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

Cuando se aplica una fuerza a un muelle lineal, la fuerza restauradora aumenta de forma uniforme con el desplazamiento. Esta relación sigue la ley de Hooke, que permite calcular el trabajo realizado a partir de la curva fuerza-desplazamiento.Para un muelle no lineal, la ley de Hooke no se aplica. Su fuerza restauradora cambia con la posición, por lo que el gráfico fuerza-desplazamiento se vuelve curvo.El trabajo total realizado en el estiramiento del muelle se determina integrando esta fuerza sobre el desplazamiento.Sin embargo, la integral es difícil de resolver directamente porque un término raíz cuadrada complica el proceso estándar de integración.Para simplificar la integral, se utiliza la regla de sustitución. Introduce una nueva variable para el término complejo. Diferenciar la variable y reorganizar la derivada nos ayuda a sustituir los otros términos dentro de la integral.Aplicar la sustitución transforma el término dentro de la raíz cuadrada en una expresión más simple, convirtiendo la integral en una sola función de potencia.Tras la integración, la nueva variable se reemplaza por la variable de desplazamiento original, preservando su interpretación física.La expresión final representa el trabajo necesario para estirar un muelle no lineal.

regla de sustitución aplicada a integrales indefinidas

Cuando se aplica una fuerza a un resorte lineal, la fuerza restauradora aumenta proporcionalmente al valor del desplazamiento. Este comportamiento se describe mediante la ley de Hooke, que permite determinar el trabajo realizado sobre el resorte directamente a partir de la relaci&#xF3;n fuerza&#x2013;desplazamiento. En este caso, la fuerza var&#xED;a de forma simple y predecible, lo que facilita el c&#xE1;lculo.Por otro lado, un resorte no lineal no cumple la ley de Hooke. Su fuerza restauradora depende de la posici&#xF3;n de forma no proporcional, lo que provoca que la relaci&#xF3;n fuerza&#x2013;desplazamiento adopte una forma curva en lugar de lineal. Un ejemplo de dicha fuerza restauradora se da en la expresi&#xF3;n:\begin{equation*}F \varpropto x \bm{\sqrt{{1 + \left(\jfrac{x}{L}\right)^2}}}\end{equation*}donde la fuerza aumenta m&#xE1;s r&#xE1;pidamente que el desplazamiento por s&#xED; solo. Esta forma muestra c&#xF3;mo los efectos no lineales se acent&#xFA;an a medida que el resorte se estira m&#xE1;s.El trabajo total realizado al estirar un resorte no lineal se determina acumulando la fuerza a lo largo del desplazamiento. Sin embargo, evaluar esta cantidad directamente es dif&#xED;cil porque el t&#xE9;rmino bajo la ra&#xED;z cuadrada complica las t&#xE9;cnicas de integraci&#xF3;n est&#xE1;ndar. Para simplificar el c&#xE1;lculo, se aplica la regla de sustituci&#xF3;n. Este m&#xE9;todo introduce una nueva variable para representar la expresi&#xF3;n dentro de la ra&#xED;z cuadrada, lo que facilita el manejo de los t&#xE9;rminos restantes.Al derivar la variable sustituida y reorganizar los t&#xE9;rminos, la expresi&#xF3;n puede reescribirse de forma m&#xE1;s sencilla, reduci&#xE9;ndola a una &#xFA;nica potencia. Tras realizar la integraci&#xF3;n, la variable sustituida se reemplaza de nuevo por la variable de desplazamiento original para conservar el significado f&#xED;sico del resultado. La expresi&#xF3;n final representa el trabajo necesario para estirar el resorte no lineal.

Regla de sustitución aplicada a integrales indefinidas

Cuando se aplica una fuerza a un resorte lineal, la fuerza restauradora aumenta proporcionalmente al valor del desplazamiento. Este comportamiento se ...

substitution-rule-applied-to-indefinite-integrals

Education

Educación

JoVE Core

Calculus

Cálculo

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Unidad

Integrals

Integrales

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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

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calculus

Work Done on a Nonlinear Spring

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