Simple Pendulum Period Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/9/2026
A simple pendulum's period is calculated with T = 2π × √(L ÷ g), where L is its length and g the acceleration of gravity. A 1 m pendulum on Earth has a period of about 2.006 seconds, close to the classic "seconds pendulum" benchmark.
Explanation
A simple pendulum is an idealized physical model: a point mass suspended from a massless, inextensible string, oscillating without friction. For small oscillation amplitudes (generally under 15-20°), its period depends on only two parameters: its length and the local acceleration of gravity — remarkably, it depends on neither the suspended mass nor the oscillation amplitude (as long as it stays small), a property that made the pendulum historically valuable for timekeeping. Doubling the pendulum's length doesn't double its period: since the length sits under a square root, the period is only multiplied by √2 (about 1.41). This is why a pendulum about 1 m long has a period close to 2 seconds on Earth, a well-known physics benchmark called the "seconds pendulum" (one one-way swing then lasts about 1 second). On another celestial body, with different gravity, a pendulum of the same length would swing at a different rate: more slowly on the Moon (weaker gravity, longer period), faster on a planet with stronger gravity. This calculator is distinct from our natural frequency of an oscillator calculator, which covers a resonant circuit rather than a pendulum: the two systems share the general principle of harmonic oscillation, but their period formulas depend on completely different physical parameters (length and gravity here, versus inductance and capacitance there) — a mechanical spring-mass system, meanwhile, follows yet another formula depending on stiffness and mass, the stiffness itself computed via our Hooke's law calculator for the spring's restoring force.
Example: a 1 m pendulum on Earth
Inputs
Length: 1 m. Gravity: 9.81 m/s² (Earth).
Calculation
T = 2π × √(1 ÷ 9.81) = 2π × 0.31934 ≈ 2.0061 s.
Result
This pendulum oscillates with a period of about 2 seconds.
Frequently asked questions
Why does the formula only work for small oscillations?
The formula T = 2π√(L/g) relies on a mathematical approximation (sin(θ) ≈ θ, valid in radians for a small angle θ) that simplifies the pendulum's equation of motion. For larger amplitudes, this approximation becomes less and less accurate, and the actual period lengthens slightly compared to what this simplified formula predicts; an exact calculation for large amplitudes requires more advanced mathematical tools (elliptic integrals).
Why doesn't the pendulum's mass factor into the calculation?
This is one of the simple pendulum's most remarkable properties: the force of gravity accelerating the mass and the inertia resisting that acceleration are both proportional to the mass, which makes the mass cancel out mathematically in the equation of motion. A heavy pendulum and a light pendulum of the same length therefore oscillate at exactly the same period, in this idealized frictionless model.
Does a real pendulum behave exactly as this formula predicts?
Approximately, but a real pendulum departs from the ideal model in several ways: the string has nonzero mass, air friction gradually reduces the oscillation amplitude, and the suspended mass is never a perfect point. These deviations generally stay small for a well-designed, small-amplitude pendulum, which is why this formula long served as the basis for pendulum clocks.