I would like to solve this question without using conservation of angular momentum(because of some reason I'll elaborate later).
So imagine that we have a pole with radius $r$ and a ball attached to a rope of length $R$.
Initially the distance between the ball and the attach point is $R$(which means the rope is tight). The initial velocity of the ball is perpendicular to the rope. Now we know $R>r$. At a later time $t$, what is the direction and magnitude of the velocity of the ball?
The reason we cannot use conservation of angular momentum is because I don't assume that the ground is not moving. Instead I assume that the ground is the Earth. So if the rope applies a torque on the Earth, the Earth's angular momentum will increase. I always see people use conservation of angular momentum to solve such a question but not conservation of energy. And if we assume that the ground is not moving, there must be one of them that is not conserved! And I would like to know if the common way of doing it is right or wrong.
However I got stuck at the very beginning... Once I figured out the initial and final energy and momentum, I should be able to solve it. Any help would be much appreciated!
Note: The pole is vertical. And we can assume no gravity. Also I guess friction is important, otherwise the rope can't be wrapped around the pole. However let's assume there is no loss of energy thru friction. There is of course tension on the rope.
IMPORTANT REMARKS: It might look really simple to you at first. But it's actually as straightforward as it seems. When we solve this kind of problem in the standard first year physics text, we always assume conservation of angular momentum. However this would imply the violation of conservation of energy if we assume the ground does not move. If we do assume that the ground moves, both angular momentum and energy will not be conserved for the pole (because the system gives angular momentum and energy to the earth). In that case, if we take the limit as mass of the earth goes to infinity, which one is more conserved? (It has to be angular momentum, otherwise we would have used conservation of energy to solve this kind of problem in first year physics textbooks. But now I just want to justify it)?
THIS IS NOT A FIRST YEAR PHYSICS TEXTBOOK PROBLEM!
Also although I added the homework tag to this question, it's not an actual homework problem(as you can see from the date I posted it). So please feel free to write down any detail that you think might help explaining your solution. Thanks!