pulling strings in physics

Factual Questions
1

My physics teacher in HS put this example to the class, wherein she suspended a 500g weight from a string, and tied another length of the same string to the bottom of it. When she jerked on the (bottom) string, the length attached to the bottom broke in half. She then tied another piece to it, and said that she was going to do the same thing in slow motion, and proceeded to pull steadily on the bottom string. Now the top string snapped in half. She said it would happen this way every time-- if jerked rapidly, the bottom string would break, and if pulled steadily, it would be the top one. (It’s possible I have these backwards, but either the top or the bottom broke when jerked, and the other one broke when pulled)

I sucked (and still do) at physics, and so never understood why this worked. Can anyone give me an explanation? Note that my math skills are also nothing to write home about.

2

WAG:

The string which breaks is going to be the one which has more stress on it.

In the case in which she pulls fast, the weight wants to maintain its position in air. That is to say, it has some amount of inertia that is keeping it from moving. So if you pull faster than inertia will allow the weight to move, it acts like you’re pulling the (lower) string from a stable object and it breaks.

When you pull slowly, however, the upper string breaks simply because it has more weight pulling on it. (Much more WAG on this one.)

3

The second situation is much more obvious to me. The bottom string only has the pulling force acting on it while the upper string has the pulling force and the weight acting on it.

4

It’s all about inertia. In the first example, the inertia of the 500g mass was enough to keep the keep the force from the jerk from transferring into the top string. You can get a grasp of this by thinking of the problem upside down. If the weight were on the bottom, and you were to pull up sharply on the string supporting it (using the same string and weight), it would likely break*.

In the second example, the top string has the same tension as the bottom string, plus a force of 500g * 9.8m/s^2 = .490 N . So, everything else being equal (most importantly the tensile strength of the strings), the top string will have more tension, and will therefore fail first when the force is applied slowly.

*This may be beyond the scope of the question, but when evaluating materials properties, a dynamic load is often given as being 1.8 to two times an equivalent static load. For example, when designing a bike, the load onto the pedal of a 180 pound rider is assumed to be 360 pounds.

5

The way I see it, the key point is that a string breaks because it is stretched to the breaking point.

If you pull on the bottom string abruptly, the bottom string breaks before the weight has moved far enough to stretch the top string to its breaking point. You can think of that as the inertia of the weight resisting motion.

If you pull gently, the weight has enough time to respond to the pulling force. The top string is always stretched more because there is always more tension on it (sum of tension on the bottom string and the weight of the weight).

6

Hm, the second example I think I understand fairly easily now. The top string has the weight of 500g on it more than the bottom string does.

The other one, I’m still not so clear on. I tried thinking of it upside down, but whether you’re pulling up on this design or down on it, it still sort of seems just as likely that the string behind the weight will snap instead of the one you’re tugging on. Maybe I’m just not too clear on inertia. I understand that you need to exert more force on the weight to cause it to move than the tensile strength of the string being used, but because the weight is hanging, isn’t it already stretching the other string’s tensile strength as well? Being equal, shouldn’t the top string feel the force of a jerk just as much as the bottom string?

Wait wait, maybe this does it. If you use a weaker string above (dental floss or something) and then jerk the bottom string, the top one will snap because it has a weaker tensile strength, and so the tension on it breaks it before snapping the tougher string. But being equal, the bottom one has to pull the weight from a standstill against an equal tensile strength, which will necessarily be greater than the tensile strength alone. Is that right?

My only remaining question is that the top string is already supporting the weight of the 500g. Shouldn’t that put the exact same strain on its tensile strength as the bottom string’s strain to overcome the inertia? Does overcoming inertia require more force than the weight of an object? And doesn’t inertia only talk about an object in motion? Otherwise the inertia is 0, because of P=mv. Gah.

7

Objects in motion stay in motion. Objects at rest stay at rest. That’s inertia.

8

“Inertia” is the tendency of an object to resist acceleration (i.e. change in velocity). And there is no threshold for “overcoming” it; if you apply a small force, the object will change velocity by a small amount. If you apply a larger force, the velocity would change by a larger amount.

9

Think of it this way:

The top string is under tension. How much tension it’s under is determined by the distance between the block and the top of the string. Move the block down and the tension increases until the string snaps.

In order to move the block you have to apply a force to it. The force produces an acceleration which, over time, gives the block some downward velocity. This downward velocity, over time, moves the block down.

Note the phrase “over time”. It takes time to accelerate the block and it takes time for the moving block to move far enough down to snap the top string. That gives us a window of opportunity to operate in. If we can increase the tension in the bottom string so rapidly that it snaps before the block has time to move, the top string won’t break.

10

So why doesn’t the time it takes to accelerate the block away from the top string equal the time it takes for the bottom string to snap?

Isn’t this only with no other forces? If I apply a tiny bit of force to the bottom string, I generate no inertia because nothing goes anywhere. Same as doing 0 work by pushing on a wall. I guess that’s what I meant by “overcoming,” though I understand I’m not using the most technical language here.

11

This is a variation of the old yank the tablecloth out and leave the dishes on the table trick.

A basic principle of inertia says that dishes at rest on top of a table want to stay where they are.

If you slowly and gently pull the tablecloth off, the friction between the tablecloth and the bottoms of the dishes (you can think of it as a very short string) will mean that the dishes will be pulled along with the tablecloth and move off the table. This happens because the force exerted by the tablecloth to overcome the inertia of the dishes is less than the force needed to break the friction between the dishes and the cloth (the string).

On the other hand, if you give the tablecloth a sharp, quick yank, the inertia of the dishes (their resistance to being dragged along) will be great enough to overcome the friction between them and the tablecloth (the string). The tablecloth will slip through and the dishes will remain where they were.

