The Sinusoidal Description of Simple Harmonic Motion

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  • 0:02 What is Simple…
  • 1:18 Sinusoidal Equations
  • 4:05 Example Problem
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Lesson Transcript
Instructor: David Wood

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

Watch this video to learn about simple harmonic motion. Learn the sinusoidal equations we use to solve problems of simple harmonic motion, and test your knowledge afterwards with a short quiz.

What is Simple Harmonic Motion

Simple harmonic motion is any motion where a restoring force is applied that is proportional to the displacement, in the opposite direction of that displacement. Or in other words, the more you pull it one way, the more it wants to return to the middle. The classic examples of simple harmonic motion include a mass on a spring and a pendulum. With a spring, the more you stretch it, the more it tugs back to where it started. And with a pendulum, the tension in the string has a similar effect.

Simple harmonic motion is also considered to be PERIODIC, or in other words, it is a pattern that repeats itself. If we ignore frictional forces, the motion of a pendulum, or a bouncing spring, repeats itself. The time to complete one full cycle is called the time period and is represented with a capital T. And the number of cycles that occur each second is called the frequency, represented by an f.


Time period is measured in seconds, and frequency is measured in Hertz. And these two values are the reciprocal of each other - to get from one to the other, you find the reciprocal. So a time period of 2 seconds will be a frequency of one over two, or 0.5 hertz. And a time period of 0.5 seconds, will be a frequency of 2 hertz.

Sinusoidal Equations

There are many equations that describe simple harmonic motion. There are energy equations, equations that only work for the mass on the spring scenario, others that only work for pendulums, and kinematics equations. Today we're going to focus on kinematics and look directly at the motion, ignoring the forces.

The motion of an object in simple harmonic motion is sinusoidal. What do we mean by that? We mean that values like displacement, velocity and acceleration vary in the shape of a sine or cosine curve. A sine curve looks like the red curve below. A cosine curve looks like the blue curve.

A sine curve and a cosine curve

A cosine curve is just a sine curve shifted to one side - the fundamental shape is the same.

If you look at the motion of a spring or a pendulum, it's easy to see why it's a sine curve. The motion left to right of a bouncing spring speeds up and slows down. In the middle it's moving fast, and at the edges it's moving more slowly. And this cycles back and forth. This matches the pattern of a sine curve - the position on the y-axis changes most rapidly as it crosses the x-axis and most slowly at the peaks and troughs.

Here are the equations we use to describe the displacement, velocity and acceleration of an object in simple harmonic motion:

Equations for Displacement, Velocity and Acceleration

In these equations, x is the displacement in meters, v is the velocity in meters per second, and a is the acceleration in meters per second squared. And they're written in terms of the amplitude of the variation, A, (otherwise known as the maximum displacement), multiplied by sine omega-t, where omega is the angular frequency of the variation, and t is the time.

These equations assume that you start your stopwatch in the middle - time t=0 is right in the middle as it swings by at full speed. But you don't have to do it this way. If you start your stopwatch at one of the outside edges, the equations remain quite similar, but the sines and cosines swap, and some of the signs change. This is because sine and cosine are really the same shape, just shifted left or right. You start your graph wherever your object happens to be at t=0.

Finally, you might be wondering: what is angular frequency? Well, angular frequency is the number of radians that are completed each second. A full 360 degrees is 2-pi radians, and that represents one full oscillation: from the middle, to one side, back to the middle, to the other side, and then back to the middle again. You can convert to regular frequency by dividing the angular frequency by 2-pi. Regular frequency just tells you the number of complete cycles per second and is measured in hertz. So we can adjust our sinusoidal equations and replace angular frequency with 2-pi-f, which changes them to look like this:

Equations with Angular Frequency Replaced

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