Which formulas does this calculator use?

Small-angle approximation: T0 = 2π√(L/g). Exact large-angle period: T = 4√(L/g) K(k) where k = sin(θ0/2) and K is the complete elliptic integral of the first kind (computed via AGM). Frequency f = 1/T.

What units should I use?

SI units: length L in meters (m), gravity g in m/s², amplitude θ0 in degrees. Results are seconds (s) and hertz (Hz).

When is the small-angle approximation valid?

It’s very good up to about 10°; at 20° it’s ~1% low, and at 60° it’s ~7% low. Use the large-angle mode for accuracy at bigger amplitudes.

Pendulum Calculator — Period & Frequency

Enter length, gravity, and amplitude to get the period and frequency. Choose small-angle (fast) or exact large-angle mode. All computation is private and client-side.

Inputs

Length L (m) Pivot to bob center of mass in meters.

Gravity g (m/s²) Earth: 9.80665. Moon: 1.62. Mars: 3.71.

Amplitude mode Small-angle ignores θ₀; large-angle uses exact elliptic integral.

Amplitude θ₀ (°) Peak deflection from vertical (0–179°). Used in large-angle mode.

Precision (digits)

Tip: Press Ctrl/Cmd + Enter to calculate. The URL updates so you can bookmark or share your inputs.

Results

Pendulum 101: Formulas, Physics & Accuracy

A pendulum is one of the most recognizable systems in physics: a mass (the bob) swinging back and forth under the influence of gravity. Despite its apparent simplicity, it provides a beautiful link between geometry, dynamics, and mathematical analysis. Understanding pendulum motion is valuable not only for homework problems but also for timekeeping, seismology, and even amusement rides.

The core question is: how long does a swing take? For a pendulum of length L under constant gravitational acceleration g, the small-angle approximation gives the classic formula: T₀ = 2π√(L/g). This assumes the swing amplitude is small (θ₀ ≲ 10°), which linearizes the restoring force. Under these conditions, the motion is almost perfectly sinusoidal and the period is independent of amplitude—a property known as isochronism that allowed pendulums to become the basis of accurate clocks.

At larger amplitudes, however, the small-angle model underestimates the true period. The exact formula involves the complete elliptic integral of the first kind: T = 4√(L/g)·K(k), with k = sin(θ₀/2). This integral has no simple closed-form expression but can be evaluated rapidly using the arithmetic-geometric mean (AGM) method. Our calculator implements this approach, ensuring accurate results up to very large amplitudes (approaching 179°).

Units: Length in meters, gravity in m/s², amplitude in degrees. Output: period in seconds (s), frequency in hertz (Hz).
Accuracy: At 10° the small-angle formula is essentially perfect. At 20° it is ~1% low, at 60° ~7% low, and near 120° the error rises to ~18%.
Scaling: The period grows with the square root of length (√L) and increases with amplitude. Doubling length increases the period by about 41%.

This distinction between the approximation and the exact result is an excellent teaching point. Students see how linearization works, why physicists love approximations, and where the limits of those approximations lie. It also highlights the role of special functions (elliptic integrals) in applied physics.

Worked Example

Suppose L = 1.0 m, g = 9.80665 m/s², and amplitude θ₀ = 60°. The small-angle estimate gives T₀ ≈ 2.006 s. The exact large-angle formula yields T ≈ 2.146 s, which is about 6.9% longer. The frequency is f ≈ 0.466 Hz, meaning the pendulum completes less than one full swing per second. This example shows how amplitude matters once you leave the small-angle regime.