Problem 42 - Thomas Calculus in SI Units

Question

An aluminum beam was brought from the outside cold into a machine shop where the temperature was held at \(18^{\circ} \mathrm{C}\). After \(10 \mathrm{min}\), the beam warmed to \(2^{\circ} \mathrm{C}\) and after another \(10 \mathrm{min}\) it was \(10^{\circ} \mathrm{C}\). Use Newton's Law of Cooling to estimate the beam's initial temperature.

Step-by-Step Solution

Answer

The beam's initial temperature is \(-14^{\circ}C\).

Step 1: Understand Newton's Law of Cooling

Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference in temperature between the object and its surroundings. The formula is \( T(t) = T_s + (T_0 - T_s) e^{-kt} \), where \( T(t) \) is the temperature at time \( t \), \( T_s \) is the surrounding temperature, \( T_0 \) is the initial temperature, and \( k \) is a constant.

Step 2: Identify Given Values

From the problem, the surrounding temperature \( T_s = 18^{\circ}C \). At \( t = 10 \) minutes, \( T(10) = 2^{\circ}C \), and at \( t = 20 \) minutes, \( T(20) = 10^{\circ}C \). We need to find the initial temperature \( T_0 \).

Step 3: Use the Temperature at 10 Minutes

The temperature after 10 minutes is \( 2 = 18 + (T_0 - 18)e^{-10k} \). Simplifying gives \( -16 = (T_0 - 18)e^{-10k} \) and \( e^{-10k} = \frac{-16}{T_0 - 18} \).

Step 4: Use the Temperature at 20 Minutes

The temperature after 20 minutes is \( 10 = 18 + (T_0 - 18)e^{-20k} \). Simplifying gives \( -8 = (T_0 - 18)e^{-20k} \).

Step 5: Relate the Equations with Exponentials

Divide the equation from Step 3 by Step 4 to eliminate \( T_0 - 18 \). \[ \frac{-16}{-8} = \frac{e^{-10k}}{e^{-20k}} \] This simplifies to \( 2 = e^{10k} \) or \( e^{10k} = 2 \).

Step 6: Solve for k

Take the natural logarithm of both sides: \[ 10k = \ln(2) \] Thus, \( k = \frac{\ln(2)}{10} \).

Step 7: Substitute to Find Initial Temperature \( T_0 \)

Substitute \( k = \frac{\ln(2)}{10} \) back into the equation from Step 3: \( -16 = (T_0 - 18)e^{-10\cdot\frac{\ln(2)}{10}} \) \( -16 = (T_0 - 18) \cdot \frac{1}{2} \) Solve for \( T_0 \): \( T_0 - 18 = -32 \) \( T_0 = -14 \).

Step 8: Final Result

The initial temperature of the aluminum beam is \(-14^{\circ}C\).

Key Concepts

Rate of Change of Temperature

Newton's Law of Cooling provides a foundation for understanding how the temperature of an object changes over time when placed in a different environment. Specifically, the rate of change of temperature is crucial. It tells us how quickly or slowly an object's temperature shifts from its initial state to match the surroundings.
The rate of change is described as being proportional to the difference between the object's current temperature and the ambient temperature. In simpler terms, this means:

• If the object is much colder than its surroundings, it will warm up faster.
• As the temperature difference decreases, the rate of temperature change slows down.

This understanding allows us to predict and estimate future temperatures at any given time using the formula. Therefore, having a grasp of this concept can help us, for example, determine the initial temperature of an object upon entering a new environment as demonstrated in the problem.

Exponential Decay

Exponential decay is a pattern often seen in nature where a quantity decreases at a rate proportional to its current value. In the context of Newton's Law of Cooling, the temperature difference between the object and its environment decreases exponentially as time passes. This means the rate at which the object's temperature changes slows down over time.
This relationship is captured by the expression \( e^{-kt} \) in the cooling formula. Here, \( k \) is a constant that needs to be determined based on observations. By analyzing temperature changes over known time intervals, as seen in the exercise, we can solve for the constant \( k \).
Exponential decay in temperature change means:

• The largest temperature change happens initially.
• As time goes on, the temperature approaches the surrounding temperature more and more closely, but changes happen at a slower pace.

Understanding exponential decay helps provide insights into how long it might take for an object to reach a desired temperature when brought into a new environment.

Temperature Differential Equation

To solve problems involving temperature change, we often use a differential equation based on Newton's Law of Cooling. This equation shares how temperature changes over time relative to another temperature—the surrounding environment in this case.
The key differential equation for this law is given by:\[\frac{dT}{dt} = -k(T - T_s)\]Here, \( T \) is the temperature of the object, \( T_s \) is the surrounding temperature, and \( k \) is a positive constant. The negative sign indicates that the temperature difference decreases over time as the object warms up or cools down toward the ambient temperature.
Solving this differential equation leads us to the exponential decay formula used in cooling problems:\[T(t) = T_s + (T_0 - T_s) e^{-kt}\]This formula, as applied in the solution for the given exercise, allows us to predict the object's temperature at any point in the future, given its initial temperature and ambient conditions. Understanding these equations is fundamental to analyzing and predicting temperature behavior over time.