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Angular Momentum Problem

  1. Dec 7, 2016 #1
    1. The problem statement, all variables and given/known data
    "The flywheel rotates with angular velocity of w=0.005theta^2 rad/s. Determine the angular acceleration after it has rotated 20 revolutions.

    2. Relevant equations
    I thought the equations were all but self-evident using the problem description! (See below.)
    alpha=dw/dt

    3. The attempt at a solution
    The angular velocity is merely a=0.01theta. I just plugged in 20*2pi for the acceleration at that time and was... wrong. Massively wrong, in fact. The book solutions agrees with me about my equations for w and a, but they actually multiple the acceleration and the velocity together!

    The book defines alpha as dw/dtheta... but then, in this solution set, states that alpha=w(dw)(dtheta), which is not something I can make *any* sense out of. They then use this value of alpha to compute the acceleration over a rotation of (40pi), complete with theta cubed.

    What happened here? I can't make heads or tails of it.
     
  2. jcsd
  3. Dec 7, 2016 #2
    Are you sure of this definition?
     
  4. Dec 7, 2016 #3

    hilbert2

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    You need to use the chain rule of differentiation when applying ##\frac{d}{dt}## on ##[\theta (t)]^2##.
     
  5. Dec 7, 2016 #4
    due to the chain rule of differentiation it is ##\frac{dw}{dt}=\frac{dw}{d\theta}\frac{d\theta}{dt}## and it is also ##\frac{d\theta}{dt}=w## so what your book says is correct.
     
  6. Dec 7, 2016 #5
    your book is wrong, it is not $$\alpha = w\times dw \times d\theta$$

    it is,
    $$\alpha = {d \omega \over dt} = \underbrace{{d\omega \over d\theta } \times {d\theta \over dt}}_{\text{chain rule}} = ?$$
     
  7. Dec 7, 2016 #6

    haruspex

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    .... assuming k2 meant alpha=w(dw)/(dtheta) and not
     
  8. Dec 7, 2016 #7
    guys I believe the post has some typos, I believe he meant to write that his book says that a=wdw/dtheta , ##a=w\frac{dw}{d\theta}## in latex :).
     
  9. Dec 7, 2016 #8
    I'll give you a screenshot if you want!

    The book defined alpha (angular acceleration, I take it) merely as $$\frac{dw}{d\theta}$$. Then is also says that $$\alpha d\theta = \omega d\omega$$.

    The exact answer has the line in it:
    $$\alpha=\omega \frac{d\omega}{d\theta}$$ and using the given (.005theta) value and its derivative, multiplies them together and posits the value derived thereof to be the answer.
     
  10. Dec 7, 2016 #9

    haruspex

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    That is a typo in the book. It should by dt in the denominator.
    That is correct, but you omitted the / when you wrote alpha=w(dw)(dtheta).
     
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