Seemingly simple kinematics exercise

[dunn]

Homework Statement

Rigid body
The plane rigid body ABCD shown in the figure has rectangular shape and
slides along the orthogonal runners X, Y. The points A and B are always in
contact with the ranners.
Data: ω=3 rad/s (angular velocity counterclockwise); \Theta=30°; AB=CD=1m;
AD=CB=0.5m.
Determine:
The A and C velocities;
The position of instantaneous rotation center;
The value of \Theta for which the horizontal velocity component of D is null.

Homework Equations

Fundamental geometric identities

The Attempt at a Solution

Without knowing the center of rotation or the relevant velocities a priori I don't know how one can be solved for the other. So really, I can't make it past the first step.

The instructor gave us this solution which to me makes no sense.

He claims that V_a = V_a/b + V_b

and that V_a/b = ω*AB

But this makes no sense since it assumes the center of rotation is at point B, or else V_a/b wouldn't be given as the product of ω times the length of segment AB.

Am I missing something or did the instructor forget to inform us that ω refers to point B?

 

[CWatters]

I may not get chance to come back to this for a few days and I'm not even 100% sure but...

Lets call the instantaneous center of rotation point O. I believe the walls must be tangential/right angles to OB and OA.

 

[dunn]

Here's the (sloppy) solution as given to us. Does this make any sense?

 

[dunn]

Hrm, I was completely overlooking the fact that in rigid bodies and angular velocity and angular acceleration are independent of the reference point of the body.

It makes sense now.