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Differences between equations of SHM

  1. Jul 23, 2015 #1
    Why is it that ## y = A\sin (\omega t + \phi) ## whereas ## x = A\cos (\omega t + \phi) ##?

    Why is it that the y function is a sine wave, whereas the x function a cosine wave? I'm sorry if this question sounds ridiculous.
  2. jcsd
  3. Jul 23, 2015 #2
    is this circular motion? for SHM it doesn't matter if you use cosine or sine. for circular motion it's because the two components have to be out of phase. if you used both cosine or both sine, then the particle would oscillate back and forth on the line y=x
  4. Jul 23, 2015 #3
    because y'' = acceleration and for SHM i needs to be directly proportional to -y(displacement)
    and it happens to be that second derivative of sine is ''-''cosine !!
    otherwise both equations are fundamentally correct
  5. Jul 23, 2015 #4


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    If you plot the 2 solutions you will see that they differ by a phase of pi / 2 . This corresponds to a difference in starting position, or the location of the "bob" when you start the clock. So mathematically these are 2 solutions to the same differential equation resulting from different initial conditions.
  6. Jul 24, 2015 #5
    The same can be said for the x function; x" = - acceleration, isn't it?

    ## x = A\cos (\omega t + \phi) ##
    ## x' = - A\omega\sin (\omega t + \phi) ##
    ## x'' = - A\omega^2\cos (\omega t + \phi) ##
    ## x'' = - \omega^2 (A\cos (\omega t + \phi)) ##
    ## x'' = - \omega^2 x ##

    No, the only thing I'm asking is that why is the y function a sine wave, and the x component a cosine wave? Is it, as Integral said, just an example of the two functions having two initial conditions that have a phase difference of ##\frac{\pi}{2}##?
  7. Jul 24, 2015 #6
    Post edited .
    Last edited: Jul 24, 2015
  8. Jul 24, 2015 #7
    There is no relation between the two equations - first is along y-axis direction and other along x-axis .

    If your question is why one uses cosine and other sine , the cos and sin are interchangeable due to the phase φ by + - π/2 ( changing φ at the same time ) .
  9. Jul 24, 2015 #8
    Right, thanks for clearing that up! :)
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