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## Homework Statement

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]

## Homework Equations

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

## 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 .