Normal-Tangential Coordinate for Ball Moving in Circle

[bio667]

Homework Statement

The problem is:

The ball B is traveling on a horizontal circle table. It is attached to the string that leads through a hole in the center of the circle. At the very beginning, the ball B is traveling around a circular path counter-clockwise, then the cord is pulled down through the hole with a constant speed. Sketch both polar coordinate and normal-tangential coordinates.

Homework Equations

The Attempt at a Solution

My issues lie with understanding what the normal and tangential coordinates would be in this case.

Would U_θ and U_t be the same in this case? Also for the U_n would it just be parallel to the axis in the center and go through the ball? Or am I thinking about this wrong and is it just a simple 2D circular motion still.

 

[carllacan]

I don't know what the normal and tangential coordinates are. Maybe you mean the tangential and normal components of velocity or acceleration?

If the string is pulled down (so the portion that goes from the hole to the ball gets shorter) the ball will move towards the hole, so its velocity will have a normal component. What will happen to its tangential component? (here normal and tangential are to be understood as "to the trajectory")

 

[bio667]

Okay so if I'm drawing the normal and tangential coordinates with respect to the motion of the ball on the table, than the normal component U_n should still point from the ball towards the center of the circle. My understanding for the tangential component U_t will be tangent to the path so can I therefore assume that the path is still a circle and therefore U_t will equal the polar coordinate U_θ?

 

[carllacan]

Yes, I would say you can assume the path is still a circle, as long as the string doesn't get shorter very fast.

 

[bio667]

This is an image of the problem for posterity's sake.