Answer first
What this calculator tells you
Calculate the period and frequency of a pendulum from its length. Find the swing time of a pendulum, a swing or a clock weight. Formula: Period = 2π × √(L ÷ g), for small swings; frequency = 1 ÷ period. At the worked-example inputs, the period (s) is 2. Holding every other input steady, moving pendulum length (m) from 0.8 to 1.2 moves the result from 1.8 to 2.2.
Transparent method
The formula
Find the swing time of a pendulum, a swing or a clock weight.
Worked example
Example inputs
How to interpret the result
A pendulum's swing time depends on its length and on gravity, and not on the weight of the bob. A pendulum a little under 1 meter long takes about 2 seconds for a full back and forth, which is why pendulums have long been used to keep time. Quadruple the length and the period doubles. The rule holds for small swings; a wide swing takes a little longer than the formula says.
At the worked-example inputs the period (s) is 2. It rises with pendulum length (m) and falls as gravity (m/s²) increases.
These are textbook formulas for ideal conditions: no air resistance, no friction, gravity of 9.81 meters per second squared and gases that behave ideally. Real results differ, so treat them as first estimates and use the units the formulas expect (meters, kilograms, seconds).
Before you rely on it
What to check
Measure the length from the pivot to the center of the bob, not to the bottom of it. A small error in length shifts the period.
The common error
Where people go wrong with pendulum period calculator
Thinking a heavier bob swings slower. Mass cancels out of the formula, so only length and gravity set the period.
Sensitivity evidence
How pendulum length (m) changes the period (s)
Holding every other input at the worked-example value, moving pendulum length (m) from 0.8 to 1.2 moves the period (s) from 1.8 to 2.2: a spread of 0.403, or 20% of the worked-example result.
| Pendulum length (m) | Period (s) | Frequency (Hz) |
|---|---|---|
| 0.8 | 1.8 | 0.557 |
| 0.9 | 1.9 | 0.525 |
| 1worked example | 2 | 0.498 |
| 1.1 | 2.1 | 0.475 |
| 1.2 | 2.2 | 0.455 |
Every input, tested
Which input moves the period (s) most
Of the 2 inputs, pendulum length (m) moves the period (s) most (0.201 across the range tested) and gravity (m/s²) moves it least (0.2).
| Input | Tested from | To | Period (s) at each end | Swing |
|---|---|---|---|---|
| Pendulum length (m) | 0.9 | 1.1 | 1.9 to 2.1 | 0.201 (10%) |
| Gravity (m/s²) | 9 | 11 | 2.1 to 1.9 | 0.2 (10.0%) |
Two variables at once
Period (s) by pendulum length (m) and gravity (m/s²)
Across the grid the period (s) runs from 1.6 to 2.4. Moving pendulum length (m) from 0.8 to 1.2 shifts it by 0.399 at the middle column, and moving gravity (m/s²) from 8 to 12 shifts it by 0.408 at the middle row, so gravity (m/s²) is the bigger lever here.
| Pendulum length (m) \ Gravity (m/s²) | 8 | 10 | 12 |
|---|---|---|---|
| 0.8 | 2 | 1.8 | 1.6 |
| 0.9 | 2.1 | 1.9 | 1.7 |
| 1 | 2.2 | 2 | 1.8 |
| 1.1 | 2.3 | 2.1 | 1.9 |
| 1.2 | 2.4 | 2.2 | 2 |
The highlighted cell is the worked example.
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Pendulum length (m) | 1 | Enter the pendulum length (m) used in this calculation. |
| Gravity (m/s²) | 9.8 | Enter the gravity (m/s²) used in this calculation. |
| Period (s) | 2 | |
| Frequency (Hz) | 0.498 | |
Inputs, definitions and assumptions
Pendulum length (m)
Enter the pendulum length (m) used in this calculation. The prefilled worked-example value is 1.
Gravity (m/s²)
Enter the gravity (m/s²) used in this calculation. The prefilled worked-example value is 9.8.
How to use this calculator
- 1Verify the inputs. Gather pendulum length (m) and gravity (m/s²) from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the period (s) at 2. Store your own version of it as Scenario A.
- 3Test one change. Start with pendulum length (m), the input with the biggest effect here: moving pendulum length (m) from 0.9 to 1.1 takes the period (s) from 1.9 to 2.1, a swing of 10% of the worked-example figure.
- 4Check the extremes. At half the example pendulum length (m) (0.5) the period (s) is 1.4; at double (2) it is 2.8.
People also ask
Frequently asked questions
How do you calculate pendulum period?
Period = 2π × √(L ÷ g), for small swings; frequency = 1 ÷ period. At the worked-example inputs the period (s) is 2.
What does the pendulum period result mean?
Find the swing time of a pendulum, a swing or a clock weight. At the worked-example inputs the period (s) is 2. It rises with pendulum length (m) and falls as gravity (m/s²) increases.
How much does pendulum length (m) change the period (s)?
Holding every other input at the worked-example value, moving pendulum length (m) from 0.8 to 1.2 moves the period (s) from 1.8 to 2.2, a spread of 0.403.
What are the limits of this pendulum period calculator?
These are textbook formulas for ideal conditions: no air resistance, no friction, gravity of 9.81 meters per second squared and gases that behave ideally. Real results differ, so treat them as first estimates and use the units the formulas expect (meters, kilograms, seconds). The tables on this page test pendulum length (m) only from 0.8 to 1.2; a value outside that range is not tabulated here.
Which input moves the period (s) most in the pendulum period calculator?
Ranked by how far each moves the period (s) across the range tested: pendulum length (m) (0.201, 10%) and gravity (m/s²) (0.2, 10.0%).
If I double pendulum length (m) in the pendulum period calculator, does the period (s) double?
Doubling it from 1 to 2 takes the period (s) from 2 to 2.8, which is 1.41 times the worked-example figure. So it grows, but by less than double. Halving it to 0.5 gives 1.4.
How much does gravity (m/s²) matter in the pendulum period calculator?
The worked example uses 9.8. With the other inputs left at the worked example, moving gravity (m/s²) from 9 to 11 takes the period (s) from 2.1 to 1.9, a swing of 10.0% of the worked-example figure.
Which inputs change the frequency (hz) in the pendulum period calculator?
At the worked-example inputs it is 0.498. Pendulum length (m) takes it from 0.525 to 0.475 and gravity (m/s²) takes it from 0.477 to 0.528.
Which units do the formulas expect?
Meters, kilograms and seconds, which give newtons, joules and watts. A speed in miles per hour or a mass in pounds needs converting first, or the result will be wrong by a fixed factor.
Sources and evidence
Free Calculators Online is independent and is not affiliated with or endorsed by the source organizations. Educational estimates only.