Simple harmonic motion chapter

[briannamorgan]

The chapter I'm reading is titled simple harmonic motion and the reference circle. I am so completely lost. It's talking about displacement, velocity, acceleration, and frequency of vibration. I understand displacement, velocity, and acceleration on a linear and angular level. I guess I'm confused about what harmonic motion is and how to find these things. Thanks for your help.

 

[ShawnD]

First off, ALWAYS do angular calculations using radians, not degrees!

Simple harmonic motion is just like the above guy said, motion where distance from a certain point (equilibrium point), velocity and acceleration all follow a sinusoidal pattern.

Let's say we have a spring moving up and down (no friction, no air, 100% efficient). Let's say we have a spring and we pull it 15cm from the equilibrium point. That spring will absolutely never go more than 15cm from the equilibrium point in either direction. If we also know the period or the rotation speed, we can figure out instantaneous distances, velocities and accelerations.
The formula for our displacement in this case is:
d = Rcos(wt)
R is the maximum displacement from equilibrium (15cm in this case), w is rotation speed and t is time in seconds.
The formula for rotation speed is w = 2pi/T
w is not actually a w, it's supposed to be omega which looks much like a w. pi is the constant pi and T is the period of oscillation (sp?).
The formula for our displacement is determined for where we consider the start point to be. Most physics teachers will want you to consider the start point to be equilibrium and think of the spring as moving up, in that case our formula is d = Rsin(wt). If it starts in the middle and is going down, it's d = -Rsin(wt). If it starts from the vertical maximum, d = Rcos(wt). If it starts from vertical minimum, d = -Rcos(wt).

No matter what our formula for distance is, the formula for velocity is always the derivative of displacement with respect to time and acceleration is always the derivative of velocity with respect to time.

If you don't know what a derivative is, the following will just confuse you so don't read it

If our formula for displacement was d = Rcos(wt) meaning it starts from the vertical maximum, the formula for velocity will be v = -Rwsin(wt) and our acceleration will be a = -Rw^2cos(wt).