Wize High School Grade 12 Physics Textbook > Oscillations

SHM as the Projection of a Uniform Circular Morion

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SHM as the Projection of a Uniform Circular Motion

The motion of an oscillator undergoing as simple harmonic motion could be mapped to a uniform circular motion. To see that better, let's look at the trigonometric unit circle:

 Let's choose angle θ\thetaθto be equal to ωt\omega tωtwhere ω\omegaωis the angular frequency of SHM. 
 
Wize Concept
When time passes, ωt\omega tωt is getting larger, so the point on the circle corresponding to this angle rotates in a counter-clockwise direction. 

From the picture, using trigonometric relations for the triangle shown at any moment:

x(t)=cos⁡(θ)=cos⁡(ωt) \boxed{x\left(t\right)=\cos\left(\theta\right)=\cos\left(\omega t\right)\ }x(t)=cos(θ)=cos(ωt) ​

y(t)=sin⁡(θ)=sin⁡(ωt) \boxed{y\left(t\right)=\sin\left(\theta\right)=\sin\left(\omega t\right)\ }y(t)=sin(θ)=sin(ωt) ​

Above equations are similar to the functions which describe the position of an oscillator in a SHM. 

Wize Concept
Since both SHM and uniform circular motion over the above circle are described by the same equation, we can describe SHM as a projection of circular motion over x or y axis.

Wize Tip
As we know, the phase constant indicated the starting point of the oscillation. For example, φ=0\varphi=0φ=0if the circular motion starts from the positive x-axis or φ=π/2\varphi=\pi/2φ=π/2 if it starts from the highest point on the positive y-axis.