2.7
La règle de la chaîne : résolution de problèmes
La dilatation thermique d’une tige métallique illustre l’application de la règle de la chaîne lorsqu’une grandeur physique dépend d’une autre qui vari…
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.
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Q1: Why is the chain rule necessary when finding the rate of change of a metal rod's length?
The chain rule is necessary because the rod's length depends on temperature, which itself depends on time. Since length is an indirect function of time through temperature, the chain rule allows us to find the instantaneous rate of change of length with respect to time by multiplying the rate of change of length with respect to temperature by the rate of change of temperature with respect to time.
Q2: How does linear thermal expansion relate to the rate of change of a metal rod's length?
In linear thermal expansion, the rod's length changes proportionally to temperature change. The rate of change of length with respect to temperature is constant, determined by the product of the initial length and the material's coefficient of thermal expansion. This constant rate becomes one component in the chain rule calculation.
Q3: What does it mean that temperature varies quadratically with time in this problem?
A quadratic temperature variation means temperature follows a second-degree polynomial relationship with time. When differentiated with respect to time using the power rule, this quadratic relationship yields a linear rate of change. This linear rate becomes the second component multiplied in the chain rule formula.
Q4: How do you calculate the instantaneous rate of change of the rod's length at a specific time?
Apply the chain rule by multiplying two derivatives: the rate of change of length with respect to temperature and the rate of change of temperature with respect to time. Then substitute the specific time value (10 seconds) into the resulting expression to obtain the instantaneous rate at that moment.
Q5: What role does the coefficient of thermal expansion play in determining elongation rate?
The coefficient of thermal expansion is a material property that determines how much the rod's length changes per unit temperature change. It directly multiplies the initial length to give the constant rate of change of length with respect to temperature, which is essential for applying the chain rule.
Q6: Why is the chain rule calculation separated into two distinct components?
Separating the calculation simplifies the chain rule by breaking it into manageable parts: first differentiating length with respect to temperature to get a constant, then differentiating temperature with respect to time using the power rule to get a linear expression. Multiplying these components yields the final rate of change.
Q7: What is the significance of evaluating the rate of change at exactly 10 seconds?
Evaluating at t = 10 seconds gives the instantaneous rate of change at that specific moment. Since temperature changes quadratically with time, the rate of elongation varies continuously. The evaluation at 10 seconds provides the precise elongation rate for the 2-meter rod at that instant.