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

Ein Metallstab verlängert sich beim Erhitzen aufgrund linearer Wärmeausdehnung. Die Länge hängt von der Anfangslänge, dem thermischen Ausdehnungskoeffizienten des Materials und der Temperaturänderung des Systems ab.Hier ändert sich die Temperatur des Stabs quadratisch mit der Zeit.Das Ziel ist es, die momentane Änderungsrate der Länge eines 2 Meter langen Metallstabs mit bekanntem thermischen Ausdehnungskoeffizienten genau von 10 Sekunden zu bestimmen.Da die Länge von der Temperatur und die Temperatur von der Zeit abhängt, wird die Kettenregel angewendet.Um die Kettenregelberechnung zu vereinfachen, wird die Differenzierung in zwei verschiedene Komponenten unterteilt.Erstens wird die Länge bezüglich der Temperatur differenziert. Diese Differenzierung ergibt einen konstanten Wert: das Produkt der Anfangslänge und des Expansionskoeffizienten.Zweitens wird die Temperatur zeitlich mittels der Potenzregel differenziert, was eine lineare Änderungsrate ergibt.Nach der Kettenregel ist die Endrate das Produkt dieser beiden Komponenten. Setzt man die bekannten Werte bei 10 Sekunden ein, erhält man die erforderliche sofortige Änderungsrate der Länge.

die kettenregel: problemlösung

The Chain Rule: Problem Solving

Die thermische Ausdehnung eines Metallstabs veranschaulicht die Anwendung der Kettenregel, wenn eine physikalische Gr&#xF6;&#xDF;e von einer anderen abh&#xE4;ngt, die sich mit der Zeit ver&#xE4;ndert. Beim Erhitzen des Stabs &#xE4;ndert sich seine L&#xE4;nge entsprechend der linearen thermischen Ausdehnung, w&#xE4;hrend die Temperatur des Systems quadratisch mit der Zeit variiert.Bei linearer thermischer Ausdehnung h&#xE4;ngt die L&#xE4;nge L des Stabs von der Temperatur T ab, sodass die &#xC4;nderungsrate der L&#xE4;nge in Bezug auf die Temperatur konstant ist:<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>wobei L_0 = 2 m die Anfangsl&#xE4;nge des Stabs ist und &#x3B1; den thermischen Ausdehnungskoeffizienten bezeichnet.Die Temperatur variiert mit der Zeit gem&#xE4;&#xDF; einer quadratischen Beziehung, sodass ihre &#xC4;nderungsrate linear ist:<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>Da die L&#xE4;nge von der Temperatur und die Temperatur von der Zeit abh&#xE4;ngen, ergibt die Kettenregel die &#xC4;nderungsrate der L&#xE4;nge in Abh&#xE4;ngigkeit von der Zeit:<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>Durch Einsetzen der obenstehenden Ausdr&#xFC;cke folgt:<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>Die Auswertung dieses Ausdrucks bei t = 10 s liefert die momentane &#xC4;nderungsrate der Stabl&#xE4;nge:<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>Dieser Ausdruck gibt die gesuchte Rate an, mit der sich der 2 Meter lange Metallstab nach 10 Sekunden verl&#xE4;ngert.

Die Kettenregel: Problemlösung

Die thermische Ausdehnung eines Metallstabs veranschaulicht die Anwendung der Kettenregel, wenn eine physikalische Gr&#xF6;&#xDF;e von einer anderen ...

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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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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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