2.7
連鎖律：問題解法
金属棒の熱膨張は、一つの物理量が、時間とともに変化する別の物理量に依存する場合に、連鎖律が適用されることを示しています。棒が加熱されると、その長さは線膨張に従って変化しますが、系の温度は時間に対して二次関数的に変化します。
線膨張の場合、棒の長さLは温度Tに依存し、温度に対する長さの変化率は一定です。
…
金属棒は加熱されると線形熱膨張により伸びます。長さは初期長さ、材料の熱膨張係数、系の温度変化に依存します。
ここで、棒の温度は時間とともに2乗ずつ変化します。
目的は、熱膨張係数が既知の2メートルの金属棒の長さの瞬時変化率を正確に10秒ごとに求めることです。
長さは温度に依存し、温度は時間に依存するため、連鎖法則が適用されます。
連鎖ルールの計算を簡単にするために、微分は2つの異なる要素に分けられます。
まず、長さは温度に関係して微分されます。この微分は定数値を与えます:初期長さと展開係数の積です。
次に、時間に対して温度を冪則を用いて微分し、変化率を線形にします。
連鎖法則によれば、最終レートはこれら2つの成分の積です。既知の値を10秒に代入すると、必要な瞬間変化率が得られます。
View the full transcript and gain access to JoVE Core videos
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.