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Thread: Equation of motion

  1. #1
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    Equation of motion

    Equation of motion-screenshot-11-.pngNeed help finding the equation of motion. Have no idea where to even start.
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  2. #2
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    Re: Equation of motion

    If you have no idea, why were you given the problem?

    Anyway, your attachment is not readable.
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    Re: Equation of motion

    I can see the attachment but it doesn't help clarify your question. The entries in the attachment look correct, at least for part (c). So, what exactly is your question?
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    Re: Equation of motion

    I do not understand how to find the equation of motion
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    Re: Equation of motion

    Quote Originally Posted by shunae95 View Post
    I do not understand how to find the equation of motion
    So type out exactly what you are asking. Why do you think that we should struggle to read your postings?
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  6. #6
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    Re: Equation of motion

    A mass of 200 grams stretches a spring 49/80 meters.
    I've found the spring constant k.
    m= 200g = .2kg
    L= 49/80 = .6125

    k= 3.2

    w^2= 16

    I've found w, T, and f for the free undamped motion.
    w= 4
    T= 1.57
    f= .637

    The general solution, I've found to be
    u(t) = c1cos(4t)+c2sin(4t)

    So now the question is: Suppose that the mass initially starts at rest in its equilibrium position. Solve for the equation of motion.
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  7. #7
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    Re: Equation of motion

    So far so good. At time t=0 the spring is in the equilibrium position, so u(0) = 0. From that it must be that the unknown constant c1 = 0, and your equation is now:

     u(t) = C_2 \sin(\omega t).

    Now take the derivative:

    u'(t) =\omega  C_2  \cos(\omega t)

    At time t = 0 you say that the mass is at rest, so u'(0) = 0. Which means c2 = 0.

    So the equation of motion is: u(t) = 0.

    In other words it doesn't move at all. If a spring/mass is in the equilibrium position and is not moving, there's nothing that's going to make it start moving. Hence u(t) = 0. Usually for these type of problems they give you an initial deflection, or velocity, or acceleration, but in this case it's simply at rest.
    Last edited by ChipB; Nov 2nd 2016 at 06:34 AM.
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  8. #8
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    Re: Equation of motion

    Thanks for you help. I tried working it out and trying your solution but that still did not give me the correct answer.
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  9. #9
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    Re: Equation of motion

    Quote Originally Posted by shunae95 View Post
    Thanks for you help. I tried working it out and trying your solution but that still did not give me the correct answer.
    Please write the problem out exactly as stated. As I noted the problem as you wrote it is a bit strange - I've never seen a problem like this where there is no initial displacement and no initial velocity. But I wonder - the problem says the mass starts at its "equilibrium position," but perhaps what they meant to say is that it starts at its "unstretched position?"
    Last edited by ChipB; Nov 2nd 2016 at 09:35 AM.
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  10. #10
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    Re: Equation of motion

    OK, I see what the problem is. Looking at the attachment it appears that this problem is a continuation of part (c), not a problem on its own. So there is an exterior force of 6 sin(wt) acting on the mass. The basic equation governing all this is:

     \sum F = ma

    so,

     6 sin(2 t) - kx = m\ddot x

    Rearrange:

     \frac {6 \sin(2t)} m = \ddot x + \frac k m x

    Using the coefficients you already worked out yields:

     \frac {6 \sin(2t)}{0.2 kg} = 12A \cos(2t) + 12B \sin(2 t)

    from which

    A = 0, \ B=\frac {30}{12} = 2.5

    You have been told that the equation of motion is x(t) = A \cos(2t) + B \sin(2t). Can you take it from here?
    Last edited by ChipB; Nov 2nd 2016 at 09:43 AM.
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