Newton's Cradle Problem: Exploring Momentum and Energy Conservation"

[7bear]

Suggest that one metal ball is removed in Newton's cradle. Like the following(thw second one is removed) :
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O OOO

Neglect the air resistance~
When we pull one ball(the left hand side one) of it to one side and then release, it will hint the third ball. Compare with the case that the second ball is hinted instead, do the two cases obtain the same result(the last ball move up to the height which is same as the ball pulled at first, total momentum and energy are conserved)?

Help me please...

 

[russ_watters]

If I understand correctly, in both cases, you pull one ball back, its just that in one case, there are more stationary balls than in the other?

So here are my questions for you: what happens to those stationary balls during the experiment? Do they move at all? If they don't move, do they have any associated energy? Where else can the energy go?

 

[BobG]

Don't you have a Newton's cradle to experiment with? First off, don't most have 5 balls instead of 4? (at least mine does)

I take it you mean you're pulling up two balls then releasing them, yes, momentum is conserved (it always is). There's two possibilities (plus an infinite number in between): Will two balls on the opposite side move the same distance as the released balls or will only one ball move, but go higher (or some combination in between).

The experimental answer is that two balls on the opposite side move the same distance as the released balls. If you drop 3 balls, three balls move on the opposite side (including one of the dropped balls). If you think about both kinetic energy and momentum, you can see why.

If you mean you're taking one ball out of the sequence, then momentum is still conserved, but ... Have you noticed how the spacing is set so the path of a ball drives right through the center of the next ball? With one ball removed, you strike the ball across the gap at an angle - not a large angle, but definitely not straight through the center.

Experimentally, the balls quickly get out of sync with each other, but, they eventually resync with all the balls going back and forth in the same direction (no neat collisions).

If you do the traditional thing of just dropping one of the end balls, eventually the system loses energy faster than it loses momentum (air resistance, friction, etc provide a small external force meaning energy and momentum can't truly be conserved with the normal desktop model), resulting in all the balls moving back and forth together, but not nearly as far (no neat collisions).

You really should either get one of these things, or at least browse through a gift or science store. That way, not only can you see the conservation of momentum, but you can see where you're losing energy (mine is kind of old and I've had the unhealthy habit of placing business cards or my bottle of "Future Glue" in the cradle - the strings vibrate, which means my balls probably arent' perfectly aligned any more). [But, the Future Glue is pretty cool. You know how if you drop your coffee cup, you can never find all the tiny little fragments to glue your cup back together very well. If you use the Future Glue on your cup the night before you drop it, you don't lose all those little fragments ]

Browse the store with you notebook and pencil, of course.

Salesperson: "Can I help you with something?"
Reply: "No, I'm just doing my homework ... Uh, wait, do you remember the formula for kinetic energy?"
Salesperson:

 

[Bob3141592]

Suggest that one metal ball is removed in Newton's cradle. Like the following(thw second one is removed) :
| | | |
| | | |
| | | |
| | | |
O OOO

Neglect the air resistance~
When we pull one ball(the left hand side one) of it to one side and then release, it will hint the third ball. Compare with the case that the second ball is hinted instead, do the two cases obtain the same result(the last ball move up to the height which is same as the ball pulled at first, total momentum and energy are conserved)?

Help me please...

In the unaltered toy, the balls are set so that they are just touching when at rest. This means that they always hit the adjacent ball exactly at the bottom part of their swing, when all the energy of the ball is kinetic and there is no potential energy.

By removing the second ball, the first ball has to travel further before it can make contact with another ball, and when it does so it will have swung upwards somewhat, trading off a bit of it's kinetic energy into potential energy. Only the kinetic energy will be transferred to the struck ball (and that not very efficiently, since it will hit off center). Some of that kinetic energy will push the struck ball horizontally, but some of it will go into trying to push the ball downwards (since the contact point was above the centerline) which will waste energy in stretching and heating the string. Meanwhile, the first ball will drop downwards, also putting it's (potential)energy into stretching and heating it's own string, dissipating more energy. The ball will probably bounce and oscillate for awhile, therefore being out of position when the ball it struck swings back to strike it in return.

The first contact would be easy enough to model with vector diagrams, but subsequent collisions will involve all sorts of feedback loops and other factors like the length of the string, the mass and diameters of the balls, the elasticity of the metal, etc, that will make it near impossible to solve the problem analytically.

Suffice it to say that after only a very few collissions the toy will be out of synch and not enjoyable to watch.

However, in this case and in all cases, energy and momentum are always conserved. If we can ignore reality and propose perfectly elastic balls, perfectly incompressible strings, a perfect vacuum, etc, in theory you'd end up with a collection of balls moving in distorted ovals up and down as well as side to side, probably in chaotic fashion, sometimes colliding and sometimes not, depending on the radius of the balls. In short, it'd be a mess.

Given the Newton's Cradles I've seen, I'd guess it'd take no more than five swings before the motion would qualify as a "mess." But you'd have to buy and destroy a toy to make the experiment. If you do, please tell me if my guess was too long or too short.

 

[BobG]

Instead of destroying the 'toy', you could just hold one of the balls up out of the way with one hand and drop one of the balls with the other. Yes, they fall out sync, but soon get back in phase with each other moving back and forth together. The amount of time it takes to 'resync' is different depending on whether the dropped ball is on the side of the gap or the side opposite the gap.

The amount of time to become a 'mess' varies depending on which side you drop the ball. In either case, they're falling out of sync right from the get go and the point where you draw the line between some semblance of order and total chaos is kind of subjective. But neither side will give you two complete swings (forward and back) and still have any significant sense of order, making it hard to count how many 'swings' it takes for the balls to transition to swinging in phase with each other (no collisions).