Suppose there is a lever arm fixed at one end, and it is parallel to the ground. There is an object resting somewhere on top of the lever arm (the object is not attached to the lever). At the moment when the lever arm is released, does the object have any angular velocity? According to my physics teacher, the answer is yes. I know the approximation $\sin y \approx y$ for $y \approx 0$ but I fail to see how this situation (from the object's perspective) is different than just a regular free fall situation.
Angular velocity is $ v/ r$ while angular acceleration is $ a/r$.
Since the block has a vertical velocity(assumption :it was at the end of the rod) after the rod/lever is released, it also has an angular velocity about the point where the lever has been pivoted. If the block is near to the pivot then for some time the block will tend to slide down the lever as well as go down, but even in this case you can find out angular velocity with respect to pivot.
I would say, from your description, that at the instant of release the answer is yes. The Instant before release the answer is no. If you think of a delta t over which you measure the positions, shrink delta t to an infinitesimal dt. Can you tell me at what point you would go from moving to not moving?
But, what level of a problem is this? It is actually much more complicated. Since the end of the arm is fixed and the arm rotates about that fixed point, the gravitational force on the portions closer to the pivot is exerting a torque on the arm. It is restrained from falling as fast as it could if unrestrained. (This is why the top of a falling tree bends so far upward. It may look like wind or air resistance, but it is the end being "whipped" downward by the torque and moving much faster than if it had fallen freely).
This means along some portion of the arm closer to the pivot your object will stay in contact with the arm since it is also restrained from free fall and it will also rotate as it lies on the arm.
If the object is near the free end of the arm, it will fall free because the arm will be torqued out from under it at a rate faster than free fall. Is your teacher addressing all this? Or is this a concept physics question about when things start to happen?