# Spring potential energy and Hooke's law review

Review the key concepts, equations, and skills for spring potential energy and Hooke's law.  Understand how to analyze a spring force vs. displacement graph.

## Key terms

Term (symbol)Meaning
SpringObject that can extend or contract and return to the original shape.
Spring constant ($k$)Measure of a spring’s stiffness, where a more stiff spring has a larger $k$. SI units of $\dfrac{\text N}{\text m}$.
Spring force ($\vec F_s$)Force applied by a spring given by Hooke’s law. SI units of $\text N$.
Elastic potential energy ($U_s$)Potential energy stored as a result of applying a force to deform a spring-like object. SI units of $\text J$.
If force is applied to a spring so that its elastic limit is exceeded, then the spring will no longer return to its original shape.

## Equations

EquationSymbolsMeaning in words
$\lvert \vec F_s \rvert = k \lvert \vec x \rvert$$\vec F_s$ is spring force, $\vec x$ is length of extension or compression relative to the unstretched length, and $k$ is spring constantThe magnitude of the force required to change the length of a spring-like object is directly proportional to the spring constant and the displacement of the spring.
$U_s = \dfrac{1}{2}k x ^2$$U_s$ is the elastic potential energyElastic potential energy is directly proportional to the square of the change in length and the spring constant.

## Hooke’s law

The force required to stretch an elastic object such as a metal spring is directly proportional to the extension of the spring for small distances. The force exerted back by the spring is known as Hooke's law
$\vec F_s= -k \vec x$
Where $F_s$ is the force exerted by the spring, $x$ is the displacement relative to the unstretched length of the spring, and $k$ is the spring constant.
The spring force is called a restoring force because the force exerted by the spring is always in the opposite direction to the displacement. This is why there is a negative sign in the Hooke’s law equation. Pulling down on a spring stretches the spring downward, which results in the spring exerting an upward force.
The displacement $x$ of a spring is equal to the stretched or compressed length of the spring $L$ minus the rest length $L_0$
$x = L - L_0$

## How to analyze a spring force vs. displacement graph

The area under the force in the spring vs. displacement curve is the work done on the spring. Figure 1 shows a plot of force on the spring vs. displacement, where displacement is $0$ when the spring is unstretched. The work done on a spring stores elastic potential energy $U_s$ in the spring until the spring goes back to its original length. Therefore, $U_s$ is equal to the work done and also to the area under the curve.
Figure 1: The work done by a force on an ideal spring. The vertical and horizontal arrows represent the rise and run, respectively. The spring constant $k$ is the slope of the line, $\dfrac{F}{x}$.
The area is a triangle with the following equation:
\begin{aligned}U_s&=\dfrac{1}{2}\,\text {base}\cdot \text{height} \\\\ &=\dfrac{1}{2} x \cdot k x \\\\ &=\dfrac{1}{2} k (x)^2\end{aligned}
Note that the spring constant $k$ is the slope of the line since $k =\dfrac{\vert \vec F \vert }{ \vert \vec x \vert }$.

## Common mistakes and misconceptions

Although the spring force is a restoring force and has a negative sign, the elastic potential energy $U_s$ cannot be negative. As soon as the spring is stretched or compressed, there is positive potential energy stored in the spring.