The Physics Philes, lesson 77: Return of Hooke’s Law
Now that we have some of the basics of periodic motion down, we can start talking about simple harmonic motion. I need to emphasize that this is just the beginning of simple harmonic motion – SHM for short. This section goes on for a while. But it’s important, as we’ll see.
A body is in SHM when the restoring force is directly proportional to the displacement from equilibrium. Think about that spring diagram from last week:
Springs that obey Hooke’s Law (remember Hooke’s Law?) will exhibit simple harmonic motion. In other words, it will be a harmonic oscillator. Basically this means that if the elongation of the spring isn’t too great, the displacement will be directly proportional to the force. In math, this turns out to be F = kx, where k is something called the force constant. This value will vary depending on the spring. (For reference, a Slinky has a force constant value of about 1 N/m.)
However, the x-component of the force the spring is exerting is actually negative, so that needs to be reflected in the equation. That means that the restoring force exerted by an ideal spring is F = -kx.
We can also figure out the acceleration of a body in SHM using Newton’s Second Law of Motion. Using that law, we know that acceleration equals force divided by mass. We can replace the force with the negative product of the force constant and the displacement x.
It’s important to note that the acceleration is not constant. We’ll need some calculus-y tools to get that done. (Don’t worry. We’ll get there. I promise.)
You may be wondering why we need to know about simple harmonic motion. I mean, how many oscillating bodies out there have a restoring force that is directly proportional to the displacement? Well, not many, actually. But that doesn’t mean this stuff isn’t important because there are a lot of systems that are almost harmonic oscillators. For example, AC electrical current and the oscillations of atoms in molecules and solids approximate SHM. That means we can use the acceleration equation to approximate simple harmonic motion. So while few things fit this definition exactly, it’s still some mad useful knowledge.
It’s a short lesson this week, but don’t get too cozy. I’ve got a lot more in the pipeline. Next week: circular motion.
Featured image credit: University of Salford via Flickr