Calculus: Project 14

This project is to try to experimentally verify Newton's law of cooling: the rate of cooling is proportional to the difference between the object temperature and ambient temperature. If we let T= the number of degrees Celsius above ambient and t= the time in minutes, Newton's law of cooling becomes

We know that if the initial temperature is T₀, the solution to the differential equation with this initial condition is

There are two questions: (1) How do we measure k? and (2) How good is Newton's law of cooling?

Three students decided to try an experiment and find out for themselves. They put hot water in a plastic cup in a room at

C and measured the following temperature differences:

Scientists often use "semilog" plots of quantities. We want you to discover why.

Preliminary "Semilog" Theory
Suppose we have a function T=T₀e^-kt. Take natural logs of both sides of this equation and show that

where is the new variable . The new "semi-log" function is linear. Where does appear on its graph? How would you measure k from its graph?

Purer Semi-log Linearity
Suppose we have a function T=T₀e^-kt. Show that

How could you measure k from a plot of versus t?

Take logs of T, the temperature differences above ambient for the data above and plot versus t. If Newton's law of cooling is correct, what should the graph look like? How can you measure k on this graph? How would you compute the best approximation to k? How well does the students' data match Newton's law of cooling? Can you think of mistakes they may have made in their experimental procedure?

Use the program CoolHelp from our website if you wish.

You can either just look at this data question or write up a good review problem of the whole sad story of Suzie's canary. Review Problem 4.1, Exercise 4.2.1, Exercise 8.2.1, Problems 8.1, and 8.2 from the main text and write up a complete explanation of Newton's law of cooling for the canary.

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