Dynamics: relative-motion analysis using rotating axis

[cherry]

Homework Statement

The disk rotates with the angular motion shown. Determine the angular velocity and angular acceleration of the slottted link AC at this instant. The peg at B is fixed to the disk.

Relevant Equations

v_b = v_a + Ω x r_b/a + (v_b/a)_xyz

Hi, I'm struggling to understand how two different coordinate systems come together.
In this question, I reached my answer for the angular velocity which was incorrect.
The correct answer is ω = 0 rad/s

I think my biggest error happen in the step where I equated v_B to each other.
When I did that should I have converted the xy coordinate system to match the XY coordinate system?

I am also confused regarding that as I have tried that for another question and I still did not get the correct answer.
What am I missing?
Help would be appreciated, thanks!

 

[Lnewqban]

I would make one of the axes of your Cartesian coordinate system coincide with the geometrical length of link AC.
Note that even the rotation centers of the disc and the link are not on the same vertical line.

The reason for the lack of angular velocity of link AC at the represented moment in time is that the tangential velocity of the peg located at B is aligned with the pivot of link AC.
Therefore, there is no component of that velocity vector that could induce an oscilation of AC at that instant.

Please, see:
https://507movements.com/mm_100.html

https://en.m.wikipedia.org/wiki/Quick_return_mechanism

Note how the angular velocity and acceleration of the link changes according to the radial position of the peg, which keeps a constant angular and tangential velocity.