The Physics Philes, lesson 81: Expanding the Reach of SHM

In case you’re just joining us, we’ve been exploring simple harmonic motion. So far the only situation we’ve looked at so far is a body attached to an idea spring on a frictionless surface. However, there are such things as vertical and angular SHM, as well. That’s what I’ll be covering today.

Remember when we first started talking about oscillation? Basically, if the system has a restoring force that is directly proportional to the displacement from equilibrium, we have a system that can perform SHM. We can express it mathematically: F = -kx. The force constant k is different in each situation, but that’s OK. We can find the force constant by looking at the net force of the system.

Let’s first take a look at vertical simple harmonic motion. It’s not completely unlike the horizontal SHM we’ve been talking about, but there are some important differences. First let’s try to visualize what we’re dealing with. Instead of the horizontal body-spring system, we have something closer to this:

In contrast to the horizontal system, this system is in equilibrium when the body hangs at rest. That is, the spring is stretched a little. It’s stretched the length Δl, which is just enough so that the spring’s upward force kΔl balances the weight of the body. Or, in math, kΔl = mg.

Let’s consider x = 0 to be the equilibrium point and the positive x-axis to be up. When the body is a distance x above the equilibrium position, the extension of the string is Δl – x. That means the upward force on the body is k(Δl – x), so the net x-component force on the body is F = k(Δl – x) + (-mg) = -kx. The product kx is negative because it’s the downward force. This is similar to when the body is below the equilibrium point of x = 0. There is a net upward force of kx. Once a body is set into vertical SHM, it will oscillate with the same angular frequency as a body in horizontal SHM:

Really, vertical SHM isn’t all that different from horizontal SHM. You just need to remember that x = 0 does not mean that spring isn’t stretched.

OK. That’s easy. What about angular simple harmonic motion? Here is where we start seeing some new stuff. Let’s think about a mechanical watch. (Did you know that oscillations of a balance wheel in the watch how mechanical watches keep time? I didn’t!) Anyway, the wheel has a moment of inertia I about its axis. There is a restoring torque τ that is proportional to the angular displacement θ from the equilibrium. We can express the torque mathematically like this:

That weird looking K is the Greek letter kappa and it stands for something called the torsion constant, which is basically the springy-ness of the torque.

Now we use the rotational analog of Newton’s second law of motion, we get:

Or:

Remember that second derivatives are basically the acceleration of a body at a particular point. This is the equation for the acceleration in simple harmonic motion, but x is replaced by θ and k/m is replaced by κ/I. We can make these replacements for the equations for angular frequency and for frequency.

Angular frequency:

Frequency:

Those are the equations for angular simple harmonic motion!

That’s all for now. Come back next week for either a discussion of molecule vibrations or a simple pendulum. I have no decided yet. You’ll just have to wait to find out.

Featured image credit: Tom Stelmach via Flickr