1. By: Jill Robinson Hands on Lab: Newton’s Law of Cooling

2. 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)

3. 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

4. Vocabulary: Differential Equation: an equation containing the derivative of one or more dependent variables with respect to one or more independent variables. (Zill)

5. 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)

6. Hypothesis: Which cup do you think will keep the water the hottest after 420 seconds? Paper Plastic Styrofoam

7. 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.

8. 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

9. Trial 1: Time (seconds) Paper Cup °C Plastic Cup °C Styrofoam Cup °C 073 60686670 12065 180636463 240606162 300586062 360565761 420545659

11. Trial 2: Time (seconds) Paper Cup °C Plastic Cup °C Styrofoam Cup °C 073 60657172 120616572 180586268 240576065 300555963 360545661 420525660

12. Average: Time (seconds) Paper Cup °C Plastic Cup °C Styrofoam Cup °C 073 6066.568.571 120636568.5 18060.56365.5 24058.560.563.5 30056.559.562.5 3605556.561 420535659.5

13. Graph of Average Temp of Paper Cup

14. 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?

15. Newton’s Law of Cooling Mathematically: Differential Equation: Solution to the Differential Equation: (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 = constant

16. 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)

17. Solution:

19. 2. What is your percent error for the paper cup at t=180 seconds?

20. Solution:

21. Problem to think about: What does a negative k value mean?

22. 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.

23. 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.