You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.
The table and graph below show some simple data we collected using the experimental setup shown above. Since it is real data, it is not as clean as one might wish. This temperature data is in degrees Celsius.
The same data appears in your CAS window.
The first model we look at as one possibility for describing cooling is called Newton's Law of Cooling or, better yet, Newton's Model of Cooling to emphasize that it is not enacted by Congress or handed down on stone tablets but is just one possible model and may or may not be a good model.
Newton's Model of Cooling says that if we make a series of temperature measurements of a cooling object at equally spaced times then the temperature changes according to the equation
A - T(i + 1) = k (A - T(i))
where A is the ambient or room temperature and k is a constant that depends on the object that is cooling. If the object is small and poorly insulated then k will be relatively large but if the object is more massive and well insulated then k will be relatively small.
If we know the ambient temperature A then we can determine the value of the constant k using any two data points by noticing that
(A - T(n + p)) / (A - T(n)) = k^p
If we don't know the ambient temperature then we can determine the values of both constants, A and k using any three equally spaced data points as shown below.
Once we have determined the constants A and k using these three data points we have a model given by
S(i) = A + k^(n - i) (T(n) - A)
where S(i) is the temperature predicted by the model at the time of the i-th data point.
Now you can use your CAS window to compare the data with various models based on Newton's Model of Cooling. Notice that each model matches the data exactly at the three points used to determine the model.
You should experiment with this data and other data, using your CAS window and curve fitting using various data points to see how well Newton's Model of Cooling describes a cooling object. If you have lab equipment available you should try several variations on the basic theme.
