Hands on Lab: Newtons Law of Cooling

By: Jill RobinsonHands on Lab: Newtons Law of CoolingNewtons 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 Newtons Law of CoolingTo apply Newtons Law using a classroom demonstrationTo comprehend differential equationsObjectives: To model real data To collect data, create graphs and draw conclusions To measure in degree CelsiusVocabulary: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 cupA Styrofoam cupA paper cupHot water (73C)Pot to boil waterA stopwatchA measuring cup in ounces (oz)

Hypothesis:Which cup do you think will keep the water the hottest after 420 seconds?

Paper Plastic Styrofoam

Procedure:This project can be done in small groups or done in the front of the class by the instructor to ensure the students safety.

Measure the temperature of the room and record that number on the third page of your worksheet under Room Temperature.

In a pan, bring water to 73C.

Have the 3 different cups (paper, plastic, Styrofoam) in a line on the table with the thermometer already in the cup.CAREFULLY pour 4oz of water into each cup.

Every 60 seconds, take the temperature and write it down in the chart on the next page. -Repeat steps 36 for another trial

Trial 1:Time(seconds)PaperCup CPlasticCup CStyrofoamCup C073737360686670120656565180636463240606162300586062360565761420545659

Trial 2:Time(seconds)PaperCup CPlasticCup CStyrofoamCup C073737360657172120616572180586268240576065300555963360545661420525660Average:Time(seconds)PaperCup CPlasticCup CStyrofoamCup C07373736066.568.571120636568.518060.56365.524058.560.563.530056.559.562.53605556.561420535659.5Graph of Average Temp of Paper CupQuestions: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 25C?

What are other applications one could use Newtons Law of Cooling?Newtons Law of Cooling Mathematically:Differential Equation:Solution to the Differential Equation:(Zill)(Zill)

Where:= rate at which temperature changes t = time k = constant of proportionality = room temperature

Where: T(t) = temperature of the object at time t C = constantProblems:At what time will the water in the paper cup be 1C above your room temperature? (In my case room temperature is 25C) (Hint: use the data from the charts you collected)Solution:

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:Newtons 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. "Newtons 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.