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Homework Help: Solving a Differential Equation from story problem

  1. May 15, 2010 #1

    TG3

    User Avatar

    The problem statement, all variables and given/known data

    A mass weighing 2 lbs stretches a spring 6 inches. If the mass is then pulled down an additional 3 inches and then released, and there is no damping, determine the position u of the mass at ay time T according to the form Acos(wT-phi).


    The attempt at a solution

    The k of the spring is 2 lbs/ 6 inches, or 1/3 lbs/inches. The mass is 2lbs/32ft/sec^2, or 1/16.

    From this I get the differential equation:
    1/16U'' +1/3U = 0

    Making U e^rt and skipping a few steps
    1/16r^2+1/3 = 0
    Using the quadratic:
    r = 0+- 8 sqroot(1/12) (-1/12 actually, but it doesn't matter here.)

    So: y = C1 cosine (8 sqroot(1/12)T) + C2 sine (8 sqroot(1/12)T)

    The initial condition of y(0) ought to be 3, since the spring was stretched 3 inches, and y'(0) ought to be 0 since it was released not pushed, correct? (I'm suspicious of this step.)

    If so, y(0) = C1 cosine (8 sqroot(1/12)T) + 0 (since the sine of 0 is 0).
    And furthermore, y(0) = C1, so C1 =3.

    y'(0) = [0] + C2 (8 sqroot(1/12)) cosine (8 sqroot(1/12)T)
    y'(0) = C2 (8 sqroot (1/12))
    0 = C2

    So y=3cosine(8sqroot(1/12)T)

    From this, translating to the form of A cosine (wT-phi) isn't hard:
    A = (3^2)^.5 = 3
    W = 8 sqroot(1/12)
    And phi is tan^-1 (0/3) = 0

    My solution is 3 cosine 8(sqroot(1/12))T.

    Unfortunately, this is wrong. Where did I goof up?
     
  2. jcsd
  3. May 15, 2010 #2

    Mark44

    Staff: Mentor

    There are also two initial conditions. You should state them with your differential equation.
    Well, really it does, since it makes the difference between complex solutions for r and real solutions.

    In any case,
    Be consistent in your variables. The dependent variable started off as u; now it's y.

    It all depends on which direction you choose to be the positive y axis. I would be more inclined to say that y(0) = -3.
    It could be that the book's answer has rationalized the radical. What answer does the book give?
     
  4. May 15, 2010 #3

    TG3

    User Avatar

    The answer given is:
    .25 cos(8t)
     
  5. May 16, 2010 #4

    Mark44

    Staff: Mentor

    Your spring constant is incorrect, which leads to an incorrect differential equation. The mass is 1/16 slug (1 slug = 32 lb/(ft/sec)^2). Your spring constant is 2lb/6 in = 4 lb/ft. The spring constant should not be in terms of inches.

    All length measures, including the initial condition, should be in terms of feet, not inches.
     
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