Determine the Angluar Velocity of the Slender Rod.

[Northbysouth]

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.

 

[TSny]

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.

 

[Northbysouth]

Do you mean something like:

x = h/tan(θ)

 

[TSny]

Yes. And 1/tanθ equals another trig function.

 

[Northbysouth]

Do you mean 1/tan(θ) = cos(θ)/sin(θ) ?

 

[TSny]

No. cotθ

 

[Northbysouth]

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?

 

[TSny]

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.