# Pendulums in Physics: Definition & Equations

## 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).

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

Next, to find *x*, we first need to calculate the angular frequency. The regular frequency is 0.25 Hz, so just multiply that by 2pi, and we'll get omega. That comes out as an angular frequency of 1.57 radians per second. Now plug numbers into the displacement equation, and making sure your calculator is in radians mode, we get a displacement of 0.08 meters.

And that's it; we have our answer.

## Lesson Summary

A **pendulum** is a weight hung from a stationary point such that it can 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.

A simple pendulum undergoes simple harmonic motion. **Simple harmonic motion** is any motion where a restoring force is applied that is proportional to the displacement and in the opposite direction of that displacement. With a pendulum (or any simple harmonic motion for that matter), the velocity is greatest in the middle, but the restoring force (and therefore the acceleration) is greatest at the outer edges.

These equations can be used to solve problems involving pendulums:

Here, *x* is the displacement of the pendulum, *A* is the amplitude, omega is the angular frequency, *t* is the time, *g* is the acceleration due to gravity (which is always 9.8 on Earth), *T* is the time period of the pendulum, *L* is the length of the string, and *f* is the regular (non-angular) frequency.

## Learning Outcomes

Once you've completed this lesson, you should be able to:

- Define pendulum
- Explain the simple harmonic motion of a pendulum
- Identify the equations used to solve pendulum problems

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

Create your account

## Build Your Own Pendulum

In this activity, students are going to build their own pendulum and calculate the period using the equation for the lesson and compare it to an actual measurement from observation of the pendulum. To do this activity you'll need a ring stand, a C-clamp, about 20cm of string and a washer.

### Student Instructions

Now that you're familiar with what a pendulum is and some of the equations, it's time to build your own. Follow the instructions below to build a pendulum, then we'll calculate *and* observe the period and compare our results.

- Attach the C-clamp to the ring stand so that it is perpendicular to the stand.
- Cut a string that is 0.2m long, or just long enough so that there is an inch between the ground and the string when it is tied. Tie the string to the C-clamp.
- Next, tie a washer to the bottom of the string.
- Measure the distance between the top of the string and the bottom of the washer. This is the length (
*L*) of the pendulum. - Use the equation in the lesson to calculate the period. Record your work and your final answer.
- Next, raise the pendulum and let it go. Record how long it takes the pendulum to complete one back and forth motion in seconds. This is the observed period.
- Next, Repeat this trial four times and take the average observed period.

### Questions

- How did the calculated period compare to the observed period? If there was a difference explain why you think that could have happened.
- How did the average observed period differ from the single observation? Why do we take an average of values in science?
- Explain how this experiment demonstrated simple harmonic motion.
- What are some applications for studying the period of a pendulum?

### Expected Results

Students should see a slight discrepancy in the actual versus observed period. The observed period may be longer because of interaction with air resistance and friction. The accuracy of students' measurements can also cause a discrepancy between calculated and observed results. The pendulum shows simple harmonic motion because as the pendulum is pulled one way, the motion when released is equal but opposite. Over time the pendulum slows down due to air resistance.

### Register to view this lesson

### Unlock Your Education

#### See for yourself why 30 million people use Study.com

##### Become a Study.com member and start learning now.

Become a MemberAlready a member? Log In

Back