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Homework Help: Simple Harmonic Motion and time period

  1. Sep 5, 2005 #1
    A particle follows SHM with its eqn.of motion being

    s(x,y)=Asinwt i + 3Asin3wt j

    What is its time period? Also find the expression for total mechanical energy wrt time.

    I tried it in the following way
    V=Awcoswt i + 3Awcos3wt j = Aw[(coswt)^2 +9(cos3wt)^2]^(1/2)
    KE=0.5m(Aw)^2[(coswt)^2 +9(cos3wt)^2]
    a= -[A(w^2){sinwt i + 9sin3wt j }]
    F=ma
    PE=F.s = -m[(Aw)^2{(sinwt)^2 + 27(sin3wt)^2}]
    Total energy=KE + PE
    = m(Aw)^2[0.5{(coswt)^2 +9(cos3wt)^2}- {(sinwt)^2 + 27(sin3wt)^2}]
    Is this correct? And how to calculate the time period?
     
  2. jcsd
  3. Sep 5, 2005 #2

    HallsofIvy

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    "V=Awcoswt i + 3Awcos3wt j = Aw[(coswt)^2 +9(cos3wt)^2]^(1/2)"

    This appears to be saying that
    cos wt+ 3cos 3wt= (cos2wt+ 9cos23wt)1/2 which is NOT true.
    [tex] (a+ b)^2 \ne a^2+ b^2[/tex]
    and
    [tex]a+ b \ne \sqrt{a^2+ b^2}[/tex].

    1. What is the period of cos wt?
    2. What is the period of cos 3wt?

    3. What is the least common multiple of those?
     
  4. Sep 5, 2005 #3

    andrevdh

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    Why don't you try the basic SHM formulae for the x and y motion separately - because energy is a scalar quantity you can add these up to get the total energy. That is
    [tex] U_x = \frac{1}{2}kx^2[/tex]
    for the potential energy and
    [tex] T_x = \frac{1}{2}mv_x^2[/tex]
    for the kinetic energy. Just remember that the
    [tex] \omega[/tex]
    differs for the two components and use the basic relation
    [tex] \omega = \sqrt{\frac{k}{m}}[/tex]
    Great - if you get the same answer ! It just means you have improved the confidence in your result. Anyway, by graphing the motion for a phase from 0 to pi for
    [tex]\omega t[/tex]
    the mass moves from 1 to 2 ... p to 5. Thereby the x motion have completed one oscillation, while the y motion have completed one oscilation at point a, another at b and another at point 5. Henceforth the motion repeats itself. The y motion therefore runs 3x faster than the x motion.
     

    Attached Files:

    Last edited: Sep 6, 2005
  5. Sep 7, 2005 #4
    While adding two vectors A and B which are mutually perpendicular
    A^2 +B^2=(A+B)^2. Hence I think I was right while finding the velocity. HallsofIvy.
    Also,I think i have figured out the answer , with help from andrevdh. The total enegy of particle=41m(Aw)^2=constant.The time period comes out to be 3w.
    But can't we find the time period of the particle by differentiating the equation for energy twice and using a=-xw^2? Can someone please tell me how can I find the time period here using this method?
     
  6. Sep 7, 2005 #5

    andrevdh

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    Homework Helper

    The period is per definition the amount of time needed to complete one oscillation. The attachment displays the motion for
    [tex]\omega t[/tex]
    ranging from
    [tex]0\ \rightarrow 2\pi[/tex]
    giving one oscillation in the x direction, but three oscillations in the y direction for the corresponding time. After this (1,2, ...,5) the motion will repeat itself. The angular frequency of the x-motion is
    [tex]\omega [/tex]
    while it is
    [tex]3\omega [/tex]
    for the y-motion. Clearly the period of the combined motion is therefore the period of the x-motion, which can be obtained from
    [tex]\omega=\frac{2\pi}{T} [/tex]
    See also 1,2 and 3 of HallsofIvy above.
     
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