1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    Svein

    User Avatar
    Science Advisor

    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

    cnh1995

    User Avatar
    Homework Helper

    Perhaps this would help.
    Animation1.gif
    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
    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)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Graphical Derivation of x = Asin(ωt)
Loading...