Florida International University
GENERAL PHYSICS
LABORATORY 1
MANUAL
Edited Fall 2019
0
Florida International University
Department of Physics
Physics Laboratory Manual for Course
PHY 2048L
Contents
Course Syllabus
Grading Rubric
Estimation of Uncertainties
2
4
5
Experiments
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Graph Matching
Ball Toss and Error Analysis
Projectile Motion
Newton’s First and Third Laws
Newton's Second Law
Atwood's Machine
Static and Kinetic Friction
Kinetic and Potential Energy
Momentum, Energy and Collisions
Conservation of Angular Momentum & Rotational Dynamics
Simple Harmonic Motion
Sound Waves and Beats
9
13
20
24
30
35
39
46
51
56
61
66
1
COURSE SYLLABUS
LAB COORDINATOR
Email: Please use Canvas Inbox
UPDATES
Updates to the lab schedule, make-up policy, etc. may be found on Canvas.
CLASS MEETINGS
• During Fall and Spring Semesters classes start the second week of the semester and end
the week prior to the final exam week.
• Students that have missed their own section may attempt to make-up by attending another
section during the time the same experiment is conducted (see PantherSoft for
available sections). Admission for make-up is granted by the Instructor on site, no
reservation, no guaranteed seating.
• Students must sign in each class meeting to verify attendance.
ACTIVE LEARNING
One of the important goals of this lab course is to strengthen your understanding of what you
have learned in the classroom. You will be working in groups and encouraged to help each other
by discussing among yourselves any difficulties or misconceptions that occur to you. Apart from
the instructor in charge, student Learning Assistants (LA) will be on hand to encourage
discussion, for example by posing a series of questions.
LAB REPORTS
You will be required to submit a lab report at the end of the class period. The format of the report
is dictated by the experiment. As you work your way through the experiment, following the
procedures in this manual, you will be asked to answer questions, fill in tables of data, sketch
graphs, do straightforward calculations, etc. You should fulfill each of these requirements as you
proceed with the experiment. Any preliminary questions could be answered before coming to the
lab, thereby saving time. This way, you will effectively finish the report as you finish the
experiment. Note that for experiments that require them, blank or partially filled in data tables
are provided on separate perforated pages in this manual at the end of the experiment. You may
carefully tear them out along the perforation and staple them to the rest of your report.
GRADES
•
The weekly lab reports and your active participation will determine your grade in the
course. Each week you will receive 30% for active participation and up to 70% for your lab
report.
•
A missed assignment or lab will receive a ZERO grade.
•
Lab reports are to be handed in before you leave the lab.
•
THERE IS NO FINAL EXAM
•
The grading system is based on the following scale although your instructor may apply a
"curve" if it is deemed necessary. In addition, “+” and “-“may be assigned in each grade
range when appropriate.
o A: 90-100%
2
o B: 75-90%
o C: 60-75%
o D: 45-60%
WHAT YOU NEED TO PROVIDE
Calculator with trig. and other math functions including mean and standard deviation.
AT THE END OF CLASS.
1. Disconnect all sensors that you have connected.
2. Report any broken or malfunctioning equipment.
3. Arrange equipment tidily on the bench.
DROPPING THE LECTURE BUT NOT THE LAB
If you find it necessary to drop the lecture course, PHY 2048 or PHY 2053, you do not also have
to drop this lab course, PHY 2048L. However, you will need to see a Physics Advisor and get a
waiver.
3
GRADING RUBRIC
Expectations for a successfully completed experiment and lab report are indicated in the
following rubric. Note that not every scientific ability in the rubric may be tested in every
experiment. Therefore the graders will determine the maximum number of points attainable for a
report (14), add on participation points (6), and indicate your score out of 20.
Grade
Scientific
ability
Attempt to
answer
Preliminary
Questions
Able to draw
graphs/diagrams
Able to present
data and tables
Able to analyze
data
Able to answer
Analysis
questions
Able to conduct
experiment as
evidenced by the
quality of results
Missing
(0 pt)
Inadequate
(1 pt)
No attempt to
answer
Preliminary
Questions
No graphs or Graphs/drawings
drawings
poorly drawn with
provided
missing axis labels
or important
information is
wrong or missing
No data or
Not all the
tables
relevant data and
provided
tables are provided
No data
analysis or
analysis
contains
numerous
errors
No Analysis
questions
answered
Data analysis
contains a number
of errors
indicating
substantial lack of
understanding
Less than half the
questions
unanswered or
answered
incorrectly
Little or no
experimental
ability as
evidenced by
poor quality
of results
Results indicate a
marginal level of
experimental
ability
Needs
improvement
(2 pt)
Graphs/drawings
have no wrong
information but a
small amount of
information is
missing
Data and tables
are provided but
some information
such as units is
missing
Adequate
(3 pt)
Answers to
Preliminary
Questions
attempted
Graphs/drawing
s contain no
omissions and
are clearly
presented
Complete set of
data and tables
with all
necessary
information
provided
Data analysis is
Data analysis is
mostly correct but complete with
some lack of
no errors
understanding is
present
Less than a
quarter of the
questions
unanswered or
answered
incorrectly
Results indicate a
reasonable level
of experimental
ability with room
for improvement
All questions
answered
correctly
Results indicate
a proficient level
of experimental
ability
4
ESTIMATION OF UNCERTAINTIES
The purpose of this section is to provide you with the rules for determining the uncertainties in
your experimental results. All measurements have some uncertainty in the results due to the fact
you can never do a perfect experiment. We begin with the rules for estimating uncertainties in
individual measurements, and then show how these uncertainties are to be combined to produce
the uncertainty in the final result.
The “absolute uncertainty” in a measured quantity is expressed in the same units as the quantity
itself. For example, length of table = 1.65 ± 0.05 m or, symbolically, L ± L. This means we are
reasonably confident that the length of the table is between 1.60 and 1.70 m, and 1.65 m is our
best estimate. If L is based on a single measurement, it is often a good rule of thumb to make L
equal to half the smallest division on the measuring scale. In the case of a meter rule, this would
be 0.5 mm. Other considerations, such as a rounded edge to the table, may make us wish to
increase L. For example, in the diagram, the end of the table might be estimated to be to be at
35.3 ± 0.1 cm or even 35.3 ± 0.2 cm.
If the same measurement is repeated several times, the average (mean) value is taken as the most
probable value and the “standard deviation” is used as the absolute uncertainty. Therefore if the
length of the table is measured 3 times giving values of 1.65, 1.60 and 1.85m, the average value
is
165 + 160 + 185

= 170 m
3
The deviations of the 3 values from the average are -0.05, -0.10 and +0.15m, and the standard
deviation
sum of squares of deviations
=
number of measurements
So now we express the length of the table as 1.7 ± 0.1 m.
Note: Your calculator should be capable of providing the mean and standard deviation
automatically. Excel can also be used to calculate these quantities.
=
0.052 + 010
. 2 + 015
. 2
3
= 01
. m
5
Generally it is only necessary to quote an uncertainty to one, or at most two, significant
figures, and the accompanying measurement is rounded off (not truncated) in the same decimal
position.
“Fractional uncertainty” or “percentage uncertainty” is the absolute uncertainty, expressed as a
fraction or percentage of the associated measurement. In the above example, the fractional
uncertainty, L/L is 0.1/1.7 = 0.06, and the percentage uncertainty is 0.06 x 100 = 6%.
Rules for obtaining the uncertainty in a calculated result.
We now need to consider how uncertainties in measured quantities are to be combined to
produce the uncertainty in the final result. There are 2 basic rules:
A)
When quantities are added or subtracted, the absolute uncertainty in the result is equal to
the square root of the sum of the squares of the absolute uncertainties in the quantities.
B)
When quantities are multiplied or divided, the fractional uncertainty in the result is equal
to the square root of the sum of the squares of the fractional uncertainties in the
quantities.
Examples
1.
In calculating a quantity x using the formula x = a + b - c, measurements give
a = 2.1 ± 0.2 kg
b = 1.6 ± 0.1 kg
c = 0.8 ± 0.1 kg
Therefore, x = 2.9 kg
Absolute error in x, x = 0.22 + 01
. 2 + 01
.2
= 0.2 kg
The result is therefore x = 2.9 ± 0.2 kg
2.
In calculating a quantity x using the formula x = ab/c, measurements give
a = 0.75 ± 0.01 kg
b = 0.81 ± 0.01 m
c = 0.08 ± 0.02 m
Therefore x = 7.59375 kg (by calculator).
 0.01  2  0.01 2  0.02  2
x
= 
Fractional uncertainty in x,
 +
 +
 = 0.25
 0.75   0.81  0.08 
x
Absolute uncertainty in x, x = 0.25  7.59375
= 2 kg (to one significant figure)
The result is therefore x = 8 ± 2 kg

Note: the value of x has to be rounded in accordance with the value of x. If x had been
calculated to be 0.003 kg, the result would have been x = 7.594 ± 0.003 kg.
3.
The following example involves both rule A and rule B.
In calculating a quantity x using the formula x = (a + b)/c, measurements give
6
a = 0.42 ± 0.01 kg
b = 1.63 ± 0.02 kg
c = 0.0043 ± 0.0004 m3
Therefore x = 476.7 kg/m3
Absolute uncertainty in a + b = 0.012 + 0.02 2 = 0.02 kg
Fractional uncertainty in a + b = 0.02 / 2.05 = 0.01
Fractional uncertainty in c = 0.0004 / 0.0043 = 0.093
Fractional uncertainty in x = 0.0932 + 0.012 = 0.094
Absolute uncertainty in x, x = 0.094 476.7 = 40 kg/m3 (to one significant figure)
The result is therefore x = 480 ± 40 kg/m3
Note that almost all of the uncertainty here is due to the uncertainty in c. One should therefore
concentrate on improving the accuracy with which c is measured in attempting to decrease the
uncertainty.
Uncertainty in the slope of a graph
Often, one of the quantities used in calculating a final result will be the slope of a graph.
Therefore we need a rule for determining the uncertainty in the slope. Graphing software such as
Excel can do this for you. Another way to do this is “by hand” as follows: In drawing the best
straight line (see figure on following page),
1.
The deviations of the data points from the line should be kept to a minimum.
2.
The points should be as evenly distributed as possible on either side of the line.
3.
To determine the absolute uncertainty in the slope:
a. Draw a rectangle with the sides parallel to and perpendicular to the best straight
line that just encloses all of the points.
b. The slopes of the diagonals of the rectangle are measured to give a maximum
slope and a minimum slope.
max slope - min slope
c. The absolute uncertainty in the slope is given by:
, where n
2 n
is the number of data points.

