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General Physics I:

Subsection 8.1.2 Hook’s Law for a Spring

Figure 8.1.5.
From Hook’s law,
\begin{equation*} \sigma\propto\epsilon \end{equation*}
\begin{equation*} \sigma =E\epsilon \end{equation*}
\begin{equation*} \frac{F}{A} =E\frac{\Delta L}{L} \end{equation*}
\begin{equation*} F =E\frac{A\Delta L}{L} \end{equation*}
Since cross-sectional area, \(A\text{,}\) length of spring,\(L\text{,}\) and elastic constant of spring material, \(E\) are constant, we can assume
\begin{equation*} k=\frac{AE}{L} \end{equation*}
which provides stiffness constant of a spring it is also called a force constant.
\begin{equation*} \therefore \quad F =k\Delta L \qquad \Rightarrow \quad F \propto x \end{equation*}
where \(x=\Delta L\text{.}\)