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Instantaneous angular speed question

  1. Jul 21, 2016 #1
    1. The problem statement, all variables and given/known data
    How do you calculate instantaneous angular speed?



    2. Relevant equations
    I have been told that it is when delta t approaches 0, so its just the derivative of delta theta over delta t.



    3. The attempt at a solution
    does it involve tangent line like normal instantaneous velocity ?
     
  2. jcsd
  3. Jul 21, 2016 #2

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    You are on the right track involving derivatives. See below.
    Yes, this is true.

    (I'm making a few assumptions about [itex] \theta (t) [/itex] not having a discontinuity at the point of t in question. But for simplicity sake, I'll just say, "yes, that's true," which it is for most cases.)
    Be careful here. It is not the derivative of [itex] \frac{\Delta \theta}{\Delta t} [/itex]. Be careful of your wording there.

    Rather, the instantaneous angular velocity [itex] \omega (t) [/itex] is the derivative of [itex] \theta (t) [/itex] with respect to t (not "over [itex] \Delta t [/itex]").

    Making the stipulations about smooth functions (not having discontinuities and so forth),

    [tex] \lim_{\Delta t \rightarrow 0} \frac{\Delta \theta}{\Delta t} = \frac{d}{dt} \{ \theta (t) \} = \omega (t) [/tex]

    That question has a simple answer, but I'll let you ponder that. If you objectively know the behavior a variable or function, say [itex] \theta(t) [/itex] which changes as a function of time, what is the instantaneous rate of change of that function? What is the derivative of a function?
     
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