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Determine the Angluar Velocity of the Slender Rod.

  • #1
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Homework Statement


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

Homework Equations





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.
 

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Answers and Replies

  • #2
TSny
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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.
 
  • #3
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Do you mean something like:

x = h/tan(θ)
 
  • #4
TSny
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Yes. And 1/tanθ equals another trig function.
 
  • #5
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Do you mean 1/tan(θ) = cos(θ)/sin(θ) ?
 
  • #6
TSny
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No. cotθ
 
  • #7
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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?
 
  • #8
TSny
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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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