# Simple Pendulum Calculator

Created by Krishna Nelaturu
Last updated: Nov 30, 2022

Welcome to our simple pendulum calculator, where you can easily calculate the period of a pendulum and its frequency from its length. You've come to the right place if you want to learn more about a simple pendulum. We shall discuss some fundamentals of a simple pendulum here, including:

• What is a simple pendulum?
• How to find its frequency and period using the simple pendulum equation?
• How to find the length of the pendulum?

If you're interested in finding the simple pendulum formula by yourself, then the may be helpful in the intermediate steps.

## How to use this simple pendulum calculator?

This simple pendulum calculator uses the simple pendulum equation to find the pendulum's period and frequency from its length:

• Enter the pendulum's length, and this simple pendulum calculator will determine the pendulum's period and frequency, using a period of a pendulum equation.
• Enter its frequency or period, and the calculator will find the pendulum's length using the pendulum length formula.
• The gravitational acceleration constant is set at $9.807 \text{ m/s}^2$. You can change it to a custom value if you wish.

## What is a simple pendulum?

A pendulum is a mass suspended from a pivot about which it can swing freely. To classify a pendulum as a simple pendulum (or simple gravity pendulum), it must satisfy these conditions:

1. The pivot must be frictionless. Air resistance must be negligible.
2. The string or rod used to suspend the mass must be weightless.

The inclined plane is another simple structure we often consider in physics problems. The inclined plane calculator is dealing specifically with this issue!

## Frequency and period of a pendulum

For a simple pendulum with small amplitudes $\theta$ such that $\sin(\theta) \approx \theta$, the period of the pendulum is independent of the mass and the amplitude. In such cases, the formula for a period of oscillation is given by:

$T = 2 \pi \sqrt{\frac{L}{g}} \kern{3em}\theta \leq 15\degree$

Where:

• $T$ is the pendulum's period;
• $L$ is the pendulum's length (length of the string suspending the pendulum mass);
• $g$ is the gravitational acceleration, equal to $9.81 \text{ m/s}^2$; and
• $\theta$ is the amplitude of the pendulum swing.

We can use this formula for the period of a pendulum to calculate its frequency:

$f = \frac{1}{T} = \frac{1}{2\pi}\sqrt{\frac{g}{L}}$

Here, $f$ is the frequency of the pendulum.

For amplitudes greater than $15 \degree$, the pendulum's period also depends on the moment of inertia of the suspended mass. In such cases, the period of a pendulum formula is given by:

$T = 2\pi \sqrt{\frac{I}{mgD}}$

Where:

• $I$ is the moment of inertia about the pivot;
• $m$ is the suspended mass; and
• $D$ is the distance from the pivot to the center of mass.

Our calculator does not perform this calculation, but we have the perfect tool for this scenario: physical pendulum calculator.

## Length of a pendulum formula

We can find the length of the simple pendulum by rearranging the formula of a period of oscillation:

\begin{align*} T &= 2 \pi \sqrt{\frac{L}{g}}\\ L &= \frac{gT^2}{4\pi^2}\\ \end{align*}

If you're now familiar with simple pendulum, why don't you try other calculators explaining fundamental problems of physics? Visit our trajectory calculator to learn the vertical distance formula of a thrown object or its angle of trajectory.

Krishna Nelaturu
T = 2π√(L/g)
Acceleration of gravity (g)
g
Pendulum length (L)
ft
Pendulum period (T)
sec
Pendulum frequency (f)
Hz
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