2.7
链式法则：问题求解
金属棒的热膨胀展示了链式法则在一个物理量依赖于随时间变化的另一个物理量时的应用。随着金属棒被加热，其长度按线性热膨胀规律随温度变化，而系统温度则随时间按二次函数关系变化。
对于线性热膨胀，金属棒的长度 L 随温度 T 变化，且长度关于温度的变化率为常数：
其中，L_0 = 2 m 为金属棒的初始长度，α…
金属棒在受热时会因线性热膨胀而伸长。其长度变化取决于初始长度、材料的热膨胀系数以及系统温度的变化。
此处，杆的温度随时间呈二次变化。
目标是求在已知热膨胀系数的情况下，一根2米长的金属杆在恰好10秒时长度的瞬时变化率。
由于长度取决于温度，而温度又取决于时间，因此应用链式法则。
为了简化链式法则的计算，将微分过程分为两个不同的部分。
首先，长度对温度求导。该求导结果为一个常数：即初始长度与膨胀系数的乘积。
其次，利用幂法则对温度关于时间求导，得到线性变化速率。
根据链式法则，最终速率是这两个分量的乘积。将 10 秒时的已知值代入，可得到长度所需的瞬时变化率。
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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.