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Simple harmonic motion diff. equation

  1. Nov 18, 2009 #1
    1. The problem statement, all variables and given/known data
    A mass of 1 slug is suspended from a spring whose spring constant is 9 lb/ft. The mass is initially released from a point 1 foot above the equilibrium position with an upward velocity of [tex]\sqrt{3}[/tex] ft/s. Find the times at which the mass is heading downward at a velocity of 3 ft/s.


    2. Relevant equations
    x(double prime) + k/m x=0
    (k/m)^(1/2)=w


    3. The attempt at a solution
    So first, m=1, k=9 and the initial conditions: x(0)=-1 and x(prime)(0)=-[tex]\sqrt{3}[/tex]
    x(double prime)+9x=0 which has roots (plus or minus)3i. The solution then looks like

    x(t)=C1cos(3t)+C2sin(3t) and then plugging in initial conditions I get C1=-1, C2=-[tex]\sqrt{3}[/tex]/3

    Then converting it to another form where A=[tex]\sqrt{C1^2+C2^2}[/tex] and phi=arctan(C1/C2)
    x=Asin(wt+phi)
    x=2[tex]\sqrt{3}[/tex]/3 sin(3t+4pi/3)
    Taking the derivative to get the velocity:
    x(prime)=6[tex]\sqrt{3}[/tex]cos(3t+4pi/3)
    setting x(prime)=3 and solving for t yields a negative value for t, t=-.97

    I know I didn't really show a lot of work in a form that is easy to read or understand, but where am I going wrong here? Thanks for any help you can provide!
     
    Last edited: Nov 18, 2009
  2. jcsd
  3. Nov 19, 2009 #2
    Cosine function gets a given value in more than one point; maybe you should solve the first positive value of t where it holds.
     
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