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Manipulating Simple Harmonic Motion Equation

  1. Apr 5, 2010 #1
    1. The problem statement, all variables and given/known data
    An object moves with simple harmonic motion. If the amplitude and the period are both doubled, the object's maximum speed is:
    A) Quartered
    B) Halved
    C) Quadrupled
    D) Doubled
    E) Unchanged

    2. Relevant equations
    x(t) = Acos(wt + [tex]\varphi[/tex])
    v(t) = -wAsin(wt + [tex]\varphi[/tex])

    3. The attempt at a solution
    Since 2 double the period (T) is to halve the frequency (f) because of (f = 1/T), and since omega (w) = 2[tex]\pi[/tex]f...then w will be halved as well (right?):

    f = 1/2T --> .5f --> 2[tex]\pi[/tex].5f = .5w

    So that gives me: v(t) = -.5wAsin(.5w + [tex]\varphi[/tex])
    And with the amplitude doubled: v(t) = -wAsin(.5w + [tex]\varphi[/tex])

    Not sure if I did any of that correctly and I'm not sure what that means for my speed...is it halved? Any help would be appreciated. Thanks.
  2. jcsd
  3. Apr 5, 2010 #2
    You are correct in your logic that [tex]\omega[/tex] is halved as well. Lets look at what the question is asking. What is the *Maxiumum* velocity? You have written the equation for velocity:

    [tex] v(t) = A_0 \omega Sin(\omega t) [/tex]

    You should know that [tex] A_0 [/tex] is the maximum amplitude.

    Because the maximum value of any Sin function is 1, that means that the maximum velocity will be given by

    [tex] v_{max} (t) = A_0 \omega [/tex]

    Now what will happen when the amplitude and period are both doubled?
  4. Apr 5, 2010 #3
    Ah. Okay...I didn't know how to deal with the whole sine thing (obviously). So based on this Vmax equation the max velocity will remain unchanged. Correct?
  5. Apr 5, 2010 #4
    Exactly. When dealing with maximum values, your sin and cosine functions will generally disappear for the reasons stated above.
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