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

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

What is a simple pendulum

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

Diagram of simple gravity pendulum, an ideal model of a pendulum.
Diagram of simple pendulum, an ideal model of a pendulum.
(Chetvorno / Public domain)

Period of a pendulum equation

Surprisingly, for small amplitudes (small angular displacement from the equilibrium position), the pendulum period doesn't depend either on its mass or on the amplitude. It is usually assumed that "small angular displacement" means all angles between -15º and 15º. The formula for the 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); and
  • g is the acceleration of gravity. On Earth, this value is equal to 9.80665 m/s² - 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

  1. Determine the length of the pendulum. For example, it can be equal to 2 m.
  2. Decide a value for the acceleration of gravity. We will use the Earthly figure of 9.80665 m/s², but feel free to check how the pendulum would behave on other planets.
  3. Calculate the period of oscillations according to the formula above: T = 2π√(L/g) = 2π * √(2/9.80665) = 2.837 s.
  4. Find the frequency as the reciprocal of the period: f = 1/T = 0.352 Hz.
  5. 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 also depends 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.
Bogna Haponiuk