A metal wire of length ${\mathrm{L}}_1$ and area of cross-section A is attached to a rigid support. Another metal wire of length ${\mathrm{L}}_2$ and of the same cross-sectional area is attached to the free end of the first wire. A body of mass M is then suspended from the free end of the second wire; If ${\mathrm{Y}}_1 \mathrm{and} {\mathrm{Y}}_2$ are the Young's moduli of the wires respectively, the effective force constant of the system of two wires is

Using the usual expression for the Young's modulus, the force constant for the wire can be written as

$\mathrm{k} = \frac{\mathrm{F}}{∆\mathrm{L}} = \frac{\mathrm{YA}}{\mathrm{L}}$

When the symbols have their usual meanings,

When the two are connected together in series, the effective force constant is given by

${\mathrm{k}}_{\mathrm{eq}} = \frac{{\mathrm{k}}_1{\mathrm{k}}_2}{{\mathrm{k}}_1+{\mathrm{k}}_2}$

Substituting the corresponding lengths, area of cross sections and the Young's moduli, we get

${\mathrm{k}}_{\mathrm{eq}} = \frac{\left({\displaystyle \frac{{\mathrm{Y}}_1\mathrm{A}}{{\mathrm{L}}_1}}\right)\left({\displaystyle \frac{{\mathrm{Y}}_2\mathrm{A}}{{\mathrm{L}}_2}}\right)}{{\displaystyle \frac{{\mathrm{Y}}_1\mathrm{A}}{{\mathrm{L}}_1}+\frac{{\mathrm{Y}}_2\mathrm{A}}{{\mathrm{L}}_2}}} = \frac{\left({\mathrm{Y}}_1{\mathrm{Y}}_2\right)\mathrm{A}}{{\mathrm{Y}}_1{\mathrm{L}}_2+{\mathrm{Y}}_2{\mathrm{L}}_1}$