Springs question - vertical versus horizontal stretching

[1ledzepplin1]

Springs question -- vertical versus horizontal stretching...

So I know hooke's law well enough and I understand the spring constant and it's vertical applications where force equals displacement times some constant.
What I am unsure of is to what degree this all applies to a horizontal stretch of a string?
For one scenario let's say our spring of constant k is bound at both ends and is stretch sideways, what's the relationship between the constants and the tensile force drawing the spring back?

For the second scenario let's say we leave one end dangling of a slinky and pull it at 1/2 the length, how does the original spring constant change between the two divisions?

Lastly, given a slinky, what effects on tension are we looking at if we twisted it? You can watch some pretty nifty tensile responses from coiling a slinky beyond what it's equilibrium state is and I am very curious about this. What does it doto the spring constant? Are there localized regions of stress for any particular reason or is it manufacturer inconsistency?

 

[256bits]

Why should there be any difference in the spring constant dependent on orientation of the spring?

 

[AlephZero]

Why should there be any difference in the spring constant dependent on orientation of the spring?

In a real horizontal spring, if you don't ignore gravity, the deformed shape will be a curve, the tension in the spring will not be uniform along its length, and the force at the ends will not obey Hooke's law.

You can ignore the above effects if the weight of the spring is small compared with the tension in it. For a slinky, you probably need to include them.

Actually, a complete model of a slinky might be even more complicated because the coils of the spring can't overlap, and therefore the maximum amount of curvature of the slinky at any point along its length under its own weight depends on the tension at that point.

 

[256bits]

I was assuming the ideal spring equation in textbooks was what the OP was asking about, where if it was a free standing (laying ) spring, the deflection due to gravity is ignored. The spring can lie upon a surface where the frictional effects can be also small, as would the case where the spring is guided by an interior rod.

In the case of a large spring constant, small diameter, and long length, the sag would be visible as a catenary ( as seen on cables on bridges ), and the spring can be modeled as a wire.

 

[AlephZero]

The "classical" model that gives the catenary shape for suspension bridges etc assumes the cable is inextensible, and is longer than the distance between the supports. If the cable can stretch along its length, the shape of the curve is different.

This (and the references in it) might be interesting: http://www.slac.stanford.edu/econf/C04100411/papers/038.PDF

 

[256bits]

Thanks for the link.
I didn't know they had it so much down to a science!

where is the OP guy.
I had this generic site for him to peruse.
Well, more like generic equations of catenary and related curves, including elastic cable.
http://www.digplanet.com/wiki/Catenary

 

[1ledzepplin1]

Thanks 256bits and AlephZero for each of your valuable links. Both contained precisely the information I was looking for and I'm pleased to see that it's as complicated and interesting as I had thought it was from direct observation. I was once told by my college physics professor that her christmas list, along with many other physicists she cited, had only wishes for more slinkys each year and I have a growing appreciation for this as I've learned more about complex tensile forces and the like.

I do however have one question still that one of you may have already answered but i didn't see it in the links.
I am looking not for just the mathematics of a stretch caternary, but i want to know anything and everything mathematical about an over-coiled spring and its caternary and curve redistribution.

If I'm not yet making sense, consider a slinky fixed at both ends and stretched to a forearms width. Now keep one end fixed and coil the other end in the direction the slinky is wound such that it is overcoiled.
The result will be a single locus of tension that, in a confined region, forms a twist resembling a cubic function.
The interesting thing is that this cubic-function stress shape moves freely across the length of the spring given nearly any form of orientational change whatsoever.
Any ideas or resources, friends?

 

[256bits]

Not too much knowledge on that from me.

If I read you right, what you are describing is a loop or kink in the slinky.

This happens when a cable is unwound from a spool. Under no tension loops can form in the cable. With sufficient tension on the cable, the loops can be taken up and removed, as long as the loops are not too small.
One sees that all the time when unraveling an electrical extension cable, or even a telephone handset cord.

Slinky probably has more in common with a telephone handset cord than a regular cable, since the telephone cord has already a preformed coil just like a slinky.