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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
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 $\frac{N}{m}$ ‍.
Spring force ( ${\vec{F}}_{s}$ ‍)	Force applied by a spring given by Hooke’s law. SI units of $N$ ‍.
Elastic potential energy ( $U_{s}$ ‍)	Potential energy stored as a result of applying a force to deform a spring-like object. SI units of $J$ ‍.

[What happens when a spring is stretched too far?]

Równania

Equation	Symbols	Meaning in words
$\| {\vec{F}}_{s} \| = k \| \vec{x} \|$ ‍	${\vec{F}}_{s}$ ‍ 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.
$U_{s} = \frac{1}{2} k x^{2}$ ‍	$U_{s}$ ‍ 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

{\vec{F}}_{s} = - k \vec{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 do we find x?]

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{aligned} U_{s} & = \frac{1}{2} base \cdot height \\ = \frac{1}{2} x \cdot k x \\ = \frac{1}{2} k (x)^{2} \end{aligned}

Note that the spring constant

k

is the slope of the line since

k = \frac{| \vec{F} |}{| \vec{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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