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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
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 start fraction, start text, N, end text, divided by, start text, m, end text, end fraction.
Spring force (F, with, vector, on top, start subscript, s, end subscript)Force applied by a spring given by Hooke’s law. SI units of start text, N, end text.
Elastic potential energy (U, start subscript, s, end subscript)Potential energy stored as a result of applying a force to deform a spring-like object. SI units of start text, J, end text.

## Equations

EquationSymbolsMeaning in words
open vertical bar, F, with, vector, on top, start subscript, s, end subscript, close vertical bar, equals, k, open vertical bar, x, with, vector, on top, close vertical barF, with, vector, on top, start subscript, s, end subscript is spring force, x, with, vector, on top 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, start subscript, s, end subscript, equals, start fraction, 1, divided by, 2, end fraction, k, x, squaredU, start subscript, s, end subscript 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
F, with, vector, on top, start subscript, s, end subscript, equals, minus, k, x, with, vector, on top
Where F, start subscript, s, end subscript 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, start subscript, s, end subscript in the spring until the spring goes back to its original length. Therefore, U, start subscript, s, end subscript 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, start fraction, F, divided by, x, end fraction.
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, equals, start fraction, \vert, F, with, vector, on top, \vert, divided by, \vert, x, with, vector, on top, \vert, end fraction.

## Common mistakes and misconceptions

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

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?
• Because elastic potential energy is Average force multiplied by the extension so (1/2kx)(x)=1/2kx^2
• hi, does anyone know the difference between a potential energy and force? I would rlly appreciate it if anyone would help me out!
• 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.
• how can i calculate the spring constant of 2 spring-like object joined together?
• 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)
• F=-Kx Then work should be = -Kx^2 why is it 1/2-Kx^2
(1 vote)
• 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.
(1 vote)