Simple pendulum period and frequency
For a simple pendulum at a small angle, period depends on effective length and local gravitational acceleration, not bob mass. Frequency is the reciprocal of period.
Measure effective length from the pivot to the bob’s center of mass. Large amplitude, distributed mass, string elasticity, drag, and pivot friction require a more detailed model.
How to use the pendulum calculator
- Choose the unknown: Select period, frequency, effective length, or local gravity.
- Enter the known values: Use a positive effective length, period, and gravitational acceleration where requested.
- Use a small amplitude: The ideal formula uses the small-angle approximation.
- Calculate: Review the default or updated result and the model assumptions.
Formula and variables
Use seconds, meters, and meters per second squared in the base equations.
T = 2π√(L/g); f = 1/T; L = g(T/2π)²; g = 4π²L/T²- T — Period
- Time for one complete oscillation (s)
- f — Frequency
- Oscillations per second (Hz)
- L — Effective length
- Pivot-to-center-of-mass distance (m)
- g — Local gravity
- Local gravitational acceleration (m/s²)
One-meter pendulum
A pendulum has an effective length of 1 m where g = 9.80665 m/s².
- Length
- 1 m
- Gravity
- 9.80665 m/s²
- T = 2π√(1/9.80665)
- f = 1/T
Result: Period is about 2.0064 s and frequency is about 0.4984 Hz.
One complete back-and-forth oscillation takes about two seconds.
Understanding your results
How length changes period
Period grows with the square root of effective length.
- A fourfold length increase doubles period.
- Greater local gravity shortens period.
- Bob mass cancels from the ideal model.
- Large amplitude makes the actual period longer than this estimate.
Assumptions
- The bob behaves as a point mass on a massless, inextensible support.
- The oscillation angle is small.
- Gravity is uniform and damping and pivot friction are negligible.
Limitations
- Does not model physical, compound, torsional, conical, driven, or damped pendulums.
- Does not include large-angle corrections.
- Experimental gravity estimates still require uncertainty analysis.
Common mistakes
- Measuring only string length instead of pivot-to-center-of-mass length.
- Timing half an oscillation as a full period.
- Using a large release angle.
- Confusing period with frequency.
Practical use cases
Physics coursework
Solve ideal small-angle pendulum relationships.
Gravity experiments
Estimate local gravity from positive measured length and period.
Frequently asked questions
Does bob mass affect pendulum period?
Not in the ideal small-angle simple-pendulum model.
Where is pendulum length measured?
From the pivot to the bob’s center of mass.
How are frequency and period related?
Frequency in hertz is the reciprocal of period in seconds.
Sources and review
- The Simple Pendulum — OpenStax College Physics. Accessed 2026-07-14.
Reviewed 2026-07-14.