Pendulum Calculator

What Is a Pendulum and How Is Its Period Calculated?

A simple pendulum is a mass (the bob) hanging from a fixed pivot on a light string or rod, swinging back and forth under gravity. Its period is the time for one complete swing out and back. For small angles the period depends only on the pendulum's length and the local gravitational acceleration — not on the bob's mass.

The small-angle pendulum formula, derived from Galileo Galilei's observations around 1602 and formalised by Christiaan Huygens in his 1673 work Horologium Oscillatorium, is: T = 2π √(L ÷ g), where T is the period in seconds, L is the length from the pivot to the bob's centre of mass in metres, and g is gravitational acceleration (9.80665 m/s² at Earth's surface). Rearranged, L = g (T ÷ 2π)² and g = 4π²L ÷ T².

A pendulum with a period of exactly 2 seconds — one second per swing, known as a "seconds pendulum" — is about 0.9936 metres (39.1 inches) long at standard gravity. This length was proposed as a basis for the metre in the 17th century before the metre was defined from the Earth's meridian in 1791.

How to Use This Pendulum Calculator

Choose what you want to solve for, then fill in the remaining fields:

• Find Period: Enter the length and gravity to get the period, frequency, angular frequency, and ticks per minute.
• Find Length: Enter the period you want — for example 2 s for a seconds pendulum — to get the length required.
• Find Gravity: Enter the measured length and period from a physics experiment to estimate local g.
• Amplitude: The release angle from vertical. Set 0° for the textbook formula, or enter a larger angle for the exact result.
• Location: Choose Earth, the Moon, Mars, Venus, Jupiter, or enter a custom gravity value.

Lengths can be entered in metres or feet; gravity is always in m/s². Standard gravity varies from about 9.78 m/s² at the equator to 9.83 m/s² at the poles, so a pendulum clock tuned in one country can run slightly fast or slow in another.

How Does Amplitude Affect a Pendulum's Period?

The formula T = 2π√(L/g) is only exact as the swing angle approaches zero. Most textbooks treat angles up to about 15° as "small". At larger angles the period grows; this calculator applies the exact correction T = T₀ ÷ AGM(1, cos(θ₀/2)), using the arithmetic-geometric mean, rather than a truncated series.

Because the period barely changes for small swings, Huygens' pendulum clock of 1656 improved timekeeping accuracy from roughly 15 minutes per day to about 15 seconds per day, according to the Royal Museums Greenwich.

Pendulum Period on Different Planets

Because period scales with 1/√g, the same pendulum swings more slowly where gravity is weaker. A 1 metre pendulum has these small-angle periods:

Frequently Asked Questions About the Pendulum Calculator

What is the formula for the period of a pendulum?

The period of a simple pendulum at small angles is T = 2π√(L ÷ g), where L is the length in metres and g is gravitational acceleration, 9.80665 m/s² on Earth. A 1 metre pendulum therefore has a period of about 2.007 seconds.

Does the mass of the bob affect the period?

No. In the simple pendulum model the bob's mass cancels out, because gravity's restoring force and the bob's inertia both scale with mass. Only length, gravity, and (slightly, at large angles) amplitude change the period.

How long is a pendulum with a 1 second period?

A pendulum with a 1 second period is about 0.248 metres (9.78 inches) long at standard gravity. A 2 second period, the classic seconds pendulum used in grandfather clocks, requires about 0.994 metres (39.1 inches).

How accurate is this pendulum calculator?

The calculator is exact for an ideal simple pendulum — a point mass on a massless, inextensible string with no friction — including large amplitudes. Real pendulums have a bob of finite size, string mass, and air drag, so measure length to the bob's centre of mass and expect small deviations.