bulb ---- Light Bulb Mathematics Introduction This lab project is a collection of four experiments using the same equipment and setup. All four experiments can be run in one class period to collect the data. Data can then be distributed to the students for their analysis and conclusions. The four experiments explore the logistic function, the sine function, the inverse square function, and the exponential function. Knowledge of these functions is a prerequisite for this lab. Setup Because the setup of all four experiments is the same, it will be described here only once. Equipment Required: CBL unit TI-83 graphics calculator with a unit-to-unit link cable TI light probe Lamp with a standard light bulb (60W to 150W) Cardboard tube Meter stick Equipment Setup Procedure 1. Connect the CBL unit to the TI-83 calculator with the unit-to-unit link cable using the I/O ports located on the bottom edge of each unit. Press the cable ends in firmly. 2. Connect the TI light probe to Channel 1 (CH1) on the top edge of the CBL unit. 3. Place the light sensor inside a cardboard tube. Position the tube and sensor so that the sensor is facing the light bulb. The purpose of the tube is to keep out extraneous light. A long (30 in.) tube is best. This type of tube may be obtained from a roll of gift wrapping paper. If necessary, a short tube will work but the room lights will have to be dimmed and the tube must be carefully aimed at the light bulb for each data collection effort. 4. Turn on the CBL unit and the calculator. The CBL system is now ready to receive commands from the calculator. EXPERIMENT 1: Discovery What happens when a light bulb is switched on? This experiment allows students to discover and investigate the mathematical curve that describes the light intensity increase of a light bulb when it is turned on. Introduction We generally refer to light intensity as "brightness." More precisely, intensity is defined as the rate at which energy is transferred per unit area, measured in Watts per square meter. When power is suddenly introduced to the filament of the light bulb, the filament heats until it glows, radiating the energy as light. We will use the TI light probe to discover the mathematical relationships of light intensity vs. time as a light bulb increases in brightness. Program Listing This experiment requires that you download or enter the BULBON.83P program into your TI-83 calculator. Experiment Procedure 1. Place the light sensor inside the cardboard tube and position the tube so that separation between the light sensor and the center of the bulb is between 20 in. and 30 in. If the sensor is placed too close to the bulb it will become saturated with light and the intensity curve will be unnaturally flattened at the high end. 2. Darken the room and turn off the light bulb or, if you have a long tube, place the end of the tube against the light bulb to block out as much light as possible. 3. Make sure the CBL is turned on. Start the program BULBON on the TI-83. The student who is positioning the light probe must carefully aim the probe at the light bulb by looking at the bulb through the tube. This position must now be held until the data is collected. If a long tube is used this care is unnecessary. When the program prompts to PRESS [ENTER] TO ZERO it will establish a baseline minimum light intensity. 4. The program will then prompt: PRESS [ENTER] TO COLLECT DATA AND THEN TURN LIGHT ON 5. After the data is collected, a plot of light intensity (in mWcm) vs. time (in seconds) appears on the calculator screen. Make a print-out of the graph using TI-GRAPH LINK or save it as a PIC variable to be printed later. Attach this print-out to your lab notebook. Be sure to include appropriate scales and axis labels on the print-out. The data is saved in lists L2 and L4. It would be prudent to save these lists to lists with new names, perhaps D1 and D2, as subsequent experiments will erase L2 and L4. 6. Notice that the data seem to show two different phenomena. The first is the long period tendency of the light to increase. The second is an apparent short period variation in the intensity with a smaller amplitude. This suggests two more experiments with the data collection parameters adjusted to highlight each of the different phenomenon in question. EXPERIMENT 2: Bulb On What is the long period behavior of the light intensity when a light bulb is switched on? This experiment allows students to discover and investigate the mathematical curve that describes the light intensity increase of a light bulb as it is turned on. Introduction The data collected in the Discovery Experiment showed an initial rapid rate of increase in light intensity followed by a decrease in the rate of increase as the light reached its full intensity. The curve is characteristic of quantities that will increase until available resources are being used as fast as they are being created. This relationship can be described by the logistic function I(t)=c/(1+a*e^(-b*x)) Consequently, the intensity of the light increases almost exponentially when it is first turned on and then the rate of increase slows as the bulb reaches full output. In this experiment you will use the light intensity sensor to verify the relationship stated above. Program Listing This experiment requires that you download or enter the BULBON.83P program into your TI-83 calculator. Experiment Procedure 1. Modify the program BULBON to collect the long period data. Press [PRGM] EDIT BULBON to edit the program listing. The variables N, representing the number of data points to collect and V, representing the time in seconds between each point are stored in the first two lines of the program for easy access. Multiply these two values and we can see that the data collected spans .08 seconds. We will modify these two variables to collect 20 data points with .008 seconds between each point. The data collected will then span .16 seconds but the longer time between points will reveal the long period behavior. Exit the program editor. 