Calculators
Simple Pendulum Calculator
Solve pendulum period, frequency, length, or gravity.
Simple Pendulum Calculator tool
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About This Tool
A simple pendulum is an ideal model of a mass swinging from a light string or rod. For small angles, its period depends mainly on pendulum length and local gravity, not on the mass. This calculator uses T = 2π√(L/g) to find period, frequency, length, or gravity and converts common classroom and lab units automatically. It is useful for physics homework, quick experiment planning, and checking whether measured swing times are close to the ideal prediction.
How To Use It
- Choose whether to solve for period, frequency, length, or gravity.
- Enter the known positive values with their units. Length means the distance from pivot to the centre of mass.
- For length mode, choose whether your known timing value is period or frequency.
- Use the result as an ideal small-angle estimate and compare real measurements with the model assumptions.
Examples
1 metre pendulum on Earth
With L = 1 m and g = 9.80665 m/s², T = 2π√(1/9.80665) ≈ 2.006 s.
Find frequency
The same 1 m pendulum has f = 1/T ≈ 0.498 Hz, or about 29.9 swings per minute.
Find length from period
A pendulum with a 2 s period near Earth gravity has L = g(T/2π)² ≈ 0.994 m.
Estimate gravity
If a 1 m pendulum has a measured period of about 2.006 s, rearranging the formula gives g ≈ 9.81 m/s².
Useful Notes
Pendulum period formula
For small swing angles, the ideal simple-pendulum period is T = 2π√(L/g), where T is period in seconds, L is length in metres, and g is gravitational acceleration in m/s².
Frequency relationship
Frequency is the number of cycles per second, so f = 1/T. A longer pendulum has a longer period and a lower frequency when gravity stays the same.
Solving for length or gravity
Rearranging the period formula gives L = g(T/2π)² and g = 4π²L/T². These forms are useful for designing a pendulum with a target period or estimating local gravity from measurements.
Small-angle assumption
The formula is most accurate when the swing angle is small, commonly around 10 degrees or less. Larger amplitudes increase the real period slightly compared with this ideal equation.
What the model leaves out
The simple-pendulum model ignores air resistance, pivot friction, string mass, rod stiffness, large-amplitude effects, and any movement of the support point.
FAQ
Does the mass affect pendulum period?
In the ideal small-angle model, mass does not affect the period. Real systems can differ if air resistance, pivot friction, or the shape of the object matters.
What length should I enter?
Use the distance from the pivot to the centre of mass of the swinging object, not just the visible string length if the object has noticeable size.
Why is the formula only for small angles?
The standard formula uses a small-angle approximation that simplifies the motion. At larger angles, the exact motion has a slightly longer period.
Can I use this as a gravity calculator?
Yes, if you know the pendulum length and measured period. The result is an estimate and depends on careful measurement and the small-angle assumptions.