Newton’s Law of Cooling

Spencer LeeVikalp Malhotra

Shankar IyerPeriod 3

History•In the 17th 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

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 tr = 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.

)( rykdtdy )( rykdtdy

The Basic ConceptIn order to get the previous equation to something that we can use, we must solve the

differential equation. The steps are given below.

1. 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.

2. Integrate each side

3. Find antiderivative of each side

4. 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.

)( rykdtdy

kdtrydy

kdtrydy

1ln Cktry

rey Ckt 1

)( rykdtdy

kdtrydy

kdtrydy

1ln Cktry

rey Ckt 1

The ProblemSpencer 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°?

How to do it

min88363.9

23995.0)ln(

37575110

)ln(

37575370

375

75450

75

75

75ln

)(

)75(

37535

))((ln

375295

375295

)1(

)0(

1

751

751

375295

1

t

t

e

k

e

e

C

Ce

Cey

ey

Ckty

kdtdy

kdtdy

ykdt

dy

t

k

k

k

kt

Ckt

y

y

It takes about 8.88363 more minutes For the object to cool to a temperature of110°

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