7
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1
4
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M
Extension(mm)
1
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0
8
6
4
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3
4
5
6
7
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(
k
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)
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What has been described above is known as “standard uncertainty theory”. In this system, a
calculated result, accompanied by its uncertainty (the standard deviation s), has the following
properties: There is a 70% probability that the “true value” lies within the ± s of the calculated
value, a 95% probability that it lies within the ± 2s, a 99.7% probability that it lies within ± 3s,
etc. We may therefore state that the “true value” essentially always lies within plus or minus 3
standard deviations from the calculated value. Bear this in mind when comparing your result
with the expected result (when this is known).
Some final words of warning
It is often thought that the uncertainty in a result can be calculated as just the percentage
difference between the result obtained and the expected (textbook) value. This is incorrect. What
is important is whether the expected value lies within the range defined by your result and
uncertainty.
Uncertainties are also sometimes referred to as “errors.” While this language is common practice
among experienced scientists, it conveys the idea that errors were made. However, a good
scientist is going to correct the known errors before completing an experiment and reporting
results. Erroneous results due to poor execution of an experiment are different than uncertain
results due to limits of experimental techniques.
8
Lab 1. Graph Matching
One of the most effective methods of describing motion is to plot graphs of position, velocity,
and acceleration vs. time. From such a graphical representation, it is possible to determine in
what direction an object is going, how fast it is moving, how far it traveled, and whether it is
speeding up or slowing down. In this experiment, you will use a Motion Detector to determine
this information by plotting a real-time graph of your motion as you move across the classroom.
The Motion Detector measures the time it takes for a high-frequency sound pulse to travel from
the detector to an object and back. Using this round-trip time and the speed of sound, the
interface can determine the distance to the object; that is, its position. It can then use the change
in position to calculate the object’s velocity and acceleration. All of this information can be
displayed in a graph. A qualitative analysis of the graphs of your motion will help you develop
an understanding of the concepts of kinematics.
board to increase
reflection
Figure 1
OBJECTIVES
Analyze the motion of a student walking across the room.
Predict, sketch, and test position vs. time kinematics graphs.
Predict, sketch, and test velocity vs. time kinematics graphs.
MATERIALS
computer
Labquest Mini
Vernier Motion Detector
board
meter stick
masking tape
9
PRELIMINARY QUESTIONS
1. Below are four position vs. time graphs labeled (i) through (iv). Identify which graph
corresponds to each of the following situations and explain why you chose that graph.
a. An object at rest
b. An object moving in the positive direction with a constant speed
c. An object moving in the negative direction with a constant speed
d. An object that is accelerating in the positive direction, starting from rest
2. Below are four velocity vs. time graphs labeled (i) through (iv). Identify which graph
corresponds to each of the following situations. Explain why you chose that graph.
a. An object at rest
b. An object moving in the positive direction with a constant speed
c. An object moving in the negative direction with a constant speed
d. An object that is accelerating in the positive direction, starting from rest
PROCEDURE
Part I Preliminary Experiments
1. Connect the Motion Detector to a digital (DIG) port of the interface. Set the
sensitivity switch to Ball/Walk.
2. Place the Motion Detector so that it points toward an open space at least 4 m long. Use short
strips of masking tape on the floor to mark the 1 m, 2 m, 3 m, and 4 m positions from the
Motion Detector.
3. Open the file “01a Graph Matching” from the Physics with Vernier folder. Monitor the
position readings. Move back and forth and confirm that the values make sense.
4. Use Logger Pro to produce a graph of your motion when you walk away from the detector
with constant velocity. To do this, stand about 1 m from the Motion Detector, hold the board
against your back to improve the reflection of the high frequency sound pulses, and have
10
your lab partner click
begin to click.
. Walk slowly away from the Motion Detector when you hear it
5. Examine the graph. Sketch a prediction of what the position vs. time graph will look like if
you walk faster. Check your prediction with the Motion Detector. NOTE When printing
graphs, save the trees by selecting only the pages that you really want to print.
Part II Position vs. Time Graph Matching
6. Open the experiment file “01b Graph Matching.” A position vs. time graph with a target
graph is displayed.
7. Decide how you would walk to produce this target graph.
8. To test your prediction, choose a starting position and stand at that point. Click
to
start data collection. When you hear the Motion Detector begin to click, walk in such a way
that the graph of your motion matches the target graph on the computer screen.
9. If you were not successful, repeat the process until your motion closely matches the graph on
the screen. Print or sketch the graph with your best attempt showing both the target graph and
your motion data.
10. Choose Clear All Data from the Data menu, and then click Generate Graph Match,
target graph is displayed. Repeat Steps 7–9 using the new target graph.
. A new
11. Answer the Analysis questions for Part II before proceeding to Part III.
Part III Velocity vs. Time Graph Matching
12. Open the experiment file “01d Graph Matching.” A velocity vs. time graph is displayed.
13. Decide how you would walk to produce this target graph.
14. To test your prediction, choose a starting position and stand at that point. Click
to
start data collection. When you hear the Motion Detector begin to click, walk in such a way
that the graph of your motion matches the target graph on the screen. It will be more difficult
to match the velocity graph than the position graph. Repeat the process until your motion
closely matches the graph on the screen. Print or sketch the graph with your best attempt
showing both the target graph and your motion data.
15. Choose Clear All Data from the Data menu, and then click Generate Graph Match,
target graph is displayed. Repeat Steps 13–14 using the new target graph.
. A new
16. Remove the masking tape from the floor.
17. Proceed to the Analysis questions for Part III.
ANALYSIS
Part II Position vs. Time Graph Matching
1. Describe how you walked for each of the graphs that you matched.
11
2. Explain the significance of the slope of a position vs. time graph. Include a discussion of
positive and negative slope.
3. What type of motion is occurring when the slope of a position vs. time graph is zero?
4. What type of motion is occurring when the slope of a position vs. time graph is constant?
5. What type of motion is occurring when the slope of a position vs. time graph is changing?
Test your answer to this question using the Motion Detector.
Part III Velocity vs. Time Graph Matching
6. Describe how you walked for each of the graphs that you matched.
7. What type of motion is occurring when the slope of a velocity vs. time graph is zero?
8. What type of motion is occurring when the slope of a velocity vs. time graph is not zero?
Test your answer using the Motion Detector.
12
Lab 2. Ball Toss and Error Analysis
Ball Toss
When a juggler tosses a ball straight upward, the ball slows down until it reaches the top of its
path. The ball then speeds up on its way back down. A graph of its velocity vs. time would show
these changes. Is there a mathematical pattern to the changes in velocity? What is the
accompanying pattern to the position vs. time graph? What would the acceleration vs. time graph
look like?
In this part of the experiment, you will use a Motion Detector to collect position, velocity, and
acceleration data for a ball thrown straight upward. Analysis of the graphs of this motion will
answer the questions asked above.
Motion Detector
Figure 1
OBJECTIVES
•
Collect position, velocity, and acceleration data as a ball travels straight up and down.
• Analyze position vs. time, velocity vs. time, and acceleration vs. time graphs.
• Determine the best-fit equations for the position vs. time and velocity vs. time graphs.
• Determine the mean acceleration from the acceleration vs. time graph.
MATERIALS
computer
Labquest Mini
Logger Pro
Vernier Motion Detector
volleyball or basketball
wire basket
PRELIMINARY QUESTIONS
1. Consider the motion of a ball as it travels straight up and down in freefall. Sketch your
prediction for the position vs. time graph. Describe in words what this graph means.
13
2. Sketch your prediction for the velocity vs. time graph. Describe in words what this graph
means.
3. Sketch your prediction for the acceleration vs. time graph. Describe in words what this graph
means.
PROCEDURE
1. Connect the Vernier Motion Detector to a digital (DIG) port of the
interface. Set the Motion Detector sensitivity switch to Ball/Walk.
2. Place the Motion Detector on the floor and protect it by placing a wire basket over it.
3. Open the file “06 Ball Toss” from the Physics with Vernier folder.
4. Collect data. During data collection you will toss the ball straight upward above the Motion
Detector and let it fall back toward the Motion Detector. It may require some practice to
collect clean data. To achieve the best results, keep in mind the following tips:
• Hold the ball approximately 0.5 m directly above the Motion Detector when you start data
collection.
• A toss so the ball moves about 0.5 m above the detector works well.
• After the toss, catch the ball at a height of 0.5 m above the detector and hold it still until
data collection is complete.
• Use two hands and pull your hands away from the ball after it starts moving so they are not
picked up by the Motion Detector.
When you are ready to collect data, click
as you have practiced.
to start data collection and then toss the ball
5. Examine the position vs. time graph. Repeat Step 4 if your position vs. time graph does not
show an area of smoothly changing position. Check with your instructor if you are not sure
whether you need to repeat the data collection.
ANALYSIS
1. Print or sketch the three motion graphs. The graphs you have recorded are fairly complex and
it is important to identify different regions of each graph. Click Examine, , and move the
mouse across any graph to answer the following questions. Record your answers directly on
the printed or sketched graphs.
a. Identify the region when the ball was being tossed but still in your hands:
Examine the velocity vs. time graph and identify this region. Label this on the graph.
Examine the acceleration vs. time graph and identify the same region. Label the graph.
b. Identify the region where the ball is in free fall:
Label the region on each graph where the ball was in free fall and moving upward.
Label the region on each graph where the ball was in free fall and moving downward.
c. Determine the position, velocity, and acceleration at specific points.
On the velocity vs. time graph, decide where the ball had its maximum velocity, just as
the ball was released. Mark the spot and record the value on the graph.
On the position vs. time graph, locate the maximum height of the ball during free fall.
14
Mark the spot and record the value on the graph.
What was the velocity of the ball at the top of its motion?
What was the acceleration of the ball at the top of its motion?
2. What does a linear segment of a velocity vs. time graph indicate? What is the significance of
the slope of that linear segment?
3. The graph of velocity vs. time should be linear. To fit a line to this data, click and drag the
mouse across the free-fall region of the motion. Click Linear Fit, .
4. How closely does the coefficient of the t term in the fit compare to the accepted value for g?
5. The graph of acceleration vs. time should appear to be more or less constant. Click and drag
the mouse across the free-fall section of the motion and click Statistics, .
6. How closely does the mean acceleration compare to the values of g found in Step4?
Error Analysis
INTRODUCTION
In this experiment, you will use different apparatus to determine the acceleration of a freely
falling object. Once you have done this, you will address the following questions:
How do I decide if the value I obtained is “close enough” to the accepted value?
