30. SCIENCEA Mug Of Hot Chocolate Is Initially At $90^{\circ} C$ At Time $t=0$. It Is Surrounded By Air At A Constant Temperature Of $20^{\circ} C$. If Stirred Steadily, Its Temperature In Celsius After $t$

Introduction

Newton's Law of Cooling is a fundamental concept in physics that describes the rate at which an object loses heat to its surroundings. This law is essential in understanding various phenomena, such as the cooling of objects, the growth of bacteria, and the spread of diseases. In this article, we will explore the mathematical model of Newton's Law of Cooling and apply it to a real-world scenario: a mug of hot chocolate cooling in a room.

Newton's Law of Cooling

Newton's Law of Cooling states that the rate of heat loss of an object is directly proportional to the difference between its temperature and the temperature of its surroundings. Mathematically, this can be expressed as:

$$\frac{dT}{dt} = -k(T - T_s)$$

where:

• $T$ is the temperature of the object at time $t$
• $T_s$ is the temperature of the surroundings
• $k$ is the cooling constant

The Cooling of a Mug of Hot Chocolate

Let's apply Newton's Law of Cooling to a mug of hot chocolate. The initial temperature of the hot chocolate is $90^{\circ} C$, and it is surrounded by air at a constant temperature of $20^{\circ} C$. We want to find the temperature of the hot chocolate after $t$ minutes.

Separation of Variables

To solve the differential equation, we can use the method of separation of variables. We separate the variables $T$ and $t$ by dividing both sides of the equation by $(T - T_s)$:

$$\frac{dT}{T - T_s} = -kdt$$

Integration

We can now integrate both sides of the equation:

$$\int \frac{dT}{T - T_s} = -k \int dt$$

Evaluating the integrals, we get:

$$\ln|T - T_s| = -kt + C$$

where $C$ is the constant of integration.

Solving for T

We can now solve for $T$ by exponentiating both sides of the equation:

$$T - T_s = e^{-kt + C}$$

$$T = T_s + e^{-kt + C}$$

Applying the Initial Condition

We know that the initial temperature of the hot chocolate is $90^{\circ} C$, so we can apply this condition to the equation:

$$90 = 20 + e^{-k(0) + C}$$

Simplifying the equation, we get:

$$70 = e^C$$

$$C = \ln 70$$

The Final Equation

We can now substitute the value of $C$ back into the equation for $T$:

$$T = 20 + e^{-kt + \ln 70}$$

Simplifying the Equation

We can simplify the equation by combining the exponents:

$$T = 20 + 70e^{-kt}$$

The Cooling Curve

The equation for $T$ is the cooling curve of the hot chocolate. We can plot this curve to visualize the cooling process.

Conclusion

In this article, we applied Newton's Law of Cooling to a real-world scenario: a mug of hot chocolate cooling in a room. We derived the mathematical model of the cooling process and solved for the temperature of the hot chocolate after $t$ minutes. The cooling curve shows that the temperature of the hot chocolate decreases exponentially over time.

References

• Newton, I. (1701). Opticks: Or, a Treatise of the Reflections, Refractions, Inflections and Colours of Light.
• Carslaw, H. S., & Jaeger, J. C. (1959). Conduction of Heat in Solids. Oxford University Press.

Appendix

The following is a Python code to plot the cooling curve:

import numpy as np
import matplotlib.pyplot as plt

# Define the parameters
T_s = 20  # Temperature of the surroundings
k = 0.1  # Cooling constant
t = np.linspace(0, 60, 61)  # Time array

# Calculate the temperature
T = T_s + 70 * np.exp(-k * t)

# Plot the cooling curve
plt.plot(t, T)
plt.xlabel('Time (minutes)')
plt.ylabel('Temperature (°C)')
plt.title('Cooling Curve of the Hot Chocolate')
plt.grid(True)
plt.show()

Introduction

Newton's Law of Cooling is a fundamental concept in physics that describes the rate at which an object loses heat to its surroundings. In our previous article, we explored the mathematical model of Newton's Law of Cooling and applied it to a real-world scenario: a mug of hot chocolate cooling in a room. In this article, we will answer some frequently asked questions about Newton's Law of Cooling.

Q: What is Newton's Law of Cooling?

A: Newton's Law of Cooling states that the rate of heat loss of an object is directly proportional to the difference between its temperature and the temperature of its surroundings.

Q: What are the variables in Newton's Law of Cooling?

A: The variables in Newton's Law of Cooling are:

• $T$: the temperature of the object at time $t$
• $T_s$: the temperature of the surroundings
• $k$: the cooling constant

Q: How do I calculate the cooling constant $k$?

A: The cooling constant $k$ can be calculated using the following formula:

$$k = \frac{\ln \left( \frac{T_0 - T_s}{T_f - T_s} \right)}{t}$$

where:

• $T_0$ is the initial temperature of the object
• $T_f$ is the final temperature of the object
• $t$ is the time over which the cooling occurs

Q: What is the significance of the cooling constant $k$?

A: The cooling constant $k$ represents the rate at which the object loses heat to its surroundings. A larger value of $k$ indicates a faster rate of cooling, while a smaller value of $k$ indicates a slower rate of cooling.

Q: Can Newton's Law of Cooling be applied to other phenomena?

A: Yes, Newton's Law of Cooling can be applied to other phenomena, such as the growth of bacteria, the spread of diseases, and the cooling of electronic devices.

Q: What are some real-world applications of Newton's Law of Cooling?

A: Some real-world applications of Newton's Law of Cooling include:

• Designing cooling systems for electronic devices
• Predicting the growth of bacteria in food
• Understanding the spread of diseases in populations
• Optimizing the cooling of buildings and homes

Q: How can I use Newton's Law of Cooling in my daily life?

A: You can use Newton's Law of Cooling in your daily life by:

• Understanding how to cool your home or office efficiently
• Predicting the temperature of a cup of coffee or tea
• Optimizing the cooling of your electronic devices
• Understanding the growth of bacteria in food

Conclusion

In this article, we answered some frequently asked questions about Newton's Law of Cooling. We hope that this Q&A guide has helped you to better understand this fundamental concept in physics.

References

• Newton, I. (1701). Opticks: Or, a Treatise of the Reflections, Refractions, Inflections and Colours of Light.
• Carslaw, H. S., & Jaeger, J. C. (1959). Conduction of Heat in Solids. Oxford University Press.

Appendix

The following is a Python code to calculate the cooling constant $k$:

import numpy as np

# Define the parameters
T_0 = 90  # Initial temperature of the object
T_s = 20  # Temperature of the surroundings
T_f = 30  # Final temperature of the object
t = 60  # Time over which the cooling occurs

# Calculate the cooling constant k
k = np.log((T_0 - T_s) / (T_f - T_s)) / t

print("The cooling constant k is:", k)

This code will calculate the cooling constant $k$ using the formula above.