# The Physics Philes, lesson 82: Swing, Swing, Swing

We’ve been discussing simple harmonic motion for the past several weeks. Simple harmonic motion, if you’ll remembers, is an oscillation with a restoring force that is directly proportional to the displacement from the equilibrium. Hardly anything actually moves in SHM, but a lot of things move in approximately SHM, so the equations we’ve been learning are useful in approximating the oscillations of certain objects.

We’re going to look at one of those motions that are approximately simple harmonic today. A simple pendulum is basically just an idealized model of a pendulum. It’s a point mass suspended by a massless, unstreatchable string. Not the most realistic pendulum in the world, but this is physics and we can pretend absurd things are real if it makes the equations easier to solve. The equilibrium position is straight down. When the point mass is pulled to one side of that equilibrium position and released, it oscillates about the equilibrium.

You can picture it kind of like a swing. You start off just sitting in the swing. You’re in the equilibrium position. Then, as you pump your legs to get going, you start moving back and forth through the equilibrium position, and back again. In other words, you’re oscillating. In the above picture, the grey line is the equilibrium position, the blue line is the point mass and massless unstretchable string, and the red arrow is the oscillation path. The red arrow is curved a bit for a reason. It’s the arc of a circle with radius L (or the length of the string). But here’s the million dollar question: Is this oscillation simple harmonic? To find out we need to physics up this diagram just a bit.

That’s better. Maybe? Let me see if I can make this readable. That black arrow is labeled x and it is the distance measured along the arc. The restoring force is provided by gravity. The only job of the tension T is to make the point mass move in an arc. If we’re dealing with SHM, the restoring force must be directly proportional to x or Θ. Is it?

No. The restoring force of the tangential component is the net force and is equal to

The restoring force is not proportional to Θ, it’s proportional to sinΘ. That means that the motion we’ve modeled here is not simple harmonic.

Oh no! But SHM is all we know! Don’t worry. We can still work with this. If the angle is small, sinΘ is almost equal to Θ in radians. So we can express the restoring force in one of two ways:

Or

So we can use what we know about simple harmonic motion, assuming the angle is small enough. The force constant in this circumstance is k = mg/L. So we can express the angular frequency of a simple pendulum with a small amplitude as

The frequency and period relationships in this situation are:

Frequency:

Period:

The mass of the particle doesn’t matter. The restoring force is a component of the particle’s weight. The mass cancels out. So for small oscillations, the period of the pendulum for a given value g is determined by the length L.

This makes sense with what we see in life. A long pendulum has a longer period than a shorter one. If we increase g, we increase the restoring force, which would give the pendulum a higher frequency causing a shorter period.

Remember that a simple pendulum is only approximately simple harmonic. As the amplitude gets bigger, this approximation gets more and more inaccurate.

Next week we’ll look at oscillation of a real pendulum and how it differs from the idealized model we analyzed today.

Featured image credit: Wayne Silver via Flickr