What Newton's Law of Cooling Is For
Newton's Law of Cooling models how the temperature of an object changes when it is placed in an environment with a different, relatively constant temperature. The key idea is simple: the farther the object's temperature is from the surrounding temperature, the faster its temperature changes.
As the object gets closer to the surrounding temperature, the temperature difference becomes smaller, so the rate of heating or cooling slows down.
An object does not usually cool at a constant rate. A hot object cools quickly at first, then more slowly as its temperature approaches room temperature. Newton's Law captures exactly this behavior.
The Differential Equation
Let represent the temperature of the object at time , and let represent the constant surrounding temperature.
Newton's Law of Cooling says that the rate of change of the object's temperature is proportional to the difference between its temperature and the surrounding temperature:
The constant determines how quickly the object approaches the surrounding temperature. For a cooling object, is negative in this formulation.
Notice the important structure: if is much larger than , then is strongly negative, meaning the object cools rapidly. As approaches , the rate approaches zero.
The Exponential Model
Solving the differential equation gives the standard Newton's Law model:
This formula is extremely useful on AP Calculus problems because it lets you work directly with temperature measurements without solving the differential equation from scratch every time.
Here:
- is the object's temperature at time .
- is the surrounding temperature.
- is the object's initial temperature.
- is the positive cooling/heating constant.
- is the elapsed time.
A Step-by-Step Method
- Identify the surrounding temperature. This is the temperature the object approaches in the long run.
- Identify the initial temperature. This is the object's temperature when .
- Write the Newton's Law model. Start with .
- Use a second temperature measurement to determine the constant .
- Use the completed model to answer questions about future temperatures or the time required to reach a particular temperature.
Worked Examples
A cup of coffee has an initial temperature of . It is placed in a room maintained at . After 10 minutes, the coffee has cooled to . Find the coffee's temperature after 20 minutes.
A metal object is initially at and is placed in a room at . After 5 minutes, its temperature is . Find the constant in the model .
A cup of tea starts at in a room at . Its temperature is modeled by . How long will it take for the tea to cool to ?
The temperature of an object is modeled by , where temperature is measured in degrees Fahrenheit and time is measured in hours.
Heating Works Too
Despite its name, Newton's Law can model heating as well as cooling. The same exponential model applies.
If the object starts below the surrounding temperature, then is positive and the object's temperature increases toward . If it starts above the surrounding temperature, the temperature decreases toward .
Because approaches zero as becomes very large, the temperature approaches the surrounding temperature:
Common Mistakes
The surrounding temperature is the long-run equilibrium temperature. It is not the initial temperature of the object and it is not another observed temperature from the experiment.
If an object drops 20 degrees during the first 10 minutes, it will generally not drop another 20 degrees during the next 10 minutes. The rate slows as the object approaches the surrounding temperature.
The exponential term multiplies , not the initial temperature itself. The model is built around the object's distance from the surrounding temperature.
In the standard model , the constant is positive and the exponent is negative. This guarantees that the temperature difference shrinks toward zero over time.
AP Calculus Connection: Differential Equations
Newton's Law of Cooling is a classic application of differential equations because the equation describes a rate in terms of the current temperature:
This is a separable differential equation. Separating variables gives:
Integrating both sides leads to an exponential relationship. The AP exam may give you the differential equation and ask you to use a particular solution, interpret a rate, determine whether the temperature is increasing or decreasing, or solve for an unknown parameter.
Practice Problems
An object initially has a temperature of and is placed in a room at . Its temperature after 10 minutes is . Which model represents its temperature minutes after it is placed in the room?
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A temperature is modeled by . Find the instantaneous rate of change of temperature when .
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A cup of coffee is modeled by , where is measured in minutes. Find the time when the coffee reaches .
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An object satisfies the differential equation . At a particular instant, the object has temperature and the surrounding temperature is . Is the object heating or cooling, and at what rate?
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Quick Reference
Remember the interpretation: the rate of temperature change is proportional to the temperature difference between the object and its surroundings. The farther away the object is from equilibrium, the faster it moves toward equilibrium.
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