Homework Help: Tetherball - Classical Mechanics

1. The problem statement, all variables and given/known data
A small ball is attached to a massless string of lenght L, the other end of which is attached to a very thin pole. The ball is thrown so that it initially travels in a horizontal circle, with the string making an angle [tex]\theta_0[/tex] with the vertical. As time goes on, the string wraps itself around the pole. Assume that the pole is thin enough so that the length of string in the air decreases very slowly, so that the ball's motion may always be approximated as a cricle. Assume that the pole has enough friction so that the string does not slide on the pole, once it touches it. Show that the ratio of the ball's final speed (right before it hits the pole) to initial speed is [tex]v_f / v_i = \sin(\theta_0)[/tex]


2. Relevant equations
Conservation of energy applies. Conservation of angular momentum doesn't (the friction provides torque).


3. The attempt at a solution
I couldn't really do much on this problem. The [tex]\Delta L[/tex] after the mass has traveled once around the pole is:

[tex]\Delta L = - \frac{2 \pi d}{\sin(\theta)}[/tex]

I had forgot about the slow descending of the lowest contact point of the string with the pole. Without considering it, finding the velocity of the mass as a function of [tex]\theta[/tex] I could find (using the tension in the string and balancing weight):

[tex]v^2 = g L \frac{(\sin(\theta))^2}{\cos(\theta)} [/tex]

Differentiating this relation, and using energy after having expressed [tex]\Delta h[/tex] as a function of [tex]\Delta \theta[/tex], after a lot of calculation I found:

[tex](v_{n+1}^2 - v_n^2) = 8 \pi g d \frac{\cos{\theta_n}}{(\sin{\theta_n})^3}[/tex]

Anyway this relation is probably false for the mistake I spoke about before, and even if it was true I didn't know how to use it to find the final velocity. I'd like to find some sort of relation with [tex]d(v)[/tex] and [tex]d(\theta)[/tex] in order to integrate that.

4. Comments.

The problem comes from "Introduction to classical mechanics" by David Morin. It has 4 stars, which means it is quite hard. Thank you for any help, I have no idea on how to solve it .

 

If angular momentum is conserved, shouldn't the velocity of the ball be infinite when the radius becomes 0? This would not agree with the thesis...

 

Thank you all for the help. I can't work on the problem today cause I have a test this afternoon, I will try again in the next days. Hopefully this problem won't be in the test (the teacher sometime chooses some of the problems from that book).

Of course [tex]d[/tex] was the radius of the pole, I just forgot to say that.

 

I think is he may be right, actually. Please correct me if I say something false.

1) It is obvious that the kinetic energy is not conserved. Anyway, the total energy (gravitational + kinetic) of the system is conserved. This is a well-known fact about tetherball.
2) If the angular momentum is conserved, when the ball is very near the pole, it will have very high speed, because radius times speed would be a constant. This means that it can't have just [tex]1/ \sin{\theta_0}[/tex] times the speed it had at the beginning. There is no need to search for singularities (of course I didn't mean to put radius = 0 in some equation, I just meant to say that the velocity grows too much when the radius becomes very little).
3) The problem is in the chapter 5 of the book, which is the chapter about the conservation of energy. Angular momentum will be defined in chapter 8. Of course this is not a Physics argument, but I think that means that we shouldn't use angular momentum to solve the problem, while energy might be useful.