Exponential growth and decay (Part 13): Newton’s Law of Cooling

In this series of posts, I provide a deeper look at common applications of exponential functions that arise in an Algebra II or Precalculus class. In the previous posts in this series, I considered financial applications. In today’s post, I’ll discuss Newton’s Law of Cooling, which describes how quickly a hot object cools in a room at constant temperature. (This law is not to be confused with Newton’s Three Laws of Motion.)

While Newton’s Law of Cooling is easy to state, not many high school teachers are aware of the physical principles from which they arise. The basic idea is that the rate at which a hot object cools is proportional to the difference between its current temperature and the surrounding temperature. After solving the appropriate differential equation, the temperature T(t) of the object is found to be

T = S + (T_0 - S) e^{-kt},

where t is the time, S is the constant surrounding temperature, and k is a constant that depends on the object.

Of course, students in Algebra II or Precalculus (or high school physics) are usually not ready to understand this derivation using calculus. Instead, they are typically given the final formula and are expected to use this formula to solve problems. Still, I think it’s important for the teacher of Algebra II or Precalculus to be aware of how the origins of this formula, as it only requires mathematics that’s only a year or two away in these students’ mathematical development.

This is the third application of exponential functions considered in this series; the previous two were continuous compound interest and radioactive decay. Unlike these previous two applications, Newton’s Law of Cooling can actually be demonstrated in class to engage students. All that’s required is the appropriate classroom technology and a standard-issue temperature probe.This stands in sharp contrast to the previous applications of exponential functions considered in this series. While students can easily envision making money via compound interest, no one will actually give them the money during class. And certainly I don’t encourage performing a real demonstration of radioactive decay with, say uranium-235, during class time! (There are ways of simulating radioactive decay using M&Ms or other manipulatives, however.)

A simple Google search yields thousands of webpages describes multiple classroom activities for Newton’s Law of Cooling. Some activities merely require collecting data and performing a regression fit to an exponential curve; such an activity would be appropriate for middle-school students. Other activities are more explicit about using Newton’s Law of Cooling. Here’s a sampling:

  1. Texas Instruments TI-Nspire: http://education.ti.com/en/us/activity/detail?id=807CCDFC77B74AC093C38E9228235057&ref=/en/us/activity/search/subject?s=75FE5490E95144559647544901BAF03C&sa=F42399B372FE47E9AFDD4B725E6E6690&t=0FFA852BE3A54A5C949F5552EE003E98
  2. Vernier: http://www2.vernier.com/sample_labs/EZ-TMP-17-newton_cooling.pdf
  3. PBasic: http://gk12.poly.edu/amps-cbri/pdf/TELesson-Newton%27s%20Law%20of%20Cooling.pdf
  4. TI-83/TI-84: http://education.ti.com/en/us/activity/detail?id=934E66B83D2B4D8E93B89D36A87863C7
  5. TI CBL: http://www-tc.pbs.org/teachers/mathline/lessonplans/pdf/hsmp/penniespressure.pdf
  6. Casio: http://www.casioeducation.com/resource/pdfs/cooling.pdf

These links are aimed at a students at a variety of levels. Indeed, it’s possible to use a graphing calculator to plot the numerical derivative T'(t) as a function of time, use linear regression to solve for the constant k, and then produce the exponential curve using this value of k. Several years ago, I saw an effective demonstration of this idea at the Joint Mathematics Meetings in which the presenters covered these aspects of Newton’s Law of Cooling in less than 10 minutes. (Naturally, additional time is needed when students perform these activities for themselves.)

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