Name: chain rule in thermal expansion
Uploaded: 2026-01-12T12:24:08.000Z
Duration: 1 min 23 s
Channel: JoVE

Una varilla metálica, al calentarse, se alarga debido a la expansión térmica lineal. La longitud depende de la longitud inicial, el coeficiente de expansión térmica del material y el cambio de temperatura del sistema.Aquí, la temperatura de la varilla cambia cuadráticamente con el tiempo.El objetivo es encontrar la tasa instantánea de cambio de la longitud de una varilla metálica de 2 metros con un coeficiente conocido de expansión térmica en el tiempo exactamente 10 segundos.Como la longitud depende de la temperatura y la temperatura del tiempo, se aplica la regla de la cadena.Para simplificar el cálculo de la regla de cadena, la diferenciación se separa en dos componentes distintos.Primero, la longitud se diferencia respecto a la temperatura. Esta diferenciación da un valor constante: el producto de la longitud inicial y el coeficiente de expansión.En segundo lugar, la temperatura se diferencia respecto al tiempo usando la regla de la potencia, lo que da una tasa de cambio lineal.Según la regla de la cadena, la tasa final es el producto de estos dos componentes. Sustituir los valores conocidos a 10 segundos da la tasa instantánea de cambio requerida de la longitud.

la regla de la cadena: resolución de problemas

The Chain Rule: Problem Solving

La dilataci&#xF3;n t&#xE9;rmica de una varilla met&#xE1;lica evidencia la aplicaci&#xF3;n de la regla de la cadena cuando una magnitud f&#xED;sica depende de otra que var&#xED;a con el tiempo. Al calentarse la varilla, su longitud cambia seg&#xFA;n la dilataci&#xF3;n t&#xE9;rmica lineal, mientras que la temperatura del sistema var&#xED;a cuadr&#xE1;ticamente con el tiempo.Para la dilataci&#xF3;n t&#xE9;rmica lineal, la longitud L de la varilla depende de la temperatura T, de modo que la velocidad de cambio de la longitud con respecto a la temperatura es constante:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:50%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19799/PT_19799_1_editor_19799.svg" alt="PT_19799_1.svg" /></figure>donde L_0 = 2&#x2009;m es la longitud inicial de la varilla y &#x3B1; es el coeficiente de dilataci&#xF3;n t&#xE9;rmica.La temperatura var&#xED;a con el tiempo seg&#xFA;n una relaci&#xF3;n cuadr&#xE1;tica, por lo que su tasa de cambio con respecto al tiempo es lineal:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:50%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19799/PT_19799_2_editor_19799.svg" alt="PT_19799_2.svg" /></figure>Dado que la longitud depende de la temperatura y la temperatura depende del tiempo, mediante la regla de la cadena se obtiene la tasa de cambio de la longitud con respecto al tiempo como:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:50%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19799/PT_19799_3_editor_19799.svg" alt="PT_19799_3.svg" /></figure>Sustituyendo las expresiones anteriores, se obtiene:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:50%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19799/PT_19799_4_editor_19799.svg" alt="PT_19799_4.svg" /></figure>Evaluando esta expresi&#xF3;n en t = 10&#x2009;s, se obtiene la tasa instant&#xE1;nea de cambio de la longitud de la varilla en ese instante:<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:50%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19799/PT_19799_5_editor_19799.svg" alt="PT_19799_5.svg" /></figure>Esta expresi&#xF3;n proporciona la tasa requerida a la que se est&#xE1; alargando la varilla met&#xE1;lica de 2 metros a los 10 segundos.

La regla de la cadena: resolución de problemas

La dilataci&#xF3;n t&#xE9;rmica de una varilla met&#xE1;lica evidencia la aplicaci&#xF3;n de la regla de la cadena cuando una magnitud f&#xED;sica depende ...

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chain rule in thermal expansion

The Chain Rule appears in thermal expansion when a rod’s length changes because its temperature changes over time. In this example, a metal rod is heated, and its length depends on temperature. The temperature itself changes with time in a quadratic way.

For linear thermal expansion, the rod’s length L depends on temperature T with a constant rate of change, using the initial length L0 = 2 m and the coefficient of thermal expansion α. The temperature follows a quadratic relationship in time, so its rate of change with respect to time is linear. Because length depends on temperature and temperature depends on time, the Chain Rule connects these two rates.

Applying the Chain Rule gives the rate of change of length with respect to time. Substituting the expressions for thermal expansion and temperature change leads to the needed formula. Evaluating that expression at t = 10 s gives the instantaneous rate at which the 2-meter metal rod is elongating at that moment.

Chain Rule in Thermal Expansion

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the chain rule: problem solving

The Chain Rule appears in thermal expansion when a rod’s length changes because its temperature changes over time. In this example, a metal rod is heated, and its length depends on temperature. The temperature itself changes with time in a quadratic way. For linear thermal expansion, the rod’s length L depends on temperature T with a constant rate of change, using the initial length L0 = 2 m and the coefficient of thermal expansion α. The temperature follows a quadratic relationship in time,...

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key terms and concepts

See how the Chain Rule links temperature change and rod length in thermal expansion, then find the rod’s elongation rate at 10 seconds.

Chain Rule in Thermal Expansion | 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

Chain Rule in Thermal Expansion

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