Simple Pendulum Calculator

Created by Krishna Nelaturu
Last updated: Jun 27, 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 calculating the maximum speed, height, or kinetic energy available in a pendulum setup, head to our kinetic energy of a pendulum calculator.

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 m/s29.807 \text{ m/s}^2. You can change it to a custom value if you wish.

What is a simple pendulum?

A mass suspended by a massless rod to make a pendulum.
A simple pendulum has three parts: a suspended mass, a weightless string or rod, and a frictionless pivot. Image source: Wikipedia.

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.

Frequency and period of a pendulum

For a simple pendulum with small amplitudes θ\theta such that sin(θ)θ\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=2πLgθ15°T = 2 \pi \sqrt{\frac{L}{g}} \kern{3em}\theta \leq 15\degree

Where:

  • TT is the pendulum's period;
  • LL is the pendulum's length (length of the string suspending the pendulum mass);
  • gg is the gravitational acceleration, equal to 9.81 m/s29.81 \text{ m/s}^2; and
  • θ\theta is the amplitude of the pendulum swing.

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

f=1T=12πgLf = \frac{1}{T} = \frac{1}{2\pi}\sqrt{\frac{g}{L}}

Here, ff is the frequency of the pendulum.

For amplitudes greater than 15°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=2πImgDT = 2\pi \sqrt{\frac{I}{mgD}}

Where:

  • II is the moment of inertia about the pivot;
  • mm is the suspended mass; and
  • DD 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:

T=2πLgL=gT24π2\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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