How the Simple Pendulum Period Formula Works
A simple pendulum is an idealized model: a point mass (the "bob") swinging from a fixed pivot on a massless, inextensible string or rod, with no friction at the pivot and no air resistance. For swings that stay within a small angle from vertical, the bob's motion is simple harmonic, and its period — the time for one complete back-and-forth swing — depends only on the pendulum's length and the local gravitational acceleration: T = 2π√(L/g), where L is the length from the pivot to the bob's center of mass (in meters) and g is the gravitational acceleration (in m/s²). This calculator also reports the frequency (f = 1/T), the angular frequency (ω = √(g/L)), and a corrected period that accounts for a larger swing angle.
Deriving the period from Newton's second law
For a bob displaced by angle θ, gravity supplies a restoring torque proportional to sin θ. Newton's second law for rotation gives the equation of motion θ'' + (g/L) sin θ = 0. For small angles, sin θ ≈ θ (in radians), which reduces this to the classic simple-harmonic-motion equation θ'' + (g/L)θ = 0, whose angular frequency is ω = √(g/L) and whose period is T = 2π/ω = 2π√(L/g). Notice that the bob's mass never appears — it cancels out of the torque and moment-of-inertia terms, which is why a heavier bob and a lighter one of the same length keep the same beat.
Working with units
- Length and gravitational acceleration must use the same unit system before you take the square root — this calculator converts the length you enter (m, cm, mm, ft, or in) to meters internally so it stays consistent with g in m/s².
- Exact length conversions used: 1 ft = 0.3048 m, 1 in = 0.0254 m, 1 cm = 0.01 m, 1 mm = 0.001 m.
- Standard gravity is 9.80665 m/s² by international convention; use a locally measured value (typically 9.78-9.83 m/s²) for precision timing or lab work.
Large-angle corrections and limits
The small-angle approximation sin θ ≈ θ is what makes T = 2π√(L/g) simple, but it is only an approximation. For larger initial swing angles θ₀, the true period lengthens according to the series T ≈ T₀[1 + θ₀²/16 + 11θ₀⁴/3072 + 173θ₀⁶/737280 + ...] (θ₀ in radians), which this calculator uses for the "corrected period" result. This model still assumes an ideal point mass, a rigid massless support, no air drag, and no damping — a real pendulum's amplitude decays over time, and a physical (extended) pendulum needs its moment of inertia and center of mass, not just a point-mass length, for exact results.
Real-world uses of the simple pendulum
- Pendulum clocks use a fixed length and a small, near-constant swing angle to keep a stable time reference.
- Physics labs use timed swings of a known length to measure the local value of g.
- Metronomes apply the same length-period relationship to set a musical tempo.
- Seismometers and the Foucault pendulum use long, slow-swinging pendulums to detect ground motion and demonstrate Earth's rotation.