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!

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
    k=16

    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

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    In all of these the "x" variable should be "t". Other than that, I see no error or even a question!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Second order DE model
  1. Second order DE (Replies: 2)

  2. Second order DE (Replies: 6)

  3. Second Order DE question (Replies: 10)

Loading...