If I were to repeat the experiment several times, within what range would I expect my values
to fall?
This experiment affords you the opportunity to understand variations in experimentally
determined data.
OBJECTIVES
In this experiment, you will
Determine the value of the acceleration of a freely falling object.
Compare your value with the accepted value for this quantity.
Learn how to describe and account for variation in a set of measurements.
Learn how to describe a range of experimental values.
MATERIALS
Vernier data-collection interface
Logger Pro or LabQuest App
Vernier Photogate
foam pad to cushion impact
Picket Fence
clamp or ring stand to secure Photogate
15
Picket
fence
Figure 1
PREAMBLE
In this experiment, you will use software which records the elapsed time between some regularly
occurring events. When the Picket Fence (a strip of clear plastic with evenly spaced dark bands)
passes through a Photogate, the device notes when the infrared beam of the photogate is blocked
by a dark band and measures the time elapsed between successive “blocked” states. The software
uses these times and the known distance from the leading edge of a dark band to the next to
determine the velocity of the picket fence as it falls through the photogate. The elapsed time
from Blocked state to Blocked state decreases as the picket fence accelerates in free fall through
the photogate.?
PROCEDURE
1. Connect the photogate to one of the digital inputs on the interface and start the datacollection program. If the photogate has a sliding door, make sure it is open.
2. Check to see if the sensor is working by passing your hand between the infrared LED and the
detector. The gate Status should change from “Unblocked” to “Blocked.”
3. Fasten the photogate to a support rod or ring stand so that the arms of the photogate are
horizontal (see Figure 1).
4. Open the file "Picket Fence" in folder Lab02 on the desktop.
5. Change the graph setup to view only the velocity vs. time graph.
6. Place something soft on the table or floor to cushion the picket fence as it strikes the surface.
7. Hold the picket fence vertically just above the photogate, start collecting data, and release the
picket fence. Make sure that it does not strike the photogate as it passes through the arms.
8. Perform a linear fit on the graph of velocity vs. time. Print or sketch a copy of your graph.
Take a moment to discuss what the slope and intercept of the line of best fit represent. When
printing, choose Print Graph under the File menu on Logger Pro. DO NOT use Print as it
will print all pages including the data file.
16
9. Based on your discussion, predict whether either of these quantities would change if you
were to drop the picket fence through the photogate from a higher point. Test your
prediction.
10. To see how repeatable the values of the slope are, repeat Steps 7 and 8 to obtain a total of 5
sets of resdings. Record your values of the slope and intercept in the table.
11. You may quit the data-collection program for now but do not disassemble your apparatus.
You will return to it later.
EVALUATION OF DATA
1. How do you account for the fact that the values of the slope were nearly the same, whereas
the values of the intercept were much more variable?
2. It is highly unlikely that you obtained identical values of the slope of the best-fit line to the
velocity vs. time graph for each of your trials. How might you best report a single value for
the acceleration due to gravity, ag, based on your results? Perform the necessary calculation.
3. How does your experimental value compare to the generally accepted value (from a text or
other source)? One way to respond to this question is to determine the percent difference
between the value you reported and the generally accepted value. Note that if you simplify
your units of slope, they will match those of the reported values of ag.
4. Your determination of the percent difference does little to answer such questions as, “Is my
average value for ag close enough to the accepted value?” or “How do I decide if a given
value is too far from the accepted value?” A more thorough understanding of error in
measurement is needed. Every time you make a measurement, there is some random error
due to limitations in your equipment, variations in your technique, and uncertainty in the
best-fit line to your data. Errors in technique or in the calibration of your equipment could
also produce systematic error. We’ll address this later in the experiment. In order to better
understand random error in measurement, you must return to your experimental apparatus to
collect more data.
5. Begin the data-collection program "Picket Fence" in folder Lab02 as you did before and drop
the picket fence through the photogate another 20 times, bringing the total number of trials to
25. Since you are now investigating the variation in the values of ag, you need only record
the value of the slope of the best-fit line to the velocity-time graph for each trial. Record the
value of the slope in the Table on page 19.
6. Launch the Logger Pro file Lab02-histogram. Enter your values of slope in the Table on page
19.
7. In your discussion, you will decide how best to configure the features of the histogram so as
to represent the distribution of your values in the most meaningful way. To do this, choose
Options>>Additional Graph Options>>Histogram Options, and adjust the settings under the
Bin and Frequency Options tab.
8. Determine the average value of ag for all 25 trials. How does this compare with the value you
obtained for the first 5 trials? In which average do you have greater confidence? Why?
9. In what range (minimum to maximum) do the middle 2/3 of your values fall? In what range
do roughly 90% of the values closest to your average fall?
17
10. One way to report the precision of your values is to take half the difference between the
minimum and maximum values and use this result as the uncertainty in the measurement.
Determine the uncertainty in this way for each range of values you determined in Step 9.
11. In what place (tenths, hundreds, thousandths) does the uncertainty begin to appear? Discuss
whether it is reasonable to report values in your average beyond the place in which the
uncertainty begins to appear. Round your average value of ag to the appropriate number of
digits and report that value plus the uncertainty.
18
Slope
(m/s2)
Intercept
(m/s)
19
Lab 3. Projectile Motion
You have probably watched a ball roll off a table and strike the floor. What determines where it
will land? Could you predict where it will land? In this experiment, you will use a projectile
launcher to shoot a ball horizontally from a table top. Using your knowledge of physics, you will
be able to determine the launch speed. You will also be able to get the launch speed by
measuring the horizontal range as a function of the launch angle.
OBJECTIVES
Use a Mini Launcher to project a ball horizontally onto the floor
Apply concepts from two-dimensional kinematics to determine the launch speed.
Use the Mini Launcher to determine the horizontal range of a ball as a function of the launch
angle.
Use an appropriate graph to obtain an average value for the launch speed..
MATERIALS
computer
Labquest Mini
Logger Pro
Mini Launcher
level
meter stick or metric measuring tape
carbon paper
steel ball
goggles
plumb bob
PRELIMINARY QUESTIONS
Balance one penny on the edge of a table. Place your index finger on a second penny, then flick
the second penny so that it travels off the table, while the first penny is gently nudged off the
edge. It may take a few practice trials to be able to do this effectively.
Figure1
1. Predict which penny will land first, the penny moving horizontally, or the one that simply
drops off the table. Explain.
2. Perform the investigation, listening for the sound of the pennies as they land. Was your
prediction supported or refuted?
3. You may believe the pennies landed just a little bit apart from each other. Try it a few more
times. Does one always land before the other?
20
4. What will happen if you increase the speed of the second penny? Predict and then give it a
try.
5. What if you increase the height from which the pennies are dropped? Your instructor may
choose to stack two tables for you to test this.
6. Based on your observations, does the horizontal speed of the flicked penny affect the impact
times of the pennies?
7. What can you then say about the time to hit the floor for each penny?
SET UP FOR FIRST EXPERIMENT
y
x
Figure 2
1. Clamp the Mini Launcher to the edge of your table. Eventually you will shoot a ball
horizontally that will travel a couple of meters, so plan for this when choosing a location.
2. Adjust the launcher so that the launch angle is zero. Check with a level to confirm.
PROCEDURE
1. Obtain and wear goggles.
2. Measure the distance y that the ball will fall. See Fig. 2.
3. Insert a steel ball into the barrel up to the second "click" setting.
4. Launch the steel ball and note approximately where it lands. Use carbon paper on top of
white paper (with the carbon side down) taped to the floor at the landing point to more
accurately determine the landing point.
5. Launch the ball five times to obtain five impact points. From the cluster of marks on the
paper, estimate the center of the cluster and use it to determine an average value of x (see
Fig. 2).
21
SET UP FOR SECOND EXPERIMENT
Figure 3
1. Clamp the Mini Launcher so that the ball will land on the table (Fig. 3). Note how to adjust
the launcher so that the launch point marked on the launcher is level with the table top.
PROCEDURE
1. Set the launch angle to 10° and make 5 measurements of the impact point using carbon paper
as before. Obtain an average value for the horizontal range, R, and enter it in Table 1.
2. Repeat step 1 increasing the launch angle in 10° steps up to 80°.
ANALYSIS
1. To determine the launch speed, vo, in the first experiment, your only data are the values of x
and y, and an assumed value for g of 9.81 m/s2. Here are some hints on how to obtain an
expression for vo: What is the ball's initial vertical component of velocity? Can you use this
velocity component, together with y, to obtain the time of flight, t? Does the horizontal
component of velocity change from its initial value, vo? Can you combine it with t to get x?
2. Once you have obtained an expression for vo, check with your instructor that it is correct,
then use your values of y and the average value of x to obtain a numerical value.
3. To determine the launch speed, vo, in the second experiment, your only data are the values of
the horizontal range, R, for different launch angles, , and an assumed value for g of 9.81
m/s2. Here are some hints on how to obtain an expression for vo: Can you obtain an
expression for the time of flight, t, based on the initial vertical component of velocity, vosin,
and the initial and final y coordinates? Does the horizontal component of velocity change
from its initial value, vocos? Can you combine it with t to get R?
4. Once you have obtained an expression for R, check with your instructor that it is correct.
5. Open the file "Range vs. Launch Angle" from folder lab 03. Use the error bar option which
can be accessed from DATA>Column Options>Range(m). Check 'Error Bar Calculations'
then select 'Use Column' for Uncertainty in Range (m). NOTE When printing graphs, save
the trees by selecting only the pages that you really want to print.
6. From the slope and your assumed value of g, calculate an average value of vo.
7. How do your two values of vo compare? Which do you trust as the more precise
measurement? Why?
22
DATA TABLE
Table 1
Launch angle, 
sin 2
Horizontal range, R
10
20
30
40
50
60
70
80
23
Lab 4. Newton’s First and Third Laws
Newton's Third Law
You may have learned this statement of Newton’s third law: “To every action there is an equal
and opposite reaction.” What does this sentence mean? This experiment will help you investigate
this question.
Unlike Newton’s first two laws of motion, which concern only individual objects, the third law
describes an interaction between two bodies. For example, what if you pull on your partner’s
hand with your hand? To study this interaction, you can use two Force Sensors. As one object
(your hand) pushes or pulls on another object (your partner’s hand), the Force Sensors will
record those pushes and pulls. They will be related in a very simple way as predicted by
Newton’s third law.
The action referred to in the phrase above is the force applied by your hand, and the reaction is
the force that is applied by your partner’s hand (or vice versa). Together, they are known as a
force pair. This short experiment will show how the forces are related.
