# 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 |
---|---|

Spring | Object 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 $\text J$. |

## Equations

Equation | Symbols | Meaning 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 constant | The 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 energy | Elastic 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

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.## 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.

The area is a triangle with the following equation:

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.

## 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.