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 (kk)Measure of a spring’s stiffness, where a more stiff spring has a larger kk. SI units of Nm\dfrac{\text N}{\text m}.
Spring force (Fs\vec F_s)Force applied by a spring given by Hooke’s law. SI units of N\text N.
Elastic potential energy (UsU_s)Potential energy stored as a result of applying a force to deform a spring-like object. SI units of J\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.


EquationSymbolsMeaning in words
Fs=kx\lvert \vec F_s \rvert = k \lvert \vec x \rvert Fs\vec F_s is spring force, x\vec x is length of extension or compression relative to the unstretched length, and kk 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.
Us=12kx2U_s = \dfrac{1}{2}k x ^2UsU_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
Fs=kx\vec F_s= -k \vec x
Where FsF_s is the force exerted by the spring, xx is the displacement relative to the unstretched length of the spring, and kk 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 xx of a spring is equal to the stretched or compressed length of the spring LL minus the rest length L0L_0
x=LL0x = 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 00 when the spring is unstretched. The work done on a spring stores elastic potential energy UsU_s in the spring until the spring goes back to its original length. Therefore, UsU_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 kk is the slope of the line, Fx\dfrac{F}{x}.
The area is a triangle with the following equation:
Us=12baseheight=12xkx=12k(x)2\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 kk is the slope of the line since k=Fxk =\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 UsU_s cannot be negative. As soon as the spring is stretched or compressed, there is positive potential energy stored in the spring.

Learn more

For deeper explanations of elastic potential energy, see our video introducing springs and Hooke's law and the video on potential energy stored in a spring.
To check your understanding and work toward mastering these concepts, check out the exercise on calculating spring force and the exercise on calculating elastic potential energy.