Pendulum Period Calculator
Swing period, frequency, and beats per minute from length.
What it does
Pendulum period grows with the square root of string length: a meter swings about two seconds round trip while a quarter meter swings about one. Clocks exploit this stability, counting beats instead of watching hands.
The tool above takes the string length in meters or feet. It returns seconds per swing with hertz and beats per minute beside them.
The default meter reads 2.01 s (0.5 Hz, 29.9 bpm). A quarter meter reads 1 s (1 Hz, 59.8 bpm): the classic seconds pendulum ticks each way per second.
The formula follows OpenStax physics pendulum derivations with NIST standard gravity behind g. Both restrict it to small angles; wide swings need elliptic corrections.
How to use
- Pick meters or feet for the string.
- Measure pivot to bob center.
- Read period first for swing seconds.
- Use bpm for metronome settings.
- Keep amplitude small for accuracy.
Formula
- L
- Pivot to bob center length.
- g
- 9.80665 m/s² standard gravity.
- 2π
- Full cycle radians per swing.
Feet convert to meters first (x 0.3048), then run metric physics, so systems agree. Air drag and string mass sit outside the ideal line.
Worked example
Example: 1 meter
Metric default: 1 m.
- Ratio: 1 / 9.80665 = 0.10197.
- Root: 0.31933.
- Period: 2π x 0.31933 = 2.0064, shown 2.01 s.
- Frequency: 1 / 2.0064 = 0.5 Hz. BPM 29.9.
- Tool shows 2.01 s, 0.5 Hz, 29.9 bpm.
Reference table
| Length | Period | BPM |
|---|---|---|
| 0.25 m | 1 s | 59.8 |
| 0.5 m | 1.42 s | 42.3 |
| 1 m | 2.01 s | 29.9 |
| 2 m | 2.84 s | 21.1 |
Limits
- Small angles only; wide arcs run slow.
- Ideal string with point bob assumed.
- Air drag ignored.
- Variable gravity across planets differs.
Common mistakes
- Timing half swings as full periods.
- Measuring to bob top instead of center.
- Using degrees in the root (none needed).
- Expecting metronome exactness from string.
- Forgetting amplitude effects on long arcs.
Try it yourself
Quick checks. Answers sit tucked under each question.
0.25 m?
2π root(0.0255) = 1 s, 59.8 bpm.
Double 1 m to 2 m?
Period x root 2: 2.01 to 2.84 s.
BPM of 2 s period?
60 / 2 = 30 bpm.
Questions
How long is a seconds pendulum?
About 0.994 m for a 2 s period (1 s each way).
Why square root?
Restoring acceleration scales with displacement over length; roots follow.
Does weight matter?
No. Mass cancels out of the ideal period.
Wide swings?
Period stretches past small angle math; keep under ~15°.
Metronome use?
Set beats from the bpm output for practice tempos.
Moon pendulums?
Slower: swap g for 1.62 and periods stretch 2.5x.