Hooke's Law

Hooke's Law is named after Robert Hooke and describes the elastic deformation of solids. A well-known example is the extension of a spring. The left spring is in equilibrium after a weight has been applied, while the right spring has been deflected by a distance $x$ by an additional force $F_A$. Since the spring strives to return to its original position, a restoring force $F_R$ acts on the center of gravity of the weight, which is equal in magnitude to the applied force but opposite in direction. In the case of elastic extension, the restoring force is directly proportional to the extension:
\begin{equation}\label{eq:hookesches_gesetz}
{F_R(s) = -ks}
\end{equation}
The minus sign indicates the direction of the force relative to the extension. The proportionality constant $k$ is called the spring constant in this context. It has the unit N/m and depends on the material, length, and cross-section of the spring. Hooke's law is applied, for example, to a spring force gauge. A linear scale is attached to the spring, which is pulled out of the dynamometer housing when a force is applied.

The following rules applies: The smaller the spring constant, the further the spring stretches per force and the smaller the reading error.

In addition to the force, the stored tension energy often plays an important role. Since the force itself depends on the extension, this must be determined by integration:
\begin{equation}
E_\mathrm{sp} = -\int_0^s F_R(s')\,\mathrm{d}s'
\end{equation}
Inserting Hooke's law and calculating the integral yields:
\begin{equation}
{E_\mathrm{sp} = \frac{1}{2}ks^2}
\end{equation}
Noteworthy here is the quadratic relationship between the energy and the deflection, as well as the constant factor 1/2. The shape is therefore similar to the previously defined kinetic energy of a body.