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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
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