Pendulum Frequency Calculator — f = (1/2π)√(g/L)
Enter the pendulum length and gravitational acceleration to find its frequency in hertz — how many complete back-and-forth swings occur per second — along with the period and angular frequency.
m
m/s²
Complete oscillations per second (small-angle approximation)
- 1
g ÷ L
9.81 ÷ 0.25 = 39.24 s⁻² - 2
Angular frequency ω = √(g/L)
√(39.24) = 6.2642 rad/s - 3
Frequency f = ω ÷ (2π)
6.2642 ÷ 6.283185 = 0.9970
How does this calculator work?
Pendulum frequency f = (1/2π) × √(g/L) in hertz. A 1-metre pendulum on Earth (g ≈ 9.81 m/s²) swings at ≈0.498 Hz with a period of ≈2.006 s. Frequency grows with stronger gravity and shorter length; the bob mass has no effect. Valid only for small swing angles (< 15°).
Formula
How this is calculated
A simple pendulum (ideal point mass on a massless string) undergoes simple harmonic motion for small swing angles (less than ~15°). Its angular frequency is ω = √(g / L) radians per second, where g is the gravitational acceleration and L is the string length. The ordinary frequency in hertz (cycles per second) is f = ω / (2π) = (1 / 2π) × √(g / L), and the period — the time for one full swing — is T = 1 / f.
Frequency is inversely related to the square root of length: doubling the length reduces the frequency by a factor of √2 ≈ 1.414, and halving the length raises the frequency by √2. This is why a grandfather clock has a long pendulum for a slow, one-second beat, while a short pendulum swings rapidly.
This formula is derived from the small-angle approximation sin(θ) ≈ θ. For swing amplitudes beyond about 15° the true frequency is slightly lower than this formula predicts. Air resistance, string mass, and pivot friction are not modelled.
Frequently asked questions
f = (1 / 2π) × √(9.81 / 1) ≈ 0.498 Hz, so it completes roughly half a swing per second, or one full oscillation every ≈2.006 seconds.
Frequency is inversely proportional to the square root of length (f ∝ 1/√L). Making the pendulum four times longer halves its frequency; making it four times shorter doubles the frequency.
No — the mass cancels out in the derivation. Two pendulums of the same length but different bob masses swing at exactly the same frequency (for the same gravity and small angles).
Also known as
TG we-Calculate Editorial Team. (2026). Pendulum Frequency Calculator — f = (1/2π)√(g/L) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/pendulum-frequency-calculator
TG we-Calculate Editorial Team. "Pendulum Frequency Calculator — f = (1/2π)√(g/L)." TG we-Calculate. 2026. https://we-calculate.com/calculator/pendulum-frequency-calculator.
TG we-Calculate Editorial Team, "Pendulum Frequency Calculator — f = (1/2π)√(g/L)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/pendulum-frequency-calculator
@misc{wecalculate_pendulum_frequency_calculator, title = {Pendulum Frequency Calculator — f = (1/2π)√(g/L)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/pendulum-frequency-calculator}}, year = {2026}, note = {TG we-Calculate} }
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