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Equations of the harmonic oscillator

  1. Mar 2, 2014 #1
    Hello, my book explains detailed the proofs of these three formulas:

    y = Asin(ωt + φo)
    v = ωAcos(ωt + φo)
    a = -ω²Asin(ωt + φo)

    Where a is acceleration, v is velocity, ω is angular velocity, A is amplitude.

    The book uses the following figures:
    Figure a) --> http://tinypic.com/view.php?pic=30m8f9j&s=8
    Figure b) --> http://tinypic.com/view.php?pic=2hrejde&s=8
    Figure c) --> http://tinypic.com/view.php?pic=a0f9te&s=8

    In the first case a) what I'm seeing is that the sine in the triangle whose hypotenuse is R and opposite side vector y is (vector y)/(vector R), in this case R is the amplitude.

    The rest b) and c) are just complicated, I don't understand them. Anyone out there to help me here?
    Thank you!!

    Edit: Sorry for posting this here, it's supposed to go to homework help, my fault.
  2. jcsd
  3. Mar 2, 2014 #2


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    Staff: Mentor

    Conceptual questions that arise from not understanding your textbook or lecturer's notes, are fine here. The homework-help forums are for getting help with solving specific exercises.

    The simplest way to derive the formulas for v and a is by using calculus. v is the derivative of y with respect to t, and a is the derivative of v with respect to t. I suppose you haven't had calculus yet?
    Last edited: Mar 2, 2014
  4. Mar 2, 2014 #3
    Hi! Thank you for replying, no I haven't done calculus yet. I thought these formulas came from looking at the geometry of these vectors.
  5. Mar 2, 2014 #4


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    Staff: Mentor

    For figure (b), focus on the small right-triangle that has ##\vec v_t## as its hypotenuse. Which trig function (sin or cos) is associated with the y-component of ##\vec v_t##, and is that component + or -?

    To get the amplitude (maximum value) of v, remember that the point moves around the circle at constant angular speed ω (radians/sec). What linear speed (m/sec) does that correspond to? (This is the magnitude of the vector ##\vec v_t##.)
    Last edited: Mar 2, 2014
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