Pendulum Period Formula (T = 2π√(L/g)): Calculator & Examples

How to calculate pendulum period with T = 2π√(L/g). Solve for length or g, with worked examples and a free pendulum period calculator.

How to Calculate Pendulum Period (Quick Answer)

For a small-angle simple pendulum, the period is T = 2π √(L/g). That is the formula people write as T = 2π√(L/g) or t = 2π√(l/g). Multiply 2π by the square root of length over g. The answer is in seconds.

  • Measure L from the pivot to the center of the bob (meters).
  • Use the paper’s g (often 9.8 or 9.81 m/s²).
  • Compute T = 2π √(L/g). Frequency is f = 1/T.
  • To find length: L = g T² / (4π²). To find g: g = 4π² L / T².

    • Need a checked number now? Open the free pendulum period calculator (T = 2π√(L/g)) — then come back for lab traps and worked examples.

    What Is T = 2π√(L/g)?

    A simple pendulum is a small bob on a light inextensible string. In the intro-physics small-angle model (about 15° or less from vertical), the motion is simple harmonic. Period depends on length and gravity — not on mass, and not on how far you pull the bob.

    L is pivot-to-center length. Doubling L multiplies T by √2, not by 2. Large angles need a more advanced formula; homework usually wants the small-angle form.

    Rearrangements

  • 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 from a lab timing.

    • Solve any of these on the free pendulum calculator.

    Where the Formula Comes From

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

    Same SHM idea as a mass on a spring — compare with the simple harmonic motion calculator.

    Units

  • Length L → meters (80 cm = 0.80 m).
  • g → m/s² (9.8 or 9.81 unless the paper says otherwise).
  • T → seconds. f → Hz.

    • Worked Examples

    Example 1: T for a 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. A “2-second pendulum” is just under one meter on Earth.

    Example 3: estimate g in the lab

    Time 50 swings totaling 100.35 s → 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.

    Check Examples 1–3 on the pendulum calculator.

    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.
  • Thinking mass changes the period in the ideal model — it does not.

    • Free Pendulum Period Calculator

    Use the free pendulum period calculator to solve T = 2π√(L/g) for period, length, or g with step-by-step working.

    SHM calculator.

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

    Potential energy (PE = mgh) — related gravity homework.

    Master T = 2π√(L/g) with L in meters — then verify every homework answer on the calculator.

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