Spring with an Initial Compression and Hookes Law

Next we describe La Coste s invention and demonstrate that it allows one to make a spring with much smaller stiffness. From the condition of equilibrium it follows that 

In such a case the period would drastically increase, as well as the mechanical sensitivity, and we obtain 

Here /q is the initial length of the spring with restoring force F. Correspondingly, the total force applied to mass and directed downward is a sum of the force equal to the restoring one and a force caused by a spring expansion. Both of them have the same direction 

Here ko is the stiffness of the spring with the restoring force and Equation (3.127) can be treated as Hooke s law for such spring. Suppose that the mass performs vibrations near a point of equilibrium. Then the equation of motion changes slightly and we have  

Now assume that the spring is made in such a clever way that the restoring force is directly proportional to the original length /q and the coefficient of proportionality is exactly equal to the elastic parameter  

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