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### Kurs: Fizyka - 11 klasa (Indie)>Rozdział 10

Lekcja 5: Energia potencjalna odkształconej sprężyny i prawo Hooka

# 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.

## Pojęcia kluczowe

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 $\frac{\text{N}}{\text{m}}$.
Spring force (${\stackrel{\to }{F}}_{s}$)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 $\text{J}$.

## Równania

EquationSymbolsMeaning in words
$|{\stackrel{\to }{F}}_{s}|=k|\stackrel{\to }{x}|$${\stackrel{\to }{F}}_{s}$ is spring force, $\stackrel{\to }{x}$ is length of extension or compression relative to the unstretched length, and $k$ is spring constantThe magnitude of the force required to change the length of a spring-like object is directly proportional to the spring constant and the displacement of the spring.
${U}_{s}=\frac{1}{2}k{x}^{2}$${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
${\stackrel{\to }{F}}_{s}=-k\stackrel{\to }{x}$
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.

## 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:
$\begin{array}{rl}{U}_{s}& =\frac{1}{2}\phantom{\rule{0.167em}{0ex}}\text{base}\cdot \text{height}\\ \\ & =\frac{1}{2}x\cdot kx\\ \\ & =\frac{1}{2}k\left(x{\right)}^{2}\end{array}$
Note that the spring constant $k$ is the slope of the line since $k=\frac{|\stackrel{\to }{F}|}{|\stackrel{\to }{x}|}$.

## Często spotykane błędy i nieporozumienia

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.

## Dowiedz się więcej

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.

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