How can we prove that spring constant is always positive?

How can we prove that spring constant is always positive??
 

Mapes

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Imagine that you had a spring with a negative spring constant. If you hooked two of them up in series (with the other ends each attached to a wall, for example) what would happen?
 
834
1
The sign of the spring constant depends on your definition of the spring force.

It may be [tex]\vec{F}=k\vec{x}[/tex], in which it is always negative, or

[tex]\vec{F}=-k\vec{x}[/tex], in which it is always positive (in these cases for ideal springs). The important part is that the spring force always tends to pull or push the spring back to equilibrium.

You cannot prove this. These force expressions are designed to fit experimental data. It is the same as trying to prove that like charges always repel. It just is that way.

Also, as Mapes pointed out, the opposite case leads to a runaway process when the spring is disturbed from (unstable) equilibrium.
 
Please give some experiment that can verify that K will always positive or always negative.
 
834
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If you have a spring available, you can easily confirm it yourself. In any case, I think that fact is incorporated in the definition of "a spring".
 
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Things other than springs can have 'spring constants' Consider something in unstable equilibrium, like a mass on a sphere. The further you displace it the higher the force is, only the force is in the other direction giving a runaway effect described above. The reason no spring behaves this way is that they weren't made to behave that way, if they were they would be masses on spheres (or some equivalent)
 
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Consider a spring with the other sign. What would happen if you moved the end a small distance from it's equilibrium?
 

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