A buoy oscillates in simple harmonic motion as waves go by. The buoy moves a total of 3.5 feet from its low point to its high point and it returns to its high point every 10 seconds.

Question

A buoy oscillates in simple harmonic motion as waves go by. The buoy moves a total of 3.5 feet from its low point to its high point and it returns to its high point every 10 seconds. 
What is the equation that describes this motion if it reaches its high point is at t=0?

Damon · Accepted Answer

Well there are lots of equations but let's take a cosine function because that is a simple one which is maximum at t = 0.
Call it y = a cos ( w t )
that has maximum a and minimum -a and is a at t = 0
now if it is to be 3.5/2 = 1.75 at t = 0 then a = 1.75 so so far:
y = 1.75 cos (w t)
now w t is zero at zero but I want it to be 2 pi or a full cycle at t = 10
so
w (10) = 2 pi
so w = 2 pi/10 = pi/5
so
y = 1.75 cos (pi t /5)

Emily · Answer

Wow thank you

Kaylee · Answer

Thanks!!! I couldn’t figure this out myself, and your explanation really helped!!!