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

Quando una forza viene applicata a una molla lineare, la forza di ripristino aumenta uniformemente con lo spostamento. Questa relazione segue la legge di Hooke, che consente il calcolo del lavoro svolto dalla curva forza-spostamento.Per una molla non lineare, la legge di Hooke non si applica. La forza di ripristino cambia con la posizione, quindi il grafico forza-spostamento diventa curvo.Il lavoro totale svolto nello stiramento della molla è determinato integrando questa forza rispetto allo spostamento.Tuttavia, l'integrale è difficile da risolvere direttamente perché un termine a radice quadrata complica il processo di integrazione standard.Per semplificare l'integrale, si usa la regola della sostituzione. Introduce una nuova variabile per il termine complesso. Differenziare la variabile e riorganizzare la derivata ci aiuta a sostituire gli altri termini all'interno dell'integrale.Applicando la sostituzione, il termine all'interno della radice quadrata si trasforma in un'espressione più semplice, convertendo l'integrale in una singola funzione di potenza.Dopo l'integrazione, la nuova variabile viene sostituita con la variabile di spostamento originale, preservandone l'interpretazione fisica.L'espressione finale rappresenta il lavoro necessario per allungare una molla non lineare.

regola della sostituzione applicata agli integrali indefiniti

Quando si applica una forza a una molla lineare, la forza elastica di richiamo aumenta proporzionalmente allo spostamento. Questo comportamento &#xE8; descritto dalla legge di Hooke, che consente di determinare direttamente il lavoro compiuto sulla molla a partire dalla relazione forza&#x2013;spostamento. In questo caso, la forza varia in modo semplice e prevedibile, rendendo il calcolo relativamente semplice.Al contrario, una molla non lineare non obbedisce alla legge di Hooke. La sua forza elastica di richiamo dipende dalla posizione in modo non proporzionale, facendo s&#xEC; che la relazione forza&#x2013;spostamento diventi curva anzich&#xE9; lineare. Un esempio di tale forza elastica di richiamo &#xE8; dato dall&#x2019;espressione\begin{equation*}F \varpropto x \bm{\sqrt{{1 + \left(\jfrac{x}{L}\right)^2}}}\end{equation*}in cui la forza aumenta pi&#xF9; rapidamente dello spostamento considerato da solo. Questa forma mostra come gli effetti non lineari divengano pi&#xF9; marcati man mano che la molla viene allungata ulteriormente.Il lavoro totale compiuto per allungare una molla non lineare si determina accumulando la forza rispetto allo spostamento. Tuttavia, valutare direttamente tale quantit&#xE0; &#xE8; difficile, poich&#xE9; il termine con la radice quadrata complica le tecniche standard di integrazione. Per semplificare il calcolo, si applica la regola della sostituzione. Questo metodo introduce una nuova variabile per rappresentare l&#x2019;espressione all&#x2019;interno della radice quadrata, rendendo pi&#xF9; gestibili i termini restanti.Differenziando la variabile introdotta con la sostituzione e riorganizzando i termini, l&#x2019;espressione pu&#xF2; essere riscritta in una forma pi&#xF9; semplice, che diventa una singola potenza. Dopo aver eseguito l&#x2019;integrazione, la variabile introdotta viene sostituita con la variabile di spostamento originale, cos&#xEC; da preservare il significato fisico del risultato. L&#x2019;espressione finale rappresenta il lavoro richiesto per allungare la molla non lineare.

Regola della sostituzione applicata agli integrali indefiniti

Quando si applica una forza a una molla lineare, la forza elastica di richiamo aumenta proporzionalmente allo spostamento. Questo comportamento &#xE8; ...

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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