Question

Suppose an object with an initial temperature of \(T_{0}>0\) is put in surroundings with an ambient temperature of \(A\) where \(A<\frac{T_{0}}{2} .\) Let \(t_{1 / 2}\) be the time required for the object to cool to \(\frac{T_{0}}{2}\) a. Show that \(t_{1 / 2}=-\frac{1}{k} \ln \left(\frac{T_{0}-2 A}{2\left(T_{0}-A\right)}\right)\) b. Does \(t_{1 / 2}\) increase or decrease as \(k\) increases? Explain. c. Why is the condition \(A<\frac{T_{0}}{2}\) needed?

Short answer

Question: Based on the given solution, derive the formula for the time required for an object to cool to half its initial temperature using Newton's Law of Cooling. Additionally, explain the effect of the constant of proportionality on this time, and justify the temperature condition given. Answer: The derived formula for the time required for an object to cool to half its initial temperature is \(t_{1/2}=-\frac{1}{k}\ln\left(\frac{T_0-2A}{2(T_0-A)}\right)\). The effect of the constant of proportionality \(k\) on this time is that as \(k\) increases, the time required for the object to cool to half of its initial temperature decreases. The condition \(A<\frac{T_0}{2}\) is necessary to ensure that the object will cool, not heat up, as assumed by Newton's Law of Cooling. If this condition is not met, the object's temperature will either remain constant or increase, violating the cooling assumption.

Step-by-step solution

Write down the Newton's Law of Cooling equation

To solve this problem, we must first understand Newton's Law of Cooling. The law is represented by the following differential equation: \(\frac{dT}{dt}=-k(T-A)\) Where \(T\) is the object's temperature, \(t\) is the time, \(k\) is the constant of proportionality, and \(A\) is the temperature of the surroundings.

Separate variable and integrate both sides

Next, we will separate variables and integrate both sides: \(\int \frac{dT}{T-A}=-k\int dt\)

Evaluate the integrals

Now, we will evaluate the integrals: \(\ln(T-A)=-kt+C\)

Determine the constant C

To determine the constant \(C\), we will consider the initial condition, which is when \(t=0\) and \(T=T_0\). Therefore, \(\ln(T_0-A)=C\)

Write the equation for T in terms of t

Now, we will rewrite the equation for \(T\) in terms of \(t\) using the constant \(C\): \(\ln(T-A)=\ln(T_0-A)-kt\) \(T-A=(T_0-A)e^{-kt}\)

Determine t when T is half of its initial value

To find \(t_{1/2}\), we substitute \(T = \frac{T_0}{2}\) into the equation, which gives: \(\frac{T_0}{2}-A=(T_0-A)e^{-kt_{1/2}}\)

Solve for t_{1/2}

Now, we will solve for \(t_{1/2}\): \(t_{1/2}=-\frac{1}{k}\ln\left(\frac{T_0-2A}{2(T_0-A)}\right)\) So, we have derived the desired formula for \(t_{1/2}\) in part (a).

Analyze the effect of k on t_{1/2}

In part (b), we have to understand how \(t_{1/2}\) changes when \(k\) changes. To do this, notice that \(k\) is in the denominator of the fraction, and the natural logarithm is always positive for the given temperature condition. As \(k\) increases, the negative exponent decreases, meaning \(t_{1/2}\) also decreases. Therefore, the time required for the object to cool to half of its initial temperature decreases as \(k\) increases.

Explain the reason for the temperature condition

In part (c), we have to discuss the necessity of the condition \(A<\frac{T_0}{2}\). The given condition ensures that the object will actually cool, not heat up. If \(A\geq\frac{T_0}{2}\), the object would not cool down to half of its initial temperature because the surroundings' temperature is equal to or more than the desired final temperature. In such situations, the object's temperature will either remain constant or increase, violating the assumption of Newton's Law of Cooling.

Key concepts

Differential Equations

Differential equations are fundamental in describing how a certain quantity changes over time. These equations involve functions and their derivatives, giving us a way to analyze how physical systems evolve. In the context of Newton's Law of Cooling, the differential equation is \( \frac{dT}{dt} = -k(T - A) \). This specific equation helps us understand how the temperature \( T \) of an object changes as it cools in a surrounding environment with temperature \( A \). Here, \( \frac{dT}{dt} \) represents the rate of change of temperature over time, which is crucial in predicting how quickly or slowly the cooling process occurs.

Understanding this differential equation is important because:

• It shows that the rate of cooling is proportional to the difference \((T - A)\) between the object's temperature and ambient temperature.
• By solving this equation, we can find how long it will take for the object to reach a certain temperature. This is done by finding the function \( T(t) \), which expresses temperature at any given time \( t \).

Once you grasp the concept of differential equations, you can apply them to various real-world problems, not just in physics but also in fields like biology, economics, and engineering.

Separation of Variables

Separation of variables is a technique used to solve certain differential equations, such as the one present in Newton's Law of Cooling. The method involves rearranging the equation so that each variable appears on a separate side of the equation, making it possible to integrate both sides individually. For Newton's Law of Cooling, the differential equation \( \frac{dT}{dt} = -k(T - A) \) is handled by separating the variables \( T \) and \( t \):

\( \int \frac{dT}{T-A} = -k \int dt \)
By integrating both sides, you can simplify the problem and find a solution.

The process of separation of variables is significant because:

• It provides a straightforward method for obtaining a general solution to the differential equation.
• It allows us to apply initial conditions to find particular solutions, which is essential for solving real-world problems.

This technique is particularly useful because it often transforms a complex differential equation into simpler integrals that can be evaluated practically, offering clear insights into the behavior of the system.

Proportionality Constant

The proportionality constant, often represented as \( k \), is a crucial component in the mathematical formulation of Newton's Law of Cooling. Its role is to dictate the rate at which the object cools, directly influencing the solution to the differential equation. In the equation \( \frac{dT}{dt} = -k(T - A) \), \( k \) scales the rate of cooling based on the difference between the object's temperature \( T \) and the ambient temperature \( A \).

Some important aspects of the proportionality constant include:

• A higher value of \( k \) indicates a faster cooling process, as it suggests that the object loses heat more quickly.
• Conversely, a smaller \( k \) implies slower cooling, meaning the temperature difference reduces more gradually.
• The constant \( k \) is essential for predicting real cooling times, such as how long it takes for an object to reach half its initial temperature, represented as \( t_{1/2} \).

Understanding the role of this constant gives you the ability to assess how environmental conditions and properties of the object are affecting the rate of temperature change, which is pivotal for accurate predictions in various scientific and engineering applications.