How is the elastic restoring force defined exactly for a spring? We know by Hooke's law that
$$F_\text{restoring} = -kx$$
but what does $F_\text{restoring}$ really mean? I thought up till now that it was the force the spring pulled with at both ends if you stretched it by a distance $x$. This definition worked pretty well until I encountered some problems when I was doing problems a little above my usual level.
I have stripped down the problem I encountered to its core (where I think my confusion arises from):
Consider a spring attached to a wall (massless, ideal) in its relaxed. If we pull it with a force $F$, clearly the spring exerts a pulls with a force $F$. However, initially the spring is unstretched. The definition fails in this case.
What is the precise definition of a restoring force in a spring in the most general case?
Best Answer
The word 'restoring' is synonymous with 'opposing' in that it matches the applied force, but in the opposite direction. But more so 'restoring' implies that energy is being stored - potential energy - which can subsequently be retrieved. The potential energy is the integral of force over the path of deflection:
$$E_p=(1/2)kx^2$$
The energy imparted by the pulling force is stored in the spring which is able to do work.
In it's relaxed state (position) one can arbitrarily assign 'zero' potential energy by defining 'x' as zero at that position. Any deflection relative to zero stores energy.
Another interpretation is the fact that springs tend to 'restore' position to the relaxed state once the net external forces are removed.