# Homework Help: Difference b/t equilibrium and unstretched length of spring?

1. Mar 24, 2017

### Vitani11

1. The problem statement, all variables and given/known data
Consider a mass attached to four identical springs, as shown in the figure. Each spring has a force constant k and an unstretched length lo, and the length of each spring when the mass is at equilibrium at the origin is a (not necessarily the same as lo). When the mass is displaced by a small distance, show that its potential energy has the form (1/2)kor2 appropriate to an isotropic harmonic oscillator. Express the constant ko in terms of k.

2. Relevant equations

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

2. Mar 24, 2017

### Grinkle

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

3. Mar 24, 2017

### 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.

4. Mar 24, 2017

### haruspex

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.

5. Mar 28, 2017

### 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.

6. Mar 28, 2017

### haruspex

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

7. Mar 28, 2017

### haruspex

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

8. Mar 28, 2017

### Vitani11

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