Introduction to Hooke’s Law
Hooke’s Law is a fundamental principle in mechanics that describes the behaviour of elastic materials, particularly springs. Named after the 17th-century British physicist Robert Hooke, it states that the force required to extend or compress a spring by some distance is proportional to that distance. This relationship is crucial for understanding various mechanical systems and the properties of materials.
Definition
Hooke’s Law states that the force needed to extend or compress a spring by some distance is proportional to that distance. In simple terms, it means the further you stretch or compress a spring, the harder it pulls back. The formula for Hooke’s Law is:
Mathematical Formulation
The mathematical expression for Hooke’s Law is:
F=−kx
Where:
F is the force applied to the spring.
k is the spring constant or stiffness of the spring (measured in N/m).
x is the displacement of the spring from its equilibrium position (measured in metres).
The negative sign indicates that the force exerted by the spring is in the opposite direction of the displacement.
Spring Constant (k)
The spring constant k is a measure of the stiffness of a spring. A higher k value indicates a stiffer spring, requiring more force to produce the same displacement. Conversely, a lower k value indicates a less stiff spring.
Graphical Representation
To illustrate Hooke’s Law, let’s consider a typical load-extension graph. On this graph, the load (force) is plotted on the vertical axis (Y-axis), and the extension (displacement) is plotted on the horizontal axis (X-axis).
Example Data and graph:
Interpretation of the Load-Extension Graph
The graph above illustrates the linear relationship between the load applied to the spring and the resulting extension, confirming Hooke’s Law. Key points to note:
Linear Relationship:
A linear relationship on a graph indicates that there is a constant rate of change between the variables plotted. In other words, as one variable increases or decreases, the other changes proportionally in a straight-line fashion. The linearity of the graph (a straight line through the origin) indicates that the relationship between the applied load and the resulting extension is linear. In other words, if you double the load, the extension doubles as well; if you triple the load, the extension triples, and so on. This linear relationship confirms that the spring is behaving elastically within the observed range.
Slope of the Line: The slope of the line represents the spring constant k. In this case, k can be calculated by taking any two points from the graph and using the formula:
Elastic behaviour and Elastic limit
Elastic behaviour means that when the load is removed, the spring returns to its original shape and size. In the context of the graph, this would mean that as you decrease the load back to zero, the extension also returns to zero, following the same straight line path in reverse.
Hooke’s Law is valid only up to the elastic limit of the material. Beyond this point, the material may no longer obey the linear relationship, and permanent deformation can occur. The elastic limit is the maximum extent to which a material can be stretched without undergoing permanent deformation.
Applications of Hooke’s Law
Mechanical Springs: Used in various devices such as automotive suspensions, mattresses, and mechanical watches.
Material Testing: Determines the elastic properties of materials in engineering and construction.
Weighing Scales: Springs in scales measure the weight based on the displacement caused by the load.
