Period of a Pendulum | Simple harmonic motion and rotational motion | AP Physics 1 | Khan Academy

So a simple pendulum is just a mass hanging from a string, and if you were to pull this mass—sometimes it's called a pendulum bob—if you were to pull it back and then let go, gravity would act as a restoring force, and this mass would swing back and forth over and over.

Because this simple pendulum is a simple harmonic oscillator, its motion, its angle as a function of time, would be accurately described by a sine or a cosine graph. In other words, if you were to pull this back initially by 15 degrees, you might get a graph that looks something like this.

Now, every simple harmonic oscillator has a characteristic period of motion. The period of motion is the time it takes to complete one full cycle. So the time it takes to swing from here all the way to there, all the way back to here, would be one full period. In this graph, I've written it as 0.5 seconds, but the period of every simple pendulum is not going to be 0.5 seconds. The period of a simple pendulum depends on the characteristics of that pendulum and the environment that it's in.

To derive this formula for the period of a pendulum, you would need calculus. So I'm just going to write it down, give you a quick tour, and compare it with the period formula for a mass on a spring. The period of a pendulum is going to be equal to 2 pi times the square root of the ratio of the length of the pendulum (L), so the length of that string here, the length (L) divided by (g) the gravitational acceleration of the planet that the pendulum is being used on.

Now, if you look at this and you've been paying attention in physics, you might be like, "Wait, that looks really similar to the formula for the period of a mass on a spring." So if you were to take a mass on a spring, displace it 15 centimeters, you'd get a similar graph, and this would also have a characteristic period of motion. If you wrote down the period for a mass on a spring, it looks like this: its period for a mass on a spring is also 2 pi, so that's identical, and it's also the square root of a ratio. But instead of L over g for the mass on a spring, it's m, the mass of the block connected to the spring, divided by k, the spring constant of the spring.

So one obvious similarity between these two formulas is just their format. They're both a 2 pi times a square root of a ratio. But another important similarity between these that might not be evident is that neither of these formulas depend on the amplitude of the motion. So, the period of a pendulum does not depend on the amplitude of the motion, and the period of a mass on a spring also does not depend on the amplitude of the motion.

What I mean by that is, if you were to pull this pendulum back instead of 15 degrees, pull it back 20 degrees and let go, it would have farther to swing. So its motion might look something like this, and it would get down farther, and it would get back up. But it would take the exact same amount of time; the period would not change. If you pulled this amplitude back farther, the same goes with this mass on a spring. Instead of pulling at 15 centimeters, let's say you pulled it 20 centimeters. Again, this would start up higher; it would get down lower, but the time it takes to complete one full cycle would not vary as you vary this amplitude.

And that might seem weird. You're like, "Wait, don't these objects have farther to go now that you pulled it back to a larger amplitude?" That's true; they'll both have farther to travel, but they'll be going faster now, and faster motion over a bigger distance is going to offset. The amplitude does not affect the period of the motion of a pendulum or of a mass on a spring.

As for the differences, well, the denominator here for the mass on a spring depends on k, and that's the spring constant k. That should make sense: more spring constant means you get more restoring force. The spring force is the restoring force; here, more restoring force means this mass is going to be moving faster. That means it's going to take less time to go through a full period. So dividing by a bigger number—bigger spring k—gives you less period. That's also true here, but it's not k; the force—that's the restoring force for a pendulum—isn't a spring; it's the force of gravity. So mg depends on the g. A bigger g would give you a bigger restoring force over here for the pendulum.

That means the pendulum could be moving faster. Moving faster means it's going to take less time to complete a full period. So dividing by a bigger g—if you took this pendulum to the surface of Jupiter or something where the g is bigger—it would swing back and forth faster. It would take less time to complete it because that restoring force is bigger. So even though these denominators are different letters, they're arriving from the same source. They're both arriving because the restoring force is larger when you increase these denominators, which increases the speed of the object.

