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Simple Pendulum Calculator

Created by Bogna Szyk
Reviewed by Steven Wooding
Last updated: Jan 18, 2024


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.

We made a video about pendulums! Watch it here:

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. (discover how it occurs with our simple harmonic motion calculator)

If you want to calculate the energy of the pendulum, make sure to use our kinetic energy calculator and potential energy calculator.

Diagram of simple gravity pendulum, an ideal model of a pendulum.
Diagram of a 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 - the 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 the 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. This type of pendulum is studied in our physical pendulum calculator, and the equation for its period 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.

FAQ

How do I calculate the time period of a simple pendulum?

To calculate the time period of a simple pendulum, follow the given instructions:

  1. Determine the length L of the pendulum.

  2. Divide L by the acceleration due to gravity, i.e., g = 9.8 m/s².

  3. Take the square root of the value from Step 2 and multiply it by .

  4. Congratulations! You have calculated the time period of a simple pendulum.

How do I determine the acceleration due to gravity using a simple pendulum?

To determine the acceleration due to gravity using a simple pendulum, proceed as follows:

  1. Measure the time period T for one oscillation using a stopwatch.

  2. Determine the length L of the pendulum.

  3. Divide the length L by the square of the time period T.

  4. Multiply the value obtained from Step 3 by 4π², and you will get the value for acceleration due to gravity, g.

How do I calculate the length of a simple pendulum?

To calculate the length of a simple pendulum, use the formula L = (T/ 2π)²×g. Where T is the time period of the simple pendulum and g is the acceleration due to gravity.

What is the length of a simple pendulum with a time period of 2 seconds?

99.36 cm. Using the formula, L = (T/ 2π)²×g, we can determine that the length of a simple pendulum with a time period of 2 seconds is 99.36 cm.

Bogna Szyk
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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