Login
Copyright

Pendulums in Physics: Definition & Equations

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Differences Between Translational & Rotational Motion

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:01 What is a Pendulum?
  • 1:30 Equations
  • 2:55 Example Problem
  • 4:26 Lesson Summary
Add to Add to Add to

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Login or Sign up

Timeline
Autoplay
Autoplay
Create an account to start this course today
Try it free for 5 days!
Create An Account

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: David Wood

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

After watching this lesson, you will be able to explain what a pendulum is, why it is an example of simple harmonic motion, and use equations to solve pendulum problems. A short quiz will follow.

What Is a Pendulum?

A pendulum is a weight hung from a stationary point in a way that allows it to swing freely back and forth. A simple pendulum is one where the pendulum bob is treated as a point mass, and the string from which it hangs is of negligible mass. Simple pendulums are interesting from a physics perspective because they are an example of simple harmonic motion, much like springs or rubber bands can be.

Simple harmonic motion is any periodic motion where a restoring force is applied that is proportional to the displacement and in the opposite direction of that displacement. Or in other words, the more you pull it one way, the more it wants to return to the middle. This is easy to imagine with a spring because you feel the increased tug as you stretch it more and more.

But what about a pendulum? Well, when you lift a pendulum to one side, the force of gravity wants to pull it back down, and the tension in the string wants to pull it left (or right). These combined forces work together to pull it back towards the middle (the equilibrium position). Ultimately, upon reaching the middle, the pendulum's velocity has increased, so it continues past the equilibrium position and off to the other side. This pattern then continues.

With a pendulum (or any simple harmonic motion), the velocity is greatest in the middle, but the restoring force (and therefore the acceleration) is greatest at the outer edges.

Equations

There are many equations to describe a pendulum. One equation tells us that the time period of the pendulum, T, is equal to 2pi times the square-root of L over g, where L is the length of the string, and g is the acceleration due to gravity (which is 9.8 on Earth).

pendulum equation

But since all simple harmonic motion is sinusoidal, we also have a sine equation:

Sine equation
pendulum equation

This one says that the displacement in the x-direction is equal to the amplitude of the variation, A (otherwise known as the maximum displacement), multiplied by sine omega-t, where omega is the angular frequency of the variation, and t is the time. In this equation, you start your mathematical stopwatch in the middle - time, t=0, is right in the middle as it swings by.

Finally, you might be wondering: what is angular frequency? Well, angular frequency is the number of radians that are completed each second. A full 360 degrees is 2pi radians, and that represents one full oscillation: from the middle to one side, back to the middle to the other side, and then back to the middle again. You can convert this angular frequency to regular frequency by dividing the angular frequency by 2pi. Regular frequency just tells you the number of complete cycles per second and is measured in hertz.

Example Problem

Okay, let's go through an example. A pendulum that is 4 meters in length completes one full cycle 0.25 times every second. The maximum displacement the pendulum bob reaches is 0.1 meters from the center. What is the time period of the oscillation? And what is the displacement after 0.6 seconds?

First of all, we should write down what we know. L, the length of the pendulum, equals 4 meters. The frequency, f, of the pendulum is 0.25, the amplitude (or maximum displacement), A, is 0.1, and the time, t, is 0.6. Also, g, as always, is 9.8. We're trying to find T and also x.

First of all, to find T, we just plug numbers into this equation and solve. 2pi times the square-root of 4 divided by 9.8. That comes out as 4.01 seconds.

To unlock this lesson you must be a Study.com Member.
Create your account

Register for a free trial

Are you a student or a teacher?
I am a teacher
What is your educational goal?
 Back

Unlock Your Education

See for yourself why 10 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back

Earning College Credit

Did you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it free for 5 days!
Create An Account
Support