# Hey, Cool-It, Newton

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Newton's Cool Law
(or Newton's Law of Cooling)
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Prerequisites: Familiarity with exponential functions, properties of logarithms and familiarity with
Newton's Law of Cooling.
Items you'll need:
TI-83Plus
CBL
Temperature Probe
1 cup hot water
1 cup ice water
Overview: In this lab activity, you'll collect temperature data and use exponential regression to confirm
Newton's Law of Cooling. You'll then use the regression results to predict the cooling behavior in a
different environment.
1. Turn on the TI-83 Plus graphing calculator.
2. Press the "APPS" button and choose "CBL/CBR."
3. After pressing "Enter" to begin the program, select "Gauge"
4. Select the following options:
PROBE: Temp
TYPE: Meter
MIN: 0
MAX: 100
UNITS: °C
DIRECTNS: On
GO <Select and Press "Enter">
5. Follow the onscreen instructions pressing "Enter" after each step until you see a temperature gauge
on the screen.
Record the room temperature here: Ts =______°C.
6. When you're done recording the room's temperature, place the probe in the hot water
7. Press "ENTER" then 2nd-QUIT.
8. Choose "DATA LOGGER" and select the following settings
PROBE: Temp
#SAMPLES:20
INTRVL (SEC): 3
UNITS: °C
PLOT:Realtime
DIRECTNS: Off
GO <DO NOT PRESS "Enter" YET!>
9. Pull the probe out of the hot water and immediately press "Enter" to start collecting data.
10. The CBL will collect temperature data and plot it on the TI-83 for 60 seconds.
11. When the CBL displays "DONE", Press "ENTER" on the TI-83, then 2nd-QUIT. Choose "QUIT".
12. Now put the probe in ice water and let it chill while you complete the next steps.
13. To use Newton's model , T  Ts  (T0  Ts )ekt we want to subtract Ts from the collected temperature
data. We'll store the time data (TTEMP) in list L1 and the temperature data, TEMP (minus Ts ) in
list L2.
Created by Burt Dixon, Foothill High School, [email protected]
a) LTTEMP  L1 (Use LIST to get LTTEMP . Use STO button to get the "store" arrow)
b) LTEMP - Ts  L2 (In place of Ts , put in your Ts value from step 5, above.
14. Press "STAT" and "EDIT" and you should see all of your data in L1 and L2.
15. To find the exponential equation that models our Time vs. T- Ts data, press STAT, and choose
CALC: ExpReg ( L1, L2 )
Record the regression equation here. y = ______________ (use 4 decimal places)
16. Go to STAT PLOT and set PLOT1 to plot L1 and L2 data.
17. Go to Y= and enter the regression equation from step15, above.
18. Choose an appropriate window and graph the data and the regression equation.
Xmin:_____, Xmax:_____,
Ymin: ______, Ymax _______
Sketch the graph and data points here.
Be sure to label axes and scale.
NOTE: Step 19 doesn't require the lab equipment and can be completed at home. Please jump to step 22
and start the next experiment.
19. Use properties of logarithms to transform the equation in step 15 to an equivalent in equation in the
form y  (T0  Ts )e kx Show your work and result here. (you may want to use your graphing
calculator to confirm that they're the same)
k = _________
20. Use your equation from 19 or the regression equation to predict what the temperature would've been
at 90 seconds (Be carefull! The variable "y" in 15 and 19 represents T  Ts , not T.).
21. Since Newton's Law of Cooling also works for objects that are warming up to room temperature,
what do you expect the equation to be if we chill the probe to near freezing, then let it warm up to
room temperature? Do you expect the same k-value? Explain.
22. Make sure the probe has been in for 1 minute and repeat the experiment above.
Created by Burt Dixon, Foothill High School, [email protected]
23. Use regression to find an equation that models this new data? NOTE: Instead of storing LTEMP - Ts
as L2, use Ts - LTEMP as L2 and use "-a" instead of "a" from the regression result. Write the
regression equation here.
y = __________________
24. Graph both your regression equation and
your prediction equation from 21 on the
same axis. Sketch the result below.
Label the axes, scale and curves.
25. What is k in this model?
26. Use the regression equation to predict what the temperature would've been at 90 seconds (Be
careful! The variable "y" represents T  Ts , not T.).
27. What do you conclude about the k values in the two experiments. In particular, when will k values
remain the same and when will they change?
28. Explain why we use regression on "Time vs. LTEMP - Ts " instead of "Time vs. LTEMP".
Created by Burt Dixon, Foothill High School, [email protected]
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