Pendulum Period Calculator — T = 2π√(L/g)
Enter the pendulum length, gravitational acceleration, and swing angle to get the period — with a large-angle correction beyond the standard small-angle formula T₀ = 2π√(L/g).
m
m/s²
°
Time for one full oscillation, corrected for swing angle
- 1
√(L ÷ g)
√(1 ÷ 9.81) = 0.319275 s - 2
Small-angle period T₀ = 2π × √(L/g)
6.283185 × 0.319275 = 2.0061 s - 3
Angle correction factor (1 + θ²/16 + 11θ⁴/3072)
1.000476θ = 0.08727 rad; factor > 1 for non-zero swing angles. - 4
Period T = T₀ × correction
2.0061 × 1.000476 = 2.0070
How does this calculator work?
Pendulum period T = 2π√(L/g) for small angles; multiply by (1 + θ²/16 + …) for larger angles (θ in radians). A 1-metre Earth pendulum (small angle) has T ≈ 2.006 s. Period depends on length and gravity only — not the bob mass.
Formula
How this is calculated
The classic small-angle period T₀ = 2π√(L/g) applies when the swing angle is small enough that sin(θ) ≈ θ (generally under ~15°). For larger angles, the actual period is longer. The exact solution requires an elliptic integral, but the power-series approximation T ≈ T₀ × (1 + θ²/16 + 11θ⁴/3072 + …) in radians is accurate to well under 0.1% for angles up to about 45°, and is used here.
The correction percentage shows how much longer the true period is compared to the small-angle value. At 15° the correction is about 0.46%; at 45° it grows to about 7.3%, which is significant for precision timekeeping. Grandfather-clock designers accounted for this by adjusting the effective pendulum length.
The gravity field lets you explore other worlds: on the Moon (g ≈ 1.62 m/s²) a 1-metre pendulum swings once every ≈4.94 s, compared to ≈2.006 s on Earth. The model assumes an ideal rigid point mass on a massless inextensible string with no air resistance or pivot friction.
Frequently asked questions
On Earth (g ≈ 9.81 m/s²): T ≈ 2.006 s. On the Moon (g ≈ 1.62 m/s²): T ≈ 4.94 s. The period scales as 1/√g, so the Moon's lower gravity makes the pendulum swing about 2.46× slower.
Under about 5° the correction is less than 0.05% — negligible for most purposes. At 15° it reaches ≈0.46%; at 30° ≈1.9%; at 45° ≈7.3%. For precision clocks even small errors matter; for school experiments under 15° the simple T₀ formula is typically fine.
Rearranging: L = g(T/2π)². For T = 2 s and g = 9.81 m/s²: L ≈ 0.9937 m, the classic "seconds pendulum". Most grandfather clocks use a pendulum close to this length.
Also known as
TG we-Calculate Editorial Team. (2026). Pendulum Period Calculator — T = 2π√(L/g) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/pendulum-period-calculator
TG we-Calculate Editorial Team. "Pendulum Period Calculator — T = 2π√(L/g)." TG we-Calculate. 2026. https://we-calculate.com/calculator/pendulum-period-calculator.
TG we-Calculate Editorial Team, "Pendulum Period Calculator — T = 2π√(L/g)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/pendulum-period-calculator
@misc{wecalculate_pendulum_period_calculator, title = {Pendulum Period Calculator — T = 2π√(L/g)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/pendulum-period-calculator}}, year = {2026}, note = {TG we-Calculate} }
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