F
o
rc
e
S
e
n
s
o
r
D
u
a
lR
a
n
g
e
Figure 1
OBJECTIVES
•
Observe the directional relationship between force pairs.
Observe the time variation of force pairs.
• Explain Newton’s third law in simple language.
•
MATERIALS
computer
Labquest Mini
Logger Pro
two Vernier Dual-Range Force Sensors
500 g mass
string
rubber band
PRELIMINARY QUESTIONS
Answer these questions as best you can. You will have a chance to revisit your answers after the
activity.
1. You are driving down the highway and a bug splatters on your windshield. Which is greater:
the force of the bug on the windshield or the force of the windshield on the bug?
24
2. Hold a rubber band between your right and left hands. Pull with your left hand. Does your
right hand experience a force? Does your right hand apply a force to the rubber band? What
direction is that force compared to the force applied by the left hand?
3. Pull harder with your left hand. Does this change any force applied by the right hand?
4. How is the force of your left hand, transmitted by the rubber band, related to the force
applied by your right hand? Write a rule, in words, for the force relationship.
PROCEDURE
1. Set the range switches of the Dual-Range Force Sensors to 50 N. Connect the two Force
Sensors to the interface.
Force Sensors measure force only along one direction; if you apply a force along another
direction, your measurements will not be meaningful. The Dual-Range Force Sensor
responds to force directed parallel to the long axis of the sensor.
2. Open the file “11 Newtons Third Law” in the Physics with Vernier folder.
3. (Optional) Because you will be comparing the readings of two different Force Sensors, it is
important that they both read force accurately. To increase the accuracy of the sensors, you
will calibrate them.
a. Choose Calibrate from the Experiment menu. Select CH1: Dual-Range Force. Click
.
b. Remove all force from the first sensor and hold it vertically with the hook pointed down.
Enter 0 (zero) in the Value 1 field, and click
. This defines the zero force condition.
c. Hang the 500 g mass from the sensor. This applies a force of 4.9 N. Enter 4.9 in the
Value 2 field, and then click
.
d. Click
to complete the calibration of the first Force Sensor.
e. Repeat the process for the second Force Sensor.
4. Zero the Force Sensors so they read the same magnitude under the same force. Hold the
sensors horizontally with no force applied, and click
. Select both sensors in the Zero
Sensor Calibrations box and click
to zero both sensors. This step makes both sensors
read exactly zero when no force is applied.
5. Click
to take a trial run of data. Pull on each Force Sensor and note the sign of the
reading. Use this to establish the positive direction for each sensor. For this activity it is
helpful to set up the two Force Sensors differently, since later you will have the sensors
positioned so that a force to the left will generate the same sign of force on each sensor.
6. Make a short loop of string with a circumference of about 30 cm. Use it to attach the hooks
of the Force Sensors. Hold one Force Sensor in your hand and have your partner hold the
other so you can pull on each other using the string as an intermediary.
7. Click
to begin collecting data. Gently tug on your partner’s Force Sensor with your
Force Sensor, making sure the graph does not go off scale. Also, have your partner tug on
your sensor. You will have 10 seconds to try different pulls. Choose Store Latest Run from
the Experiment menu.
25
8. What do you predict would happen if you used the rubber band instead of the string? Would
some of the force get “used up” in stretching the band? Test your prediction by repeating
Steps 6–7 using the rubber band instead of the string.
ANALYSIS
1. Examine the two data runs. What can you conclude about the two forces (your pull on your
partner and your partner’s pull on you)? How are the magnitudes related? How are the signs
related?
2. How does the rubber band change the results—or does it change them at all?
3. While you and your partner are pulling on each other’s Force Sensors, do your Force Sensors
have the same positive direction? What impact does your answer have on the analysis of the
force pair?
4. Is there any way to pull on your partner’s Force Sensor without your partner’s Force Sensor
pulling back? Try it.
5. Reread the statement of the third law given at the beginning of this activity. The
equal
 phrase

and opposite
must
be
interpreted
carefully,
since
for
two
vectors
to
be
equal
(
)
and
A
=
B
 


opposite ( A = −B ) then we must have A = B = 0 ; that is, both forces are always zero. What is
really meant by equal and opposite? Restate Newton’s third law in your own words, not
using the words “action,” “reaction,” or “equal and opposite.”
6. Re-evaluate your answers to the preliminary questions.
Newton's First Law
Newton’s first law tells us that for a body to remain at rest, or to move with constant velocity,
when acted on by a number of forces, the net force acting on it must be zero. Therefore if we
sum the components of the individual forces in each of two perpendicular directions, e.g., along
the x and y directions, the law may be expressed as:
Fx = 0 (the sum of the x components of the forces = 0)
and
Fy = 0 (the sum of the y components of the forces = 0)
OBJECTIVES
•
Understand the vector nature of force.
• Practice finding the components of a force.
• Explain Newton’s first law in simple language.
MATERIALS
Force table
assorted masses
mass hangers
26
PRELIMINARY QUESTIONS
1. What is the difference between saying that no forces act on a body and saying that the net
force acting on the body is zero?
2. How would you answer someone who says that you need to have a net force acting on a body
to keep it moving at constant velocity, because otherwise it would slow down?
APPARATUS
The circular “force table” shown below is the principal component used in this experiment. A
small circular ring represents the “body” to which forces will be applied. These forces are
produced by hanging masses connected to the ring by strings passing over pulleys clamped to the
edge of the table. The directions of the forces are measured using a circular scale, calibrated in
degrees, around the perimeter of the table. This scale only has meaning when the ring is precisely
at the center of the table.
PROCEDURE
1.
2.
3.
4.
5.
The first part of the experiment is to note the forces produced on the ring by the hanging
masses. These forces will be the weights of the masses, transmitted to the ring via the
tensions in the strings.
Check out a set of masses: 100g  2, 50g  4, 20g  5, 10g  2, 5g  2 (total 15 pieces).
Note: Return all masses to their correct boxes. Failure to do so will mean you forfeit all
your points!
Check with a level that the force table is horizontal in two perpendicular directions, adjusting
the leveling screws if necessary.
Set two of the pulleys at 0° and 270°. Move the pulley by releasing the lock and holding its
base. Never force the pulley! Then hang masses totaling 100 g and 200 g over each of them
respectively. The mass must include that of the mass hanger.
Remove the fourth mass hanger and then adjust the angle and mass on the third mass hanger
to obtain equilibrium with the ring at the center of the table. The strings should all be
pointing directly towards the center of the table.
Calculate the magnitude of each force (weight of each mass) and complete the first 3
columns of the first table.
Now set three of the masses to 100 g, 150 g, and 120 g. Set the angle between the 100 g and
150 g to 60° and between the 120 g and 150 g to 85°. Complete the first 3 rows of the second
table having chosen a convenient x-y coordinate system. Now make an "educated guess" of
27
the necessary components of a fourth force (in the fourth row) that will result in equilibrium.
From these components, calculate the magnitude and direction of the fourth force. Proceed to
test this calculation by adjusting the mass on the fourth hanger and the position of the pulley.
6. Set four masses of 200 g, 50 g, 100 g, and 150 g in either clockwise or counter-clockwise
order and fix the angle between the 200 g and 50 g at 45°. Then adjust the directions of the
100 g and 150 g to achieve equilibrium. Complete the third table.
ANALYSIS
1. Draw a free body force diagram for the ring for each of the three force systems. Choose
convenient x-y coordinate systems (you have already done this for Procedure step 5) and
draw the axes on your diagrams.
2. Calculate the x and y components of the forces with respect to your chosen coordinate
system and write the values in the tables.
3. Calculate the net x and net y components for each of the three force systems.
4. Do your results indicate that Newton's first law is true? If there is a discrepancy, can it be
accounted for by experimental uncertainties?
5. Would it have made a difference if you had chosen a differently oriented x-y coordinate
system? Explain.
28
DATA TABLES FOR NEWTON'S FIRST LAW
Hanging
mass, g
Magnitude
of
force, N
Direction,
degrees
100
0
200
270
x comp.
of force,
N
F
x
Hanging
mass, g
Magnitude
of
force, N
Direction,
degrees
100
0
150
60
120
145
F
=
y
x comp.
of force, N
F
x
Hanging
mass, g
Magnitude
of
force, N
Direction,
degrees
y comp.
of force,
N
=
y comp.
of force, N
F
=
y
x comp.
of force, N
=
y comp.
of force, N
200
150
100
50
Angle between 100 g and 150 g =
F
x
=
F
y
=
29
Lab 5. Newton’s Second Law
How does a cart change its motion when you push and pull on it? You might think that the
harder you push on a cart, the faster it goes. Is the cart’s velocity related to the force you apply?
Or, is the force related to something else? Also, what does the mass of the cart have to do with
how the motion changes? We know that it takes a much harder push to get a heavy cart moving
than a lighter one.
A Force Sensor and an Accelerometer will let you measure the force on a cart simultaneously
with the cart’s acceleration. The total mass of the cart is easy to vary by adding masses. Using
these tools, you can determine how the net force on the cart, its mass, and its acceleration are
related. This relationship is Newton’s second law of motion.
Figure 1
OBJECTIVES
•
Collect force and acceleration data for a cart as it is moved back and forth.
• Compare force vs. time and acceleration vs. time graphs.
• Analyze a graph of force vs. acceleration.
• Determine the relationship between force, mass, and acceleration.
MATERIALS
computer
Logger Pro
Vernier Dynamics Track
Vernier Dynamics Cart
Labquest, Dual-Range Force
Sensor, and Low-g Accelerometer
0.50 kg mass
Weighing balance
PRELIMINARY QUESTIONS
Use a tennis ball and a flexible ruler to investigate these questions.
1. Apply a small amount of force to the ball by pushing the flat end of the
ruler against the ball. Maintain a constant bend in the ruler. You may
need a lot of clear space, and you may need to move with the ruler. Does
the ball move with a constant speed?
2. Apply a larger force and keep a constant larger bend in the ruler. Does
the ball move with a constant speed?
Figure 2
30
3. What is the difference between the movement when a small force is applied versus a large
force?
PROCEDURE
Check out a weight set consisting of one 50 g mass and five 20 g masses.
PART 1
Trial I
1. Set up the sensors and Logger Pro for data
collection.
Using Dual-Range Force Sensor and Accelerometer
a. Set the range switch on the Dual-Range Force
Sensor to 10 N.
b. Attach the Force Sensor to a Dynamics Cart so
you can apply a horizontal force to the hook,
Figure 3
directed along the sensitive axis of the sensor
(see Figure 3).
c. Attach the Accelerometer so the arrow is horizontal and parallel to the direction that the
cart will roll. Orient the arrow so that if you pull on the Force Sensor the cart will move in
the direction of the arrow.
d. Use a scale (in lab) to measure the mass of the cart with the Force Sensor and
Accelerometer attached. Record the mass in the data table.
e. Connect the Force Sensor and Accelerometer to the Vernier computer interface.
