How to use it
- Type the Length of the string, m, for example 1 for a metre-long pendulum.
- Keep the Gravitational acceleration, m/s² at 9.80665, or type the value for your place.
- Read the Period, s and the Frequency, Hz, then the steps of the working.
The formula of the simple pendulum
A simple pendulum is a small mass hung from a light string that swings to and fro. For a small swing its period depends only on the length of the string and on the gravitational acceleration (OpenStax, University Physics Volume 1, Section 15.4).
T = 2π × √(L ÷ g)
f = 1 ÷ T
T = period, s (one full swing there and back)
f = frequency, Hz
L = length of the string, m
g = gravitational acceleration, m/s²- The mass of the bob does not change the period. The source says the period is completely independent of mass.
- The period grows with the square root of the length, so four times the length doubles the period.
- On a stronger gravity the pendulum swings faster: at four times the gravity the period is half as long.
The value of g
The default is standard gravity, 9.80665 m/s², the conventional value that is exact by definition. The acceleration of gravity differs a little from place to place on Earth, so type a local value when the accuracy matters.
What the calculator does not include
- Large swings. The source says the period is nearly independent of amplitude below about 15°; the calculator does not correct the period for larger swings.
- Friction, air resistance, and a string that stretches or a bob that is not small.
- A physical pendulum, such as a rod or the pendulum of a clock, whose period depends on how its mass is spread out.
Worked example: a 75.000 cm pendulum
OpenStax, Example 15.3, takes a simple pendulum of length 75.000 cm, with g = 9.8281 m/s², and finds a period of 1.7357 s. Type 0.75 and 9.8281 to check it: T = 2π × √(0.75 ÷ 9.8281) = 1.7357 s, and the frequency is 1 ÷ 1.7357 = 0.5761 Hz.
| Figure | Value |
|---|---|
| Period | 1.7357 s |
| Frequency | 0.5761 Hz |
Frequently asked questions
Why does a longer pendulum swing more slowly?
A longer string makes each swing larger, and the period grows with the square root of the length. Four times the length gives twice the period.
Does the mass of the bob change the period?
No. The source says the period is completely independent of mass, so a heavy and a light bob on the same string swing together.
How large a swing should I keep to?
Keep the swing to about 15 degrees or less. The source says the period is nearly independent of the amplitude below that angle, which is the basis of the formula.
Why is the frequency the reciprocal of the period?
The period is the time of one complete swing, and the frequency counts the complete swings in one second, so the frequency is 1 divided by the period.
Is my data sent to a server?
No. The calculation runs in your browser, and nothing you type is sent to us or stored.
Sources
- OpenStax, University Physics Volume 1, Section 15.4 Pendulums (Eq. 15.19, Example 15.3)
- Wikipedia, Standard gravity (9.80665 m/s², exact by definition)
Figures and text last checked on Wednesday, November 11, 2026.