Simple-pendulum period and frequency on Earth or any planet, with amplitude correction.
Small-angle approximation: T = 2π√(L/g). The corrected value adds the standard θ²/16 series term and is accurate for amplitudes up to ~30°.
The simple pendulum is the canonical mechanical oscillator: a point mass on a massless string, swinging under gravity. Its period — the time for one full back-and-forth swing — depends only on the length of the string and the local gravitational acceleration, and is famously independent of the mass of the bob. That counter-intuitive fact, first nailed down by Galileo, drove the design of pendulum clocks for three hundred years; it is still used in physics demonstrations to introduce simple harmonic motion, in geophysics to estimate local g, and in engineering to size structural pendula for skyscraper damping. The same machinery underlies metronomes, Foucault pendula in science museums, and the swing of a child's playground swing.
The standard small-angle formula T = 2π√(L/g) is taught in every introductory physics class, but it carries an implicit assumption — that the amplitude is small (under 10°). Beyond that, the period grows nonlinearly with amplitude, and the small-angle formula understates it. This calculator provides both the small-angle period and an amplitude-corrected period using the standard series expansion, so you can see the divergence. It also lets you swap the gravity preset for the Moon, Mars, or any other planet — handy for physics-class problems.
For a simple pendulum of length L (measured to the center of mass of the bob) under gravity g, the small-angle period is:
T = 2π · √(L / g)
The frequency f = 1 / T (Hz) and angular frequency ω = √(g / L) (rad/s) are direct corollaries.
For finite amplitudes θ₀, the exact period involves a complete elliptic integral of the first kind. The standard series approximation up to fourth order in θ₀ is:
T_corr = T_small · (1 + θ₀² / 16 + 11 · θ₀⁴ / 3072 + …)
with θ₀ in radians. At θ₀ = 30° (= 0.524 rad), the correction is +1.74 %; at θ₀ = 60° it is +7.3 %; at θ₀ = 90° it is +18 %. The series breaks down past ~80°.
Gravity by planet: g = 9.81 m/s² on Earth, 1.62 on the Moon, 3.71 on Mars, 3.70 on Mercury, 8.87 on Venus, 24.79 on Jupiter, 10.44 on Saturn (cloud-top).
Enter the length L of the pendulum in meters (typical 0.5–1.5 m for a clock pendulum, 67 m for the Foucault pendulum at the Panthéon). Pick a planet preset to autofill gravity, or pick "custom" and enter your own g. Enter the amplitude in degrees (typical 5° for a clock, 10°–30° for hands-on demos, smaller for precise time-keeping). The result panel shows the corrected period as the headline, plus the small-angle period (no amplitude correction), the frequency, the angular frequency, and the gravity used. The chart plots the angular displacement as a sinusoid over two full periods so you can see the swing visually.
Standard one-meter pendulum on Earth, 5° amplitude.
Foucault pendulum (L = 67 m on Earth, 3°):
Same one-meter pendulum on the Moon (g = 1.62):
Length is to center of mass, not to attachment point. For a heavy bob on a thin string, the difference is small. For a wooden ruler swinging on a pin (a "physical pendulum"), the relevant length is L_cm — the distance to the center of mass — and the formula needs the radius of gyration. Use the physical pendulum variant.
Air drag and string mass. The simple pendulum assumes no air drag, no string mass, no flex. Real pendula damp; the period increases by ~0.1 % per percent of energy loss per swing (small in practice for a heavy bob).
Temperature and length. Metal pendulum rods expand with temperature. For a precision clock, a 1 °C temperature change is a 1.2 × 10⁻⁵ relative length change, or about 0.5 s per day. Compensated pendula (Riefler, invar) reduce this.
Local g varies. At sea level g is 9.81 m/s² ± 0.05 depending on latitude (9.78 at equator, 9.83 at poles). Altitude further reduces g by ~3 × 10⁻⁶ per meter. Old pendulum clocks were calibrated to local gravity; moving them changed their rate.
Large-amplitude isochronism breakdown. The "isochronism" of the pendulum (same period regardless of amplitude) is only true for small angles. At 30°+ amplitude, the period changes measurably with amplitude — early clockmakers struggled with this, leading Huygens to invent the cycloidal cheek to enforce isochronism geometrically.
Driven pendulum. A pendulum kept swinging by an escapement is no longer a free oscillator — the period depends on the impulse mechanism. Real clock pendula keep time better than free pendula because the escapement re-locks them.
Coupled pendulums. Two pendulums of similar period mounted on the same flexible support exchange energy and drift in and out of phase (Huygens' "sympathy of clocks"). The simple model breaks for coupled systems.
Inverted pendulum. The pendulum hanging up from a pivot has imaginary period — it's unstable. Stabilizing it (Segway, rocket, robot balance) is a classic control-theory problem.
Damping. Real pendula lose energy to friction; the small-angle formula assumes none. With damping, oscillations decay exponentially; the period changes by less than 1 % until damping is severe.