2. Open the file “09 Newtons Second Law” from the Physics with Vernier folder.
3. To zero the sensors, place the cart on the Dynamics Track on a level surface. Verify the cart
is not moving and click
. Check that both Accelerometer and Force are selected and
click
.
4. You are now ready to collect force and acceleration data. Grasp the Force Sensor or WDSS
hook. Click
and roll the cart back and forth along the track covering a distance of
about 20 cm. Vary the motion so that both small and large forces are applied. Your hand
must touch only the hook and not the sensors or cart body. Only apply force along the track
so that no frictional forces are introduced.
5. Note the shape of the force vs. time and acceleration vs. time graphs. How are the graphs
similar? How are they different?
6. Click Examine, , and move the mouse across the force vs. time graph. When the force is
maximum, is the acceleration maximum or minimum? To turn off Examine mode, click
Examine, , again.
7. The graph of force vs. acceleration should appear
to be a straight line. To fit a straight line to the
data, click the graph, then click Linear Fit, .
31
Figure 4
Record the equation for the regression line in the data table, Table 1.
8. Print or sketch copies of each graph. NOTE When printing graphs, save the trees by
selecting only the pages that you really want to print.
Trial II
9. Attach four 125 g masses to the cart. Record the total mass of the cart, sensors, and additional
mass in the data table, Table 2.
10. Repeat Steps 4–8.
PART 2
Cart with force sensor
and accelerometer
Figure 5
a) Constant cart mass and changing hanging weight
1. To investigate the effect of a constant force acting on the cart, set up the apparatus as shown
in Fig. 5. The cart should still have the four 125 g masses attached. The string should be long
enough that the cart can run approximately the full length of the track. Make sure there is an
"endstop" to prevent the cart running into the pulley.
2 Before attaching a hanging mass, clear the data and set the sensors to zero. This should be
done before every subsequent measurement.
3. Starting with a hanging mass of 0.05 kg, start data collection while holding the cart stationary
for about 1 second, then let go the cart.
4. Select appropriate parts of the Force graph and Acceleration graph, click the Statistics button
to find their average values, then evaluate F/a and record the value in Table 3.
5. Repeat steps 2, 3, and 4 for the other hanging masses shown in Table 3.
6. Open the Logger Pro file "Table 3" from folder Lab 05. Click on page 2 and input the values
of F/a and the hanging mass from Table 3, then create a graph of F/a versus hanging mass.
b) Constant hanging mass and changing cart mass
7. Open the Logger Pro file "Table 4" from folder Lab 05. For each of the added cart masses
shown in Table 4, and keeping the hanging mass at 0.1 kg, perform the steps necessary to
obtain a graph of F/a versus the total cart mass (cart plus added masses).
32
ANALYSIS
PART 1
1. Are the net force on an object and the acceleration of the object directly proportional?
Explain, using experimental data to support your answer.
2. What are the units of the slope of the force vs. acceleration graph? Simplify the units of the
slope to fundamental units (m, kg, s).
3. For each trial, compare the slope of the regression line to the mass being accelerated. What
does the slope represent?
4. Write a general equation that relates all three variables: force, mass, and acceleration.
PART 2
5. For a), describe the plot. What is the value of the slope? Considering the scatter in the data
points, do you think the slope is significantly different from zero? If it is zero, explain why?
Is there any significant feature of the plot that relates to the value of some physical quantity
in the experimental setup?
6. For b), describe the plot. What are the important features? What is the value (plus units) of
the slope? Explain why the slope has this value.
33
DATA TABLES
PART 1
Trial 1
Table 1
Mass of cart with sensors (kg)
Regression line for force vs. acceleration data
Trial II
Table 2
Mass of cart with sensors and additional mass (kg)
Regression line for force vs. acceleration data
PART 2
Table 3
Trial
1
2
3
4
5
6
7
Mass of cart =
Trial
1
2
3
4
5
F/a
(Ns2/m)
Hanging mass (kg)
0.05
0.07
0.09
0.110
0.130
0.150
0.170
kg
F/a (Ns2/m)
Table 4
Hanging mass (kg)
Mass added to cart (kg)
0.1
0
0.1
0.125
0.1
0.250
0.1
0.375
0.1
0.500
Total cart mass (kg)
34
Lab. 6 Atwood’s Machine
A classic experiment in physics is the Atwood’s machine: two masses on either side of a pulley
connected by a light string. When released, the heavier mass accelerates downward while the
lighter one accelerates upward at the same rate. The acceleration depends on the difference in the
two masses, as well as the total mass.
In this lab, you will determine the relationship between the two factors that influence the
acceleration of an Atwood’s machine using a Photogate to record the machine’s motion.
Figure 1
OBJECTIVES
Use a Photogate to study the acceleration of an Atwood’s machine.
Determine the relationships between the masses on an Atwood’s machine and the
acceleration.
Determine a value for the acceleration due to gravity, g.
MATERIALS
computer
Labquest Mini
Logger Pro
Vernier photogate
mass set
string, 1.2 m long
35
PRELIMINARY QUESTIONS
1. If two objects of equal mass are suspended from either end of a string passing over a light
pulley, as in Figure 1, what kind of motion do you expect to occur? Why?
2. Draw a free-body diagram of the left-side mass, m1 in Figure 1. Draw another of the rightside mass, m2. Include all forces acting on each mass.
3. Do the two masses have the same magnitude acceleration? Why?
4. Since a massless pulley merely alters the direction of the tension in the string, we can draw
an equivalent one-dimensional system as shown in Figure 2. Only the external forces are
shown and it is assumed that m1 > m2.
m2g
m2
m1
m1g
Figure 2
What is the magnitude of the net external force? What is the mass of the system? Apply
Newton's second law and obtain an expression for the acceleration. Have your instructor
check your result. It should include the total mass, mT (= m1 + m2) and the mass difference,
mdiff (= m1 - m2).
PROCEDURE
Part I Constant Total Mass
For this part of the experiment you will keep the total mass used
constant, but move weights from one side to the other. The
difference in masses is what will be changing.
Check out a set of masses: 100g  2, 50g  1, 20g  4, 10g  3 (total
10 pieces). Note: Return all masses to their correct boxes. Failure
to do so will mean you forfeit all your points!
1. Set up the Atwood’s machine apparatus as shown in Figures 1
and 3. Your design should ensure that the lighter mass does not
crash into the photogate and that the two masses do not crash
into each other.
2. Connect the photogate to a digital (DIG) port of the interface and
launch the data collection file '10 Atwood Machine' in the folder
Lab 06.
Figure 3
3. Arrange a collection of masses totaling 200 g for m2 and 200 g for m1. What is the
acceleration of this combination? Record your values for mass and acceleration in the data
table.
4. Move 10 g from m2 to m1. (You will be doing this step repeatedly, so plan your moves with
the available masses now.) Record the new masses in the data table.
5. To measure the acceleration of this system, steady the masses so they are not swinging. Click
to start data collection. After a moment, release the smaller mass, catching the falling
mass before it strikes the floor or the other mass strikes the pulley.
36
6. Click Examine, , and select the region of the graph where the velocity was increasing at a
steady rate. Click Linear Fit, , to fit the line y = mt + b to the data. Note the slope, which is
the acceleration, then repeat step 7 two more times and record the average of the three
accelerations in the data table.
7. Continue to move masses from m2 to m1 in 10 g increments, changing the difference between
the masses, but keeping the total mass constant. Repeat Steps 5–6 for at least five different
mass combinations.
Part II Constant Mass Difference
For this part of the experiment you will keep the difference in mass between the two sides of the
Atwood’s machine constant and increase the total mass.
8. Use 160 g for m1 and 150 g for m2.
9. Repeat Steps 5–6 to collect data and determine the acceleration.
10. Add mass in 10 g increments to both sides, keeping a constant difference of 10 g. (You will
be doing this step repeatedly, so plan your moves with the available masses now.) Record
the resulting masses for each combination in the data table. Collect motion data and
determine the acceleration for at least five different mass combinations.
ANALYSIS
1. For each trial, calculate the difference between m1 and m2. Enter the result in the column
labeled mdiff.
2. For each trial, calculate the total mass in grams. Enter the result in the column labeled mT.
Note that for Part II, there is a column in the table for 1/mT.
3. Disconnect all sensors and choose New from the File menu. Plot a graph of acceleration vs.
mdiff, using the Part I data. Based on your analysis of the graph, what is the relationship
between the mass difference and the acceleration of an Atwood’s machine? NOTE When
printing graphs, save the trees by selecting only the pages that you really want to print.
4. Similarly, plot a graph of acceleration vs. 1/mT, using the Part II data. Based on your analysis,
what is the relationship between total mass and the acceleration of an Atwood’s machine?
5. Are your results consistent with the theoretical expression that you derived for the
acceleration?
6. Use the slope of the graph in Part I, together with the total mass, mT to obtain the acceleration
due to gravity, g. Also obtain g using the slope of the graph in Part II. How do your values
compare with each other and with the "textbook value"? If there are differences, what factors
do you think might be responsible?
37
DATA TABLES
Part I Constant Total Mass
m1
(g)
Trial
mT,
m1+m2
(g)
m2
(g)
Acceleration
(m/s2)
mdiff,
m1–m2
(g)
1
2
3
4
5
Part II Constant Mass Difference
Trial
m1
(g)
m2
(g)
mdiff,
m1–m2
(g)
Acceleration
(m/s2)
mT,
m1+m2
(g)
1/mT
(g-1)
1
2
3
4
5
38
Lab 7. Static and Kinetic Friction
If you try to slide a heavy box resting on the floor, you may find it difficult to get the box
moving. Static friction is the force that counters your force on the box. If you apply a light
horizontal push that does not move the box, the static friction force is also small and directly
opposite to your push. If you push harder, the friction force increases to match the magnitude of
your push. There is a limit to the magnitude of static friction, so eventually you may be able to
apply a force larger than the maximum static force, and the box will move. The maximum static
friction force is sometimes referred to as starting friction. We model static friction, Fstatic, with
the inequality Fstatic  s N where s is the coefficient of static friction and N is the normal force
exerted by a surface on the object. The normal force is defined as the perpendicular component
of the force exerted by the surface. In this case, the normal force is equal to the weight of the
object.
