2.7
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 t…
A metal rod, when heated, elongates due to linear thermal expansion. The length depends on the initial length, the material’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.
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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.