2.7
Die Kettenregel: Problemlösung
Die thermische Ausdehnung eines Metallstabs veranschaulicht die Anwendung der Kettenregel, wenn eine physikalische Größe von einer anderen abhängt, di…
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.
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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.