In nature, what’s something that you see *all the time?* Is it light? (don’t be a smart ass)

Is it… I’ll just tell youse what I’m getting at. It’s *wiggles*!

Yes, wiggling is everywhere. Your Jello in the morning, my Jello in the afternoon, and everyone and there mother’s Jello too. It just seems to keep going!

For a body to wiggle properly, it’s gotta go from side to side. That means it’s velocity will * accelerate* towards some

**equilibrium**

*position,**indicating the presence of a Force that’s always pointing towards this equilibrium. Simple Harmonic Motion (SHM) is the simplest kind of oscillation (hence the*

*simple*) – in which this “restoring force” (and hence, the acceleration) is

**directly proportional**to the displacement from the equilibrium.

— As you can see, this acceleration (in 1D, but 1’s fun, 1’s good ’nuff) is **not constant**. Lemme repeat: * not constant*. It’s got a dependence on that there displacement “x” from the equilibrium. The further away, the greater the acceleration.

Stated another way, the situation is: for some scalar constant “k”. This is the same form that we see when introduced to **Hooke’s Law** for ideal springs, that “k” being Hooke’s **Constant of Proportionality** (more on that at **Elasticity and Plasticity**).

Simple Harmonic Motion describes many things that oscillate back and forth, including Waves, the swinging of a pendulum, and situations of **Circular Motion **(like planets orbiting (really, they’ve got the same equations, different constants and stuff)).

Take a look at this GIF.

I’ll leave the more in-depth examination of the equations to the post on circular motion later on, but I’ll quote a few of the results here.

The **reference circle** is the circle whose **radius** is equal in magnitude to the **amplitude** of the oscillation in SHM. A point **P** on this circle moves around the circle with a constant **angular speed** ω. The **Vector** pointing from the center of the circle to the point makes some angle θ to the x-axis, and rotates along with the rotating point. This particular vector is called a **Phasor**.

From **Circular Motion**, we know that with a constant ω we have a constant **Centripetal Acceleration** pointing to the center of the circle that produces the circular motion. This acceleration is given by or you could replace the radius there with the amplitude and get

Now, if you think about it, the x-component of the phasor (corresponding to the x displacement in SHM) is given by , and the x-component of the acceleration in circular motion is given by

And as this corresponds to SHM, which has we find that -> for SHM.

From one view, ω is the **angular speed** and from another, it’s the **angular frequency** **of oscillation**.

## 2 thoughts on “Simple Harmonic Motion”