Now maybe the biggest difference here is that the numerator here for the mass on a spring depends on mass, but nowhere is mass to be found in this period of a pendulum. The period of a pendulum does not depend on the mass. This is kind of interesting. So, you know, a really big heavy person gets on a swing, swings back and forth. A very light child gets on the same swing; they will take the same amount of time to complete a full cycle. The mass does not factor in here to the period of a pendulum, but it does for the mass on a spring.

Why is that? Well, a bigger mass on a spring gives you more inertia in the system. If you have more inertia in the system, it's more sluggish to movement; it's going to go slower. That means it's going to take more time to complete a full cycle. Now, you might think, "Wait a minute, doesn't that hold true up here? Look, if we have a bigger mass pendulum bob, that should increase the rotational inertia, and so that should make this take longer. It should be more sluggish to movement; it should take longer to go through a full period." But look; the restoring force is also for a pendulum proportional to mass. So if you increase the mass of this pendulum, you do get more inertia, but you're getting more restoring force because the restoring force is gravity. Those completely offset.

Mass does not end up showing up in this pendulum formula even though it does down here. So the spring force is not proportional to mass; the spring force is kx. Increasing the mass of this block on the end does not increase the spring restoring force, so this mass stays in the numerator here, but it does not affect the period of a pendulum.

So why does this L show up then? Why is it length of the pendulum in the numerator? Well, the rotational inertia does get increased when you increase the length of a pendulum. But increasing that length does not increase the force of gravity. If you want to get technical, rotational inertia is proportional to length squared, but the torque would only be proportional to length. That's why only one L shows up here. Long story short, if you increase the length of a pendulum, you're going to increase the inertia of that pendulum.

That's going to make it take longer to go through a full cycle. This is why I like going to the park and finding the long swings. The longer the swing, the longer it actually takes, the more time it actually takes to swing back and forth. I think that's more fun than the little short swings that go back and forth really quick.

So let's try a sample problem to see how this period formula works. Let's say you went to the park; you're 60 kilograms, you're swinging on a swing, and your friend pulls you back 20 degrees on a swing that's a one meter in length. Let's find the period of the motion. So in other words, the time it takes to go all the way to here and then all the way back to there.

We use the period formula for a pendulum: it's 2 pi root L over g. So we would do 2 pi times the square root—the length here is the length of the string here, so one meter. Technically, you'd be to wherever your center of mass is, but we're going to assume our center of mass is right here at the end, divided by g. Well, we're on Earth, I'm assuming, so 9.8 meters per second squared.

If you solve all that, you get a period of about two seconds exactly. So this swing would have a period of about two seconds. Now notice we did not use this 20 degrees; that's the amplitude. This period does not depend on the amplitude. We also did not use the fact that I was 60 kilograms; the period of a pendulum also does not depend on the mass of the bob at the end of the pendulum.

We only use the length, and if you're on Earth, the denominator here is always going to be 9.8. Now one thing I should be clear about is that a pendulum is technically not a perfect simple harmonic oscillator; it's only approximately a simple harmonic oscillator. So this formula here is only approximately correct, but for small angles, it's almost perfect.

So at 20 degrees, this formula is only going to be off; the error is only going to be about one percent, which isn't bad. If you get this back to like 70 degrees, even then the error is only around 10 percent. So this is a really, really good approximation if you're in small angles like around 20 to 30 degrees. The bigger the angle gets though, the worse the approximation gets. For small angles, most physicists just treat a pendulum as if it's a perfect simple harmonic oscillator. As you get to those higher angles, though, you got to be careful; this can start deviating more significantly from the actual value.

So to recap, the period of a pendulum depends on the length of the pendulum and the surface gravity of the planet that you're on. It does not depend on the amplitude or the mass of the pendulum, and the form it takes is very similar to the period of a mass on a spring, where the numerator increases due to increased inertia and the denominator increases due to increased restoring force.