Pendulum Calculator
This pendulum calculator finds the period and frequency of a simple pendulum from its length, using T = 2π √(L / g). Enter the length and the pendulum calculator returns how long each swing takes and how many swings per second. It is the quick way to work a pendulum period problem.
Results
The Pendulum Calculator is part of our Motion & Force collection. Browse more motion & force to size, convert, and check your work alongside this tool.
What the pendulum calculator shows
A simple pendulum swings back and forth at a rate set almost entirely by its length. This pendulum calculator finds the period, the time for one full swing, and the frequency, the number of swings per second. What surprises people is that the mass of the bob and the size of the swing do not matter for small angles, so the pendulum calculator needs only the length and gravity.
The pendulum period calculator formula
The pendulum period calculator uses the simple pendulum formula:
L is the length and g is gravity, and the frequency is 1 divided by the period. Because the length sits under a square root, quadrupling the length only doubles the period. The mass and the swing size drop out for small angles, which is why this formula is so simple.
Pendulum calculator example
A 1 m pendulum on Earth. The pendulum calculator gives:
The frequency is about 0.50 swings per second. A 1 m pendulum is close to a one second half swing, which is why grandfather clocks use roughly that length.
How to use the pendulum calculator
Enter the pendulum length in metres and the gravity value, then the pendulum calculator returns the period in seconds and the frequency in hertz. Use 9.81 m/s² for Earth. The length is measured from the pivot to the centre of the bob. As a pendulum length calculator you can also work backward: a target period sets the length you need.
Reading the pendulum calculator result
The period is the time for one complete back and forth swing, and the frequency is its inverse. Only the length and gravity change these; a heavier bob or a wider swing does not, as long as the swing stays small, under about 15 degrees. For larger swings the real period is a little longer than the formula gives. Use the pendulum calculator to tune a clock or to find the length for a target swing rate.
When to use the pendulum calculator
- Finding the swing period with the pendulum calculator
- Tuning a clock to a one second beat
- Working a pendulum period calculator problem in physics
- Sizing a length for a target frequency
Common pendulum calculator mistakes
- Expecting mass to matter. The period depends on length and gravity, not the bob mass.
- Measuring to the wrong point. Use the length from the pivot to the centre of the bob.
- Using a wide swing. The formula is for small angles; large swings run a little slow.
Tips for pendulum problems
- Measure the length to the centre of the bob.
- Keep the swing small for the formula to hold.
- Quadruple the length to double the period.
The pendulum period calculator and length
The pendulum period calculator rests on a neat result: for small swings, the period depends only on the length and gravity, not on how heavy the bob is or how far it swings. That is why a pendulum keeps such steady time and why it ran the best clocks for centuries. Because the length is under a square root, the relationship is not linear: to double the period you must quadruple the length, so a 2 second clock pendulum is about 1 m long while a 1 second one is only about 0.25 m. Working backward, a pendulum length calculator finds the length for a target period, which is exactly how a clockmaker sets the beat.
Using the pendulum calculator with angle
The simple formula assumes a small swing, so a pendulum calculator with angle matters when the swing is wide. As the amplitude grows past about 15 degrees, the real period stretches slightly beyond T = 2π√(L/g), because the restoring pull is no longer perfectly proportional to the displacement. For most clocks and lab pendulums the swing is kept small on purpose, so the simple formula is accurate. Gravity is the other factor: the same pendulum swings slower on the Moon, where g is lower, so the gravitational force calculator helps explain why the period changes with the body you are on.
Pendulum length calculator reference
Period and frequency by length on Earth, from the pendulum length calculator:
| Length | Period | Frequency |
|---|---|---|
| 0.25 m | 1.00 s | 1.00 Hz |
| 0.5 m | 1.42 s | 0.70 Hz |
| 1 m | 2.01 s | 0.50 Hz |
| 2 m | 2.84 s | 0.35 Hz |
Pendulum Calculator FAQ
What is the pendulum period formula?
The period formula is T = 2 pi root(L / g), length over gravity under a square root. The pendulum calculator applies it and gives the frequency as its inverse.
Does the mass affect the period?
No. For a simple pendulum at small angles the period depends only on the length and gravity, not the mass of the bob or the size of the swing.
What length gives a one second period?
About 0.25 m gives a 1 second period on Earth. A 1 m pendulum has a period near 2 seconds, close to a one second half swing.
Does the swing angle matter?
For small swings under about 15 degrees, no. For larger swings the real period is slightly longer than the pendulum calculator formula gives.