Name: work done on a nonlinear spring
Uploaded: 2026-01-12T12:24:06.000Z
Duration: 1 min 27 s
Channel: JoVE

Wenn eine Kraft auf eine lineare Feder ausgeübt wird, steigt die Rückwärtskraft gleichmäßig mit der Verschiebung. Diese Beziehung folgt dem Hookeschen Gesetz, das die Berechnung der Arbeit aus der Kraft-Verschiebungskurve ermöglicht.Für eine nichtlineare Feder gilt das Hookesche Gesetz nicht. Seine Rückstellenkraft ändert sich mit der Position, sodass das Kraft-Verschiebungsdiagramm gekrümmt wird.Die Gesamtarbeit beim Dehnen der Feder wird durch die Integration dieser Kraft über die Verschiebung bestimmt.Das Integral ist jedoch schwer direkt zu lösen, da ein Quadratwurzelterm den Standardintegrationsprozess erschwert.Um das Integral zu vereinfachen, wird die Substitutionsregel verwendet. Sie führt eine neue Variable für den komplexen Term ein. Das Differenzieren der Variablen und das Umordnen der Ableitung hilft uns, die anderen Terme innerhalb des Integrals einzusetzen.Die Anwendung der Substitution verwandelt den Term innerhalb der Quadratwurzel in einen einfacheren Ausdruck, wodurch das Integral in eine einzelne Potenzfunktion umgewandelt wird.Nach der Integration wird die neue Variable durch die ursprüngliche Verschiebungsvariable ersetzt, wodurch ihre physikalische Interpretation erhalten bleibt.Der Endausdruck stellt die Arbeit dar, die erforderlich ist, um eine nichtlineare Feder zu dehnen.

anwendung der substitutionsregel auf unbestimmte integrale

Wird eine Kraft auf eine lineare Feder ausge&#xFC;bt, nimmt die R&#xFC;ckstellkraft proportional zur Auslenkung zu. Dieses Verhalten wird durch das Hookesche Gesetz beschrieben, das es erm&#xF6;glicht, die an der Feder verrichtete Arbeit direkt aus der Kraft-Weg-Beziehung zu bestimmen. In diesem Fall variiert die Kraft auf einfache und vorhersagbare Weise, wodurch die Berechnung relativ einfach ist.Eine nichtlineare Feder hingegen gehorcht nicht dem Hookeschen Gesetz. Ihre R&#xFC;ckstellkraft h&#xE4;ngt nicht proportional von der Auslenkung ab, sodass die Kraft-Auslenkungs-Beziehung gekr&#xFC;mmt statt linear verl&#xE4;uft. Ein Beispiel f&#xFC;r eine solche R&#xFC;ckstellkraft ergibt sich aus:\begin{equation*}F \varpropto x \bm{\sqrt{{1 + \left(\jfrac{x}{L}\right)^2}}}\end{equation*}Dabei steigt die Kraft schneller an als die Auslenkung allein. Diese Form zeigt, wie nichtlineare Effekte st&#xE4;rker hervortreten, wenn die Feder weiter gedehnt wird.Die gesamte Arbeit, die beim Dehnen einer nichtlinearen Feder verrichtet wird, ergibt sich durch Aufsummieren der Kraft &#xFC;ber die Auslenkung. Die direkte Berechnung ist jedoch schwierig, da der Wurzelterm die Anwendung g&#xE4;ngiger Integrationsverfahren erschwert. Zur Vereinfachung der Berechnung wird die Substitutionsregel angewendet. Dabei wird eine neue Variable eingef&#xFC;hrt, die den Ausdruck unter der Quadratwurzel repr&#xE4;sentiert, wodurch die verbleibenden Terme leichter behandelbar werden.Durch Differenzieren der substituierten Variable und Umstellen der Terme l&#xE4;sst sich der Ausdruck vereinfachen und auf eine einzelne Potenzfunktion reduzieren. Nach der Integration wird die substituierte Variable wieder durch die urspr&#xFC;ngliche Auslenkungsvariable ersetzt, sodass die physikalische Bedeutung des Ergebnisses erhalten bleibt. Der resultierende Ausdruck stellt die Arbeit dar, die zum Dehnen der nichtlinearen Feder erforderlich ist.

Anwendung der Substitutionsregel auf unbestimmte Integrale

Wird eine Kraft auf eine lineare Feder ausge&#xFC;bt, nimmt die R&#xFC;ckstellkraft proportional zur Auslenkung zu. Dieses Verhalten wird durch das ...

substitution-rule-applied-to-indefinite-integrals

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Substitution Rule Applied to Indefinite Integrals

work done on a nonlinear spring

Work done on a nonlinear spring can be found by using integration and the substitution rule. A linear spring follows Hooke’s law, so its restoring force increases in direct proportion to displacement. That makes the force-displacement graph a straight line and the work calculation straightforward.

A nonlinear spring does not obey Hooke’s law. Its restoring force changes with position in a non-proportional way, so the force-displacement graph is curved instead of linear. In the example shown, the force increases more rapidly than displacement alone, especially as the spring is stretched farther.

To find the total work, the force must be accumulated over the displacement. Direct integration is difficult because the square root term makes the expression harder to solve with standard methods. The substitution rule simplifies the integral by replacing the expression inside the square root with a new variable.

After the new variable is chosen, it is differentiated and rearranged so the integral becomes a simpler power expression. The calculation is then completed with integration. Finally, the substituted variable is changed back to the original displacement variable, and the result gives the work required to stretch the nonlinear spring.

Work Done on a Nonlinear Spring

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Work done on a nonlinear spring can be found by using integration and the substitution rule. A linear spring follows Hooke’s law, so its restoring force increases in direct proportion to displacement. That makes the force-displacement graph a straight line and the work calculation straightforward. A nonlinear spring does not obey Hooke’s law. Its restoring force changes with position in a non-proportional way, so the force-displacement graph is curved instead of linear. In the example shown,...

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Learn how integration and the substitution rule are used to calculate the work required to stretch a nonlinear spring when the force is not linear.

Work Done on a Nonlinear Spring | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Work Done on a Nonlinear Spring

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