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# 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 ( | Measure of a spring’s stiffness, where a more stiff spring has a larger |

Spring force ( | Force applied by a spring given by Hooke’s law. SI units of |

Elastic potential energy ( | Potential energy stored as a result of applying a force to deform a spring-like object. SI units of |

## Equations

Equation | Symbols | Meaning in words |
---|---|---|

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. | ||

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 $k={\displaystyle \frac{|\overrightarrow{F}|}{|\overrightarrow{x}|}}$ .

## 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.

## Want to join the conversation?

- Why in the Spring Potential Energy equation is x^2 and not just x?(11 votes)
- Because elastic potential energy is Average force multiplied by the extension so (1/2kx)(x)=1/2kx^2(5 votes)

- hi, does anyone know the difference between a potential energy and force? I would rlly appreciate it if anyone would help me out!(7 votes)
- Hi there! Potential energy is stored energy in an object due to its situation/position that can be converted into other kinds of energy, such as kinetic energy, while a force is a type of influence on an object that can cause/change the motion of the object.(14 votes)

- how can i calculate the spring constant of 2 spring-like object joined together?(4 votes)
- A system of two springs with spring constants
`k₁`

and`k₂`

linked together is the same as a larger spring with spring constant`k`

, which can be found with the equation`k=(k₁k₂)/(k₁+k₂)`

.(2 votes)

- how do you find k without f and only x(3 votes)
- you can look at graph privided or they might give you two example like when doubling k, the displacement is y times original. then you can use this to know to know k. Or you can use potential energy equation to know the answer.(1 vote)

- F=-Kx Then work should be = -Kx^2 why is it 1/2-Kx^2(2 votes)
- Work is force times distance, yes. But, the work done by stretching (compressing) a spring is not constant, since more force is required as the displacement increases. So, a better way to look at it is "Work from x1 to x2 = F * (x2 - x1)". Or, Delta W = F * Delta x. The total work done by stretching from x0 to xn is Sum(Delta W) = Sum(F* Delta x), and as Delta x -> 0, this becomes W = the integral of F dx, from x=x0 to x=xn. The integral of F = kx is (1/2) k x^2.(3 votes)

- What is K as an approximate value?(1 vote)
`k`

is the spring constant for a specific spring and differs for different springs (as a value for its stiffness); therefore, there is no specific value for it.(4 votes)

- Although the spring force is a restoring force and has a negative sign, the elastic potential energy cannot be negative

when the spring come back to its original length , it release the elastic potential energy which is stored in it , can this energy be negative as an indication of releasing ?(1 vote) - How do we find the spring constant (k) if the initial problem doesn't show it (i.e is the an equation to find k)?(1 vote)
- When a spring is pulled away from the its resting position, is the restorative force negative?(1 vote)
- It depends on which direction is defined to be positive/negative. Stretching a spring will result in a restorative force with the opposite sign of the restorative force caused by compressing the spring.(1 vote)

- Why does a spring do more work from X=A to X=A/2 than X=A/2 to X=0?(1 vote)
- Work is equal to force times distance, w=fd. For a spring, f=-kx. So a stretched out or compressed spring will exert more work when x is higher.(1 vote)