Tetherball wrapping around a pole

[opticaltempest]

Hello,

I am trying to analyze the following situation:

A tetherball is kicked with velocity of $$v$$ meters per second at time $$t=0$$ seconds. The length of the string attaching the tetherball to the pole is $$l$$ meters. The radius of the pole is $$r$$. Assume no gravity and no air resistance so that the ball wraps around the pole in the plane in which it is initially kicked. In other words, when the ball makes one complete revolution around the pole the length of the string is reduced by $$2\pi r$$.

1. I would like to set up a differential equation that describes the length of the string at any time $$t$$.

2. My ultimate goal is to analyze the angular velocity and linear velocity at any time when the ball is wrapping inward towards the pole without using the conservation of angular momentum.

This isn't a homework problem so I might have left out information needed to complete the problem.

I am having trouble on all approaches. Any help will be greatly appreciated.

 

[berkeman]

I don't think you need to use any DiffEq for this. Just write an equation for the length of string versus time, and use that to express he position of the ball mass versus time...

 

[opticaltempest]

I think I see how to express tangential velocity and angular velocity in terms of how many revolutions were made.

How would I express the tangential velocity and angular velocity in terms of time?

From the relationship between tangential velocity $$v_{t}$$ and
angular velocity $$\omega$$

$$v_{t}=r\omega$$

Where $$r$$, in our case, would be the length of the string.

The length of the string $$l$$ at any revolution is

$$l=l-2\pi r \gamma$$

Where $$\gamma$$ is the number of revolutions.

The tangential velocity is then

$$v_{t}=(l-2\pi r \gamma)\omega$$

and

$$\large \omega = \frac{v_{t}}{l-2\pi r \gamma}$$

 

[bulfinb]

You have probably forgoten about this post by now. I have been recently looking at this problem. The magnitude of the linear velocity is constant. Angular momentum as it is not conserved in this system. If you let the centre of the pole be a fulcrum then there is a torque due to the tension in the string which decreases the angular momentum as it wraps around.

So to do the problem you let the velocity stay constant and energy is conserved. angular velocity will increase and angular momentum will decrease.