Two Compressed Springs -> Unstable Equilibrium

AI Thread Summary
The discussion focuses on the instability of a system with two identical compressed springs arranged on a frictionless table. A slight vertical displacement of the midpoint leads to a change in the springs' lengths, which affects the potential energy of the system. The potential energy is expressed as U(x) = (1/2) k x^2 for both springs, but calculating the exact compressed length x requires geometric considerations. The conversation explores how to derive the relationship between the displacement and spring lengths, ultimately questioning the net force acting on the system when displaced. The conclusion emphasizes that if the net force exceeds zero in either direction, the system is deemed unstable.
brentd49
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I will make a crude visualization of this system:

|-------------O--------------|
<-----a------><------a------>

Identical springs: k1=k2=k
Natural Length: l > a

The problem is to prove that the system is unstable.

Obviously, a slight movement directed off the horizontal axis will cause the springs to unstretch to a natural position vertically above or below the current position. The setup is arranged on a frictionless horizontal table.
I know that the second derivative of the potential energy will tell me about the stability, so I am trying to write down the potential energy. My problem is how to write down the 'x' for the two springs, i.e.

U(x) = \frac{1}{2} k x^2_1 + \frac{1}{2} k x^2_2 , x_1=x_2
U(x) = k x^2

I suppose it is just a geometry question, but I'm not sure to find that compressed length x.
 
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Let's say that the 'midpoint' is displaced upwards by some distance d. Can you calculate the length the springs would have then?
 
so, if I take my orgin at the far left, with l_o the natural length.

x = l_f - l_o
x = \sqrt{a^2 + d^2} - l_o

so, now I need to replace d, right?

d = \sqrt{(l_o + x)^2 - a^2}

but that can't be right, because I would have x = x(x). I must be missing something.
 
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anyone.............
 
Presumably if the net force is greater than zero, in either up or down then the system is unstable.

What is the net force if O is displaced upward by d?

If the springs were constrained in the horizontal, then one could establish an equation for SHM with one spring a+x(t) and the other a-x(t).
 
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