Relationship between Force and Extension for a Spring and other Simple Systems

Hooke’s Law

Hooke’s law states that the extension of a spring is directly proportional to the force applied to it, as long as the elastic limit is not exceeded.
It is given by the formula F = kx, where F is the force applied, k is the spring constant, and x is the extension of the spring.
The spring constant (k) is a measure of the stiffness of the spring. A high value of k indicates a stiff spring, while a lower value indicates a more flexible spring.
The elastic limit is the maximum extent to which a spring can be extended or compressed and still return to its original length once the force is removed.

#Graphs and Hooke’s Law

The relationship between force and extension is usually shown on a force-extension graph.
If Hooke’s Law is being obeyed, the graph line should be a straight line passing through the origin indicating a linear relationship.
The slope of the graph line represents the spring constant (k). A steeper slope indicates a stiffer spring.
When the extension is beyond the elastic limit, the relationship between force and extension is no longer linear. The spring will not return to its original length and is said to be overstretched or plastic deformation has occurred.

#Work Done on Springs

When you stretch or compress a spring, the work done is stored as elastic potential energy.
The work done (W) in stretching or compressing the spring is given by the formula W = 0.5 kx², where k is the spring constant and x is the extension or compression of the spring from its original length.
If the elastic limit is not exceeded, all the work done on the spring can be gotten back, generally as kinetic energy.
If the elastic limit is exceeded, not all the work done on the spring can be recovered. Some energy has gone into permanently deforming the spring.

#Investigating Elastic Materials

When conducting an experiment for elasticity in materials, important factors that might affect results need to be controlled. These can include the material type, length, cross-sectional area, and temperature.
Different types of materials will behave differently when subjected to forces - some may closely follow Hooke’s Law, while others may not.
Elasticity is the property of a material to return to its original shape after the deforming force is removed, while plasticity is the property to retain deformation even after removal of the force.
Limit of Proportionality is the point beyond which Hooke’s Law is no longer obeyed in a force-extension graph.