How to Calculate Pendulum Period (T = 2π√(L/g))

Find a simple pendulum’s period with T = 2π√(L/g), or solve for length and g. Worked examples and a free pendulum calculator.

What a Simple Pendulum Asks

A simple pendulum is a small bob on a light inextensible string that swings in a vertical plane. Intro physics uses the small-angle model: the restoring force is proportional to displacement, so the motion is simple harmonic.

The period does not depend on mass or on how far you pull the bob, as long as the angle stays small (about 15° or less from vertical). It depends on length and g.

If you already have L and g and only need a checked period, open the free simple pendulum calculator first, then return here for rearrangements and lab traps.

The Formula

  • T = 2π √(L/g) — period in seconds.
  • f = 1/T — frequency in hertz.
  • L = g T² / (4π²) — length in meters.
  • g = 4π² L / T² — local gravity in m/s².

    • L is measured from the pivot to the center of the bob, not to the knot or the bottom of a large sphere.

    Where It Comes From

    For small θ in radians, sinθ ≈ θ. The tangential restoring force is −mg sinθ ≈ −mg (s/L), where s is arc length. Then a = −(g/L) s, which is SHM with ω² = g/L, so T = 2π/ω = 2π √(L/g).

    That is the same SHM period idea as a mass on a spring, just with a different ω. Compare with the simple harmonic motion calculator.

    Units

  • Length L → meters.
  • g → m/s² (often 9.8 or 9.81).
  • T → seconds. f → Hz. ω → rad/s.

    • Worked Examples

    Example 1: 1.0 m pendulum

    L = 1.0 m, g = 9.8 m/s². T = 2π √(1/9.8) ≈ 2.007 s. f ≈ 0.498 Hz.

    Example 2: length for T = 2.0 s

    L = g T² / (4π²) = 9.8 × 4 / (4π²) = 9.8/π² ≈ 0.993 m.

    Example 3: estimate g in the lab

    Time 50 swings totaling 100.35 s, so T = 2.007 s, with L = 1.00 m. g = 4π² L / T² ≈ 9.80 m/s². Time many cycles; divide total time by the count.

    Common Mistakes

  • Leaving L in centimeters while g is in m/s².
  • Assuming T doubles when L doubles (T scales with √L).
  • Using the small-angle formula for a very wide swing without saying so.
  • Measuring string length only, and skipping the radius of a large bob.

    • Check It on the Calculator

    Simple pendulum calculator.

    Simple harmonic motion calculator.

    Hooke's law calculator (spring SHM uses T = 2π√(m/k)).

    Learn T = 2π√(L/g) and check every homework number.

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