In this example, the friction is like the bottom string. We could imagine that the dishes were not only sitting on the table, but also tied to the wall with a very light thread (the top string in your example). If you were to yank the tablecloth off, the “bottom string” friction bond could break, but the top thread could remain in place. On the other hand, if you slowly pull the tablecloth, you could exert enough force through the friction on the bottom of the dishes so that you could break the thread tying the dishes to the wall before the friction “thread” would break.

(Professional driver on a closed course. Do not attempt at home. Not responsible for any broken tableware.)

12

Pochacco’s explanation is essentially correct, but it may be a little hard to follow. Let me recast the argument by going back to some fundamentals.

Fundamental One: All materials are elastic. What that means is when you pull on a material, it will stretch. Doesn’t matter what the material is. Sometimes you can see this with the naked eye (pull on a rubber band), sometimes you can’t (pull on a steel bar), but it’s always true. And it’s most certainly true for a string. You can test this yourself: tie a string around a chair leg and pay out six to ten feet, then gently tug on the end.

Fundamental Two: Unbalanced forces produce acceleration. Your computer monitor stays in one place because all the forces on it are balanced; i.e., everything cancels out. If you give it a stiff shove, the force from your hand will overcome any resisting friction. This *unbalanced * force causes acceleration.

Fundamental Three: Accelerations cause a change in velocity; velocities cause a change in position. Perhaps this is obvious, but when you shove your monitor, the acceleration causes the monitor’s velocity to change (it used to be zero, right?). That velocity causes the monitor’s position to change (now it’s in a different location on your desk).

Fundamental Four: Mass (or inertia, if you like) is the resistance of an object to being accelerated. This is just another way of saying Newton’s law: F=ma. With a certain force, a small mass will be accelerated a lot, and a large mass will be accelerated only a little.

OK, now put those together:

If you yank on the lower string in your example, what you’re essentially doing is rapidly stretching the lower string. That rapid stretch is equivalent to a force in the string, and the stretch (and force) keeps increasing until the string breaks.

The same force is applied to the 500g weight. But remember, that force causes the weight to accelerate And here’s the key: If the force is only applied briefly, then the acceleration is applied only briefly, and the velocity if the mass is small, and thus the change in location of the mass is small. Or, succinctly: The mass will not move as fast or as far as the end of the string in your physics teacher’s hand does.

If the change in location of the mass is small, then the upper string is only stretched a little bit. If the upper string is only stretched a little bit, then it won’t stretch enough to break.

13

Acceleration equals force divided by mass. For a given force, big objects accelerate more slowly than little ones.

The bottom string is very light. When you jerk it, it speeds up very quickly. The block is heavy. The same force takes longer to get it going. So the bottom string stretches and snaps before the block has time to move very far.

14

That’s not correct - if you apply a small amount of force on the bottom string, the weight will move down by a tiny amount, causing the top string to stretch more.

By the way, if you assume perfectly inelastic everything (including perfectly non-stretchable strings), the experiment won’t work. The top string will always break first because the tension on the bottom string is instantly transmitted to the top string, adding to the pre-existing tension (created by the weight). It only works in the real world because strings stretch before they break. Like I said, if the force on the bottom string is large enough, the bottom string will break before the weight has moved far enough to stretch and break the top string.

15

Would it help if you thought of the example on its side? Try imagining a tow strap that can tow 2500 pounds, attached to a car that weighs 2000 lbs. Now immagine the back end of that car attached, with an identical tow strap, to an immovable object. If the back strap was taut (or even not taut), and the towing vehicle pealed out and took off, which strap would snap first? If inertia prevents the car from transferring any of the force to the back strap before the front strap’s tension exceeds its yield strength, it’s the same situation, just from a different angle.

500g is a measure of mass. Since we know F=ma, it does not become weight until it’s accelerated by gravity. In other words, weight is a force.

16

When you go home tonight and are sitting on the can/john/throne, pull out a length of toilet paper from the roll about 6-7 squares long. Holding onto the roll with the other hand, give the TP a good, quick jerk; it will nearly always break at the perforation nearest the roll. Now repeat, but this time slowly stretch the TP length until it breaks. The TP will break at just about any perforation (the experiment works better with a newer roll).

The “jerk” case IMO relies somewhat on the inertia of the roll, but a more important factor is that the force of the jerk travels down the length of TP in a pulse wave, hits the roll and reflects back. For a short time in the small area near the roll, the wave is folded over, thereby doubling the amount of tension this section of paper is subjected to. Naturally, it is more likely to break there. In the “slowly stretch” case, the break occurs at whatever perforation is the weakest, which could be any of them.

Applying this to the original experiment with the strings and 500 lb. weight, the “jerk” case explains why the quick pull not only snaps the string below the weight, but snaps it at a point very near the weight; by applying the force in a quick pulse, it is partially reflected like the wave at the point where the string is attached. Thus, a portion of the string nearest the weight has almost double the tension on it as the wave is reflected, and so is more likely to break. The “slowly stretch” case allows the tension to be applied more evenly (reflections are smaller and stretched over a longer period of time). However, the upper section has an additional 500 lb. of tension to support the weight, so it is thus more likely to break.

Bathroom physics–gotta love it.

17

Ah, ok-- the whole string being lighter and therefore accelerating much faster finally made it clear to me. CJJ*, thanks for the explanation about the wave. That’s an interesting additional dimension to the experiment that wouldn’t have occurred to me.

It all actually seems really clear and simple now, which is a really weird feeling for me about a physics concept. Thanks everyone. Where the hell were you during HS? If only this board had existed a few years earlier…