Circular motion and simple harmonic motion

In this video clip, you can see the relationship between a circular motion on the left and a simple harmonic motion on the right. For this circular motion, the red point starts moving from (x, y)-coordinates of (1,0). We set this angle as 0 degree (or 0 radian). If the red point moves around the origin and comes back to the starting point in T [s], we express this as the period of one cycle is T [s]. When T = 2[s], it takes 0.25 s that the red point travels for the 1/8 cycle, that is 45 degrees. Likewise, it takes 0.5s for the 1/4 cycle, that is 90 degrees, and 1 s for the 1/2 cycle, that is 180 degrees.

A simple harmonic motion is one you can observe when a mass is attached to a spring and set into vibration. This motion can be expressed by a sine or cosine function. Furthermore, this motion can also be considered as the side view of a circular motion as the video above. Therefore, the position of the red point in the simple harmonic motion corresponds to the rotation angle of the circular motion is expressed as the phase of a sinusoidal function (its unit is degree or radian).

Sometimes, it is convenient to say that the rotation or vibration was done with certain cycles per a certain time period. In case of circular motion, the number of rotations per second is used (the unit is Hertz, or Hz). Likewise, for simple harmonic motion, cycle per second, or frequency is used (the unit is also Hertz, or Hz).