What You Will Do:
- Watch the Khan Academy video: "Newton's Law of Cooling"
- Collect temperature data in the Desmos Graphing Calculator for a cup of cooling hot water.
- Use this mathematical model to describe how the temperature of hot water changes over time:
- Observe that the rate of cooling is proportional to the difference in temperature between the hot object and the surrounding room.
- Complete the Google Docs worksheet: "Modeling Newton's Law of Cooling" and turn it in according to your teacher's directions.
$$
f(x) = (T_{\text{initial}} - T_{\text{room}})\, e^{\,k(x - x_{\text{offset}})} + T_{\text{room}}
$$
- Initial Temperature (Tinitial) : the temperature of the hot object at the moment cooling begins.
- Room Temperature (Troom): the surrounding ambient temperature that the object is cooling toward.
- Temperature Difference: the difference between the object’s temperature and the room temperature.
- Cooling Rate Constant (k): a value that describes how quickly the object cools; larger values mean faster cooling. Physically, this constant depends on the material and environment surrounding the sensor.
- X offset (Xoffset): a horizontal translation of the graph to the right. This aligns the moment cooling actually begins with the moment data collection starts.
- Click to watch this Khan Academy video to learn more about these concepts.
- Click this link Modeling Newton's Law of Cooling Worksheet to open the worksheet in a new browser tab. Click ‘Make a copy’ to save your version to your Google Drive.
- Click the Show Directions button in the upper-right corner to learn how to collect temperature data for this activity.
Directions:
- Use a USB-C cable to plug the Observe temperature sensor into your computer's USB port.
- Select a Sample Rate of 5 Hz.
- Click the Connect button at the top-left corner of this page.
- Select the USB serial port (COM X) and click Connect. X varies by computer and is not important.
- Confirm that the status in the top-right corner says “ready (5 Hz)”.
- Set the sensor on the table and do not touch the metal part. Note the temperature shown in the status bar; this is the ambient room temperature. In the Desmos window below, adjust the Troom slider in the expression list to match this value.
- Carefully fill a cup with hot water and place the sensor into it. Note the temperature shown in the status bar; this is the Tinitial. Adjust the Tinitial slider in the expression list to match this value.
- Remove the sensor from the hot water and quickly click Start Collection to log data as the sensor cools.
- Click Stop Collection after the sensor has cooled to room temperature.
- Clear Graph resets; Capture Graph copies; Export / Import saves or opens a CSV data file.
- You may want to clear your graph and repeat a few times until you are satisfied with your data.
- Click the Show Instruction button in the upper-right corner to continue the activity.
- When finished measuring, click the Zoom to Data button.
- Adjust the k and Xoffset sliders until the model best-fits your data.
- Add five more points, at 20 second intervals, of temperatue difference and rate of cooling to the data table in the expression list. Notice the first point at time = 0 has already been entered into the table.
- When finished, click Capture Graph to copy your graph and paste it into your Google Docs worksheet.
- Scroll back up to the graph and click the Clear Graph button. You may want to export the data before clearing it if you plan to import and analyze it further in the future.
- Collect a second data set, but this time warm the sensor instead of cooling it. Review the steps in the directions.
- Wrap the sensor in a flexible frozen cool-pack and chill it until it drops below zero. Remove the sensor from the cool-pack and click the Start button. Hold the sensor in the air and allow it to warm. Do not touch the metal part. Collect data until the cold sensor warms to room temperature.
- When collection is finished, repeat the steps above in the Analyzing Your Data section.
- When finished, click Capture Graph to copy your graph and paste it into your worksheet.