2. Now follow the experiment procedure steps 1 through 4 described in the Discovery Experiment. 3. After the data is collected, a plot of light intensity (in mW/cm^2) vs. time (in seconds) appears on the calculator screen. Make a print-out of the graph using TI-GRAPH LINK or save it as a PIC variable to be printed later. Attach this print-out to your lab notebook. Be sure to include appropriate scales and axis labels on the print-out. The data is saved in lists L2 and L4. It would be prudent to save these lists to lists with new names, perhaps O1 and O2, as subsequent experiments will erase L2 and L4. 6. Notice that the data shows the phenomena of the increase of light intensity without the short period variation. Analysis and Conclusion We will use the statistical features of the TI-83 to match the data to a logistic function. 1. The time data is stored in list L2 and the light intensity data is stored in list L4. To determine the relationship between these variables, select Logistic from the [STAT] CALC menu. Enter the appropriate regression command on the home screen, Logistic L2,L4,Y1 2. Record the regression equation and correlation coefficient in your lab notebook. Does this equation agree with the mathematical model relating intensity and time that was described in the introduction section? 3. Press [GRAPH] to see the scatter plot and regression curve together. Make a print-out of this graph using TI-GRAPH LINK and attach it to your lab notebook. 4. Repeat this experiment using a different type of light source. If the source is significantly brighter, you may need to start at a separation greater than 20 in. Record all relevant data in your lab notebook, as before. 5. Truncate your data sets by deleting the last 14 points from each list L2 and L4. There is no danger of losing your original data if you already saved it to new list names. Now try to fit the data to an exponential function. Select ExpReg from the [STAT] CALC menu. Enter the appropriate regression command on the home screen, ExpReg L2,L4,Y1 6. Record the regression equation and correlation coefficient in your lab notebook. Does this equation seem to indicate that the initial increase in light intensity follows an exponential curve? Can you explain this behavior from what you know about exponential functions? EXPERIMENT 3: Bulb Glow What is the short period behavior of the light intensity of a glowing light bulb? This experiment allows students to discover and investigate the mathematical curve that describes the steady state variation in light intensity of a glowing light bulb. Introduction The data collected in the Discovery Experiment seemed to show a short period sinusoidal variation in light intensity that appeared to ride on the overall logistic increase in light intensity. This short period phenomenon will be investigated by collecting data from a glowing light bulb and checking the fit of the data to the function I(t) = A*sin(Bt+C) You will use the light intensity sensor to verify the relationship stated above. Program Listing This experiment requires that you download or enter the BULBGLOW.83P program into your TI-83 calculator. Experiment Procedure 1. The program BULBGLOW to collects short period data. The variables N, representing the number of data points to collect and V, representing the time in seconds between each point are stored in the first two lines of the program for easy access and may be modified easily if desired. We will collect 40 data points with .0005 seconds between each point. The data collected will span .02 seconds and the short time between points will reveal the short period behavior. The equipment will not handle V set to a time period shorter than .0001 seconds. 1. Place the light sensor inside the cardboard tube and position the tube so that separation between the light sensor and the center of the bulb is between 20 in. and 30 in. If the sensor is placed too close to the bulb it will become saturated with light and the intensity curve will be unnaturally flattened at the high end. 2. Darken the room and turn the light bulb on or, if you have a long tube, place the end of the tube against the light bulb to block out as much outside light as possible. 3. Make sure the CBL is turned on. Start the program BULBGLOW on the TI-83. The student who is positioning the light probe must carefully aim the probe at the light bulb by looking at the bulb through the tube. This position must now be held until the data is collected. If a long tube is used this care is unnecessary. 4. The program will prompt: PRESS [ENTER] TO COLLECT DATA Hold the probe steady until the data is dispayed on the TI-83. 5. After the data is collected, a plot of light intensity (in mW/cm^2) vs. time (in seconds) appears on the calculator screen. Make a print-out of the graph using TI-GRAPH LINK or save it as a PIC variable to be printed later. Attach this print-out to your lab notebook. Be sure to include appropriate scales and axis labels on the print-out. The data is saved in lists L2 and L4. It would be prudent to save these lists to lists with new names, perhaps G1 and G2, as subsequent experiments will erase L2 and L4. 