How to Write Simple Harmonic Motion Equations

Credit: Public Domain

By Andrew Bennett

Making models to describe the world around us is a huge part of science. One way to model something is to write an equation that describes it. For example, we can write equations that describe the position of objects in simple harmonic motion.

Since a simple harmonic oscillator (such as a pendulum or a mass on a spring) goes back and forth again and again, we need to model this using a function that does the same thing. As long as the requirements for simple harmonic motion are met, the motion can be modeled with a sine or cosine function. (SHM is periodic motion caused by a restoring force that is proportional the object's distance from equilibrium. See this post for more information.)

Simple Harmonic Motion Equation

If we were to graph Y = sin(x) and Y = cos(x), we would see that both functions have a maximum value of 1, a minimum value of -1 (so the amplitude of each function is 1), and a period of 2ℼ radians (360 degrees). Since there is no requirement that a simple harmonic oscillator have those same values for amplitude and period, we need to modify the equations to match our object's motion. By convention, we typically use "t" instead of "x" in an equation like this, because we are modeling the position as a function of time. If you need to put one of these into a calculator and graph it, use x rather than t.

How to Modify the Amplitude

Modifying amplitude is straightforward. If we put a coefficient in front of the sine or cosine function, the amplitude of the function will be equal to the coefficient we put in. For example, the function Y = 3sin(t) has an amplitude of 3 instead of 1. If we noticed that our oscillator has an amplitude of 12 cm, we would simply make the coefficient in front of sine or cosine be 12 cm.

How to Modify the Period

Modifying the period of the function can be done by adding a coefficient inside the sine or cosine function, right next to the t. It is most common to use radians for these functions, so we'll do that here. Using the equation Y = sin(t), we could see that the graph begins to repeat once t gets to 2ℼ.  Except it doesn't have to be t that gets to 2ℼ, it just has to be the value inside the parentheses that reaches 2ℼ.  So, if we instead used the equation Y = sin(2t), the function will repeat when 2t gets to 2ℼ, meaning t only has to get up to a value of ℼ before it repeats. Since the other function repeated after getting to twice the value of t, we have cut the period of the function in half by adding the coefficient of 2.

How to Determine the Coefficient

Starting with a known value of the period of the oscillator, we can set up an equation to determine what the coefficient in the sine or cosine function should be. Namely, we know that when t is equal to the measured period, the whole inside of the parentheses must be equal to 2ℼ.

Simple Harmonic Motion Video

This video takes you through the process of writing an equation to model the position of a simple harmonic oscillator as a function of time. With an equation like this written, you could then make predictions of where the object will be at a certain moment in the future by plugging in for time.

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