Physics · Oscillations
Pendulum period calculator.
Find the period and frequency of a simple pendulum from its length, including the correction for large swings, or of a mass on a spring.
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Pendulum and spring period formulas
For small swings a simple pendulum has period T = 2π√(L ÷ g), which does not depend on mass or amplitude. OpenStax treats angles below about 15° as small, where the small-angle approximation is reasonable (sin θ and θ differ by about 1% at 15°). A 1 m pendulum on Earth has T = 2.006 s.
Large swings. The exact period is T = 2π√(L ÷ g) ÷ M(1, cos(θ0 ÷ 2)), where M is the arithmetic-geometric mean (Wikipedia). At 60° the same pendulum takes 2.153 s, 1.0732 times the small-angle value. At 10° the factor is only 1.0019.
Mass on a spring. T = 2π√(m ÷ k). A 0.5 kg mass on a 20 N/m spring has T = 0.9935 s (1.007 Hz).
These are point-mass, frictionless, massless-string and massless-spring models, so real clocks, swings and springs differ. They are for study and not for timing equipment design.
Frequently asked questions
What is the formula for the period of a pendulum?
T = 2π√(L ÷ g) for small angles. With L = 1 m and g = 9.80665 m/s² the period is 2.006 s.
Does the mass of the bob change the period?
No. Mass cancels out for a simple pendulum, as does the amplitude for small swings.
How accurate is the small-angle formula?
Within about 1% below 15°. At 60° the true period is 1.0732 times larger, which is why the large-angle option exists.
How long is a pendulum with a 2 second period?
About 0.9936 m on Earth, from L = gT² ÷ 4π².
What is the period of a mass on a spring?
T = 2π√(m ÷ k). For 0.5 kg and 20 N/m that is 0.9935 s.
Sources
Checked
- OpenStax University Physics Volume 1, 15.4 Pendulums
- Wikipedia: Pendulum (mechanics)
- OpenStax University Physics Volume 1, 15.1 Simple Harmonic Motion
- NIST CODATA: standard acceleration of gravity (9.806 65 m/s2, exact)
- NASA NSSDC Planetary Fact Sheet (surface gravity row)
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Scope
Idealised textbook models. Not for clock, instrument or structural design. Results are arithmetic from the formulas shown, for study and general interest, and not professional or engineering advice.