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

When a force is applied to a linear spring, the restoring force increases evenly with displacement. This relationship follows Hooke&#8217;s law, which enables the calculation of work done from the force&#8211;displacement curve.For a nonlinear spring, Hooke&#8217;s law does not apply. Its restoring force changes with position, so the force&#8211;displacement graph becomes curved.The total work done in stretching the spring is determined by integrating this force over displacement.However, the integral is difficult to solve directly because a square root term complicates the standard integration process.To simplify the integral, the substitution rule is used. It introduces a new variable for the complex term. Differentiating the variable and rearranging the derivative helps us substitute the other terms inside the integral.Applying the substitution transforms the term inside the square root into a simpler expression, converting the integral into a single power function.After integration, the new variable is replaced with the original displacement variable, preserving its physical interpretation.The final expression represents the work required to stretch a nonlinear spring.

substitutieregel toegepast op onbepaalde integralen

Wanneer op een lineaire veer een kracht wordt uitgeoefend, neemt de herstellende kracht evenredig toe met de verplaatsing. Dit gedrag wordt beschreven door de wet van Hooke, die het mogelijk maakt de op de veer verrichte arbeid rechtstreeks te bepalen uit de kracht&#x2013;verplaatsingsrelatie. In dit geval varieert de kracht op een eenvoudige en voorspelbare wijze, waardoor de berekening relatief eenvoudig is.Een nietlineaire veer daarentegen voldoet niet aan de wet van Hooke. De herstellende kracht is op een niet-evenredige manier afhankelijk van de positie, waardoor de kracht&#x2013;verplaatsingsrelatie krom is in plaats van lineair. Een voorbeeld van een dergelijke herstellende kracht wordt beschreven door de uitdrukking\begin{equation*}F \varpropto x \bm{\sqrt{{1 + \left(\jfrac{x}{L}\right)^2}}}\end{equation*}waarbij de kracht sneller toeneemt dan de verplaatsing zelf. Deze vorm illustreert hoe nietlineaire effecten steeds uitgesprokener worden naarmate de veer verder wordt uitgerekt.De totale arbeid die wordt verricht bij het uitrekken van een nietlineaire veer wordt bepaald door de kracht over de verplaatsing te accumuleren. Het rechtstreeks bepalen van deze grootheid is echter moeilijk, omdat de wortelterm de toepassing van standaardintegratietechnieken bemoeilijkt. Om de berekening te vereenvoudigen, wordt de substitutieregel toegepast. Deze methode voert een nieuwe variabele in die de uitdrukking binnen de wortel representeert, waardoor de resterende termen eenvoudiger te behandelen zijn.Door de ingevoerde variabele te differenti&#xEB;ren en de termen te herschikken, kan de uitdrukking worden herschreven in een eenvoudiger vorm die wordt gereduceerd tot &#xE9;&#xE9;n macht. Na het uitvoeren van de integratie wordt de ingevoerde variabele terugvervangen door de oorspronkelijke verplaatsingsvariabele, zodat de fysische betekenis van het resultaat behouden blijft. De uiteindelijke uitdrukking geeft de arbeid weer die nodig is om de nietlineaire veer uit te rekken.

Substitutieregel toegepast op onbepaalde integralen

Wanneer op een lineaire veer een kracht wordt uitgeoefend, neemt de herstellende kracht evenredig toe met de verplaatsing. Dit gedrag wordt beschreven ...

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.

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Work Done on a Nonlinear Spring

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