Hooke's law states that the force needed to extend or compress a spring by some distance is proportional to that distance. Mathematically, it is expressed as $F = kx$, where $F$ is the force applied, $k$ is the spring constant, and $x$ is the displacement from equilibrium.
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The spring constant $k$ is a measure of the stiffness of the spring.
Hooke's law can be integrated to find the work done in stretching or compressing a spring: $$W = \int_0^x kx \, dx = \frac{1}{2}kx^2$$.
The graph of Hooke's law is a straight line passing through the origin with slope equal to $k$.
Hooke's law applies only within the elastic limit of the material; beyond this limit, permanent deformation occurs.
In problems involving Hooke's law, integration techniques are often used to calculate potential energy stored in a spring.
Review Questions
What does the constant $k$ represent in Hooke's law?
How do you calculate the work done on a spring using integration?
What happens when a material exceeds its elastic limit according to Hooke's law?
Related terms
Elastic Potential Energy: The energy stored in an elastic object, such as a spring, when it is stretched or compressed. It is given by $$U = \frac{1}{2}kx^2$$.
Spring Constant: A parameter that measures how stiff and resistant to deformation a spring is. Represented by $k$ in Hooke’s Law.
Elastic Limit: The maximum extent to which a material can be stretched or compressed without undergoing permanent deformation.