Geneva wheel problem

1. The problem statement, all variables and given/known data
I’ll attach the screenshot, it has everything.

2. Relevant equations
This is in the section of absolute motion analysis.

3. The attempt at a solution
I actually was able to solve the problem correctly, but it was annoyingly complicated and I’m curious as to whether there’s a better way.

So, first off, I wanted to find a way to write [itex]\varphi[/itex] in terms of [itex]\theta[/itex].

So, recognize that tan[itex]\varphi[/itex] = Dsin[itex]\varphi[/itex]/Dcos[itex]\varphi[/itex]

The D*sin[itex]\varphi[/itex] is always going to be equal to 100sinθ.

For cos[itex]\varphi[/itex] it was a bit more tricky, but it’s easy to recognize the relationship because the distance stays constant.

100√2 – 100cos(θ) = D*cos[itex]\varphi[/itex]

So now we have:

tan[itex]\varphi[/itex] = 100sinθ/(100√2 – 100cosθ)

And now I take the derivative. This is a quotient rule, and the derivative is going to be with respect to time, so through the chain rule we will have our angular velocities pulled out. Note that this derivative has been simplified from the raw form just a little.

sec2[itex]\varphi[/itex]*ωphi = ωθ (√2*cosθ – 1)/(√2 – cosθ)2

So now the next conundrum is how to get rid of the damn sec2[itex]\varphi[/itex]

Well, we recongnize that sec = Hypotenuse / Adjacent side,

And we’ve solved for the adjacent side already, it is equal to the cos[itex]\varphi[/itex] equation, which is:

100√2 – 100cosθ

But what is the hypotenuse? Well, by the pythagorean theorem:

H2 = (100sinθ)2 + (100√2 – 100cosθ)2

So, now we have our values, and since it is sec2[itex]\varphi[/itex], we will use:

H2 / B2

Which gives (after some more excessive simplifying):

ωphi(3 – 2√2 cosθ)/(√2 – cosθ)2 = ωθ (√2*cosθ – 1)/(√2 – cosθ)2

Now notice the denominators are the same, they cancel out. Then just divide to solve for the angular velocity with respect to [itex]\varphi[/itex]

You GET:

ωphi = ωθ (√2 cosθ – 1)/(3 – 2√2 cosθ)

And that IS the correct answer, it’s just that it took a really long time and a ton of failed attempts before I finally solved it. I’m just wondering if there’s a better way! Thanks!

Attached Images
File Type: png physics question.png (190.9 KB)

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