# Simple Harmonic Motion Reference Circle

by daveyavey
Tags: circle, harmonic, motion, reference, simple
 HW Helper P: 1,454 When one uses a reference circle to derive the SHM equations it is assumed that the radius vector rotates at a constant angular velocity $$\omega$$. Usually one takes it to rotate anticlockwise from some reference line. If the reference line is the horizontal line running throught the diameter of the reference circle one would start the time measurement when the radius vector passes the three o'clock position. This would then give the formula $$y = a \sin(\theta)$$ for the displacement where $$theta$$ is the angle that the radius vector is making with the reference line. If one chooses to start the time measurement when the radius vector is at the twelve o'clock position the formula would use the cosine rather than the sine of the angle (which is then measured from the y-axis). In this case a is the amplitude of the motion which is 0.14 m. For a complete cycle we have that $$\theta = 2 \pi$$ radians which means that for one eight of a cycle the angle would be $$\theta = \frac{\pi}{4}$$ radians which is $$45^o$$ which means that the displacement would be $$y = 0.14\sin(\frac{\pi}{4})$$ meters after one eight of a cycle.