Movement of a Slinky

[mooeypoo]

Hey guys,

 

I was talking to a friend of mine and the subject of a slinky came up, specifically if and how the movement of a slinky can be represented with physical formula.

 

I thought it could be an awesome challenge, but I'm not quite sure where to start. I figured this will probably be best by starting with Lagrangians - last semester we managed to represent relatively complex systems with them - But we did that with either rigid objects (which a slinky isn't) or with ropes and strings that had a relatively simplistic movement (either extended downwards or sideways). A slinky, well.. not sure how to start.

 

Also, obviously it starts with some momentum at its top and continues taking advantage of its "falling" to gather momentum towards the next steps.

 

We can decide on a simple set of rules where the stairs are all the same and the slinky is not affected by anything other than its movement (no air resistence) and that it's ONLY moving in a straight-line (no movement sideways).

 

Any ideas?

[moth]

i don't know anything about lagrangians(a song by zz top wasn't it?) but i've had a couple beers and you said "any ideas"

if you picture the slinky sitting on a flat surface in an inverted "U" configuration with more slinky on one side than the other, the side with less coils will try to pull some coils off the side with more coils.the motion will (eventually )stop when there is an equal mass of slinky on both sides of the "U". on a ramp or on steps as you pointed out, the slinky gains momentum from going downhill which allows it to pull all the coils off the upper side of the "U",flip the upper coil over the top of the lower coil, and continue down the slope.

the problem seems to be about equilibrium in a spring.

[mooeypoo]

i don't know anything about lagrangians(a song by zz top wasn't it?) but i've had a couple beers and you said "any ideas"

if you picture the slinky sitting on a flat surface in an inverted "U" configuration with more slinky on one side than the other, the side with less coils will try to pull some coils off the side with more coils.the motion will (eventually )stop when there is an equal mass of slinky on both sides of the "U". on a ramp or on steps as you pointed out, the slinky gains momentum from going downhill which allows it to pull all the coils off the upper side of the "U",flip the upper coil over the top of the lower coil, and continue down the slope.

the problem seems to be about equilibrium in a spring.

Right. Now hwo do we represent this mathematically, is the big problem.

[moth]

maybe Hooke's law? F=-kx to get started

http://en.wikipedia.org/wiki/Hooke's_law

[mooeypoo]

Okay, seriously, I'm asking about how to represent the movement of a slinky.

 

Hooke's law is a used to see how much force a spring exherts on an object connected to it.. it's not the same.

[iNow]

In other words, don't post if you don't know. Moo is quite capable of speculating all by herself, but she's looking for a real answer, not a guess.

[mooeypoo]

I'd love to have help in formulating the answer and figuring out what comes where, but seriously, posting a one-liner links about forces - without even the shred of explanation on where and how it might be used in this formula I'm trying to get - is utterly unhelpful.

 

The movement of the slinky is non linear, so it's hard, but that's why I was thinking lagrangians. I think my main problem here is that along with the movement I have a changing "mass" -- as the slinky "falls" from one step to the other, we're transferring its bulk from the top stair to the bottom stair, and I"m not entirely sure how to do that.

 

I need to rummage through my notes from last semester - we learned something about changing mass but it was more with the relation of a rocket burning its fuel; I wonder if I can use that to the slinky too... then, perhaps integrating the dm/dt of the mass that is "falling" from the top stair in relationship to gravity (so I know how fast it takes to "finish" the movement) -- but anyone has any clue how do I represent the continued movement to the NEXT step? It doesn't just drop off from one step to another, it's supposed to have some momentum forwards that allows it to CONTINUE the movement..

 

Hm.

[Vortigon]

You could use non-linear torque/force calculations.

 

That would allow you to calculate changing momentum, and drag and also have a dx/dt (dm/dt) component, so you would have a nice graph of results leading to an eventual zero momentum.

 

The step would be a problem, as the above should replicate a constant slope but not sure if you could incorporate the step changes or not.

 

Not used them in a while.

[CaptainPanic]

I would describe it as if it is 2 objects that are connected (like two vessels of water, connected with a pipe). The mass of the higher object (the higher foothold of the slinky) is decreasing, while the lower one increases in mass. The speed at which this "mass transfer" proceeds depends on the spring parameters of the slinky... and I have no suggestion how to model that, but perhaps an empirical non-dimensional spring constant might work? .

Once the mass of the higher object is below a certain threshold, the object willl move, first to the position of the lower object, and then another step (of similar size) to a lower position.

 

There are 2 formulas needed I think, because there is a period where both ends of the slinky are stationary, and a period where one end is moving.

 

Therefore, I propose the use of two formulas: one where mass of both ends is a function of time, the other where the position of one of the ends is a function of time.

[Lan(r)12]

This is a really interesting problem

 

I wont be of any help, I can barely do Hooke's Law lol. I just cant wait to see what you all come up with.

[mooeypoo]

I would describe it as if it is 2 objects that are connected (like two vessels of water, connected with a pipe). The mass of the higher object (the higher foothold of the slinky) is decreasing, while the lower one increases in mass. The speed at which this "mass transfer" proceeds depends on the spring parameters of the slinky... and I have no suggestion how to model that, but perhaps an empirical non-dimensional spring constant might work? .

Hey that's a really neat idea!

 

I was thinking of something simliar - to divide the problem into two "systems", the inner one (mass exchange between the upper step and lower step) and then the movement down the stairs -- but this might be better, in a way. It won't represent the entire slinky, but it might be good enough to start.

 

I'm going to think about it, and try to post a description of the system so maybe I can try (with help, hopefully) representing it mathematically.

[moshy]

I am also trying to solve this problem.

 

What if you represent each coil of the slinky as a mini spring, and they're all connected like this

 

O--------O--------O--------O--------O--------O--------O

 

Each node is of equal mass, and you calculate the force on each: gravity, left spring, right spring

[mooeypoo]

Right, I have emailed my physics professor and asked for help about this, and tried to see if it would be wise to use lagrangians to represent 2 "subsystems" in the overall system. He wrote me back that it might work, but we will need to meet and work on it together (so I dont' yet have the actual math).

 

But my idea was that I should represent 2 systems -- one is the movement the slinky from top stair to the next lower stair, using mass balances. The second is the movement of the entire slinky down teh stairs. Combining the two "sub systems" will give me a whole movement.

 

The reason I picked lagrangians is because I think it will be a MESS representing it with "regular" Newtonian math.. F=ma and such will result in a huge mess. Lagrangians seem to be much easier. I'll try to work on it a bit, at least present a basic concept and post it here, see if we can take it from there.

 

~moo