Pendulum Period Calculator

A simple pendulum consists of a point mass (the bob) suspended from a fixed pivot by a massless, inextensible string or rod. When displaced from its equilibrium position and released, gravity provides a restoring force that causes the bob to oscillate. For small angular displacements — typically less than about 15° from vertical — the restoring force is approximately proportional to the displacement, making the motion simple harmonic. This small-angle approximation is what allows the elegant closed-form solution for the period.

The period is given by T = 2π√(L/g), where T is the time in seconds for one complete oscillation (one full swing out and back), L is the effective length of the pendulum in meters from the pivot to the center of mass of the bob, and g is the local gravitational acceleration in m/s². From this formula, the frequency f = 1/T (oscillations per second, measured in Hertz) and the angular frequency ω = √(g/L) (radians per second) are immediately derivable. Notably, neither the mass of the bob nor the amplitude of swing (for small angles) appears in the formula — only length and gravity determine the period.

Pendulum dynamics have enormous practical relevance. Historically, pendulum clocks were the most accurate timekeepers for centuries, exploiting the isochronous (constant-period) nature of small oscillations. Today, the formula is used to measure local gravitational acceleration with high precision, to calibrate seismometers, to design Foucault pendulums that demonstrate Earth's rotation, and to study resonance phenomena in structures and mechanical systems. The simple pendulum also serves as a foundational model for understanding more complex oscillators in engineering and physics.

The formula T = 2π√(L/g) relies on the small-angle approximation (sin θ ≈ θ), which is accurate to within 1% only for amplitudes below roughly 7° and within about 4% up to 15°. For larger swings, the true period is longer and requires an elliptic integral correction: T = 2π√(L/g) × [1 + (1/16)θ₀² + ...], where θ₀ is the initial angle in radians. The model also assumes a point mass bob and a massless, rigid rod; real pendulums with distributed mass require the physical pendulum formula using the moment of inertia. Air resistance and pivot friction are neglected, both of which cause amplitude to decay over time (damped oscillations) and slightly shift the effective period. The formula further assumes a uniform, constant gravitational field — it does not account for the slight variation in g with altitude, latitude, or subsurface density anomalies, though these effects are small for most practical purposes. Finally, for pendulums with very long lengths (approaching planetary scales) or operating in non-inertial reference frames, relativistic or rotational corrections may be necessary.