How to use the pendulum period calculator
- Choose whether to find the period from the length or the length from a period.
- Pick a length unit and enter the length from the pivot to the center of the weight, or the period you want in seconds.
- Optionally enter a swing angle to see the exact period for a wide swing.
- Open More options to change gravity if you are not at the usual surface value.
Formula
period = 2 x pi x sqrt(length / gravity)
length = gravity x period squared / (4 x pi squared)
exact period = small-angle period / AGM(1, cos(angle / 2))
The first two lines are the small-swing formulas. The exact line uses the arithmetic-geometric mean, which gives the full period for any starting angle below 180 degrees. Gravity is 9.80665 meters per second squared unless you change it.
A 1 meter pendulum
For a 1 meter pendulum, the period is 2 x pi x sqrt(1 / 9.80665) = 2.0064 seconds, so it swings back and forth about 30 times a minute. A 1 foot pendulum has a period of 1.1077 seconds.
A wider swing takes a little longer. Released from 60 degrees, the same 1 meter pendulum takes 2.1532 seconds, about 7.3 percent more. To find the length that gives a 2 second period, use length = 9.80665 x 4 / 39.478 = 0.9936 meter, and a 1 second period needs 0.2484 meter.
What the formula assumes
The simple formula treats the weight as a point on a string with no mass, no friction and no air drag, and it holds for small swings. For swings up to about 10 or 15 degrees the error is small. For wider swings, fill in the angle, and the page uses the exact method.
Notice that the mass of the bob does not appear. Two pendulums of the same length have the same period whatever they weigh. What matters is the length from the pivot to the center of the bob, so measure that carefully.
Gravity and units
Gravity changes a little with place and height, and a different planet or moon needs its own value. You can change it in More options. Related physics tools include the Kinematics Calculator and the Projectile Motion Calculator, and you can convert lengths with the Unit Converter.
Assumptions and limits
- The pendulum is a point mass on a weightless string, with no friction or drag.
- Gravity is 9.80665 meters per second squared unless you change it.
- The exact wide-swing period uses the arithmetic-geometric mean and applies up to 170 degrees.
- Nothing is uploaded. The arithmetic runs in your browser.
Frequently asked questions
How long is the period of a 1 meter pendulum?
About 2.006 seconds for small swings. That is 2 x pi x the square root of 1 divided by 9.80665, and it means roughly 30 swings a minute.
Does the weight of the bob change the period?
No. In the formula the mass cancels out, so only the length and gravity matter. Measure from the pivot to the center of the bob.
How long should a pendulum be for a 2 second period?
About 0.9936 meter, or 39.1 inches. Choose Length from period, enter 2 seconds and the page gives the length in the unit you pick.
Why does a wide swing take longer?
The small-swing formula is an approximation. As the angle grows, the restoring force grows more slowly than the angle, so the swing slows and the period gets longer. Enter the angle to see the exact value.