Simple Pendulum Calculator

A simple pendulum is an idealized point mass suspended from a fixed pivot by a massless, inextensible line. When displaced and released, gravity provides a restoring torque that draws the bob toward its lowest position. For sufficiently small angles, the motion is approximately simple harmonic and repeats with a period determined by pendulum length and local gravitational acceleration. This model is a classic way to study oscillation, timing, and gravity. The small-angle period is two pi times the square root of length divided by gravity. Frequency is the reciprocal of period, while angular frequency is the square root of gravity divided by length. Length is measured from the pivot to the bob's center of mass, not merely the visible string length. A longer pendulum moves more slowly, and stronger gravity makes it oscillate faster. Bob mass does not appear because gravitational force and inertia scale equally with mass in the ideal equation. This calculator accepts length in meters and gravity in meters per second squared. The Earth preset uses standard gravity of 9.80665 m/s², while the Moon preset uses 1.62 m/s². Local Earth gravity varies slightly with latitude, elevation, and geology, so precision experiments should use a measured or location-specific value. Entering the same length under lower lunar gravity produces a longer period because the restoring acceleration is weaker. The approximation is most accurate at small release angles, commonly below about ten degrees. At larger amplitudes, the exact nonlinear period is longer and depends on release angle. Air drag, friction at the pivot, string elasticity, bob size, and motion outside a single plane also cause real behavior to differ. A physical pendulum with distributed mass requires its moment of inertia and pivot distance rather than this point-mass equation. Pendulums have historically regulated clocks and measured gravity. They also provide useful demonstrations of energy exchange: gravitational potential energy is greatest at an endpoint, while kinetic energy is greatest at the bottom. Although period does not depend on mass in the simple model, damping can depend on bob shape and size. For a practical measurement, time many complete oscillations and divide by the cycle count to reduce reaction-time error. Keep the swing small, measure length to the center of mass, and avoid pushing the bob when releasing it. Use the result as an ideal baseline for classroom problems, timing estimates, and comparisons between gravitational environments. When high accuracy or large angles matter, apply a finite-amplitude correction and model losses explicitly.