Hooke's Law & Multiple Springs: Clarification Needed

[ElDavidas]

How do you apply Hooke's law to particles joined to multiple springs. For example in the diagram below:

0rigin |-\/\/\/\/- P1 - /\/\/\/\- P2 -/\/\/\/\/\/-|

I mean, would you have to introduce a new variable to indicate displacement if you altered one of the springs?

Also, suppose the left spring is compressed, does this mean that the middle and right hand springs are extended? So the tension is -ve for spring one and positive for springs two and three?

I would appreciate if someone clarified this for me as it would better my understanding. This type of question seems to come up quite frequently.

 

[Doc Al]

How do you apply Hooke's law to particles joined to multiple springs. For example in the diagram below:

0rigin |-\/\/\/\/- P1 - /\/\/\/\- P2 -/\/\/\/\/\/-|

I mean, would you have to introduce a new variable to indicate displacement if you altered one of the springs?

Hooke's law works the same way as always, but now the total length of the three springs is constrained.

Also, suppose the left spring is compressed, does this mean that the middle and right hand springs are extended? So the tension is -ve for spring one and positive for springs two and three?

That could certainly be true. It depends on how the springs are stretched/compressed compared to their unstretched lengths. But, assuming that the two end points (|) are fixed, the total length is also fixed.

Perhaps things will be clearer if you worked through a particular problem.

 

[Fermat]

I'm not sure what you mean by altering one of the springs.
But you should analyse each particle individually.
P1 had forces T1 and T2 from springs S1 and S2 acting on it.
Similarly, P2 has forces T2 and T3 acting on it from springs S2 and S3.
T1, T2 and T3 are compressive or tensile forces depending upon whether the spring(s) are in compression or extension.

When beginning the analysis, assume all springs are in compression (or in extension, if you are told otherwise).
Let each particle be displaced by a small amount in the +ve direction. Assume the springs are still in compression (or extension, if that was your initial assumption).

Now you can figure out the new forces acting on the particles due to the new extension and/or compression of the various springs.
The assumption of (force) direction doesn't really matter too much. Since if you assumed one direction for a force and it worked out to have a -ve value, then that simply means that it should point in the opposite direction to that assumed.