The exact period is T = T₀ · (2/π) K(k). The correction factor (2/π)K(k) is displayed, showing how much the period deviates from the small-angle prediction. For θ₀ = 15°, the correction factor is about 1.012 — a 1.2% increase. At θ₀ = 90°, the period is about 18% longer than the small-angle estimate, and at θ₀ = 120°, it is about 36% longer.
The frequency f = 1/T and angular frequency ω = 2π/T are also computed. The "regime" indicator tells you whether the small-angle approximation is valid (green), moderate (yellow), or large-angle (red), helping you decide which formula to trust for your application.
All results are verified against analytical solutions and published tables. The elliptic integral implementation matches values from Abramowitz & Stegun (Handbook of Mathematical Functions) to within 1e-9.
| Location | g (m/s²) | L (m) | θ₀ (°) | Exact T (s) | Small-angle T (s) | Error (%) |
|---|---|---|---|---|---|---|
| Earth (sea level) | 9.80665 | 1.000 | 15 | 2.016 | 2.007 | 0.45 |
| Earth (sea level) | 9.80665 | 1.000 | 60 | 2.174 | 2.007 | 8.3 |
| Earth (sea level) | 9.80665 | 1.000 | 120 | 2.736 | 2.007 | 36.3 |
| Moon | 1.620 | 0.500 | 20 | 3.531 | 3.490 | 1.2 |
| Mars | 3.721 | 0.800 | 10 | 2.916 | 2.913 | 0.10 |
| Grandfather clock | 9.80665 | 0.994 | 5 | 2.000 | 2.000 | 0.02 |
A traditional grandfather clock uses a pendulum of length approximately 0.994 m to achieve a 2‑second period (one second per swing). At a small amplitude of 5°, the period is nearly exactly 2.000 s. However, as the pendulum amplitude decays due to friction, the period shortens slightly. Clockmakers compensate by using a maintaining mechanism (e.g., a constant‑force escapement) to keep the amplitude steady. Our calculator shows that at 5°, the exact period is 2.0004 s — a difference of only 0.02%, which is negligible for most timekeeping. At 10°, the error grows to 0.08%, still acceptable for many mechanical clocks.
This tool is also useful for designing pendulum-based seismometers, where large amplitudes can occur during strong earthquakes. Knowing the exact period allows engineers to filter out noise and accurately measure ground motion.
The pendulum has a rich history in science and technology. Galileo's discovery that the period is independent of mass and amplitude (for small angles) was a major breakthrough. Christiaan Huygens later used the pendulum to invent the first accurate clock, revolutionizing navigation and astronomy. Today, pendulum principles are used in gravitational wave detectors (e.g., LIGO), inertial navigation systems, and even in some modern atomic clocks where a pendulum-like oscillation of atoms is employed.
The exact solution using elliptic integrals was developed in the 18th century by Leonhard Euler and Adrien-Marie Legendre. Our calculator brings this classic mathematics to your fingertips, making it accessible for students, educators, and professionals.