Difference b/t equilibrium and unstretched length of spring?

[Vitani11]

Homework Statement

Consider a mass attached to four identical springs, as shown in the figure. Each spring has a force constant k and an unstretched length l_o, and the length of each spring when the mass is at equilibrium at the origin is a (not necessarily the same as l_o). When the mass is displaced by a small distance, show that its potential energy has the form (1/2)k_or² appropriate to an isotropic harmonic oscillator. Express the constant k_o in terms of k.

Homework Equations

The Attempt at a Solution

I just need help with interpreting this. The question didn't specify, it only told me they were different.

 

[Grinkle]

I interpret unstretched length as corresponding to the equilibrium length of the spring with no mass attached.

 

[Dr.D]

The problem statement references a figure, but there is no figure.

The free length is the length with no strain in the spring material. This usually means that the spring is not even supporting its own weight.

The equilibrium length would be the length of the spring when under some sort of load, perhaps due to gravity or some other force. The term suggests that there is some amount of strain in the spring material.

 

[haruspex]

Homework Statement

Consider a mass attached to four identical springs, as shown in the figure. Each spring has a force constant k and an unstretched length l_o, and the length of each spring when the mass is at equilibrium at the origin is a (not necessarily the same as l_o). When the mass is displaced by a small distance, show that its potential energy has the form (1/2)k_or² appropriate to an isotropic harmonic oscillator. Express the constant k_o in terms of k.

Homework Equations

The Attempt at a Solution

I just need help with interpreting this. The question didn't specify, it only told me they were different.

I assume the four springs go from the origin to the corners of a square, and that this is all in the horizontal plane.
Consider first that there is no mass, just the four springs. If a ≠ l₀ then they are either under tension or under compression - it won't matter which. Adding the mass changes nothing immediately since we are ignoring gravity. When the mass is displaced a small distance from the origin (you will need to make suitable approximations here) the tensions/compressions change, leading to a net restoring force.

 

[Dr.D]

Was there ever a figure attached by the OP? I have not seen any such, so I really don't know what system we are talking about here.

 

[haruspex]

I interpret unstretched length as corresponding to the equilibrium length of the spring with no mass attached.

The problem statement denies that interpretation. The unstretched length is l₀ but the equilibrium length is a.

 

[haruspex]

Was there ever a figure attached by the OP? I have not seen any such, so I really don't know what system we are talking about here.

The OP "liked" my interpretation in post #4, so I presume that is correct.

 

[Vitani11]

Finished, thank you for the help. I now know the difference between equilibrium and unstretched length.