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Homework Help: Simple Harmonic Motion of a sewing machine

  1. Feb 12, 2010 #1
    1. The problem statement, all variables and given/known data
    A sewing machine needle moves up and down in simple harmonic motion with an amplitude of 1.27cm and a frequency of 2.55Hz. How long does it take to travel 11.43cm?


    2. Relevant equations
    x=A*cos(w*t)
    w=2[tex]\pi[/tex]*f
    w=2[tex]\pi[/tex]/T
    T=1/f


    3. The attempt at a solution
    I can't determine the proper way to solve for t, since that's what the question is asking for. Every time I try to solve for t, the calculator gives me a 'math error' message and won't let me compute.

    The answer is 0.878s, but I need to know how to get it.
     
  2. jcsd
  3. Feb 12, 2010 #2
    Every time the the needle goes through one full period it travels 4 amplitudes.
     
  4. Feb 12, 2010 #3

    tiny-tim

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    Welcome to PF!

    Hi rspbrrylmnd! Welcome to PF! :smile:

    (have a pi: π and an omega: ω :wink:)

    Show us what you tried, including your wave equation. :smile:
     
  5. Feb 12, 2010 #4
    Ok. I tried first to solve the equation for t before putting in any numeric values. Every time I would do that, though, I'd get the same equation.

    x=A*cos(w*t)
    x/A=cos(w*t)

    From here, I assumed that the next logical step would be to take the inverse cosine of (x/A)

    cos[tex]^{-1}[/tex](x/A)=w*t

    Then I divided the entire thing by w to get:

    [cos[tex]^{-1}[/tex](x/A)]/w=t

    This is the same equation I come up with to solve for the time it takes for the needle to travel 11.43 cm. Whenever I plug in the numeric values, though, I can never calculate this equation because 11.43/1.47=9, and one cannot take the inverse cosine of anything larger than 1.

    Is there a different relationship between these specific variables that I'm not aware of that would allow me to solve for the time? I'm simply at a loss and/or I have a mental block that is just not allowing me to see this problem from a different angle than by the process I have already tried.
     
  6. Feb 12, 2010 #5
    x is the position and not the distance traveled.
    x is always between -1.27 and + 1.27 cm.
    You may start by calculating how many full periods (actually full half-periods) are required and then us the formula for the remainder. See post by Jebus_Chris too.
     
  7. Feb 12, 2010 #6

    tiny-tim

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    :wink:
     
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