Sine Up to Learn All About Simple Harmonic Motion Functions

Photo Credit: Zátonyi Sándor/Wikimedia Commons

By Andrew Bennett

In this blog post, you will learn how to model an object's position as it undergoes simple harmonic motion using sine or cosine functions.

Simple Harmonic Motion Refresher

Simple harmonic motion is defined as periodic motion caused by a restoring force that is proportional to the object's displacement from equilibrium. This definition might seem arbitrarily complicated, but there are good reasons for separating this type of motion. For this post, it is important to note that the position, velocity, and acceleration of any object undergoing simple harmonic motion can be described using the sine and cosine functions. We might not be able to model other types of repeated motions with simple equations like this.

Sine Function for SHM

A general form frequently used here for the sine function is:

x(t) = A*sin (ꞷt)

Where:

x = the position at some moment in time
A = the amplitude of the motion
ꞷ (omega) = the angular frequency of the motion
t = the time

YouTube Screenshot (https://www.youtube.com/watch?v=e1b5Dkk8cYc)

Note that there is sometimes a "plus phi (+ɸ)" inside the parenthesis. This shifts the whole graph left or right. Since we can decide when to start our time measurements, we'll just choose to have zero time correspond with a moment when the oscillator is passing through equilibrium and moving in the positive direction.

Everything in the video below is done using the sine function but works equally well with cosine. The only difference is the starting position. If your oscillator is starting at its maximum displacement at time zero (and you don't have the option to change the meaning of time zero), cosine would be a logical choice.

How to Adjust the SHM Equation

In the video below, I'll show you how to determine the values for the coefficients to tailor this equation to the oscillator you are dealing with. Once you have an equation like this, you can do all sorts of nifty tricks with it, such as taking the derivative with respect to time to find a velocity function and once more to find the acceleration function.

You could also substitute these equations into some of the energy equations. For example, you can replace the speed term in the kinetic energy equation to get an equation for kinetic energy as a function of time.

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