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Determine the Angluar Velocity of the Slender Rod.

  1. Mar 30, 2013 #1
    1. The problem statement, all variables and given/known data
    Calculate the angular velocity w of the slender bar Ab as a function of the distance x and the constant angular velocity w0 of the drum.

    I have attached an image of the question

    2. Relevant equations

    3. The attempt at a solution

    x = √(x2+h2)cos(θ)

    x' = -√(x2+h2)sin(θ)θ'

    θ' = -x'/√(x2+h2)sin(θ)

    θ' = -x'/h

    But I'm not sure where to go from here. I'm having trouble dealing with x'.

    Any advice would be appreciated.

    Attached Files:

  2. jcsd
  3. Mar 30, 2013 #2


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    When taking the time derivative, you neglected the time dependence of x on the right side of the equation. It will be easier if you start over and use a different trig function than cosine.
  4. Mar 30, 2013 #3
    Do you mean something like:

    x = h/tan(θ)
  5. Mar 30, 2013 #4


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    Yes. And 1/tanθ equals another trig function.
  6. Mar 30, 2013 #5
    Do you mean 1/tan(θ) = cos(θ)/sin(θ) ?
  7. Mar 30, 2013 #6


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    No. cotθ
  8. Mar 30, 2013 #7
    With this information I've managed to get:

    x = h/tan(θ)

    x = hcot(θ)

    x' = -hcsc(θ)

    θ' = -x'/hcsc(θ)

    θ' = -x'sin2(θ)/h

    And I know that h = √(x2+h2)sin(θ)

    θ' = (-x'sin2(θ))/√(x2+h2)sin(θ)

    Which simplifies to:

    θ' = -x'sin(θ)/√x2+h2)

    I also recognized that sin(θ) = h/√(x2+h2)

    θ' = -x'h/(x2+h2)

    At this point I'm a little unsure of the x' and what to substitute it with. I know that:

    v = wXr = rwcos(θ)

    I'm unsure if what I've written here for v is correct. Particularly, as the given answer does not have a cos(θ) in it.

    Could someone clarify this for me?
  9. Mar 31, 2013 #8


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    x'= v = tangential speed of rim of drum = rω

    Note: from the equation θ' = -x'sin2(θ)/h it would be easier to substitute for sinθ rather than substitute for h.
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