newtonian mechanics - Understanding which string breaks when one pulls on a hanging block from below

Intro:

In completing Walter Lewin's 6th lecture on Newton's Laws, he presents an experiment (go to 42:44) which leaves me baffled.

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Experiment:

(I recommend watching the video; see link above.)

• There is a $2$ kg block with 2 identical strings attached to it: one at the top, the other at the bottom.


• The top string is attached to a "ceiling", and the bottom to a "floor".


• Professor Lewin "stretches" the system (by pulling on the bottom string) with the block not accelerating.


• One string snaps.

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Prediction:

• Initially, the top string has a tension of approximately $20$ N, to counter the force of gravity. The bottom string has no tension at all.


• Then, when Lewin pulls the bottom string, it gains some tension $n$ N. To counter act the force exerted by the bottom string, the top string exerts now $20 + n$ N.


• I assume that the string with more force will give out sooner, leading me to conclude that the top string will break.

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Results:

(This was conducted by Lewin, not me; see link above.)

• Trial 1: Bottom string breaks.


• Trial 2: Top string breaks.


• Trial 3: Bottom string breaks.

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Additional Notes:

The results don't seem consistent. If I was right, I'd expect all 3 experiments to be right; conversely, if I was wrong, I'd expect all 3 experiments wrong, with one exception: the results are more-less random and one result isn't preferred over the other.

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Question:

• Why was my prediction incorrect?


• Was there a flaw in my logic?


• Why were the results inconsistent?

Answer

Your predictions of the forces adding up is correct, if nothing accelerates. Because, think about it... you are adding up forces, right? That is what you do in Newton's 1st law. Which is the law that only applies when nothing accelerates.

What if you were told that you can't use Newton's 1st law in the second case? Is something accelerating in the second case?

Or in other words, is the string trying to accelerate something in the second case?

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Solution

If something should accelerate, we are in Newton's 2nd law. If not, Newton's 1st law. Let's write it out with the forces from each string and weight $w$ present:

$$- F_{up} +F_{down} + w=0\qquad \qquad - F_{up} +F_{down} + w=ma$$

(I hope it's okay I've put the y-direction downwards.)

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  If you pull slowly down, no significant speeding up happens of the box. $F_{down}$ has some constant value. It all balances out. The 1st law.

  
  



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  If you pull fast down, the box tries to speed up fast to follow along. That means large $a$. That requires large force to cause it. And the force, that tries to cause is the $F_{down}$.

  
  

Look at those two equations again. In the first case $F_{up}=F_{down}+w$, so the upper string breaks. In the second case $F_{up}=F_{down}+w-ma$. Hmm, here is being subtracted the part $ma$...

So, is $F_{up}$ becoming smaller? No, of course not, it has it's tension and only grows as you pull downwards. Rather $F_{down}$ becomes larger. Because it tries to cause the $a$.

And as you see, it tries to but simply can't apply enough force to cause that acceleration. The necessary force in the lower string is more than the strength of the string, so it breaks.