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

Metalowy pręt, po nagrzewaniu, wydłuża się z powodu liniowej rozszerzalności termicznej. Długość zależy od długości początkowej, współczynnika rozszerzalności cieplnej materiału oraz zmiany temperatury układu.Tutaj temperatura pręta zmienia się kwadratowo z czasem.Celem jest ustalenie chwilowej szybkości zmiany długości metalowego pręta o średnicy 2 metrów o znanym współczynniku rozszerzalności cieplnej dokładnie 10 sekund.Ponieważ długość zależy od temperatury, a temperatura od czasu, stosuje się zasadę łańcucha.Aby uprościć obliczenia reguły łańcuchowej, różniczkowanie dzieli się na dwa odrębne składniki.Po pierwsze, długość jest różniczkowana względem temperatury. To różniczkowanie daje stałą wartość: iloczyn długości początkowej i współczynnika rozprężności.Po drugie, temperatura różniczkuje się względem czasu za pomocą reguły potęgowej, co daje liniową szybkość zmian.Zgodnie z regułą łańcuchową, końcowa szybkość jest iloczynem tych dwóch składowych. Podstawienie znanych wartości po 10 sekundach daje wymaganą natychmiastową szybkość zmiany długości.

reguła łańcuchowa: rozwiązywanie zadań

The Chain Rule: Problem Solving

Rozszerzalno&#x15B;&#x107; cieplna pr&#x119;ta metalowego ilustruje zastosowanie regu&#x142;y &#x142;a&#x144;cuchowej, gdy jedna wielko&#x15B;&#x107; fizyczna zale&#x17C;y od innej, kt&#xF3;ra zmienia si&#x119; w czasie. Wraz z nagrzewaniem pr&#x119;ta jego d&#x142;ugo&#x15B;&#x107; zmienia si&#x119; zgodnie z liniow&#x105; rozszerzalno&#x15B;ci&#x105; ciepln&#x105;, podczas gdy temperatura uk&#x142;adu zmienia si&#x119; kwadratowo w czasie.W przypadku liniowej rozszerzalno&#x15B;ci cieplnej d&#x142;ugo&#x15B;&#x107; L pr&#x119;ta zale&#x17C;y od temperatury T w taki spos&#xF3;b, &#x17C;e tempo zmian d&#x142;ugo&#x15B;ci wzgl&#x119;dem temperatury jest sta&#x142;e:<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>gdzie L_0 = 2&#x2009;m to pocz&#x105;tkowa d&#x142;ugo&#x15B;&#x107; pr&#x119;ta, a &#x3B1; to wsp&#xF3;&#x142;czynnik rozszerzalno&#x15B;ci cieplnej.Temperatura zmienia si&#x119; w czasie zgodnie z zale&#x17C;no&#x15B;ci&#x105; kwadratow&#x105;, wi&#x119;c jej tempo zmian wzgl&#x119;dem czasu jest liniowe:<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>Poniewa&#x17C; d&#x142;ugo&#x15B;&#x107; zale&#x17C;y od temperatury, a temperatura zale&#x17C;y od czasu, regu&#x142;a &#x142;a&#x144;cuchowa wyra&#x17C;a tempo zmian d&#x142;ugo&#x15B;ci wzgl&#x119;dem czasu nast&#x119;puj&#x105;co:<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>Podstawienie powy&#x17C;szych wyra&#x17C;e&#x144; prowadzi do:<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>Po podstawieniu t = 10&#x2009;s otrzymujemy chwilowe tempo zmian d&#x142;ugo&#x15B;ci pr&#x119;ta w tej chwili:<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>To wyra&#x17C;enie podaje wymagan&#x105; szybko&#x15B;&#x107;, z jak&#x105; metalowy pr&#x119;t o d&#x142;ugo&#x15B;ci 2 m wyd&#x142;u&#x17C;a si&#x119; w chwili t = 10 s.

Reguła łańcuchowa: rozwiązywanie zadań

Rozszerzalno&#x15B;&#x107; cieplna pr&#x119;ta metalowego ilustruje zastosowanie regu&#x142;y &#x142;a&#x144;cuchowej, gdy jedna wielko&#x15B;&#x107; ...

Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Differentiation Rules

Zasady różniczkowania

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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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application of rates of change

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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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