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Mechanics, PE to position function

  1. May 14, 2009 #1
    1. The problem statement, all variables and given/known data
    A 3 kg object is moving along the x-axis where U(x) = 4x2. At x = -.5, v = +2. Find the object's position and KE as functions of time. Assume x = 0 at time t = 0. All forces acting on the object are conservative.

    2. Relevant equations
    ME = U + K
    K = (1/2)mv2
    F = dU/dx
    F = ma

    3. The attempt at a solution
    Using initial conditions, ME = 4.
    F = dU/dx = 8x
    F = ma
    8x = (3)d2x/dt2
    This is where I got stuck. I was attempting to solve for x(t), find v(t), then use that to find K(t). Assuming everything else is correct, how do you solve a second order differential equation like this? Otherwise, please correct me.
     
  2. jcsd
  3. May 14, 2009 #2

    rock.freak667

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    Just solve

    [tex]\frac{d^2x}{dt^2}- \frac{8}{3}x=0[/tex]

    Do you know how to solve a second order differential equation with constant coefficients?


    EDIT: http://www.sosmath.com/diffeq/second/constantcof/constantcof.html" [Broken]
     
    Last edited by a moderator: May 4, 2017
  4. May 14, 2009 #3
    I am still having trouble solving for x(t).
    I got [tex]x=c_1e^{\sqrt{\frac{8}{3}}t}+c_2e^{-\sqrt{\frac{8}{3}}t}[/tex]
    and [tex]c_1+c_2=0[/tex]
    but since there is no initial value associating time and velocity, I can't find the constants.
     
  5. May 14, 2009 #4

    rock.freak667

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    How did you get ME=4 by chance?

    Also F=-dU/dx not F=+dU/dx
     
  6. May 14, 2009 #5

    Delphi51

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    Can you use the "At x = -.5, v = +2" condition?
    Would it work to begin with
    ME = U + K
    4 = 4x^2 + 1/2*mv^2 (which includes the x = -.5 condition)
    4 = 4x^2 + 1.5(dx/dt)^2
    The solution to this differential equation would have only one constant, which you should be able to get using the x=0 at t=0 condition.
     
  7. May 14, 2009 #6
    My bad, ME = 7. I forgot to square. (Is is correct to assume that ME is constant?)
    Anyway, when I retried solving the differential equation with the initial conditions, I ended up getting 0 = 0 while solving for the constants. ????
     
  8. May 14, 2009 #7
    How would you solve this differential equation? The (dx/dt)^2 term throws me off.
     
  9. May 15, 2009 #8

    Delphi51

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    dx/dt = sqrt(2/3)*sqrt(4 - 4x^2)
    sqrt(2/3) dt = dx/sqrt(4 - 4x^2)
    Integrate both sides. Doesn't look bad - trig substitution if I'm not mistaken.
     
  10. May 15, 2009 #9
    Yes! I got
    [tex]x=\frac{\sqrt{7}}{2}\sin{\sqrt{\frac{4}{3}}\,t}[/tex]
    That was much simpler than what I was doing.
    Thanks Delphi!
     
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