Pendulum Frequency Calculator

Created by Davide Borchia
Reviewed by Łucja Zaborowska, MD, PhD candidate
Last updated: Oct 21, 2022

Our frequency of a pendulum calculator is a handy tool with one purpose: to teach you how to find the frequency of oscillation of a pendulum in the easiest way possible.

If you already know the equation for the frequency of a pendulum, head directly to our tool. But if you are in for some science, keep reading our article. Here you will learn:

• What is a pendulum, and how does it operate;
• How to calculate the frequency of a pendulum: formula for small oscillations;
• The problems of bigger oscillations;
• Examples of frequencies of a pendulum.

What is a pendulum? How does a pendulum work?

Pendulums are simple devices made of — mostly — a weight and a rope, free to swing around a pivot. From the moment scientists developed an interest in pendulums, the devices showed that in such a simple construction, we can find traces of many physical phenomena and, not to forget, the key to keeping track of time before electronics became our choice.

How to calculate the frequency of a simple pendulum: frequency of oscillation

A simple pendulum only considers the force of gravity acting on the device. In the first analysis of such a pendulum, we often assume that the device completes small oscillations. For small oscillations, we use the approximation $\sin{(x)}\approx x$, where $x$ is an angle $x\ll1\ \text{rad}$.

In this approximation, the formula for the frequency of a pendulum is:

$f = \frac{1}{2\!\cdot\!\pi}\!\cdot\!\sqrt{\frac{g}{l}}$

Where:

• $f$ is the frequency of the pendulum, measured in hertz;
• $g$ is the acceleration due to gravity; and
• $l$ is the length of the swing.

The story changes a bit when you swing your pendulum far from the resting position.

The frequency of a pendulum formula for greater initial amplitudes

You know how to find the frequency of oscillations of a pendulum for small initial angles. However, when the angle at which you release the swing is bigger than, let's say, $10\degree$, the small angle approximation fails.

If you want to know how to find the frequency of a pendulum for any value of the initial angle, you need to apply the following formula:

\begin{align*} f & \!=\!\frac{1}{2\!\cdot\!\pi}\!\cdot\!\sqrt{\!\frac{g}{L}}\!\cdot\!\left(\sum_{n=0}^{\infty}\left(\frac{(2\!\cdot\! n)!}{\left(2^{ n}\!\cdot\! n!\right)^2}\!\right)^2\!\!\cdot\!\right.\\ &\left.\sin{\left(\frac{\theta_0}{2}\right)^{2\cdot n}}\right)^{-1} \end{align*}

This is not a straightforward formula, and it's rarely used outside of the academic world. We included it in our pendulum frequency calculator for completeness, but as a general rule, consider the approximate result.

A couple of examples of frequencies of a simple pendulum

You know how to find the frequency of oscillation of a pendulum: let's put this knowledge to test with a couple of examples.

Build a pendulum with swing length $1\ \text{m}$: what will be the frequency of oscillation of such device?

\begin{align*} f& = \frac{1}{2\!\cdot\!\pi}\!\cdot\!\sqrt{\frac{g}{l}} \\ \\ &= \frac{1}{2\!\cdot\!\pi}\!\cdot\!\sqrt{\frac{9.81\ \frac{\text{m}}{\text{s}^2}}{1\ \text{m}}}=0.5\ \text{Hz} \end{align*}

This is, for sure, one hell of a coincidence! An $1\ \text{m}$ long pendulum completes an oscillation in almost exactly $2\ \text{s}$. The length of the swing in pendulum clocks doesn't determine the time tracking capabilities of the device since the internal gearing can be engineered to obtain, at one point, a "ticking" at the desired $1\ \text{s}$. However, if we can have a tick... tock synchronized to the time we are tracking, we all feel better!

Using the equation for the frequency of a pendulum to find pi

Pendulums are versatile, but did you know that you can use them to measure $\pi$ and only use a stopwatch? Let's think about how to perform this experiment. First, which length of a pendulum's swing would return a frequency $f = \tfrac{1}{\pi}$ — corresponding to a complete oscillation in $\pi$ seconds?

\begin{align*} f&= \frac{1}{\pi} = \frac{1}{2\!\cdot\!\pi}\!\cdot\!\sqrt{\frac{g}{l}}\\ \\ 1&=\frac{1}{2}\!\cdot\!\sqrt{\frac{g}{l}}\\ \\ 2^2 &= \frac{g}{l}\\ \\ l& = \frac{g}{4}\approx 2.45\ \text{m} \end{align*}

Use our pendulum frequency calculator if you want to check the math, or grab a rope and measure your own $\pi$!

More than the frequency: pendulum calculators for all your needs

Calculating the frequency of a pendulum is just one of the possible calculations you can perform on a pendulum: visit our other pendulum calculators:

FAQ

How do I find the frequency of a pendulum?

To find the frequency of a pendulum in the small angle approximation, use the following formula:
f = (1/2π) × sqrt(g/l)
Where you can identify three quantities:

1. $f$ — The frequency;
2. $g$ — The acceleration due to gravity; and
3. $l$ — The length of the pendulum's swing.

What are the factors affecting the frequency of a pendulum?

A pendulum's frequency depends on three factors. In order of importance, these are:

1. The length of the swing. The longer the pendulum, the lower the frequency;
2. The acceleration due to gravity. A stronger acceleration reduces the period; and
3. The initial angle of oscillation.

For small initial amplitudes, we can exclude the effect of the angle on the duration of the swing. Moreover, g varies at most of 1% between any pair of points. This leaves us only with the length of the swing.

What is the frequency of a pendulum with swing length 25 cm?

Approximately 1 Hz, which means that the pendulum completes a back-and-forth movement in a second. To calculate the result, apply the formula for the frequency of a pendulum:
f = (1/2π) × sqrt(g/l) = (1/2π) × sqrt(g/0.25) = 0.997 Hz

Why does the mass doesn't affect the frequency of a pendulum?

Surprisingly, the mass of a pendulum's bob doesn't appear in the formulas: the situation resembles the one we study in a free-fall motion, where the acceleration remains constant, but the force acting on falling bodies varies greatly with mass.

Davide Borchia
T = 2π√(L/g)
Acceleration of gravity (g)
g
Pendulum length (L)
ft
Initial angle (θ₀)
deg
Pendulum frequency (f) — small angles
Hz
Pendulum frequency (f) — nonlinear
Hz
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