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

A metal rod, when heated, elongates due to linear thermal expansion. The length depends on the initial length, the material&#8217;s coefficient of thermal expansion, and the change in temperature of the system.Here, the temperature of the rod changes quadratically with time.The objective is to find the instantaneous rate of change of the length of a 2-meter metal rod with a known coefficient of thermal expansion at time exactly 10 seconds.Because length depends on temperature, and temperature depends on time, the chain rule is applied.To simplify the chain rule calculation, the differentiation is separated into two distinct components.First, the length is differentiated with respect to temperature. This differentiation gives a constant value: the product of the initial length and the expansion coefficient.Second, the temperature is differentiated with respect to time using the power rule, giving a linear rate of change.According to the chain rule, the final rate is the product of these two components. Substituting the known values at 10 seconds gives the required instantaneous rate of change of the length.

the chain rule: problem solving

The Chain Rule: Problem Solving

The thermal expansion of a metal rod shows the application of the Chain Rule when one physical quantity depends on another that varies with time. As the rod is heated, its length changes according to linear thermal expansion, while the temperature of the system varies quadratically with time.For linear thermal expansion, the length L of the rod depends on temperature T such that the rate of change of length with respect to temperature is constant:<figure class="image image-style-align-center" style="display:table;margin-left:auto;margin-right:auto;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19799/PT_19799_1_editor_19799.svg" alt="Thermal expansion equation \(dL/dT = L_0\alpha\), key concept in physics, formula representation." data-altupdatedat="1775223722000" data-alt="PT_19799_1.svg" /></figure>where L0 = 2&#8201;m is the initial length of the rod, and &#945; is the coefficient of thermal expansion.The temperature varies with time according to a quadratic relationship, so its rate of change with respect to time is linear:<figure class="image image-style-align-center" style="display:table;margin-left:auto;margin-right:auto;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19799/PT_19799_2_editor_19799.svg" alt="Differential equation formula diagram, dT/dt = 2at + b, illustrating rate of change concept." data-altupdatedat="1775223722000" data-alt="PT_19799_2.svg" /></figure>Because length depends on temperature and temperature depends on time, the Chain Rule gives the rate of change of length with respect to time as<figure class="image image-style-align-center" style="display:table;margin-left:auto;margin-right:auto;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19799/PT_19799_3_editor_19799.svg" alt="Calculus chain rule formula, &#8706;L/&#8706;t = &#8706;L&#8706;T/&#8706;T&#8706;t, mathematical concept diagram." data-altupdatedat="1775223722000" data-alt="PT_19799_3.svg" /></figure>Substituting the expressions above results in<figure class="image image-style-align-center" style="display:table;margin-left:auto;margin-right:auto;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19799/PT_19799_4_editor_19799.svg" alt="Differential equation for dL/dt in physics, showcasing rate of change." data-altupdatedat="1775223722000" data-alt="PT_19799_4.svg" /></figure>Evaluating this expression at t = 10&#8201;s yields the instantaneous rate of change of the rod&#8217;s length at that moment:<figure class="image image-style-align-center" style="display:table;margin-left:auto;margin-right:auto;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19799/PT_19799_5_editor_19799.svg" alt="Differential equation dL/dt at t=10, illustrating rate change, educational formula." data-altupdatedat="1775223722000" data-alt="PT_19799_5.svg" /></figure>This expression gives the required rate at which the 2-meter metal rod is elongating at 10 seconds.

The thermal expansion of a metal rod shows the application of the Chain Rule when one physical quantity depends on another that varies with time. As the ...

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Calculus

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Differentiation Rules

KEY TERMS AND CONCEPTS

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 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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