# Pendulum Frequency Calculator

Created by Davide Borchia
Reviewed by Łucja Zaborowska, MD, PhD candidate
Last updated: Aug 20, 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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