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!

Differential Equations - Simple Harmonic Oscillation

  1. Jun 28, 2007 #1
    1. The problem statement, all variables and given/known data

    Consider y''(t)+(k/m)*y = 0 for simple harmonic oscillation

    A) Under what conditions on Beta is y(t)=cos(Beta*t) a solution?

    B) What is the period of this solution?

    C) Sketch the solution curve in the yv-plane associated with this solution (Hint: y^2 + (v/Beta)^2)

    For A, I had:

    dy/dt = v

    dv/dt = -(k/m)*y

    Found y' and y''

    y'(t) = -Beta*sin(Beta*t) y''(t)=-Beta^2*cos(Beta*t)

    Plugged those into the given equation and found Beta = +/- sqrt(k/m)

    y(t) = cos (+/-sqrt(k/m)*t) -> answers for A

    For B, I found the period to be T = 2*pi / Beta

    For C, I found that y^2 + (v/Beta)^2 = cos^2(Beta*t) + Beta^2*sin^2(Beta*t)/Beta^2

    My problem now is drawing this. I think, from remembering equations of shapes, that this is an ellipse, stretching in the v direction.

    I am not sure, but I think the max and min points of the ellipse are:
    y = 0, v = +/- sqrt(k/m)
    v = 0, y = +/- 1

    I also think that the direction of the field is counter-clockwise.

    I don't know if part B and C were done totally right and am a bit confused about finding the direction of the phase field/solution.

  2. jcsd
  3. Jun 28, 2007 #2


    User Avatar
    Staff Emeritus
    Science Advisor

    Pretty good- since cosine is an evern function, you don't need the "+/-", both signs give the same function, but, yes, Beta= sqrt(k/m).

    Wouldn't it be better to replace Beta by the value you found in (a)?

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?