Ball on a string thrown around a spool

[kekpillangok]

Homework Statement

A cylindrical spool A, with radius R, is fixed with its axis positioned horizontally. Along the same horizontal line that passes through the axis of the spool, a nail is attached to its periphery. A light, inextensible string of length L (where L > πR) is tied to this nail. At the other end of the string, a ball B is suspended. The figure illustrates the situation.

The ball B is given an initial horizontal velocity v.

Determine the range of values of v for which the string will certainly become slack at some point during the subsequent motion of the ball B.

Relevant Equations

a_cp = v^2 / R
Ef = E0

I'm having a hard time understanding the conditions under which the string will become slack in this problem. Maybe I'm just bad at visualising the situation and playing the film in my head, but the only situation I could imagine, at first, where I was 100% sure the string would become slack is when the string is pointing vertically upwards and the ball has zero speed, as in this drawing:

In this situation, the string will surely become slack and it is easy, through conversation of energy, to find the initial speed you'd need to give the ball so that it reaches that point with speed 0.
The answer, however, is $\sqrt{2 g {\left( L -R \right) }}\leq v \leq \sqrt{g {\left( 5 L -3 \pi R \right) }}$ and it refers to these two positions:

i.e. the string will certainly become slack if the ball has enough speed to make it to that first position but not enough for it to get to the second position with sufficient speed to keep the string taut. ($u$ is such that the tension from the string is zero and can be found by considering the centripetal net forces.) The less-than-or-equal sign on the left implies that if the ball has just enough speed to make it to the first position (so it doesn't move at all above the horizontal line of the string), then the string will become slack. How come? Why should the string become slack in this position? Can't we just reverse the motion of the ball while keeping the string taut?

I can see that, if the ball has a bit of excess speed when it reaches the first position, it will continue to wrap the string round the spool and the string will fall on itself. But I can't see why the string should become slack in the first position with the ball having speed 0.

My question is, how can one conclude logically that the string will become slack in the first position (string lying horizontally, ball with speed 0)?

If anyone can help me think more systematically and logically about when the string becomes slack, I will be very thankful.

 

[Lnewqban]

What is increasing or reducing the velocity of the ball?
What forces are acting on it?

 

[Steve4Physics]

But I can't see why the string should become slack in the first position with the ball having speed 0.

You might want to think about a simple pendulum with angular amplitude $90^o$: at each extremum the string is horizontal with the bob instantaneously at rest.

The net force, $\vec {F_{net}}$, on the pendulum bob is the vector-sum of the bob’s weight and the string's tension.

The centripetal force is the radial (along the string) component of $\vec {F_{net}}$ and has magnitude $\frac {mv^2}r$.

Put the above facts together and consider what happens to the magnitude and direction of $\vec {F_{net}}$ when the string is horizontal and $v=0$.

 

[jbriggs444]

The less-than-or-equal sign on the left implies that if the ball has just enough speed to make it to the first position (so it doesn't move at all above the horizontal line of the string), then the string will become slack. How come? Why should the string become slack in this position? Can't we just reverse the motion of the ball while keeping the string taut?

It seems that you are concerned with the situation for exact equality.

We can agree that the tension in the string will be momentarily zero if the ball comes to a momentary stop at the first position, exactly level with the bottom of the spool.

Does this mean that the "string has gone slack"? In my opinion, it does not. A straight string with zero tension is not "slack" in my book. A section of string [free from tension and pulleys] where the separation between the endpoints is strictly less than the unstressed length of the string is "slack" in my book.

Accordingly, I agree with you that the first less than or equal to sign is not correct.

Of course as a practical matter exact equality is an engineering impossibility that we need not worry about. Before we can get there, various idealizations that we made in calculating a solution will turn out to have been invalid.