Once the box starts to slide, you must continue to exert a force to keep the object moving, or
friction will slow it to a stop. The friction acting on the box while it is moving is called kinetic
friction. In order to slide the box with a constant velocity, a force equivalent to the force of
kinetic friction must be applied. Kinetic friction is sometimes referred to as sliding friction. Both
static and kinetic friction depend on the surfaces of the box and the floor, and on how hard the
box and floor are pressed together. We model kinetic friction with Fkinetic = k N, where k is the
coefficient of kinetic friction.
In this experiment, you will use a Dual-Range Force Sensor to study static friction and kinetic
friction on a wooden block. A Motion Detector will also be used to analyze the kinetic friction
acting on a sliding block.
Mass
Wooden block
Dual-Range
Force Sensor
Pull
Figure 1
OBJECTIVES
Use a Dual-Range Force Sensor to measure the force of static and kinetic friction.
Determine the relationship between force of static friction and the weight of an object.
Measure the coefficients of static and kinetic friction for a particular block and track.
Use a Motion Detector to independently measure the coefficient of kinetic friction and
compare it to the previously measured value.
Determine if the coefficient of kinetic friction depends on weight.
MATERIALS
computer
Labquest Mini
Logger Pro
Vernier Motion Detector
Vernier Dual-Range Force Sensor
string
block of wood with hook
balance or scale
mass set
39
PRELIMINARY QUESTIONS
1. In everyday life, you often experience one object sliding against another. Sometimes they
slip easily and other times they do not. List some things that seem to affect how easily
objects slide.
2. Consider a box sitting on a table. It takes a large force to move it at constant speed. List at
least two ways you could reduce the force needed to move the box at constant speed.
3. In pushing a heavy box across the floor, is the force you need to apply to start the box
moving greater than, less than, or the same as the force needed to keep the box moving? On
what are you basing your answer?
PROCEDURE
Part I Starting Friction
Check out a set of masses: 500g  2, 100g  2 (total 4 pieces). Note: Return all masses to their
correct boxes. Failure to do so will mean you forfeit all your points!
1. Measure the mass of the block and record it in the data table.
2. Set the range switch on the Dual-Range Force Sensor to 10 N. Connect the Force Sensor to
Channel 1 of the interface.
3. Open the file “12a Static Kinetic Frict” from the Physics with Vernier folder.
4. Tie one end of a string to the hook on the Force Sensor and the other end to the hook on the
wooden block. The larger wood surface of the block should be in contact with the table.
Place a total of mass of 1 kg on top of the block. Before you collect data, practice pulling the
block and masses with the Force Sensor using a straight-line motion: Slowly and gently pull
horizontally with a small force. Very gradually, taking one full second, increase the force
until the block starts to slide, and then keep the block moving at a constant speed for another
second.
5. Sketch a graph of force vs. time for the force you felt on your hand. Label the portion of the
graph corresponding to the block at rest, the time when the block just started to move, and
the time when the block was moving at constant speed.
6. Hold the Force Sensor in position, ready to pull the block, but with no tension in the string.
Click
to set the Force Sensor to zero.
7. Click
to begin collecting data. Pull the block as before, taking care to increase the
force gradually. Repeat the process as needed until you have a graph that reflects the desired
motion, including pulling the block at constant speed once it begins moving. Print or copy
the graph for use in the Analysis portion of this activity. NOTE When printing graphs,
save the trees by selecting only the pages that you really want to print.
Part II Peak Static Friction and Kinetic Friction
In this part, you will measure the peak static friction force and the kinetic friction force as a
function of the normal force on the block, as shown in Figure 1. In each run, you will pull the
block as before, but by changing the masses on the block, you will vary the normal force on the
block.
8. Remove all masses from the block.
40
9. Click
to begin collecting data and pull as before to gather force vs. time data.
10. Examine the data by clicking Statistics, . The maximum value of the force occurs when the
block started to slide. Read this value of the maximum force of static friction from the
floating box and record the number in your data table.
11. Drag across the region of the graph corresponding to the block moving at constant velocity.
Click Statistics, , again and read the average (or mean) force during the time interval. This
force is the magnitude of the kinetic frictional force.
12. Repeat Steps 9–11 for two more measurements and average the results to determine the
reliability of your measurements. Record the values in the data table.
13. Add masses totaling 500 g, then 1000 g to the block. Repeat Steps 9–12 for each, recording
values in the data table.
14. Repeat steps 9-13 using first the small wood surface on the table surface, and then the large
cloth surface on the rough (carpet) surface.
Part III Kinetic Friction Again
In this part, you will measure the coefficient of kinetic friction a second way and compare it to
the measurement in Part II. Using the Motion Detector, you can measure the acceleration of the
block as it slides to a stop. This acceleration can be determined from the velocity vs. time graph.
While sliding, the only force acting on the block in the horizontal direction is that of friction.
From the mass of the block and its acceleration, you can find the frictional force and finally, the
coefficient of kinetic friction.
Wooden block
Push
Figure 2
15. Connect the Motion Detector to a digital (DIG) port of the Vernier
computer interface. Set the Motion Detector sensitivity switch to Track.
16. Open the file “12b Static Kinetic Frict” in the Physics with Vernier folder.
17. Place the Motion Detector on the lab table 1–2 m from a block of wood, as shown in
Figure 2. The large wood surface should be in contact with the table. Position the Motion
Detector so that it will detect the motion of the block as it slides toward the detector.
18. Practice sliding the block toward the Motion Detector so that the block leaves your hand and
slides to a stop. Minimize the rotation of the block. After it leaves your hand, the block
should slide about 1 m before it stops and it must not come any closer to the Motion Detector
than 0.15 m.
19. Click
to start collecting data and give the block a push so that it slides toward the
Motion Detector. The velocity graph should have a portion with a linearly decreasing section
corresponding to the freely sliding motion of the block. Repeat if needed.
41
20. Select a region of the velocity vs. time graph that shows the decreasing speed of the block.
Choose the linear section. The slope of this section of the velocity graph is the acceleration.
Drag the mouse over this section and determine the slope by clicking Linear Fit, . Record
this value of acceleration in your data table.
21. Repeat Steps 19–20 two more times.
ANALYSIS
1. A reliable indication that the experiments are of good quality is that results are consistent,
meaning that different Trials of the same experiment give consistent results. Inspect your
Data Tables to see if any of your experiments needs to be repeated to improve the quality.
2. Inspect your force vs. time graph from Part I. Label the portion of the graph corresponding to
the block at rest, the time when the block just started to move, and the time when the block
was moving at constant speed.
3. Still using the force vs. time graph you created in Part I, compare the force necessary to keep
the block sliding compared to the force necessary to start the slide. How does your answer
compare to your answer to Preliminary Question 3?
4. The coefficient of friction is a constant that relates the normal force between two objects
(blocks and table) and the force of friction. Based on your graph (Run 1) from Part I, would
you expect the coefficient of static friction to be greater than, less than, or the same as the
coefficient of kinetic friction?
5. For Part II, calculate the normal force of the table on the block alone and with each
combination of added masses. Since the block is on a horizontal surface, the normal force
will be equal in magnitude and opposite in direction to the weight of the block and any
masses it carries. Fill in the Normal Force entries for all three Part II data tables.
6. Plot graphs of the maximum (peak) static friction force (vertical axis) vs. the normal force
(horizontal axis). Use either Logger Pro or graph paper.
7. Since Fmaximum static = s N, the slope of the proportional curve fit for this graph is the
coefficient of static friction s. For Proportional Curve Fit, click: Analyze > Curve Fit and
choose 'Proportional'. The Proportional Curve Fit passes through the origin.
8. In a similar graphical manner, find the coefficient of kinetic friction k. Create plots of the
average kinetic friction forces vs. the normal force. Recall that Fkinetic = k N.
9. Your data from Part III also allow you to determine k. Draw a free-body diagram for the
sliding block. The kinetic friction force can be determined from Newton’s second law,
F = ma. From the mass and acceleration, find the friction force for each trial, and enter it in
the data table.
10. From the friction force, determine the coefficient of kinetic friction for each trial and enter
the values in the data table. Also, calculate an average value for the coefficient of kinetic
friction for the block.
11. Do µs and/or µk depend strongly on the materials of the contacting surfaces? Explain using
your experimental data.
42
12. Do µs and/or µk depend strongly on the area of the contacting surfaces? Explain using your
experimental data.
13. Does the coefficient of kinetic friction depend on speed? Explain, using your experimental
data.
14. Does the force of kinetic friction depend on the weight of the block? Explain.
15. Does the coefficient of kinetic friction depend on the weight of the block?
16. Compare your coefficients of kinetic friction determined in Part III to that determined in
Part II. Discuss the values. Do you expect them to be the same or different? Which one do
you think is more precise (with smaller uncertainty)? Justify your answer using your
experimental data.
43
DATA TABLES
Part I Starting Friction
Mass of block
kg
Part II
(A) Peak Static Friction and Kinetic Friction (large wood surface on table surface)
Total
mass
(kg)
Normal
force
(N)
Total
mass
(kg)
Normal
force
(N)
Peak static friction
(N)
Trial 1
Trial 2
Trial 3
Kinetic friction force
(N)
Trial 1
Trial 2
Trial 3
Average
peak static
friction
(N)
Average
kinetic friction
(N)
(B) Peak Static Friction and Kinetic Friction (small wood surface on table surface)
Total
mass
(kg)
Normal
force
(N)
Total
mass
(kg)
Normal
force
(N)
Peak static friction
(N)
Trial 1
Trial 2
Trial 3
Kinetic friction force
(N)
Trial 1
Trial 2
Trial 3
Average
peak static
friction
(N)
Average
kinetic friction
(N)
44
(C) Peak Static Friction and Kinetic Friction (large cloth surface of wood block on rough surface)
Total
mass
(kg)
Normal
force
(N)
Total
mass
(kg)
Normal
force
(N)
Peak static friction
(N)
Trial 1
Trial 2
Trial 3
Kinetic friction force
(N)
Trial 1
Trial 2
Trial 3
Average
peak static
friction
(N)
Average
kinetic friction
(N)
Part III Kinetic Friction (large wood surface on table surface)
Data: Block with no additional mass
Trial
Acceleration
(m/s2)
Kinetic friction force
(N)
k
1
2
3
Average coefficient of kinetic friction:
45
Lab 8. Kinetic and Potential Energy
When a juggler tosses a bean ball straight upward, the ball slows down until it reaches the top of
its path and then speeds up on its way back down. In terms of energy, when the ball is released it
has kinetic energy, KE. As it rises during its free-fall phase it slows down, loses kinetic energy,
and gains gravitational potential energy, PE. As it starts down, still in free fall, the stored
gravitational potential energy is converted back into kinetic energy as the object falls.
