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Homework Help: Second order DE model

  1. Aug 8, 2008 #1
    1. The problem statement, all variables and given/known data
    A 32 pound weight stretches 2 feet. Determine the amplitude and period of motion if the weight is released 1 foot above the equilibrium position with an initial velocity of 2 ft/s upward. How many complete vibrations will the weight have completed at the end of 4 pi seconds?

    3. The attempt at a solution

    Here is the solution I have come up with:

    32 = 2k

    x(0)=1 ft
    x'(0) = -2 ft/s
    m= 32/32 = 1 slug

    [tex] \frac{dx^2}{d^2t} +16x = 0[/tex]

    [tex] m^2+16[/tex]

    solution to xc:

    [tex] x(t)= A \cos{4x} + B\sin{4x}[/tex]

    with initial conditions:

    [tex] x(t) = \cos{4x} - \frac{1}{2} \sin{4x}[/tex]

    therefore amplitude= [tex] \sqrt{1+\frac{1}{4}} = \frac{\sqrt{5}}{2}[/tex]

    and [tex] \tan{\phi} = 3[/tex]

    [tex] \phi = -1.1071 + \pi = 2.034 [/tex]

    so my equation for the model is

    [tex] x(t) = \frac{\sqrt{5}}{2} \sin{(4x+2.034)}[/tex]

    and I know for a complete vibration to occur I have to have [tex] 2n\pi + \frac{\pi}{2}[/tex]

    So [tex] 48.23 = \frac{(x-6)\pi}{2}[/tex]
    x is approximatly 25 which is about 6 revolutions.

    in 4 pi seconds, the system has undergone approximatly 6 revolutions and the differential equation describing the system is

    [tex] x(t) = \frac{\sqrt{5}}{2} \sin{(4x+2.034)}[/tex]
  2. jcsd
  3. Aug 8, 2008 #2


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    Science Advisor

    In all of these the "x" variable should be "t". Other than that, I see no error or even a question!
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