Transcript Hands on Lab: Newton’s Law of Cooling

Hands on Lab:
Newton’s Law of Cooling
By: Jill Robinson
 Newton’s Law of cooling “can be used to model the “growth”
or “decay” of the temperature of an object over time. In
particular, this law states that the rate at which the
temperature of an object changes over time is proportional
to the difference between the temperature of the object and
the temperature of the surroundings.” (Edwards)
Goals:
 To understand Newton’s Law of Cooling
 To apply Newton’s Law using a classroom demonstration
 To comprehend differential equations
Objectives:
• To model real data
• To collect data, create graphs and draw conclusions
• To measure in degree Celsius
Vocabulary:
Differential Equation: an equation containing the derivative of one
or more dependent variables with respect to one or more
independent variables. (Zill)
Materials:
 Three thermometers (either °C or °F)
 A plastic cup
 A Styrofoam cup
 A paper cup
 Hot water (73°C)
 Pot to boil water
 A stopwatch
 A measuring cup in ounces (oz)
Hypothesis:
 Which cup do you think will keep the water the hottest after
420 seconds?
Paper
Plastic
Styrofoam
Procedure:
1.
This project can be done in small groups or done in the
front of the class by the instructor to ensure the student’s
safety.
2.
Measure the temperature of the room and record that
number on the third page of your worksheet under “Room
Temperature.”
3.
In a pan, bring water to 73°C.
4.
Have the 3 different cups (paper, plastic, Styrofoam) in a
line on the table with the thermometer already in the cup.
5.
CAREFULLY pour 4oz of water into each cup.
6.
Every 60 seconds, take the temperature and write it down
in the chart on the next page.
-Repeat steps 3–6
for another trial
Trial 1:
Time
(seconds)
0
60
120
180
240
300
360
420
Paper
Cup °C
73
68
65
63
60
58
56
54
Plastic
Cup °C
73
66
65
64
61
60
57
56
Styrofoam
Cup °C
73
70
65
63
62
62
61
59
Trial 2:
Time
(seconds)
0
60
120
180
240
300
360
420
Paper
Cup °C
73
65
61
58
57
55
54
52
Plastic
Cup °C
73
71
65
62
60
59
56
56
Styrofoam
Cup °C
73
72
72
68
65
63
61
60
Average:
Time
(seconds)
0
60
120
180
240
300
360
420
Paper
Cup °C
73
66.5
63
60.5
58.5
56.5
55
53
Plastic
Cup °C
73
68.5
65
63
60.5
59.5
56.5
56
Styrofoam
Cup °C
73
71
68.5
65.5
63.5
62.5
61
59.5
Graph of Average
Temp of Paper Cup
80
Water temperature in °C
75
70
65
60
55
50
45
0
50
100
150
200
250
Time in Seconds
300
350
400
450
Questions:
 What do you notice about the graphs?
 Eventually over time what temperature is the water going to
level off at if the air temperature is 25°C?
 What are other applications one could use Newton’s Law of
Cooling?
Newton’s Law of Cooling
Mathematically:
Differential Equation:
Where:
= rate at which temperature changes
t = time
k = constant of proportionality
= room temperature
(Zill)
Solution to the Differential Equation:
Where:
T(t) = temperature of the object at time t
C = constant
(Zill)
Problems:
1.
At what time will the water in the paper cup be 1°C above your
room temperature? (In my case room temperature is 25°C)
(Hint: use the data from the charts you collected)
Solution:
2.
What is your percent error for the paper cup at t=180
seconds?
Solution:
Problem to think about:
 What does a negative k value mean?
Conclusion:
 Newton’s Law of Cooling can be introduced in a high school
pre-calc or calculus class. It can also be taught at the college
level such as in a differential class
 If I were to do this lesson in class, I would tweak it a little so
it would be more realistic. For one, I would not be able to
boil water in a classroom, so instead I would grab three cups
of coffee from the cafeteria. I chose to boil water, because it
was easier for me.
References:

Edwards, C. C. "Newton’s Law of Cooling." Thesis Coastal Carolina University, Conway, SC, Dissertations and
Theses. Web. 26 April 2012.

Zill, Dennis G. A First Course in Differential Equations with Modeling Applications. Belmont, CA: Brooks/Cole,
2009.