If there is no work done by frictional forces, the total mechanical energy (KE + PE) remains
constant. In this experiment, we will see if this is true for the toss of a ball. We will study these
energy changes using a Motion Detector.
Motion Detector
Figure 1
We will also study the energy changes that occur when a small cart is projected up a ramp and
allowed to return to the starting position.
OBJECTIVES
Measure the change in the kinetic and potential energies as a ball moves in free fall.
See how the total energy of the ball changes during free fall.
Repeat these objectives for a small cart projected up a ramp.
MATERIALS
computer
Labquest Mini
Logger Pro
Vernier Motion Detector
basketball
wire basket
Vernier Dynamics Track
Vernier Dynamics Cart
PRELIMINARY QUESTIONS
For each question, consider the free-fall portion of the motion of a ball tossed straight upward,
starting just as the ball is released to just before it is caught. Assume that there is very little air
resistance.
46
1. What form or forms of energy does the ball have while momentarily at rest at the top of the
path?
2. What form or forms of energy does the ball have while in motion near the bottom of its path?
3. Sketch a graph of velocity vs. time for the ball.
4. Sketch a graph of kinetic energy vs. time for the ball.
5. Sketch a graph of potential energy vs. time for the ball.
6. If there are no frictional forces acting on the ball, how is the change in the ball’s potential
energy related to the change in kinetic energy?
PART 1. BALL TOSS
PROCEDURE
Check out a set of masses: 200g  1, 100g  2 (total 3 pieces). Note: Return all masses to their
correct boxes. Failure to do so will mean you forfeit all your points!
1. Measure and record the mass of the ball you plan to use in this experiment.
2. Connect the Motion Detector to a digital (DIG) port of the interface. Set the
Motion Detector sensitivity switch to Ball/Walk. Place the Motion Detector
on the floor and protect it by placing a wire basket over it.
3. Open the file “16 Energy of a Tossed Ball” from the Physics with Vernier folder.
4. In this step, you will toss the ball straight upward above the Motion Detector and let it fall
back toward the Motion Detector. This step may require some practice.
a. Hold the ball directly above and about 0.25 m from the Motion Detector. Use two hands.
b. Have your partner click
to start data collection.
c. Wait one second, then toss the ball straight upward. Move your hands out of the way after
you release it. A toss of 0.5 to 1.0 m above the Motion Detector works well. You will get
the best results if you catch and hold the ball when it is about 0.5 m above the Motion
Detector.
d. Verify that the position vs. time graph corresponding to the free-fall motion is parabolic in
shape, without spikes or flat regions, before you continue. If necessary, repeat data
collection until you get a good graph. When you have good data on the screen, proceed to
the Analysis section.
ANALYSIS
1. Click Examine, , and move the mouse across the position or velocity graphs of the motion
of the ball to answer these questions.
a. Identify the portion of each graph where the ball had just left your hands and was in free
fall. Determine the height and velocity of the ball at this time. Enter your values in your
data table.
b. Identify the point on each graph where the ball was at the top of its path. Determine the
time, height, and velocity of the ball at this point. Enter your values in your data table.
47
c. Find a time where the ball was moving downward, just before it was caught. Measure and
record the height and velocity of the ball at that time.
d. Choose two more points approximately halfway in time between the three recorded so far.
e. For each of the five points in your data table, calculate the Potential Energy (PE), Kinetic
Energy (KE), and Total Energy (TE). Use the position of the Motion Detector as the zero of
your gravitational potential energy.
2. How well does this part of the experiment show conservation of energy? Explain.
3. Calculate the ball's kinetic and potential energy.
a. Logger Pro can graph the ball’s kinetic energy according to KE =
1
mv 2 if you supply the
2
ball’s mass. To do this, adjust the mass parameter.
b. Logger Pro can also calculate the ball’s potential energy according to PE = mgh. Here, m is
the mass of the ball, g is the free-fall acceleration, and h is the vertical height of the ball
measured from the position of the Motion Detector. The same mass parameter will be used
to find PE.
c. Go to the next page of Logger Pro by clicking Next Page, .
4. Inspect your kinetic energy vs. time graph for the toss of the ball. Explain its shape and print
or sketch the graph.
5. Inspect your potential energy vs. time graph for the free-fall flight of the ball. Explain its
shape and print or sketch the graph.
6. Compare your energy graph predictions (from the Preliminary Questions) to the real data for
the ball toss.
7. Logger Pro will also calculate Total Energy (TE), the sum of KE and PE, for plotting.
Record the graph by printing or sketching.
8. What do you conclude from this graph about the total energy of the ball as it moved up and
down in free fall? Does the total energy remain constant? Should the total energy remain
constant? Why? If it does not, what sources of extra energy are there or where could the
missing energy have gone?
PART 2. CART ON RAMP
PROCEDURE
1. Measure and record the mass of the cart.
2. Connect the Motion Detector to a digital (DIG) port of the interface. Set the
Motion Detector sensitivity switch to Cart. Elevate one end of the Track to
form a ramp and then place the Motion Detector at the bottom of the ramp.
3. Open the file "Energy on an inclined plane” from folder Lab 08.
4. Measure the inclination angle of the ramp and enter the value in the LogerPro file, and also
enter the mass of the cart.
48
5 Practice giving the cart a gentle shove so that it reaches close to the top of the ramp. Then
have your partner click
to start data collection and once again launch the cart up the
ramp.
6. Verify that the position vs. time graph is parabolic in shape, without spikes or flat regions,
before you continue. If necessary, repeat data collection until you get a good graph. When
you have good data on the screen, proceed to the Analysis section.
ANALYSIS
1. Repeat the analysis steps 1 through 8 for Part 1 Ball Toss, entering your data in the second
table.
2. In the plot of Kinetic Energy versus Position of the cart, is there a loss of kinetic energy
when the cart returns to exactly the same position? According to the work-energy theorem, is
it positive or negative work that has been done on the cart? Calculate the magnitude of the
force that is responsible for this energy loss.
49
DATA TABLES
Mass of the ball (kg)
Time
(s)
Position
Height
(m)
Velocity
(m/s)
PE
(J)
KE
(J)
TE
(J)
Height
(m)
Velocity
(m/s)
PE
(J)
KE
(J)
TE
(J)
After release
Between release and top
Top of path
Between top and catch
Before catch
Mass of the cart (kg)
Position
Time
(s)
After shove
Between shove and top
Highest point on track
Between top and catch
Before catch
Effective frictional force on cart (N)
50
Lab 9. Momentum, Energy, and
Collisions
The collision of two carts on a track can be described in terms of momentum conservation and,
in some cases, energy conservation. If there is no net external force experienced by the system of
two carts, then we expect the total momentum of the system to be conserved. This is true
regardless of the force acting between the carts. In contrast, energy is only conserved when
certain types of forces are exerted between the carts.
Collisions are classified as elastic (kinetic energy is conserved), inelastic (kinetic energy is lost)
or completely inelastic (the objects stick together after collision). Sometimes collisions are
described as super-elastic, if kinetic energy is gained. In this experiment, you can observe elastic
and inelastic collisions and test for the conservation of momentum and energy.
Figure 1
OBJECTIVES
•
Observe collisions between two carts, testing for the conservation of momentum.
• Measure energy changes during different types of collisions.
• Classify collisions as elastic, inelastic, or completely inelastic.
MATERIALS
computer
Labquest Mini
Logger Pro
two Vernier Motion Detectors
Vernier Dynamics Track
two Vernier Dynamics Carts
with magnetic and hook-and-pile
strip bumpers
PRELIMINARY QUESTIONS
1. Consider a head-on collision between two identical billiard balls. Ball 1 is initially in motion
toward ball 2, which is initially at rest. After the collision, ball 2 departs with the same
velocity that ball 1 originally had. Disregard any friction between the balls and the surface.
What happens to ball 1? What happens to ball 2?
2. Sketch a position vs. time graph for each ball in Preliminary Question 1, starting with the
time before the collision starts and ending a short time after the collision.
3. Based on your graph from Preliminary Question 2, is momentum conserved in this collision?
Is kinetic energy conserved?
51
PROCEDURE
1. Measure the masses of the Dynamics Carts and record the values in Table 1. Label the carts
as cart 1 and cart 2. Note, it is very important to do this accurately, so carefully balance the
scale before making measurements.
2. Set up the Dynamics Track so that it is horizontal. Test this by releasing a cart on the track
from rest. The cart should not move.
3. Practice creating a gentle collision. Position cart 2 at rest in the middle of the track, and
release cart 1 so it rolls toward cart 2, magnetic bumper toward magnetic bumper. The carts
should smoothly repel one another without physically touching.
4. Place a Motion Detector at each end of the track, allowing for the 0.15 m
minimum distance between detector and cart, as shown in Figure 1.
Connect the Motion Detectors to the digital (DIG) ports of the interface.
Set the Motion Detector sensitivity switches to Track.
5. Open the file “Momentum Energy Coll” from the Lab 09 folder.
6. Place the two carts at rest in the middle of the track, with their hook-and-pile bumpers toward
one another and in contact. Keep your hands clear of the carts and click
. Select both
sensors and click
. This procedure will establish the same coordinate system for both
Motion Detectors. Verify that the zeroing was successful by clicking
and allowing the
still-linked carts to roll slowly across the track. The graphs for each Motion Detector should
be nearly the same. If not, repeat the zeroing process. Circle which is the positive direction
for each cart in Table 1. (It is convenient, but not necessary, to adjust the settings of the two
Motion Detectors so that the positive direction is the same for both.)
7. Click
to begin taking data. Repeat the collision you practiced above and use the
position graphs to verify that the Motion Detectors can track each cart properly throughout
the entire range of motion. You may need to adjust the position of one or both of the Motion
Detectors.
Part I Magnetic bumpers
8. Reposition the carts so the magnetic bumpers are facing one another. Click
to begin
taking data and repeat the collision you practiced in Step 3. Keep your hands out of the way
of the Motion Detectors after you push the cart.
9. From the velocity graphs, you can determine a mean velocity before and after the collision
for each cart. To measure the mean velocity during a time interval, drag the cursor across the
interval. Click Statistics, , to read the mean value. Measure the mean velocity for each cart,
before and after collision, and enter the four values in Table 2. Close the statistics box.
10. Repeat Steps 8–9 as a second run with the magnetic bumpers, recording the values in the data
table.
