Pendulum Calculator
Period from length — and how wrong the formula goes.
Work out Pendulum. Period from length — and how wrong the formula goes. Written for the problem set you are stuck on.
The angle from vertical. The standard formula assumes this is small.
9.807 on Earth, 1.625 on the Moon, 3.721 on Mars.
The standard formula assumes a small swing. Above about 15° the real period is longer than it reports.
Period of one full swing
2.0064 s
0.4984 Hz — small-angle approximation holds
Mass does not appear anywhere in the formula. A lead bob and a cork bob on the same string swing at exactly the same rate, which is the same fact as everything falling at the same rate. T = 2π√(L/g) is a small-angle approximation, and it is taught almost universally without that being said. The first correction is (1 + θ²/16): about 0.2% at 15°, 1% at 23°, and 4% at 45°. A grandfather clock swings only a few degrees for exactly this reason. Because the period depends on g, a pendulum is a gravimeter. The same clock runs measurably slower at the equator than at the poles, and slower again at altitude.
How the Pendulum Calculator works
Enter a length for the period and frequency, with gravity adjustable for other planets. The page also reports how far the small-angle approximation has drifted at your chosen swing, which almost no textbook does.
Also known as: pendulum period calculator · simple pendulum formula · t = 2pi sqrt(l/g) · pendulum length for 1 second
Frequently asked questions
How do I calculate a pendulum's period?
T = 2π√(L/g). A 1 m pendulum on Earth swings with a period of 2.006 seconds, and a 0.248 m pendulum gives exactly 1 second.
Does the mass of the bob matter?
Not at all. Mass appears nowhere in the formula, so a lead bob and a cork bob on identical strings swing at identical rates — the same fact as everything falling at the same rate.
How accurate is the small-angle formula?
It understates the period, by about 0.2% at 15°, 1% at 23° and 4% at 45°. That is why a grandfather clock swings only a few degrees, and why this page reports the error rather than hiding it.
Why does a pendulum clock run differently in different places?
Because the period depends on g, which varies with latitude and altitude. A clock keeping perfect time in London runs measurably fast at the equator, and slow again up a mountain.
Put this calculator on your own site
Free to use, on any site, commercial or not. Paste this where you want it to appear. It is a plain iframe, so it works in WordPress, Squarespace, Wix, Webflow, Ghost and anything else that accepts HTML.
<iframe src="https://www.thecalclibrary.com/embed/pendulum-calculator" width="100%" height="640" style="border:1px solid #e2e8f0;border-radius:12px" loading="lazy" title="Pendulum Calculator"></iframe>
<p style="font:13px/1.5 system-ui,sans-serif;margin:6px 0 0;color:#64748b">Powered by <a href="https://www.thecalclibrary.com/pendulum-calculator" style="color:#64748b">Pendulum Calculator</a> from The Calc Library</p>The only condition is that the credit line below the frame stays in place. That one line is what pays for the tool being free — it is how anyone else finds it.
Related calculators
Simple Harmonic Motion Calculator
Period, frequency and energy for a mass on a spring.
OpenFree Fall Calculator
Fall speed and time, with the air resistance caveat stated.
OpenGravitational Force Calculator
Newton's law of gravitation, with field strength.
Open