Instantaneous Center of Zero Velocity for a Yo-Yo Unravelling

[theBEAST]

Homework Statement

Here is the question along with the solution:

I was wondering why the ICZV is at point A. Is it because the string is not slipping and thus the yo-yo is essentially "rolling" on the surface of the rope? Kind of like a bike wheel rolling on the ground?

 

[Basic_Physics]

Correct. You would get exactly the same motion if you would wrap some string around a wheel and roll it along a horizontal surface. The string would unwrap and lie on the road behind it. There is a one to one relationship between the distance the wheel travels along the road and the amount by which the string unwraps. The wheel is dropping the string unto the road as it rolls along. In this case the unwrapping is just downwards. A point on the string just below A is approaching the vertical line from the right-hand side in this case and will be stationary the rest of the time once it is unwrapped from the yo-yo. The ICZV is traveling downwards along the sting as COM of the yo-yo goes down. In reallity I think the COM of the yo-yo will be below the support at the top (the point where the top end of the string is tied), so the sting will make a slight angle with the vertical that will decrease as it goes down.