Newton's Law of Cooling // Separable ODE Example

Summary

In this educational video, the host explores Newton's Law of Cooling by using a practical example of cooling tea. After setting the ambient temperature at 20°C (68°F), they record the tea's initial temperature at 71°C (161°F) and measure its temperature drop over time. By applying a separable differential equation, they model the cooling process, determine constants, and solve for the temperature function. The model's prediction is compared with experimental data, highlighting the approximation nature of models in real-world scenarios. The video concludes with a discussion on the validity and usefulness of the model.

Takeaways

• 📚 The video discusses Newton's Law of Cooling, a mathematical model that describes how the temperature of an object changes over time when it is not in thermal equilibrium with its surroundings.
• ☕ The presenter uses a cup of tea to demonstrate the concept, noting the ambient temperature of their house is 20°C or 68°F, which is the temperature they will compare against.
• 🔍 Initial data is collected by recording the tea's temperature at the start (161°F) and after two minutes (153.7°F), showing the tea is cooling over time.
• 📉 The rate of temperature change is modeled as proportional to the difference between the object's temperature and the ambient temperature, leading to a differential equation.
• 🔑 The differential equation is separable, allowing the variables to be separated and integrated on both sides to find a general solution for the temperature over time.
• ✅ The general solution is simplified using logarithms and exponential functions, resulting in a formula that includes constants a, d, and k, which are determined from the initial conditions and measurements.
• 🔍 The constant d is found to be 93 by using the initial temperature of the tea at time t=0.
• 🔢 The constant k is determined by taking a second temperature measurement at t=2 minutes, which allows for solving the equation involving k through logarithmic calculations, resulting in k=0.0409.
• 📉 The final temperature function is derived, showing the temperature of the tea as a function of time, which is a combination of the ambient temperature and the cooling effect described by the constants d and k.
• 📈 An additional data point is collected at t=5 minutes to test the model's accuracy, with the tea's temperature recorded as 142.7°F.
• 🤔 The model's prediction for t=7 minutes is calculated to be 137.8°F, which is close but not exactly the same as the experimental value of 142.7°F, indicating a reasonable but not perfect fit.
• 💭 The video concludes by emphasizing that models are approximations and are not expected to be perfectly accurate, but are useful for understanding and predicting phenomena like Newton's Law of Cooling.

Q & A

What is Newton's Law of Cooling?

Newton's Law of Cooling is a principle that describes how the rate of cooling of an object is proportional to the difference between its own temperature and the ambient temperature of its surroundings.

What is the ambient temperature of the house mentioned in the script?

The ambient temperature of the house is set at 20 degrees Celsius, which is equivalent to 68 degrees Fahrenheit.

What is the initial temperature of the tea mentioned in the video?

The initial temperature of the tea is approximately 161 degrees Fahrenheit.

Why does the rate of temperature change depend on the difference between the object's temperature and the ambient temperature?

The rate of temperature change depends on this difference because when the object's temperature is significantly higher than the ambient temperature, the rate of cooling is faster, and vice versa.

What type of differential equation is used to model the cooling process in the video?

A separable differential equation is used to model the cooling process, which allows for the separation of variables to solve the equation.

How does the script differentiate between the temperature of the tea (T) and the ambient temperature (a)?

The script denotes the temperature of the tea at time t as T(t) and the ambient temperature as a constant 'a', which is 68 degrees Fahrenheit.

What is the significance of the second temperature measurement taken at t = 2 minutes?

The second measurement helps to determine the value of the constant 'k' in the differential equation, which represents the rate of cooling.

How is the constant 'd' in the model determined using the initial condition?

The constant 'd' is determined by plugging in the initial condition (T(0) = 161 degrees Fahrenheit) into the model and solving for 'd', which turns out to be 93.

What is the process to find the value of 'k' in the model?

The value of 'k' is found by using a non-zero time point (t = 2 minutes), comparing the model's prediction with the measured temperature, and then solving the resulting equation using logarithms.

How does the script evaluate the accuracy of the model?

The script evaluates the model's accuracy by comparing its prediction at t = 7 minutes (137.8 degrees Fahrenheit) with the actual measured temperature (142.7 degrees Fahrenheit), noting a reasonable but not perfect match.

What is the final form of the temperature function as a function of time 't' according to the model?

The final form of the temperature function is T(t) = a + d * e^(-kt), where 'a' is the ambient temperature, 'd' is the initial difference between the tea's temperature and the ambient temperature, and 'k' is the cooling constant.