Pendulum

Period of a simple pendulum

About this calculator

The Pendulum Calculator finds the period of a simple pendulum from its length, gravity, and swing amplitude. Beyond the small-angle formula T₀ = 2π·√(L/g), it applies the exact large-amplitude correction T = T₀ / AGM(1, cos(θ₀/2)) using the arithmetic–geometric mean, and reports the frequency, angular frequency, and the amplitude factor T/T₀. Length and gravity must be positive and the amplitude below 180°.

How to use the Pendulum calculator

  1. Enter the quantities you know, using the units shown on each field.
  2. Read the computed results — they update instantly as you type.
  3. Use the visual explanation to see how the quantities relate.
  4. Copy the page URL to share the exact calculation.

Common examples

  • L = 1 m, g = 9.80665 m/s², small swing → T ≈ 2.006409 s (the classic one-metre 'seconds-ish' pendulum)
  • Same pendulum at θ₀ = 30° → T ≈ 2.041338 s, amplitude factor 1.017409
  • L = 0.25 m at 5° → T ≈ 1.003682 s: quartering the length halves the period
  • L = 1 m on the Moon (g = 1.62 m/s²) → T ≈ 4.936537 s
  • Swinging to 90° → T ≈ 2.368246 s, about 18% longer than the small-angle value

Frequently asked questions

How does the Pendulum calculator work?

Enter the known quantities and the calculator applies the governing physical law or scientific formula to find the unknowns, stating the equation it used and the units of every result. Computations run at high precision.

When would I use the Pendulum calculator?

It suits physics and chemistry homework, lab work, engineering estimates, and any time you want to explore how changing one quantity affects the others in a well-defined scientific relationship.

How accurate are the results?

The formula itself is evaluated at 30 significant digits of precision. Keep in mind that idealized models — such as neglecting friction, air resistance or measurement error — bound the real-world accuracy more than the arithmetic does.

Can I share a specific calculation with someone else?

Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.