Suppose you have just poured a cup of freshly brewed coffee...

Suppose you have just poured a cup of freshly brewed coffee with temperature $95^{\circ} \mathrm{C}$ in a room where the temperature is $20^{\circ} \mathrm{C}$. (a) When do you think the coffee cools most quickly? What happens to the rate of cooling as time goes by? Explain. (b) Newton's Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings, provided that this difference is not too large. Write a differential equation that expresses Newton's Law of Cooling for this particular situation. What is the initial condition? In view of your answer to part (a), do you think this differential equation is an appropriate model for cooling? (c) Make a rough sketch of the graph of the solution of the initial-value problem in part (b).

Suppose you have just poured a cup of freshly brewed coffee with temperature $95^{\circ} \mathrm{C}$ in a room where the temperature is $20^{\circ} \mathrm{C}$.
(a) When do you think the coffee cools most quickly? What happens to the rate of cooling as time goes by? Explain.
(b) Newton's Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings, provided that this difference is not too large. Write a differential equation that expresses Newton's Law of Cooling for this particular situation. What is the initial condition? In view of your answer to part (a), do you think this differential equation is an appropriate model for cooling?
(c) Make a rough sketch of the graph of the solution of the initial-value problem in part (b).

Key Concepts

Newton's Law of Cooling

This law states that the rate at which an object cools (or warms) is proportional to the difference between its temperature and that of its surroundings. It serves as a foundational principle in thermodynamics and heat transfer, and it applies as long as the temperature difference is not too extreme.

Differential Equations in Thermal Modeling

Differential equations are used to mathematically describe the rate of change of the temperature of an object over time. In the context of cooling, the differential equation typically expresses the relationship between the temperature difference (object vs. room) and the cooling rate, making it a vital tool in modeling dynamic thermal processes.

Initial Value Problem (IVP)

An initial value problem consists of a differential equation along with a specified initial condition. In thermal problems, the initial temperature of the object serves as the starting point for finding the unique solution that describes the temperature evolution over time.

Exponential Decay Behavior

The solution to the cooling model governed by Newton's Law of Cooling typically exhibits an exponential decay behavior. This means that the temperature difference decreases rapidly at first when the difference is large, and then more slowly as the object approaches the ambient temperature, ultimately leading to a gradual leveling off.

Temperature Difference Impact on Cooling Rate

The cooling rate is directly linked to the magnitude of the temperature difference between the object and its environment. Consequently, when the difference is large, the rate of cooling is high; as the difference narrows over time, the cooling rate diminishes, illustrating the dynamic interplay between the object's temperature and the ambient temperature.

Transcript

00:01 In this video, we're going to be looking at newton's law of cooling.

00:04 So newton's law of cooling states that the rate of change of temperature of an object is proportional to the different between its temperature and the temperature of its surroundings.

00:15 So the example we'll look at is a cup of coffee that is 95 degrees celsius cooling down in a room that is 20 degrees celsius.

00:25 So if we first just intuitively think about what newton's law of cooling is saying, we know that the temperature of our hot object in our cool room is going to decrease, and that temperature is going to temp film.

00:42 It's going to decrease the fastest when the difference between it and surroundings is greatest.

01:16 And that makes sense.

01:17 If we put two different cups of coffee in the same 20 degree room, with one of them already at 50 degrees celsius, if we were to measure the temperature a few seconds later in both cups, the hotter cup would have cooled off more.

01:35 Just intuitively, that makes sense to us.

01:38 So let's go ahead and try and write a different equation modeling this.

01:42 So we know the rate of change of the coffee's temperature is before.

01:46 Proportional to the temperature difference.

01:49 So rate of change sets off derivative bills in our heads right away.

01:53 So we know that dt, dt, big t being temp and little t being time, it's going to be equal to, let's write that term that it's proportional to first, t minus 20 degrees in this case.

02:13 And these are not equal, they're proportional, so we need a proportionality constant.

02:16 And we know that our temperatures is decreasing, so this needs to be negative...

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