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

Una barra metallica, quando riscaldata, si allunga a causa dell'espansione termica lineare. La lunghezza dipende dalla lunghezza iniziale, dal coefficiente di espansione termica del materiale e dal cambiamento di temperatura del sistema.Qui, la temperatura della barra cambia quadraticamente nel tempo.L'obiettivo è trovare la velocità istantanea di variazione della lunghezza di una barra metallica di 2 metri con un coefficiente di espansione termica noto al tempo esattamente di 10 secondi.Poiché la lunghezza dipende dalla temperatura, e la temperatura dipende dal tempo, viene applicata la regola della catena.Per semplificare il calcolo della regola della catena, la differenziazione è suddivisa in due componenti distinte.Innanzitutto, la lunghezza è differenziata rispetto alla temperatura. Questa differenziazione dà un valore costante: il prodotto della lunghezza iniziale e del coefficiente di espansione.In secondo luogo, la temperatura viene differenziata rispetto al tempo usando la regola della potenza, dando un tasso di variazione lineare.Secondo la regola della catena, il tasso finale è il prodotto di queste due componenti. Sostituendo i valori noti a 10 secondi si ottiene la velocità istantanea richiesta di variazione della durata.

la regola della catena: risoluzione dei problemi

The Chain Rule: Problem Solving

La dilatazione termica di una barra metallica illustra l&#x2019;applicazione della regola della catena quando una grandezza fisica dipende da un&#x2019;altra che varia nel tempo. Quando la barra viene riscaldata, la sua lunghezza cambia secondo la dilatazione termica lineare, mentre la temperatura del sistema varia quadraticamente nel tempo.Per la dilatazione termica lineare, la lunghezza L della barra dipende dalla temperatura T in modo tale che la velocit&#xE0; di variazione della lunghezza rispetto alla temperatura sia costante:<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>dove L_0 = 2 m &#xE8; la lunghezza iniziale della barra e &#x3B1; &#xE8; il coefficiente di dilatazione termica.La temperatura varia nel tempo secondo una relazione quadratica, quindi la sua velocit&#xE0; di variazione rispetto al tempo &#xE8; lineare:<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>Poich&#xE9; la lunghezza dipende dalla temperatura e la temperatura dipende dal tempo, la regola della catena fornisce la velocit&#xE0; di variazione della lunghezza rispetto al tempo come:<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>Sostituendo le espressioni precedenti si ottiene:<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>Valutando questa espressione per t = 10 s si ottiene la velocit&#xE0; di variazione istantanea della lunghezza della barra in quell&#x2019;istante:<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>Questa espressione fornisce la velocit&#xE0; richiesta con cui la barra metallica di 2 m si allunga a t = 10 s.

La regola della catena: risoluzione dei problemi

La dilatazione termica di una barra metallica illustra l&#x2019;applicazione della regola della catena quando una grandezza fisica dipende da ...

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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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