# Simple harmonic motion test

1. Nov 2, 2006

### smileandbehappy

H, I missed the class so may be asking something really dumb her but have got a test today and want to do well so here goes:

in what cases should you use either of these formulas? (we have been told to remember them but not told when either is more relavent that the other, and my book dosnt say either):

s = A cos 2 pi ft

s = A sin 2 pi ft

Again sorry if im asking a rediculous question, but better to be safe than sorry. Also thanks in advance for any help given.

2. Nov 3, 2006

### OlderDan

What do you want s to be when t = 0?

3. Nov 3, 2006

### smileandbehappy

Would that not depend on which one of the equations you used? But how do you know which equation to use? If I get a question asking me to find the displacement will I just have to guess which equation to use?

4. Nov 3, 2006

### OlderDan

Either equation can be chosen to represent the displacement. It depends on what you choose to call the displacement at time zero. If you start an oscillator at maximum displacement at time zero, you would use cosine. If you start it with no displacement and some intial velocity you would use sine for the displacement; velocity would then be a cosine.

The most general representation is a mix of the two, corresponding to an initial displacement with an initial velocity.

s = A sin 2 pi ft + B cos 2 pi ft

with

v = 2 pi f (A cos 2 pi ft - B sin 2 pi ft)

where A and B must be chosen to satisfy the initial displacement and velocity situation. In mathematics, this is referred to as the boundary conditions.

It can be shown that the displacement can also be expressed as

s = C sin(2 pi ft + ֹφ) OR s = C cos(2 pi ft + ֹφ)

φ is called the phase angle and can be chosen to match the boundary conditions for either the sine or the cosine representation of the motion. φ will be different depending on which representation you want to use.