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Rotation of a fan

  1. May 2, 2010 #1
    Rotation of a Fan - Unsolved

    1. The problem statement, all variables and given/known data
    A ceiling fan with 85cm diameter blades is turning at 66 rpm. Suppose the fan coasts to a stop 26 s after being turned off.

    What is the speed of the tip of a blade 10 sec after the fan is turned off?

    2. Relevant equations
    [tex]\alpha=\frac{w_{2}-w_{1}}{\Delta t}[/tex]
    [tex]v=r\omega[/tex]

    3. The attempt at a solution

    First I converted 66 rpm to rad/sec, resulting in [tex]6.912[/tex] (http://www.wolframalpha.com/input/?i=66+rpm+to+rad/sec)

    Then I found the angular acceleration using [tex]\alpha=\frac{w_{2}-w_{1}}{\Delta t}[/tex] as [tex]\alpha=\frac{0-6.912}{26}[/tex], which resulted in [tex]\alpha=-0.265846[/tex].

    I then (this is where I think I have made a mistake) plugged this new alpha value back into that same equation to come up with this: [tex]-0.265846=\frac{w_{2}-6.912}{10}[/tex], since I figured I need [tex]\omega[/tex] to use in the [tex]v = r\omega[/tex] equation. That resulted in [tex]\omega=0.425[/tex]/

    Now I used [tex]v = r\omega[/tex] as [tex]v = (.85)(0.425)[/tex], resulting in [tex]v = 0.36[/tex], which was determined to be incorrect. I'd start doing some trial and error, but the homework system cruelly deducts points for each incorrect answer.

    I also tried the following, but it also failed. I used the equation of motion [tex]v=v_{0}+at[/tex] as [tex]\omega=r\omega_{0}+(\alpha * t)[/tex] with the values of [tex]\omega=0.36125 + (-0.265846)(10)[/tex], resulting in [tex]4.3[/tex]. This answer is incorrect.
     
    Last edited: May 2, 2010
  2. jcsd
  3. May 2, 2010 #2

    rock.freak667

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    What exactly are you supposed to find? You didn't really state it.
     
  4. May 2, 2010 #3
    My apologies, post edited.
     
  5. May 2, 2010 #4

    rock.freak667

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    Well your diameter is 85cm. So your radius wouldn't be 85cm now would ?
     
  6. May 2, 2010 #5
    Well the question states that the blades are 85 cm in diameter, so the diameter of the entire fan, I'm assuming, would be [tex]f(x) = (85*2)+x[/tex] where [tex]x[/tex] is the width of the round base in the center from which the fan blades rotate.
     
  7. May 2, 2010 #6

    rock.freak667

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    Not sure where you got that formula from, but my line of thinking was that the distance from the center of rotation to the tip of a blade should be the radius of the fan.
     
  8. May 2, 2010 #7
    Here, I'll draw a picture to explain my thoughts.

    After drawing it, I should reformulate my previous equation to a simple [tex]d = r + r[/tex], where d is the diameter and r is the length of a blade.

    [PLAIN]http://mikederoche.com/temp/fan.png [Broken]

    The center of rotation, of course, would be where the two blades meet in the center of that big black dot. Each blade is 83 cm, so the diameter of the entire fan would be 83+83=166. So the center of rotation would lie at (just undoing the previous equation...) 166/2 = 83 cm.

    I also tried the following, but it also failed. I used the equation of motion [tex]v=v_{0}+at[/tex] as [tex]\omega=r\omega_{0}+(\alpha * t)[/tex] with the values of [tex]\omega=0.36125 + (-0.265846)(10)[/tex], resulting in [tex]4.3[/tex]. This answer is incorrect.
     
    Last edited by a moderator: May 4, 2017
  9. May 2, 2010 #8

    ehild

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    Gold Member

    If fan looks as you draw, they would speak about the "the length of the blades" That fan probably is made without that black dot, just screwing two or more blades together. The blades cover a circle when revolving, the diameter of the circle is 85 cm. Trust in rock.freak and try to use the half for r, and see if it is the correct result. Otherwise, everything else you did is fine.

    ehild
     
  10. May 2, 2010 #9
    Hm, well I did try what rock.freak suggested, but the answer was still marked incorrect.

    [tex]v = (.85/2)(0.425) = 0.18[/tex].
     
  11. May 2, 2010 #10
    Nevermind, I got it. Ended up being off by a power of 10... the answer was 1.8, not .18!

    Thanks for your help, everyone.
     
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