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

The force and extension of a spring or other simple systems such as rubber bands or elastic materials, are interconnected through Hooke’s Law.
Hooke’s Law states that the force (F) extended on a spring or a stretchable material is directly proportional to the extension (x), provided that the elastic limit is not exceeded.
The equation for Hooke’s Law is: F = kx, where F is the force, x is the extension and k is the spring constant.
The spring constant, denoted as ‘k’ in the formula, is a measure of the stiffness of the spring or elastic material. The larger the spring constant, the stiffer the spring or material is.
The unit for the spring constant ‘k’ is measured in Newtons per metre (N/m).
When graphing force versus extension for a spring or stretchable material, a straight line is observed until the ‘elastic limit’ is reached. Beyond this point, the material will deform permanently and Hooke’s Law will no longer apply.
The elastic limit refers to the maximum extension a material can undergo while still being able to return to its original shape once the force is removed. Beyond this limit, the material would undergo plastic deformation.
The amount of work done on a spring can be calculated as 1/2 times the spring constant times the square of the extension. This work is stored as elastic potential energy.
Energy transfer occurs in these systems. When a spring is stretched or compressed, kinetic energy is transferred into elastic potential energy. When the spring reverts back to its original shape, this stored energy is converted back into kinetic energy.
An ideal spring or elastic material, with no energy losses due to heat or sound, would be able to keep the energy in this system constant. This conservation of energy is a key principle in physics.
Real-world springs and elastic materials are not ideal and often have energy losses due to heat, air resistance, sound, and other factors. The concept of efficiency is introduced to account for these losses, and is the ratio of the useful output energy to the total input energy.
The study of the relationship between force and extension in a spring or other simple systems forms a basic foundation for understanding more complex systems in physics.