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.
This figure shows two identical springs with an attached weight $F_G$.
The left spring is in equilibrium after a weight has been applied, whereas 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.
Hooke's Law
In the case of elastic extension, the restoring force is directly proportional to the extension:
\begin{equation}
{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.
Hooke's law is widely used for different types of springs, for example inside so-called force-meters as shown in the figure below. Because of the linear correlation between the elongation $x$ and the applied force $F$, the scale is linear as well.
Tension Energy
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}
Tension Energy of Spring
If a string with a spring constant $k$ is elongated by the length $s$, the stored tension energy is given as
\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.
Experiment: Spring with Weights
To demonstrate Hooke's Law, a vertically suspended spring is loaded with weights of varying weights, and the spring's deflection is measured. When plotting the measurement curve, a linear relationship between the two quantities is revealed.