Pendulum calculator

How do I find the period of a pendulum?

T = 2π√(L ÷ g) for small swings. A 1-meter pendulum with g = 9.80665 m/s² has a period of 2.0064 s.

How long is a pendulum with a 2-second period?

L = g × (T ÷ 2π)² = 0.9936 m — about 1 meter, which is why it is sometimes called a seconds pendulum (one second each way).

Does the weight of the bob matter?

No — for an ideal simple pendulum the period depends only on the length and gravity.

Why only small swings?

The formula uses the small-angle approximation; larger swings take slightly longer.

Can I use a different gravity?

Yes — enter g for another place or planet; on the Moon (about 1.62 m/s²) a pendulum swings much more slowly.

How can I measure g with a pendulum?

Rearrange the period formula: g = 4π²L ÷ T². OpenStax University Physics (Volume 1, Example 15.3) uses a pendulum 75.000 cm long with a period of 1.7357 s and gets g = 9.8281 m/s². Time many swings and divide to get an accurate period, and keep the swing small; OpenStax suggests a maximum displacement below about 0.5° for precision.

What if the swinging object is a rod or a bat, not a weight on a string?

Then it is a physical pendulum, with period T = 2π√(I ÷ mgL), where I is the moment of inertia about the pivot and L the distance from the pivot to the center of mass (OpenStax, section 15.4, Eq. 15.21). For a point mass on a string I = mL², and the formula reduces to the one used here. The moment of inertia calculator gives I for common shapes.

How long should a pendulum clock’s pendulum be?

For a period of 2 s, one second each way, the length is L = g(T ÷ 2π)² = 0.9936 m, about 3.26 ft. Halving the period to 1 s needs a pendulum a quarter as long, 0.248 m, because the length grows with the square of the period.

The formula and what it assumes

The period of a simple pendulum is T = 2π√(L ÷ g) (OpenStax University Physics, Volume 1, section 15.4, Eq. 15.19). L is the distance from the pivot to the center of the bob, with the whole mass treated as a point on a massless string. The formula needs a small swing: OpenStax calls angles under 15° (about 0.26 rad) small, where sin θ and θ differ by less than 1%. Within that range the period does not depend on the mass or on the size of the swing. The length goes in meters; the calculator also shows swings per second and per minute.

Worked example: a 3 ft pendulum

Convert the length: 3 ft × 0.3048 = 0.9144 m. Then T = 2π√(0.9144 ÷ 9.80665) = 1.919 s, about 31.3 full swings per minute. With the Moon’s gravity, 1.62 m/s², the same pendulum has T = 4.72 s; enter the new g in the gravity box. Typing 3 as if it were meters would give 3.48 s, nearly double the true value.

Common mistakes

The length is in meters, so a pendulum measured in feet or centimeters must be converted first. Measure L to the center of the bob, not to its top or to the end of the string. The period is one complete swing, out and back to the starting point; counting each one-way pass as a swing halves it. And a wide swing takes slightly longer than the formula says, so for a clock or a lab measurement keep the angle well under 15°.

Sources: OpenStax University Physics, Volume 1, section 15.4 (pendulums, Eq. 15.19 and 15.21; Example 15.3); standard gravity 9.80665 m/s² is exact (NIST SP 811, Appendix B.8) and the Moon’s 1.62 m/s² is from the NASA Planetary Fact Sheet. Related: moment of inertia calculator, Hooke’s law calculator, potential energy calculator and weight on other planets calculator.