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Small-Angle Pendulum Period & Frequency

Calculate period, frequency, length or gravity for an ideal simple pendulum at small amplitude.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

At standard gravity, a pendulum 9.80665 m long has an ideal small-angle period of 2π seconds, approximately 6.28319 s, and frequency about 0.159155 Hz.

Period T = 2π√(L/g); frequency f = 1/T

Worked example

Enter these known values and leave the other values blank.

Gravitational acceleration
1 g
Pivot-to-bob-center length
9.80665 m
Full oscillation period
6.283 sec
Oscillation frequency
0.15915 Hz

Assumptions and limitations

  • Gravity, length, period and frequency must be positive. One standard g is exactly 9.80665 m/s²; actual local gravity can differ.
  • This is the small-amplitude approximation for a point-like bob on a massless, inextensible support. It omits damping, support motion and finite-bob rotational inertia.
  • Period means a complete back-and-forth cycle. At larger angular amplitudes the true period is longer, and amplitude is not an input in this model.

Common questions

Does a heavier bob change the ideal period?

No. Bob mass cancels from the ideal simple-pendulum model. Geometry and gravity determine its small-angle period.

How does length affect the period?

At fixed gravity, quadrupling the length doubles the period and halves the frequency.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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