Newton’s Law Of Cooling

Math proj

1. Newton’s Law of Cooling
Spencer Lee
Vikalp Malhotra
Shankar Iyer
Period 3

2. History
•In the 17
th
century, Isaac Newton studied the nature of cooling
•In his studies he found that if there is a less than 10 degree
difference between two objects the rate of heat loss is proportional
to the temperature difference
•Newton applied this principle to estimate the temperature of a
red-hot iron ball by observing the time which it took to cool from a
red heat to a known temperature, and comparing this with the
time taken to cool through a known range at ordinary
temperatures.
•According to this law, if the excess of the temperature of the body
above its surroundings is observed at equal intervals of time, the
observed values will form a geometrical progression with a
common ratio
•However, Newton’s law was inaccurate at high temperatures
•Pierre Dulong and Alexis Petit corrected Newton’s law by clarifying
the effect of the temperature of the surroundings

3. What is it?
•Newton's Law of Cooling is used to model the temperature change of an
object of some temperature placed in an environment of a different
temperature. The law states that:
y = the temperature of the object at time t
r = the temperature of the surrounding environment (constant)
k = the constant of proportionality
•This law says that the rate of change of temperature is proportional to the
difference between the temperature of the object and that of the
surrounding environment.
)(ryk
dt
dy
-= )(ryk
dt
dy
-=

4. The Basic Concept
In order to get the previous equation to something that we can use, we must solve the
differential equation. The steps are given below.
•Separate the variables. Get all the y's on one side and the t on the other side. The constants
can be on either side.
5.Integrate each side
7.Find antiderivative of each side
10.Leave in the previous form or solve for y
We now have a useful equation. When you are working with Newton's Law of Cooling, remember
that t is the variable. The other letters, R, k, C, are all constants. In order to find the
temperature of the object at a given time, all of the constants must first have numerical
values.
)(ryk
dt
dy
-=
kdt
ry
dy
=
-
òò
=
-
kdt
ry
dy
1ln Cktry +=-
rey
Ckt
+=
+1
)(ryk
dt
dy
-=
kdt
ry
dy
=
-
òò
=
-
kdt
ry
dy
1ln Cktry +=-
rey
Ckt
+=
+1

5. The Problem
Spencer and Vikalp are cranking out math problems at Safeway. Shankar is at home
making pizza. He calls Spencer and tells him that he is taking the pizza out from
the oven right now. Spencer and Vikalp need to get back home in time so that
they can enjoy the pizza at a warm temperature of 110°F.
The pizza, heated to a temperature of 450°F, is taken out of an oven and placed in a
75°F room at time t=0 minutes. If the pizza cools from 450° to 370° in 1 minute,
how much longer will it take for its temperature to decrease to 110°?

6. How to do it
min88363.9
23995.0)ln(
37575110
)ln(
37575370
375
75450
75
75
75ln
)(
)75(
375
35
))((ln
375
295
375
295
)1(
)0(
1
75
1
75
1
375
295
1
=
-=
+=
=
=
+=
=
+=
+=
=-
+=-
=
=
-=
+
-
-
òò
t
t
e
k
e
e
C
Ce
Cey
ey
Ckty
kdtdy
kdtdy
yk
dt
dy
t
k
k
k
kt
Ckt
y
y
It takes about 8.88363 more
minutes
For the object to cool to a
temperature of
110°

7. Real Life Applications
•To predict how long it takes for a hot
cup of tea to cool down to a certain
temperature
•To find the temperature of a soda
placed in a refrigerator by a certain
amount of time.
•In crime scenes, Newton’s law of
cooling can indicate the time of
death given the probable body
temperature at time of death and
the current body temperature