== Hooke's law in conventional form ==

Hooke's law is known as $F = -kx$.  Here, $F$ is the spring force, and $x$ is the amount of compression or elongation.  The negative sign in the expression for $F$ means a “restoring force.”

== Easy come, easy go ==

That seemed easy, but it is not quite useful in a more general setting.  The above expression is useful only when one of the spring is fixed.  The above expression can be misused very easily in other general situations when both ends of the spring are free.  As a concrete example of a general situation, imagine a fun problem like the following: a spring is compressed at both ends and then thrown into air while it is set to spin in a certain direction—assuming that the spring is stiff, what is the resultant motion of the spring?

To get the more correct expression that is valid in such a more general setting, we need to make some more fundamental observations.  Here they are.

  1. When a spring is compressed, it tries to elongate, by pushing on both sides.
  1. When a spring is elongated, it tries to compress, by pulling on both sides.
  1. For the previous two properties, it does not matter where the (mean) position of the spring is.
  1. When a spring is compressed or elongated '''uniformly,''' Hooke’s law applies to the spring as a whole: the ''magnitude'' of the restoring force that pushes on either side (case 1) or pulls on either side (case 2) is given by $k |\Delta L|$, where $\Delta L$ is the relative displacement of one end to the other.

Let us call $F_L$ the Hooke’s law force exerted by the spring on the left end, and $F_R$ the Hooke’s law force exerted by the spring on the right end.  Then, by the above properties 1–4, one concludes that $F_L = -F_R$ when the spring is compressed or elongated uniformly.

== Difficult, but more useful, expression ==

Suppose the left end of the spring is longitudinally displaced by $D_L$ and the right end by $D_R$.  Defining the right direction to be the positive direction, we then get

 $$ F_L = k (D_R - D_L), \; F_R = - F_L. $$

These two equations are valid, including the signs of $F_L$ and $F_R$, no matter what values of $D_R$ and $D_L$ are, including their signs.  Please convince yourself of this fact.  '''This is Hooke’s law!'''