# Tetherball wrapping around a pole

• opticaltempest
In summary, The conversation is about analyzing a situation where a tetherball is kicked with a certain velocity and the length of the string attached to it decreases as it wraps around a pole. The goal is to find a differential equation for the length of the string at any time and analyze the angular and linear velocities without using the conservation of angular momentum. The relationship between tangential velocity and angular velocity is v_{t}=r\omega and the length of the string 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 \omega = \frac{v_{t}}{l
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.

Last edited:
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...

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

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

Last edited:
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.

## 1. How does the tetherball wrap around the pole?

When playing tetherball, the ball is attached to a rope which is then attached to the top of a tall pole. As players hit the ball back and forth, it will wrap around the pole due to the centrifugal force created by the swinging motion.

## 2. Does the direction of the swing affect the tetherball's wrapping?

Yes, the direction of the swing does affect how the tetherball wraps around the pole. If the ball is hit in a clockwise direction, it will wrap around the pole in a clockwise direction. The same is true for a counterclockwise swing.

## 3. What causes the tetherball to eventually stop wrapping around the pole?

The tetherball will eventually stop wrapping around the pole due to friction. As the rope rubs against the pole, it creates friction which slows down the ball's rotation and causes it to eventually stop wrapping.

## 4. Can the tetherball wrap around the pole more than once?

Yes, the tetherball can wrap around the pole more than once. This is typically seen in games where players have a strong swing and the ball wraps around the pole multiple times before coming to a stop.

## 5. Does the size of the pole affect the tetherball's wrapping?

Yes, the size of the pole can affect the tetherball's wrapping. A thicker or taller pole will have a larger circumference, meaning the ball will have to travel a greater distance to complete one full rotation. This can make it more difficult for the ball to wrap around the pole multiple times.

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