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Homework Help: Finding the tangential component of acceleration

  1. Jan 31, 2010 #1
    EDIT: I meant radial in the title.
    1. The problem statement, all variables and given/known data
    A ball is going around in a circle of radius 4 m.

    It goes with a constant angular velocity of (13 rad/s)[tex]\hat{k}[/tex] for 0.5 s. After that, it takes 4 s to come to a complete stop.

    Find the radial component of the ball's acceleration at 2 s.

    2. Relevant equations



    3. The attempt at a solution
    My book says that to use the formula ar= w2r. However, w is changing, so I don't see how I can use that!

    The only thing I can think of is to find the angular acceleration:
    [tex]\alpha[/tex] = w0 + [tex]\alpha[/tex]0(t)
    0 = (13 rad/s) + [tex]\alpha[/tex]0(4 s). Solving for [tex]\alpha[/tex] gives -3.25 rad/s2[tex]\hat{k}[/tex]

    Then I use another formula to find the angular velocity at 2 s:
    wfinal = winitial + [tex]\alpha[/tex](t)
    wf = (13 rad/s) + (-3.25 rad/s2)(2 s)
    wf = 6.5 rad/s [tex]\hat{k}[/tex]

    Then use that first formula:
    ar = (6.5 rad/s)2(4 m)
    ar = (169 rad/sm)[tex]\hat{k}[/tex]

    Are those units correct? Really the formula for ar = dVt / dt, but is what I did ok?

    Also, as an aside, the TANGENTIAL part of the angular acceleration would stay the same all the time, right? If I calculated it at 1 s, 2s, ... 4.3 s, it would not change?
     
    Last edited: Jan 31, 2010
  2. jcsd
  3. Jan 31, 2010 #2

    tiny-tim

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    Hi jumbogala! :smile:

    (have an alpha: α and an omega: ω :wink:)
    Your calculations are fine, except that you've misread the question …

    you only have 1.5 s of acceleration at 2s. :wink:

    You're right to be worried … the units in the formula v = ωr are cm/s = rad/s times cm … and in the formula a = ω2r are cm/s2 = rad2/s2 times cm … the radians are dimensionless, and they just drop out. :wink:
    Not following this. :redface:

    "tangential part of the angular acceleration" makes no sense.

    Do you mean the tangential part of the ordinary acceleration (ie, the tangential acceleration)?

    If so, then yes, you're correct … for fixed radius, that's simply dv/dt, the derivative of the speed (= r dω/dt = rα). :smile:
     
  4. Jan 31, 2010 #3
    Thank you!

    I'm confused about those units still, though. Why are we using cm, if the radius is given in m? Is that just a convention?

    Also, the rad drops out for a, but if I just want to write ω, can I still write rad/s? (Instead of 1/s).
     
  5. Jan 31, 2010 #4

    tiny-tim

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    oh, I made a mistake … I thought the question used cm. :redface:
    Yes, ω is rad/s. :smile:
     
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