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Nonuniform Circular Motion - Find Maximum Total Acceleration

  1. May 22, 2013 #1
    1. The problem statement, all variables and given/known data
    A turntable is rotating at a constant angular velocity of ω = 4.0 rad/s in the direction of a clockwise fashion. There is a ten-cent coin on the turntable, at a distance of 5 cm from the axis of rotation.

    Consider the time interval during which the turntable is accelerated initially from rest to its final angular velocity (ωf = 4.0 rad/s) . This is achieved with a constant angular acceleration (α) for 0.5 s.

    Find an expression for the magnitude of total acceleration of the coin in terms of α and ω and use this to determine the maximum total acceleration experienced by the coin.


    2. Relevant equations
    [itex]a=\sqrt{a_{r}^2+a_{t}^2}[/itex]
    [itex]a_{r}=ω^2r[/itex]
    [itex]a_{t}=αr[/itex]


    3. The attempt at a solution
    Using those equations, I find an expression for the total acceleration:
    [itex]a=r\sqrt{ω^4+α^2}[/itex]
    I don't understand why the problem is asking for a maximum total acceleration. I am already given the values of r, ω, and α, can I just substitute them in to find the answer? Another thing that came into my mind when the word "maximum" pops up is finding the derivative of a function of a(t) with respect to time, but apparently I do not have that function here...

    Any help would be greatly appreciated.
    LovePhys
     
  2. jcsd
  3. May 22, 2013 #2

    Doc Al

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    Staff: Mentor

    Well, since ω varies over the given interval, you have to determine its maximum value. But I suspect you can handle it. :smile:
     
  4. May 22, 2013 #3
    @Doc Al: Thanks for the hint. I think the maximum angular velocity is also the final angular velocity ω=4rad/s, isn't it?
    [itex]α=\frac{Δω}{Δt}=\frac{4}{0.5}=8 (rad/s^2) [/itex]
    So finally maximum [itex] a=0.05\sqrt{4^4+8^2}≈0.894(m/s^2) [/itex]
     
  5. May 22, 2013 #4

    Doc Al

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    Staff: Mentor

    Exactly.

    Looks good to me.
     
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