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The torsional oscillator and the pendulum are two easy examples of simple harmonic motion. This type of motion, described by the same equations we have derived, comes up in molecular theory, electricity and magnetism, and even astronomy. The same method we applied in this section can be applied to any situation in which harmonic motion is involved.

Relation Between Simple Harmonic and Uniform Circular Motion

Through our study of simple harmonic oscillations we have used sine and cosine functions, and talked about angular frequency. It seems natural that there should be some connection between simple harmonic motion and uniform circular motion. In fact, there is an astonishingly simple connection that can be easily seen.

Consider a particle traveling in a circle of radius R centered about the origin, shown below:

Figure %: A particle, starting at point P, travelling in uniform circular motion with a radius of R, and angular velocity σ.
What is the x coordinate of the particle as it goes around the circle? The particle is shown at point Q, at which it is inclined an angle of θ from the x-axis. Thus the position of the particle at that point is given by:

x = R cosθ

However, if the particle is traveling with a constant angular velocity σ, then we can express θ as: θ = σt. In addition, the maximum value that x can take is at the point (R,0), so we can state that xm = R. Substituting these expressions into our equation,

x = xmcos(σt)    

This is the exact form as our equation for displacement of a simple harmonic oscillator. The similarity leads us to a conclusion about the relation between simple harmonic motion and circular motion:
Simple harmonic motion can be seen as the projection of a particle in uniform circular motion onto the diameter of the circle.

This is an astonishing statement. We can see this relation through the following example. Place a mass on a spring such that its equilibrium point is at the point x = 0. Displace the mass until it is at the point (R,0). At the same time that you release the mass, set a particle in uniform circular motion from the point (R,0). If the two systems have the same value for σ, then the x coordinate of the position of the mass on the spring and the particle will be exactly the same. This relation is a powerful application of the concepts of simple harmonic motion, and serves to increase our understanding about oscillations.

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