How does force and tension affect the outcome of the inertial ball experiment?

[mindboggling]

How does the inertial ball experiment work?

A classic and well-known lecture experiment, the inertial ball demonstration serves to demonstrate Newton’s laws of motion. The experiment consists of a heavy ball, suspended by a string, and an identical string that hangs below the ball. If the lower string is given a slow and steady pull, the upper string will eventually break under the additional force exerted by the hanging weight. However, if the lower string is given an abrupt jerk, the lower string will be the one to break.

I’ve been trying to analyze and gain a greater understanding about this experiment, however I have I do not fully grasp how the system works.

as far as i can understand:

The phenomena of the ‘inertia ball experiment’ can be explained in terms of Newton’s three laws of motion. The 3rd law of motion states that whenever object A exerts a force on another object B, object B will simultaneously exert a force on object A with the same magnitude in the opposite direction . Hence, when a force is applied to the lower string, the force is transferred along the string and exerts a force on the heavy ball. Assuming that the heavy ball is a stationary point, it will exert an equal and opposite force on the string. These two forces, exerted by the physical tug and exerted by the heavy ball on the string) causes tension in the string (as illustrated in the diagram). Similarly, the upper string experiences the tug at the lower string, but since the upper string has to carry the weight of the heavy ball, the tension in the upper string is greater than the lower string by the mass of the heavy ball multiplied by the acceleration of gravity.

However, different rates of forces can influence the rate of tension in the upper and lower string. Newton’s second law of motion states that the acceleration of an object is proportional to the net forces applied and inversely proportional to the mass of the object. Therefore, because the piece of string has far less inertia than the heavy ball, the lower string accelerates. As a result, the upper string does not experience as much stretch as the lower string and the tension does not increase as much.

Also, how can I model and describe this experiment using mathematics? What equations can I use to explain “ how does the rate of force applied at the lower string affect which of the two strings break first”. More specifically if I were to graph the rate of force vs the tension in the upper and lower string, what questions can I use?

 

[cesiumfrog]

If you're interested in the detail, you'll have to replace the strings with springs (assume both to have the same, large, spring constant).

 

[rcgldr]

It might help to think of the strings as very strong and somewhat elastic, for example, they won't break until tension just exceeds 10 times the weight of the ball is exerted on the string, and that they can stretch about 10% before breaking. Currently the upper string is only streched by 1%, and has another 9% to go before it breaks. Also the inertia of the ball is much greater than that of the strings, maybe 100 times as much.

The explanation you quoted is wrong.

The key issue is that that tension in the upper string is related to how far downwards the ball has moved from it's initial position. In my example, the ball's downward movement has to stretch the upper string by an additional 9% beyond its current 1% stretch before the upper string will break.

You apply a quick downward jerk to the lower string. Initially there is little force as there is little inertia in the string and it's just stretching. The tension in the string is equal to the quickly increasing downwards force applied to the string. The tension and the force on the lower part of the ball are also equal. Most of the opposing force to this tension in the lower string will be the reaction force of the ball accelerating downwards, with the remainder of the opposing force being due to the tension in the upper string, which is relatively small at this point. Only as the ball moves downwards will tension in the upper string increase. However the force on the lower string is increasing rapidly due to the jerk, and it breaks before the ball moves downwards enough to stretch the upper string enough to break it.

 

[mindboggling]

oh okay, i see what you mean

are there any equations, that relate the rate of force and the tension in the strings?

 

[Loren Booda]

The ball's weight adds to tension in the upper string for the slow pull, and reinforces tension in the lower string for the quick pull.

I would guess that for the slow pull, the resonances in the string above the ball would be of lesser wavelength (greater tension), so that the critical force there would be exceeded more readily. For the quick pull, I would guess the resonances in the string below the ball would be of lesser wavelength (greater tension), so that the critical force there would be exceeded more readily.

In the former case we discourage resonances in the string below by slowly tightening (dampening) the string, and in the latter we encourage them by setting up a sharply defined wave with many components.

 

[rcgldr]

Are there any equations, that relate the rate of force and the tension in the strings?

Ignoring the inertia of the lower string, tension in the lower string equals the force applied (assuming the string can also stretch instantly). Tension in the upper string is related to how far the upper string is stretched, which depends on how far downwards the ball has moved.

 

[RottnJP]

Old thread found during a google search on how to set up this classroom experiment.

I'm assuming the OP has long since gotten his answer, but for completeness and to help out some other poor schlub with the same question:

Yes, this problem can be modeled mathmatically. The equations you start with here are quite basic, which makes this an interesting problem for introductory calculus students. The experiment makes a nice intro, and sets up the brain teaser in terms of how you model the experiment.

Once you figure out that the strings must be described as springs, you can use F = K(x) where the force F applied to a spring with spring constant K results in a displacement x.

Recalling that F = (m)(a), where force F is equal to the product of mass m and the accelleration, you can now do the things OP has asked about, in terms of studying the effects of the force application rate, masses of the strings and weight, graphing the forces involved, and so forth.

Have fun!