Why does mass not affect the period?

A heavier bob has more gravitational force pulling it but also more inertia resisting acceleration. These effects cancel exactly, just as all objects fall at the same rate in a vacuum regardless of mass.

What are the limitations of the simple pendulum formula?

The formula is accurate only for small angles (less than about 15 degrees). At larger amplitudes, the period increases slightly. For a 90-degree swing, the period is about 18% longer than predicted.

How were pendulums used to measure gravity?

By measuring the period of a known-length pendulum, scientists could calculate local g. Variations in g across the Earth's surface revealed information about underground geological formations and the shape of the Earth.

Can a pendulum swing forever?

No. Air resistance and friction at the pivot gradually convert kinetic energy to heat, causing the amplitude to decrease. Pendulum clocks use an escapement mechanism to add small energy pulses each swing to overcome these losses.

Pendulum Calculator

Calculate period of a simple pendulum

Pendulum Period

T = 2π × √(L/g)

Length (m)

Formula

T = 2π × √(L/g)

What Is a Pendulum?

A simple pendulum consists of a mass (bob) attached to a string or rod that swings freely under gravity. When displaced from its equilibrium position and released, the pendulum oscillates back and forth in a periodic motion. For small angles (< 15°), this motion is approximately simple harmonic.

Pendulums have been used for timekeeping since Galileo's experiments in the 1580s. Galileo observed that the period of a pendulum depends only on its length and the local gravitational acceleration — not on the mass of the bob or the amplitude of swing (for small angles). This isochronous property made pendulums the basis of accurate clocks for centuries.

How to Use the Pendulum Calculator

Enter the length of the pendulum (L) in meters — measured from the pivot to the center of mass of the bob.
Enter the gravitational acceleration (g) — 9.81 m/s² on Earth's surface (use 1.62 on the Moon, 3.72 on Mars).
Click Calculate to get the period (T), frequency (f), and angular frequency (ω).
Adjust length to tune the pendulum to a desired frequency — doubling the length multiplies the period by √2.

Formula & Explanation

Period:     T = 2π √(L/g)
Frequency:  f = 1/T = (1/2π) √(g/L)
Angular:    ω = 2πf = √(g/L)

T  = period (seconds)
L  = pendulum length (m)
g  = gravitational acceleration (m/s²)
f  = frequency (Hz)
ω  = angular frequency (rad/s)

This formula is valid only for small angles (θ < 15°). For large amplitudes, the period increases and requires elliptic integrals for exact calculation.

Worked Examples

Grandfather Clock Pendulum

A traditional grandfather clock uses a 1-meter pendulum on Earth (g = 9.81 m/s²). T = 2π √(1/9.81) = 2π × 0.319 = 2.006 s ≈ 2 seconds. Each half-swing (tick-tock) takes exactly 1 second — ideal for a seconds-pendulum clock.

Pendulum on the Moon

The same 1 m pendulum on the Moon (g = 1.62 m/s²). T = 2π √(1/1.62) = 2π × 0.786 = 4.94 s. The period is 2.47× longer on the Moon because gravity is weaker. Moon clocks would run slow compared to Earth clocks.

Designing a 1 Hz Pendulum

To get f = 1 Hz (T = 1 s) on Earth: L = g/4π² = 9.81/39.48 = 0.248 m ≈ 24.8 cm. A pendulum of just 24.8 cm swings once per second — useful for metronomes and physics demonstrations.

Frequently Asked Questions

Does mass affect pendulum period?▾

No. The period T = 2π√(L/g) has no mass term. A heavy and a light bob on pendulums of equal length swing in perfect synchrony. This was one of Galileo's key observations and was later explained by the equivalence of inertial and gravitational mass.

What is the seconds pendulum?▾

A seconds pendulum has a period of exactly 2 seconds (1 second per half-swing). On Earth it is approximately 99.4 cm long. It was historically used to define the meter — the original definition was that 1 meter = half the length of a seconds pendulum.

Why do pendulum clocks need to be leveled?▾

A pendulum clock must be perfectly vertical so the pendulum swings symmetrically. Even a slight tilt causes the bob to swing in an arc that is not a true vertical plane, altering the effective length and making the clock run fast or slow.

What is a Foucault pendulum?▾

A Foucault pendulum is a large pendulum free to rotate in any direction. As it swings, the Earth rotates beneath it, causing the apparent plane of swing to rotate slowly. It was the first direct visual demonstration of Earth's rotation, demonstrated by Léon Foucault in 1851.

How does a pendulum store energy?▾

At the top of each swing, the pendulum has maximum potential energy and zero kinetic energy. At the bottom, potential energy is minimum and kinetic energy is maximum. The total mechanical energy is conserved (ignoring air resistance and friction at the pivot).