# Simple Pendulum Calculator

This simple pendulum calculator is a tool that will let you calculate the period and frequency of any pendulum in no time. Read to to learn the period of a pendulum equation and use it to solve all of the pendulum swing problems.

## What is a simple pendulum

First of all, a simple pendulum is defined to be a point mass (taking up no space) that is suspended from a weightless string. Such a pendulum moves in a harmonic motion - the oscillations repeat regularly, and kinetic energy is transformed into potential energy, and vice versa.

## Period of a pendulum equation

Surprisingly, for small amplitude (small angular displacement from the equilibrium position) the period of a pendulum doesn't depend neither on its mass nor on the amplitude. It is usually assumed that "small angular displacement" means all angles between -15º and 15º. The formula for pendulum period is

`T = 2π * √(L/g)`

where:

`T`

is the period of oscillations - time that it takes for the pendulum to complete one full back-and-forth movement.`L`

is the length of the pendulum (of the string from which the mass is suspended).`g`

is the acceleration of gravity. On Earth, this value is equal to 9.80665 m/s^2 - this is the default value in the simple pendulum calculator.

You can find the frequency of the pendulum as the reciprocal of period:

`f = 1/T = 1/2π * √(g/L)`

## How to analyze a pendulum in swing

- Determine the length of the pendulum. For example, it can be equal to 2 m.
- Decide what is the acceleration of gravity. We will use the Earthly value of 9.80665 m/s^2, but feel free to check how the pendulum would behave on other planets.
- Calculate the period of oscillations according to the formula above:
`T = 2π * √(L/g) = 2π * √(2/9.80665) = 2.837 s`

. - Find the frequency as the reciprocal of the period:
`f = 1/T = 0.352 Hz`

. - You can also let this simple pendulum calculator perform all calculations for you!

## Other pendulums

For a pendulum with angular displacement higher than 15º, the period depends also on the moment of inertia of the suspended mass. Then, the period of a pendulum equation has the form of

`T = 2π * √(I/mgD)`

where `m`

is the mass of the pendulum, `I`

is the moment of inertia of the mass, and `D`

is the distance from the center of mass to the point of suspension.