Simple Harmonic Motion: Of Special Interest to Physics C Students

Practice problems for these concepts can be found at: Simple Harmonic Motion Practice Problems for AP Physics B & C

The Sinusoidal Nature of SHM

Consider the force acting on an object in simple harmonic motion: F_net = –kx. Well, F_net = ma, and acceleration is the second derivative of position. So the equation for the motion of the pendulum becomes

This type of equation is called a differential equation, where a derivative of a function is proportional to the function itself.

You don't necessarily need to be able to solve this equation from scratch. However, you should be able to verify that the solution x = A cos(ωt) satisfies the equation, where

(How do you verify this? Take the first derivative to get dx/dt = –Aω sin(ωt); then take the second derivative to get –Aω² cos(ωt). This second derivative is, in fact, equal to the original function multiplied by –k/m.)

What does this mean? Well, for one thing, the position–time graph of an object in simple harmonic motion is a cosine graph, as you might have been shown in your physics class. But more interesting is the period of that cosine function. The cosine function repeats every 2π radians. So, at time t = 0 and at time t = 2π/ω, the position is the same. Therefore, the time 2π/ω is the period of the simple harmonic motion. And plugging in the ω value shown above, you see that—voila!—

Practice problems for these concepts can be found at: Simple Harmonic Motion Practice Problems for AP Physics B & C