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Simple Harmonic Motion (will be the death of me)

  1. Dec 21, 2009 #1
    1. The problem statement, all variables and given/known data
    The motion of an object is simple harmonic with equation of motion x = (27.1 m) sin(16.0 t /s + 0.7). At what time after t = 0 will the displacement reach its first maximum (where velocity equals zero)?

    2. Relevant equations

    (displacement)=Asin(freq.*t+phase)

    3. The attempt at a solution
    "(27.1)(sin((16x)+.7))" into calculator. Where y=displacement and x=time.
    I calculate the first maximum to be at (5.58,27.1). Since the question asks for the time my answer would be 5.58 seconds. But this is wrong, and I'm not sure where I went wrong..

    Any help greatly appreciated, Thanks!
     
  2. jcsd
  3. Dec 21, 2009 #2

    rl.bhat

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    Hi -Chad-, welcome to PF.
    At the maximum the velocity is zero. So find dx/dt and equate it to zero. From that find arc cos to find t.
     
  4. Dec 21, 2009 #3

    rock.freak667

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    How did you get t=5.58s ? If I put that value into the equation for x, I don't get 27.1. (Unless my calculator is messing up)
     
  5. Dec 21, 2009 #4
    Does 3.196 s sound like a better answer?
    I changed my calculator from degrees to radians. (if this fixes my problem, sorry for the stupid mistake)

    If not, rl.bhat, I do not understand what you want me to do. If I take the derivative at the maximum, where velocity is zero, it will also be zero.
     
  6. Dec 21, 2009 #5

    rl.bhat

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    In the problem dx/dt = (27.1m)(16.0)cos(16*t + 0.7) = 0.
    Now at what angle cosθ = 0?
    Equate that angle to 16*t + 0.7 to find t.
     
  7. Dec 21, 2009 #6

    ideasrule

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    How did you get 3.196 s? Derive x = (27.1 m) sin(16.0 t /s + 0.7) and you'll get the equation for velocity. Set the velocity to 0 and you'll be able to solve for t.
     
  8. Dec 21, 2009 #7
    ?? ok, first rl.bhat, cos90=0 90=16t+.7 89.3=16t 5.58=t (same answer i already got)

    ideasrule, i get the same answer when doing what you want me to do. I'll write out my derivation just for thoroughness.

    ________________d/dt[(27.1)(sin(16t+.7))]
    product rule______(27.1)(d/dt[sin(16t+.7)])+(d/dt(27.1))(sin(16t+.7))
    d/dt(sinx)=cosx___(27.1)(cos(16t+.7))+(0)(sin(16t+.7))
    ________________27.1cos(16t+.7)

    27.1cos(16t+.7)=0
    cos(16t+.7)=0
    cos^-1(cos(16t+.7))=cos^-1(0)
    16t+.7=90
    16t=89.3
    t=5.58
     
  9. Dec 21, 2009 #8
    Try using radians, 16t+0.7 = Pi/2
     
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