Simple Pendulum Calculator

Formula

Small-angle period = 2π√(pendulum length ÷ gravitational acceleration).

Worked example

With Pivot-to-bob length: 1 m, Local gravitational acceleration: 9.81 m/s², small-angle period is 2.006067 s.

Inputs and calculation

Measure the length from the pivot to the bob's center of mass. Divide by local gravitational acceleration, take the square root and multiply by two pi. The displayed relationship is Small-angle period = 2π√(pendulum length ÷ gravitational acceleration). Keep the fields' displayed units attached when reproducing this relationship by hand.

Worked numerical example

For example, enter pivot-to-bob length 1 m, local gravitational acceleration 9.81 m/s². Those inputs produce small-angle period of 2.0061 s. The same values also give oscillation frequency 0.4985 Hz.

Holding the other example values fixed, change pivot-to-bob length from 1 to 1.1 m. The main output becomes 2.104 s. This comparison isolates the effect of that one input.

How to read and use the result

The period is seconds per full back-and-forth oscillation; frequency is its reciprocal in cycles per second. Increasing length slows the pendulum by a square-root relationship. Explore oscillation timing in a basic mechanics demonstration.

Practical context

A one-meter pendulum at 9.81 meters per second squared takes about two seconds for a complete small-angle cycle. Timing ten cycles and dividing by ten reduces the effect of reaction-time error compared with timing one. A pendulum twice as long does not take twice as long; the square-root law gives about 1.41 times the period.

Assumptions and scope

The bob is treated as a point mass on a light, rigid string with a small swing angle and little friction. Large amplitudes have a longer period than this approximation predicts. Replace the example defaults with measured inputs for your situation and account separately for factors outside these fields.

Checks and common mistakes

Do not measure only the visible string if the bob's center adds length. Timing a half-swing and calling it a full period halves the measured value incorrectly.

Common questions

What should I verify before using the Simple Pendulum Calculator result?

Air resistance gradually reduces amplitude, while very large initial angles make the small-angle period formula less accurate even with careful timing. In a pendulum clock, the period is determined primarily by effective length and gravity in this small-angle model, not by bob mass. A heavier bob can reduce relative air-resistance effects, but mass does not enter the ideal formula.