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

Une tige métallique, lorsqu’elle est chauffée, s’allonge à cause de la dilatation thermique linéaire. La longueur dépend de la longueur initiale, du coefficient de dilatation thermique du matériau et du changement de température du système.Ici, la température de la tige change quadratiquement avec le temps.L’objectif est de déterminer le taux de variation instantané de la longueur d’une tige métallique de 2 mètres avec un coefficient de dilatation thermique connu à un temps exactement de 10 secondes.Comme la longueur dépend de la température, et que la température dépend du temps, la règle de la chaîne est appliquée.Pour simplifier le calcul de la règle de chaîne, la différentiation est divisée en deux composantes distinctes.Premièrement, la longueur est différenciée par rapport à la température. Cette différentiation donne une valeur constante : le produit de la longueur initiale et du coefficient d’expansion.Deuxièmement, la température est différenciée par rapport au temps à l’aide de la règle de puissance, ce qui donne un taux de variation linéaire.Selon la règle de la chaîne, le taux final est le produit de ces deux composantes. En substituant les valeurs connues à 10 secondes, on obtient le taux de variation instantané requis de la durée.

la règle de la chaîne : résolution de problèmes

The Chain Rule: Problem Solving

La dilatation thermique d&#x2019;une tige m&#xE9;tallique illustre l&#x2019;application de la r&#xE8;gle de la cha&#xEE;ne lorsqu&#x2019;une grandeur physique d&#xE9;pend d&#x2019;une autre qui varie avec le temps. Lorsque la tige est chauff&#xE9;e, sa longueur &#xE9;volue conform&#xE9;ment &#xE0; la loi de la dilatation thermique lin&#xE9;aire, tandis que la temp&#xE9;rature du syst&#xE8;me varie selon une loi quadratique en fonction du temps.Dans le cas d&#x2019;une dilatation thermique lin&#xE9;aire, la longueur L de la tige d&#xE9;pend de la temp&#xE9;rature T, de sorte que le taux de variation de la longueur par rapport &#xE0; la temp&#xE9;rature est constant :<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>o&#xF9; L_0 = 2&#x2009;m est la longueur initiale de la tige et &#x3B1; d&#xE9;signe le coefficient de dilatation thermique lin&#xE9;aire.La temp&#xE9;rature varie avec le temps selon une relation quadratique, de sorte que son taux de variation par rapport au temps est lin&#xE9;aire :<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>Puisque la longueur d&#xE9;pend de la temp&#xE9;rature et que la temp&#xE9;rature d&#xE9;pend du temps, la r&#xE8;gle de la cha&#xEE;ne donne le taux de variation de la longueur par rapport au temps<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>En substituant les expressions ci-dessus, on obtient<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>L&#x2019;&#xE9;valuation de cette expression pour t = 10&#x2009;s fournit le taux de variation instantan&#xE9; de la longueur de la tige &#xE0; cet instant :<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>Cette expression repr&#xE9;sente le taux d&#x2019;allongement recherch&#xE9; de la tige m&#xE9;tallique de 2 m pour t = 10 s.

La règle de la chaîne : résolution de problèmes

La dilatation thermique d&#x2019;une tige m&#xE9;tallique illustre l&#x2019;application de la r&#xE8;gle de la cha&#xEE;ne lorsqu&#x2019;une grandeur ...

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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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