6. Notice that the data shows the short period variation phenomena of the light intensity. Analysis and Conclusion We will use the statistical features of the TI-83 to match the data to a sine function. 1. The time data is stored in list L2 and the light intensity data is stored in list L4. To determine the relationship between these variables, select SinReg from the [STAT] CALC menu. Enter the appropriate regression command on the home screen, SinReg L2,L4,Y1 2. Record the regression equation and correlation coefficient in your lab notebook. Does this equation agree with the mathematical model relating intensity and time that was described in the introduction section? What is the source of this oscillation? 3. Press [GRAPH] to see the scatter plot and regression curve together. Make a print-out of this graph using TI-GRAPH LINK and attach it to your lab notebook. 4. Repeat this experiment using a different type of light source. Is the short period behavior of the light intensity apparent in a flourescent light? 5. Repeat the experiment by holding the light probe directly against the face of a computer monitor on a white region of the screen. What is happening with this light source? Try gathering the data again for this source using the program BULBCOMP.83P. Measure the length of the period by tracing along your graph from peak to peak. Now compute the frequency of the oscillation in cycles per second (Hz). Is this frequency what you might expect from the source postulated in step 2 above? Record all relevant data in your lab notebook. EXPERIMENT 4: Bulb Off What happens when a light bulb is switched off? This experiment allows students to discover and investigate the mathematical curve that describes the light intensity decrease of a light bulb when it is turned off. Introduction When power is suddenly removed from the filament of the light bulb, as it is when a light is turned off the filament immediately loses its power source and quickly radiates away it's remaining light energy. We would expect that the energy would dissipate in direct proportion to the remaining energy. This would give us a exponential decay in the light intensity. We will use the TI light probe to discover the mathematical relationship of light intensity v.s. time as a light bulb decreases in brightness. Program Listing This experiment requires that you download or enter the BULBOFF.83P program into your TI-83 calculator. Experiment Procedure 1. Place the light sensor inside the cardboard tube and position the tube so that separation between the light sensor and the center of the bulb is between 20 in. and 30 in. If the sensor is placed too close to the bulb it will become saturated with light and the intensity curve will be unnaturally flattened at the high end. 2. Darken the room and turn on the light bulb or, if you have a long tube, place the end of the tube against the light bulb to block out as much outside light as possible. 3. Make sure the CBL is turned on. Start the program BULBOFF on the TI-83. The student who is positioning the light probe must carefully aim the probe at the light bulb by looking at the bulb through the tube. This position must now be held until the data is collected. If a long tube is used this care is unnecessary. When the program prompts to PRESS [ENTER] TO ZERO it will establish a baseline maximum light intensity. 4. The program will prompt: PRESS [ENTER] TO COLLECT DATA AND THEN TURN LIGHT OFF 5. After the data is collected, a plot of light intensity (in mWcm) vs. time (in seconds) appears on the calculator screen. Make a print-out of the graph using TI-GRAPH LINK or save it as a PIC variable to be printed later. Attach this print-out to your lab notebook. Be sure to include appropriate scales and axis labels on the print-out. The data is saved in lists L2 and L4. It would be prudent to save these lists to lists with new names, perhaps F1 and F2, as subsequent experiments will erase L2 and L4. Analysis and Conclusion We will use the statistical features of the TI-83 to match the data to an exponential function. 1. The time data is stored in list L2 and the light intensity data is stored in list L4. To determine the relationship between these variables, select ExpReg from the [STAT] CALC menu. Enter the appropriate regression command on the home screen, ExpReg L2,L4,Y1. 2. Record the regression equation and correlation coefficient in your lab notebook. Does this equation agree with the mathematical model relating intensity and time that was described in the introduction section? 3. Press [GRAPH] to see the scatter plot and regression curve together. Make a print-out of this graph using TI-GRAPH LINK and attach it to your lab notebook. 4. Let's experiment with the domain of the data set. Enter the command L2+4*V [STO] L2 on the home page and press [ENTER]. Now try to fit the data to the exponential function again. How does the fit compare to the original data set? Graph the data and regression curve together. Make a print out of this graph. 5. Restore your original data set by entering the commands F1 [STO] L2 and F2 [STO] L4. Now shift the data to the right by entering the command L2-4*V [STO] L2. Fit this data to the power function using PwrReg L2,L4,Y1. Is the fit good? What is the exponent? Press [Y=] to get into the equation editor. Modify the exponent in equation Y1 to be -2. Now graph the data set together with the regression equation. Does this fit seem reasonable? Which fit is best? Scott Campbell campbel7@fnbnet.net