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

Uma haste metálica, quando aquecida, alonga-se devido à expansão térmica linear. O comprimento depende do comprimento inicial, do coeficiente de expansão térmica do material e da variação de temperatura do sistema.Aqui, a temperatura da haste muda quadráticamente com o tempo.O objetivo é encontrar a taxa instantânea de variação do comprimento de uma haste metálica de 2 metros com coeficiente conhecido de expansão térmica no tempo exatamente 10 segundos.Como o comprimento depende da temperatura, e a temperatura depende do tempo, a regra da cadeia é aplicada.Para simplificar o cálculo da regra da cadeia, a diferenciação é separada em dois componentes distintos.Primeiro, o comprimento é diferenciado em relação à temperatura. Essa diferenciação fornece um valor constante: o produto do comprimento inicial pelo coeficiente de expansão.Segundo, a temperatura é diferenciada em relação ao tempo usando a regra da potência, fornecendo uma taxa linear de variação.De acordo com a regra da cadeia, a taxa final é o produto desses dois componentes. Substituir os valores conhecidos em 10 segundos obtém a taxa instantânea de variação necessária do comprimento.

a regra da cadeia: resolução de problemas

The Chain Rule: Problem Solving

A expans&#xE3;o t&#xE9;rmica de uma barra met&#xE1;lica ilustra a aplica&#xE7;&#xE3;o da Regra da Cadeia quando uma quantidade f&#xED;sica depende de outra que varia com o tempo. &#xC0; medida que a barra &#xE9; aquecida, seu comprimento muda de acordo com a expans&#xE3;o t&#xE9;rmica linear, enquanto a temperatura do sistema varia quadraticamente com o tempo.Na expans&#xE3;o t&#xE9;rmica linear, o comprimento L da barra depende da temperatura T, de modo que a taxa de varia&#xE7;&#xE3;o do comprimento em&#xA0;<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>em que L_0 = 2&#x2009;m &#xE9; o comprimento inicial da barra e &#x3B1; &#xE9; o coeficiente de expans&#xE3;o t&#xE9;rmica.A temperatura varia com o tempo de acordo com uma rela&#xE7;&#xE3;o quadr&#xE1;tica, ent&#xE3;o sua taxa de varia&#xE7;&#xE3;o em rela&#xE7;&#xE3;o ao tempo &#xE9; linear:<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>Como o comprimento depende da temperatura e a temperatura depende do tempo, a Regra da Cadeia permite obter a taxa de varia&#xE7;&#xE3;o do comprimento em rela&#xE7;&#xE3;o ao tempo da seguinte forma:<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>Substituindo as express&#xF5;es acima, obt&#xE9;m-se:<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>Avaliando essa express&#xE3;o em t = 10&#x2009;s, obt&#xE9;m-se a taxa instant&#xE2;nea de varia&#xE7;&#xE3;o do comprimento da barra naquele momento:<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>Essa express&#xE3;o fornece a taxa necess&#xE1;ria com que a barra met&#xE1;lica de 2 metros se alonga em 10 segundos.

A Regra da Cadeia: Resolução de Problemas

A expans&#xE3;o t&#xE9;rmica de uma barra met&#xE1;lica ilustra a aplica&#xE7;&#xE3;o da Regra da Cadeia quando uma quantidade f&#xED;sica depende de ...

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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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