NEWTON'S LAW OF COOLING
As soon as a hot cup of coffee is poured, it begins to
cooling process is rapid at first, then levels off. After a long period
of time, the temperature of the coffee eventually reaches room
temperature. Temperature variations for such cooling objects were
summarized by Newton. He stated that the rate at which a warm body cools
is approximately proportional to the temperature difference between the
temperature of the warm object and the temperature of its surroundings.
where y represents the object's
temperature, x represents time, C is the surrounding temperature, and k
is a proportionality
constant. The following data was collected with the CBL system by heating a
piece of aluminum foil and allowing it to cool over a period of three
minutes (temperature was measured in degrees Celsius and time in
- CBL unit
- TI-85 graphics calculator with a unit-to-unit link cable
- TI temperature probe
- Hair dryer
- 6"x10" piece of aluminum foil
- Program COOL.85P
Equipment Setup Procedure:
- Connect the CBL unit to the TI-85 calculator with the unit-to-unit
link cable using the I/O ports located on the bottom edge of each unit.
Press the cable ends in firmly.
- Connect the temperature probe to Channel 1 (CH1) on the top edge of
the CBL unit.
- Prepare the solution of object to be cooled.
- Turn on the CBL unit and the calculator.
The CBL system is now ready to receive commands from the calculator.
- Fold the 10" side of the aluminum foil in half. Place the
temperature probe between the layers of he foil in the middle of one
side. Flatten the foil tightly around the probe.
- Make sur the CBL is turned on. Start the HEAT program on the TI-85.
Enter 3 at the "HOW MUCH TIME BETWEEN POINTS..." prompt.
- At the "HOW MUCH TIME FOR TOTAL DATA..." prompt, enter 3. The
program records 60 points (one point every 3 seconds for 3 minutes).
- Record room temperature in degrees Celsius. Enter the room
temperature at the "ROOM TEMP=?" prompt. After you enter room
temperature, wait to press ENTER until told to do so.
- Turn the hair dryer on high and heat the center of the foil. If the
foil blows around in circles, place the nozzle of the dryer closer to the
foil. Be sure to keep the hand holding the probe away from the heat.
- When the foil is hot (about 70°C), press ENTER on the TI-85 to start
collecting data. Turn the hair dryer off and allow the foil to cool. A
collected data point is displayed in real-time once every three seconds
for three minutes.
- Analyze the data.
Analysis and Conclusion:
- Note that the difference equation given by Newton's Law of Cooling is
Solve this equation for k.
- A natural question to ask is whether your data does indeed satisfy
this equation. To find out, use the data you collected to find several
values for k over one second intervals. What do you notice about these
values of k?
- The program has also calcuated the value of k over one second
intervals. These values are stored in the list DC. You can scroll
through this list and compare those values to the values you found in
Step 2. The average of all of the values in DC is stored under the name
CON. Recall this average by selecting 2ND RCL CON on the home screen of
your calculator. How does this average compare to the k you found?
- Discuss the expected shape of the curve representing the collected
data. Will the data points ever reach y=0? What is the lowest
temperature that will be recorded?
- In this experiment, time data is stored in the list L3, and
temperature data is stored in L4. Lower all the recorded temperatures by
the room temperature recorded from the classroom thermometer. To do
this, go to the home screen and type L4-C STR L5, where C is the room
temperature in degrees Celsius. Your new temperature data is now stored
in the list L5.
- Go to STAT CALC and select lists L3,L5. then choose EXPR to perform
an eponential regression. Choose STREG and enter y! at the prompt to
store the regression in your first y equation.
- Because you subtracted room temperature off of all of your data
points, you must now add that back in to your equation. Go to the
graphing editor and select the equation y1. Add room temperature to the
- The resulting equation is of the form y=A*Bx+C. Convert this
eqauation to the form y=A*ekx+C. How does the k in this equation
compare to the k you found in Part I?
- Graph your regression equation along with the Scatter plot of L3,L4.
Newton's Law of Cooling is represented by the differential equation
where y(0)=y0>C and y(x1)=y1.
Solve the differential equation to get an equation of the form
Looking at Part II and Part III, what meaning do you think the
variables A and k have in the cooling of an object?
What kind of function should be used to model the above data?