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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,

| |
(5) |

where *i* is the spring number,
*F*_{i} is the tension, *k*_{i} is the ``spring constant'' (stiffness),
and 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:

| |
(6) |

Second, the individual displacements of all the springs add to give the
total displacement:
| |
(7) |

To solve the problem we write the additive term as a function of the globally constant tension *F*,

| |
(8) |

and sum over all the springs to find the total displacement
| |
(9) |

Comparing equation (9) with (8)
we see that springs in series behave like a single spring with
a stiffness determined by the equation
| |
(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

| |
(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.

** Next:** The spring group
** Up:** Dellinger & Muir: Dix
** Previous:** INTRODUCTION
Stanford Exploration Project

11/17/1997