Simple Pendulum Period Calculator
Find the period of a simple pendulum from its length and local gravity, or solve for length or gravity given a target period. A live 3D swinging bob and charts show how T = 2π√(L/g) plays out.
Reviewed by the ToolNestr Editorial Team — July 2026
Two ideas that trip students up
1. The bob swings through an arc
The same pendulum shown at its two extreme swing positions, tracing the arc between them. It speeds up toward the bottom and slows to a stop at each extreme — mass never enters the period.
2. Longer string, longer period
Three pendulums of different length hanging side by side. Only the length (and gravity) changes the period — a longer string always swings slower back and forth.
Pendulum period graphs
How it works
The core idea in one line: a simple pendulum's period depends only on its length and local gravity — never on its mass or (for small swings) how hard it was pushed.
T = 2π√(L/g)
period — small-angle approximation (≲15°)
f = 1/T
frequency, swings per second (Hz)
ω = 2π/T = √(g/L)
angular frequency (rad/s)
Rearranged, the formula solves any of the three quantities: L = g(T/2π)² gives the length needed for a target period, and g = L(2π/T)² recovers the local gravitational acceleration from a measured period — the classic method used to measure g with nothing but a string, a bob and a stopwatch.
Worked example 1 — a 1.0 m pendulum on Earth
Given: A simple pendulum has length L = 1.0 m, swinging near Earth's surface (g = 9.81 m/s²). Find its period.
Worked example 2 — designing a grandfather clock pendulum
Given: A clockmaker wants a pendulum with an exact period of T = 2.0 s on Earth (g = 9.81 m/s²). Find the required length.
This is close to the classic "seconds pendulum" length of about 0.994 m used in real pendulum clocks with a 2-second tick-tock period.
Pendulum period by length (at Earth gravity, g = 9.81 m/s²)
Because T ∝ √L, quadrupling the length only doubles the period.
| Length L | Period T | Frequency f |
|---|---|---|
| 0.25 m | 1.003 s | 0.997 Hz |
| 0.50 m | 1.419 s | 0.705 Hz |
| 1.00 m | 2.006 s | 0.499 Hz |
| 2.00 m | 2.837 s | 0.352 Hz |
T = 2π√(L/g), small-angle approximation. Doubling L multiplies T by √2 ≈ 1.414, not by 2.
Where the pendulum period actually matters
🕰️ Pendulum clocks
From the 17th century onward, pendulum clocks used the near-constant period of a small swing to keep time to within seconds a day — the "seconds pendulum," about 0.994 m long, was even briefly proposed as a natural length standard.
🌍 Seismometers
Early mechanical seismometers used a heavy pendulum whose inertia kept it nearly stationary while the ground (and the instrument frame) moved beneath it, recording the relative motion as an earthquake trace.
🌀 Foucault pendulum
A very long, freely swinging pendulum appears to slowly rotate its swing plane over the course of a day — direct visible proof that the Earth is rotating beneath it, independent of the pendulum's period formula itself.
🛝 Playground swings
A swing is a large-amplitude pendulum, so the small-angle formula is only a rough guide — but the everyday intuition still holds: a swing with a longer chain takes noticeably longer to go back and forth than a short one.
Common misconceptions
"A heavier pendulum bob swings slower."
False — mass cancels out of the period formula entirely. T = 2π√(L/g) has no mass term, because both the inertia resisting motion and the gravitational restoring force scale with mass in exactly the same way.
"The formula T = 2π√(L/g) works for any swing angle."
False — it is the small-angle approximation, accurate to within about 0.5% only for amplitudes up to roughly 15°. For larger swings, the true period is longer than the formula predicts, and the difference grows with amplitude.
"A harder push makes the pendulum swing faster (shorter period)."
False for small angles — the starting push only changes the amplitude of the swing, not how long each swing takes. Only length and gravitational acceleration set the period in the small-angle regime.
"Pendulum period depends on where you start measuring from."
False — the period is the same whether you time from the highest point, the lowest point, or anywhere else in the cycle, as long as you measure one full back-and-forth swing.
Formula sources & further reading
The formulas here are standard, traceable to:
- • OpenStax, University Physics Volume 1 — Chapter 15, "Oscillations," pendulum section (free, peer-reviewed). openstax.org
- • Halliday, Resnick & Walker, Fundamentals of Physics — Chapter 15, Oscillations, simple and physical pendulums.
- • Serway & Jewett, Physics for Scientists and Engineers — Chapter 15, Oscillatory Motion.
T = 2π√(L/g) is the small-angle (≲15°) approximation for a simple pendulum. Results are rounded for display.
How to use this calculator
Choose the unknown
Solve for Period, Length, or Gravity depending on what you know.
Enter the two known values
Fill in the two remaining fields; the third solves live, along with frequency and ω.
Watch the swing
Use the L and g sliders to see the 3D bob swing faster or slower as the period changes.
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Frequently asked questions
What is a simple pendulum?
A simple pendulum is an idealised model: a point mass (the bob) hanging from a massless, inextensible string of length L, swinging under gravity with no friction or air resistance. Real pendulums only approximate this, but the model predicts real behaviour remarkably well for small swings.
Does the period depend on the mass of the bob?
No. Mass cancels out completely in the derivation — a heavier bob has more inertia but also more gravitational force pulling it, and the two effects exactly offset. A bowling ball and a marble on strings of the same length swing with the same period.
Does the period depend on the swing amplitude?
Only weakly, and only for larger swings. The formula T = 2π√(L/g) is the small-angle approximation, accurate to within about 0.5% for amplitudes under 15°. Beyond that, the true period grows slightly longer than the formula predicts as amplitude increases — this is why old pendulum clocks used small, controlled swings.
Why does length matter more than mass for the period?
Length sets the geometry of the swing — a longer string means the bob travels a longer arc but also experiences a smaller restoring effect from gravity per unit displacement, and the balance of these works out to T ∝ √L. Mass, by contrast, drops out of Newton's second law entirely (F = ma and the restoring force are both proportional to mass), so it never enters the period at all.
How is angular frequency related to the period?
Angular frequency ω = 2π/T = √(g/L), measured in rad/s. It describes how fast the phase of the oscillation advances, and it is the same ω that appears in the pendulum's position equation θ(t) = θ₀cos(ωt) for small-angle swings.