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Simple harmonic motion

  1. Apr 28, 2003 #1
    "simple" harmonic motion

    Trying to solve a harmonic motion problem (mass oscillating on a spring), given:
    amplitude=.100 m
    k=8.00 N/m
    mass=.5 kg
    to solve for time elapsed for the mass to travel from x=0.00 to x=.080 m.
    It's easy to solve for w (omega) = sqrt(8/.5) = 4, and using the equation for shm from my physics textbook I got this equation:
    .08 = .1 cos(4t-pi/2)
    and then
    t = ((arccos 0.8) + pi/2)/4 = .554 sec. which unfortunately is wrong. This turns out to be the time it takes for the mass to go past .08 m, past its maximum displacement of .1 m, and BACK to .08 m.

    Now, this equation is equivalent:
    .08 = .1 sin(4t)
    Check it...the graphs are exactly the same. But this one solves to
    t = .232 sec. which is the answer given by the book, and tracing the graph confirms that this is the correct answer.

    So, there must be a way to solve the first equation [.08 = .1 cos(4t-pi/2)] to get the correct answer of .232 sec., but I can't see it.

    What am I missing?
  2. jcsd
  3. Apr 29, 2003 #2

    Claude Bile

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    Science Advisor

    When using inverse trig operations, be aware that there is more than one answer, for example: -

    sinx = 0
    x = arcsin(0)
    x = n(pi) where n is an integer.

    When applying such maths to physics, it is necessary to choose which answer you require, which may, or may not be obvious.

    In your question, you assumed that the only solution to arccos(0.8) is 0.6435... when in fact, -0.6435 is an equally valid solution. (This is a consequence of cos(x) = cos(-x)). Substituting this in to your equation gives the answer provided by the book.

    If you are uncomfortable with having to choose which answer is correct without using educated guesswork or drawing graphs, use the equation for the velocity of the object to find out whether each solution corresponds to a positive or negative velocity.
  4. Apr 29, 2003 #3
    Thanks Claude, that was very helpful.
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