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Homework Help: Graphical Derivation of x = Asin(ωt)

  1. Dec 18, 2015 #1
    1. The problem statement, all variables and given/known data
    Deriving the equation for simple harmonic motion, x = Asinωt, graphically.

    2. Relevant equations
    ω = 2πf, where f = 1/T

    2. The attempt at a solution
    Take a sine curve as the simple harmonic motion (displacement, x, on y-axis; time, t, on x-axis), then transform it.

    The min/max is the amplitude, so we can stretch the graph to say that x = Asin(t).

    However, I can't quite get my head around where the ω comes from - I realise that there is a horizontal stretch which must be somehow related to the time period, but I can't quite see why it is 2πf.
  2. jcsd
  3. Dec 18, 2015 #2


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    Imagine a circle with radius 1 (m, km, take whatever dimension you want). Walk around that circle f times. How far have you walked?
  4. Dec 18, 2015 #3
    2πf units (and therefore 2πf radians covered). I think I understand what angular frequency is; I just don't seem to be able to relate it to the graph/equation.
    Last edited: Dec 18, 2015
  5. Dec 18, 2015 #4


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    Perhaps this would help.
    SHM as projection of uniform circular motion...
  6. Dec 18, 2015 #5
    Okay, so y = sin(kx) stretches the graph by a factor of (1/k), right? (compresses it by a factor of k).

    So stretching it by T would actually be a transformation of x = sin(t/T), which is x = sin(ft).

    Then you want to 'undo' the pi-ness of the x-axis to make the units seconds (and not have the pi scale hanging around), so you want to stretch by 1/(2π)? That means the whole transformation would be x = sin(2πft). And then you add on the amplitude: x= Asin(2πft).

    Does that make sense at all?
    Last edited: Dec 18, 2015
  7. Dec 18, 2015 #6

    Mister T

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    Why not just introduce ##\omega## at this point and not be bothered with ##\pi##?
  8. Dec 19, 2015 #7
    Because I still can't fit ω into it in my head! I was trying to reason it through so that it made intuitive sense to me, and the use of ω straight off just doesn't click!
  9. Dec 19, 2015 #8
    The sine function requires an angle, eg Sin(Θ), ω is not an angle, it is an angular velocity. The angle is (ωt)
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