SPRINGS IN SERIES

A standard high-school physics problem asks the student to find the effective ``spring constant'' of several springs in series. Although this problem is trivial it serves to illustrate in canonical form the fundamental features of all equivalent layered-medium problems.

Each spring obeys Hooke's law,  

$$F_i = k_i \; \Delta x_i ,$$ (5)

where i is the spring number, F_i is the tension, k_i is the ``spring constant'' (stiffness), and $\Delta x_i$ is the displacement from the equilibrium position.

We get two more equations from the way the springs are connected. First, the tension is the same in all the springs:  

$$F_i \equiv F .$$ (6)

Second, the individual displacements of all the springs add to give the total displacement:  

$$\Delta x_{\mbox{\rm\scriptsize total}} = \sum \Delta x_i .$$ (7)

To solve the problem we write the additive term $\Delta x$as a function of the globally constant tension F,  

$$\Delta x_i = {1 \over k_i} \;\; F ,$$ (8)

and sum over all the springs to find the total displacement  

$$\Delta x_{\mbox{\rm\scriptsize total}} = \sum \Delta x_i = \... ...i} \;\; F \Biggr) = \Biggl( \sum {1 \over k_i} \Biggr) \;\; F .$$ (9)

Comparing equation (9) with (8) we see that springs in series behave like a single spring with a stiffness $k_{\mbox{\rm\scriptsize total}}$ determined by the equation

$${1 \over k_{\mbox{\rm\scriptsize total}}} = \sum {1 \over k_i} .$$ (10)

The equivalence of form between equations (8) and (9) is clearly the key to this problem. Written this way it is clear that we can add springs in series by summing their compliances 1/k_i. The coefficient on the constant term in equation (8), 1/k_i, provides an alternative way of representing the spring properties, the ``spring-group'' representation. There is really no reason except convention which prevented us from starting this derivation by writing Hooke's law as  

$$\Delta x = c \; F ,$$ (11)

with the compliance c = 1/k called the ``spring constant''. If Hooke's law were normally written this way, generations of beginning physics students could have answered ``when springs are connected in series the spring constants add'' and actually have been correct.