Part II Hook-and-pile bumpers
11. Change the collision by turning the carts so the hook-and-pile bumpers face one another. The
carts should stick together after collision. Practice making the new collision, again starting
with cart 2 at rest.
52
12. Click
to begin taking data and repeat the new collision. Using the same procedure as
in Step 9, measure and record the cart velocities in Table 2.
13. Repeat the previous step as a second run with the hook-and-pile bumpers.
Part III Hook-and-pile to magnetic bumpers
14. Face the hook-and-pile bumper on one cart to the magnetic bumper on the other. The carts
will not stick, but they will not smoothly bounce apart either. Practice this collision, again
starting with cart 2 at rest.
15. Click
to begin data collection and repeat the new collision. Using the procedure in
Step 9, measure and record the cart velocities in Table 2.
16. Repeat the previous step as a second run with the hook-and-pile to magnetic bumpers.
ANALYSIS
1. For each run, determine the momentum (mv) of each cart before the collision, after the
collision, and the total momentum before and after the collision. Calculate the ratio of the
total momentum after the collision to the total momentum before the collision. Enter the
values in Table 3.
1 2
mv ) for each cart before and after the
2
collision. Calculate the ratio of the total kinetic energy after the collision to the total kinetic
energy before the collision. Enter the values in Table 4.
2. For each run, determine the kinetic energy ( KE =
3. If the total momentum for a system is the same before and after the collision, we say that
momentum is conserved. If momentum were conserved, what would be the ratio of the total
momentum after the collision to the total momentum before the collision?
4. If the total kinetic energy for a system is the same before and after the collision, we say that
kinetic energy is conserved. If kinetic energy were conserved, what would be the ratio of the
total kinetic energy after the collision to the total kinetic energy before the collision?
5. Inspect the momentum ratios in Table 3. Even if momentum is conserved for a given
collision, the measured values may not be exactly the same before and after due to
measurement uncertainty. The ratio should be close to one, however. Is momentum
conserved in your collisions?
6. Repeat the preceding question for the case of kinetic energy, using the kinetic energy ratios
in Table 4. Is kinetic energy conserved in the magnetic bumper collisions? How about the
hook-and-pile collisions? Is kinetic energy consumed in the third type of collision studies?
Classify the three collision types as elastic, inelastic, or completely inelastic.
7. You may have learned that for elastic collisions, "approach speed" equals "separation speed."
Check this by completing Table 5.
8. If the last column in Table 5 contained a ratio greater than 1.0, it would imply that the
combined kinetic energies of the carts had increased! What is the maximum value you
found?
53
DATA TABLES
Table 1
Mass of cart 1 (kg)
Mass of cart 2 (kg)
Positive direction for cart 1: Left
Right
Positive direction for cart 2: Left
Right
Table 2
Run
number
Bumper type
PART I:
PART II:
PART III:
Velocity of
cart 1
before
collision
(m/s)
Velocity of
cart 2
before
collision
(m/s)
Magnetic
1
0
Magnetic
2
0
Hook-and-pile
3
0
Hook-and-pile
4
0
Both
5
0
Both
6
0
Velocity of
cart 1 after
collision
(m/s)
Velocity of
cart 2
after
collision
(m/s)
Table 3
Run
number
Momentum
of cart 1
before
collision
(kg•m/s)
Momentum
of cart 2
before
collision
(kg•m/s)
1
0
2
0
3
0
4
0
5
0
6
0
Momentum
of cart 1
after
collision
(kg•m/s)
Momentum
of cart 2
after
collision
(kg•m/s)
Total
momentum
before
collision
(kg•m/s)
Total
momentum
after
collision
(kg•m/s)
Ratio of
total
momentum
after/before
54
Table 4
Run
number
KE of
cart 1
before
collision
(J)
KE of
cart 2
before
collision
(J)
1
0
2
0
3
0
4
0
5
0
6
0
KE of
cart 1
after
collision
(J)
KE of
cart 2
after
collision
(J)
Total KE
before
collision
(J)
Total KE
after
collision
(J)
Approach
speed
before
collision
(m/s)
Separation
speed
after
collision
(m/s)
Ratio of
total KE
after/before
Table 5
Part # :
run #
Velocity of
cart 1
before
collision
(m/s)
Velocity of
cart 2
before
collision
(m/s)
Part I:1
0
Part 1:2
0
Part II:3
0
Part II:4
0
Part III:5
0
Part III:6
0
Velocity of
cart 1
after
collision
(m/s)
Velocity of
cart 2
after
collision
(m/s)
Ratio of
speeds
separation/
approach
55
Lab 10. Conservation of Angular
Momentum & Rotational Dynamics
INTRODUCTION
In your study of linear momentum, you learned that, in the absence of an unbalanced external
force, the momentum of a system remains constant. In this experiment, you will examine how
the angular momentum, as well as the angular acceleration, of a rotating system responds to
changes in the moment of inertia, I.
OBJECTIVES
In this experiment, you will
•
•
•
•
Collect angle vs. time and angular velocity vs. time data for rotating systems.
Analyze the -t and -t graphs both before and after changes in the moment of inertia.
Determine the effect of changes in the moment of inertia on the angular momentum of
the system.
Explore Newton's 2nd law in its rotational form.
MATERIALS
Labquest Mini
Logger Pro or LabQuest App
Vernier Rotary Motion Sensor
Vernier Rotary Motion Accessory Kit
ring stand or vertical support rod
balance
metric ruler
PRELIMINARY QUESTIONS
1. What is the difference between mass and moment of inertia? Do they have the same
dimensions?
2. What is the total moment of inertia of a system of several objects rotating together?
3. What are the expressions for the moments of inertia for a point mass, a thin rod rotating
about a central axis, a disk, and an annular disk (disk with hole in middle)?
PART 1. CONSERVATION OF ANGULAR MOMENTUM
PROCEDURE
1. Mount the Rotary Motion Sensor to the vertical support rod. Place the 3-step Pulley on the
rotating shaft of the sensor so that the largest pulley is on top. Measure the mass and diameter
of the aluminum disk with the smaller hole. Mount this disk to the pulley using the longer
machine screw sleeve (see Figure 1).
56
Figure 1
2. Connect the sensor to the data-collection interface and open the file
"Rotational Dynamics" file from the folder lab 10.
3. Spin the aluminum disk so that it is rotating reasonably rapidly,
then begin data collection. Note that the angular velocity gradually
decreases during the interval in which you collected data. Consider
why this occurs. Store this run (Run 1).
4. Obtain the second aluminum disk from the accessory kit; determine
its mass and diameter. Position this disk (cork pads down) over the
sleeve of the screw holding the first disk to the pulley. Practice
dropping the second disk onto the first so as to minimize any
torque you might apply to the system (see Figure 2).
5. Begin the first disk rotating rapidly as before and begin collecting
data. After a few seconds, drop the second disk onto the rotating
disk and observe the change in both the -t and -t graphs. Store
this run (Run 2).
Figure 2
6. Repeat Step 5, but begin with a lower angular velocity than before.
Store this run (Run 3).
7. Find the mass of the steel disk. Measure the diameter of both the
central hole and the entire disk. Replace the first aluminum disk
with the steel disk and hub and tighten the screw as before (see
Figure 3).
8. Try to spin the steel disk about as rapidly as you did the aluminum
disk in Step 3 and then begin collecting data. Store this run (Run 4).
9. Repeat Step 5, dropping the aluminum disk onto the steel disk after
a few seconds. Store this run (Run 5) and save the experiment file in
case you need to return to it.
ANALYSIS
Figure 3
1. Use a text or web resource to find an expression for the moment of inertia for a disk;
determine the values of I for your aluminum disks. With its large central hole, the steel disk
57
should be treated as a cylindrical tube. Using the appropriate expression, determine the value
of I for the steel disk.
2. Examine the -t graph for your runs with the single aluminum disk (Run1) and the steel disk
(Run 4). Determine the rate of change of the angular velocity, , for each disk as it slowed.
Account for this change in terms of any unbalanced forces that may be acting on the system.
Explain the difference in the rates of change of  (aluminum vs. steel) in terms of the values
you calculated in Step 1.
3. Examine the -t graph for Run 2. Determine the angular acceleration before you dropped the
second disk onto the first.
4. Record the angular velocity just before and just after you increased the mass of the system.
Determine the time interval (t) between these two velocity readings.
•
In Logger Pro, drag-select the interval between these two readings. The x in the lower
left corner gives the value of t.
5. The angular momentum, L, of a system undergoing rotation is the product of its moment of
inertia, I, and the angular velocity, .
L = I
Determine the angular momentum of the system before and after you dropped the second
aluminum disk onto the first. Calculate the percent difference between these values.
6. Use the rate of change in , as 
determined in step 3, and the time interval between your two
readings to determine  due to friction alone. What portion of the difference in the angular
speed before and after you increased the mass can be accounted for by frictional losses?
7. Repeat the calculations in Steps 3–6 for your third and fifth runs.
PART 2. ROTATIONAL DYNAMICS
Newton's 2nd law in its rotational form is expressed by  = I . In the following experiment, the
torque  will be exerted on a thin rotating rod with attached point-like masses. Refer to Figure 4
for the set-up.
PROCEDURE
1. Measure the mass of the mass hanger and the average mass
of the two short cylindrical masses. Measure the radius of the
largest of the 3-step pulley. Record the values in the Data
Table.
2. Remove the disk from the Rotary Motion Sensor, securely
fasten the thin rod to it, then replace it on the Rotary Motion
Sensor. Attach a piece of string to the center of the rotating
shaft. Wind the string around the largest pulley 3 to 4 times
and attach the other end of the string to the mass hanger.
Figure 4
58
3. Attach the Smart Pulley to the Rotary Motion Sensor and adjust its position so that the string
passes over the Smart Pulley as shown in Figure 4.
4. Attach the two cylindrical masses to the rod symmetrically at distances r of about 7 cm from
the center. You may need to adjust the position of the Smart Pulley so that the rod with
masses can rotate freely.
5. Practice a few test runs by allowing the hanging mass to fall and collecting angular velocity
data in order to determine the angular acceleration.
6. Choose 5 to 7 values for the distance r between 7 and 17 cm, and measure the angular
acceleration in each case using the linear curve fit for the angular velocity curve. Record the
angular accelerations in the Data Table.
ANALYSIS
1. Examine your data. Does the angular acceleration, , increase or decrease with r?
2. Derive the equation for the angular acceleration, , by applying Newton's 2nd Law both to
the falling mass and, in its rotational form, to the rotating system: MgR = ( I 0 + MR 2 + 2mr 2 )  .
M is the hanging mass, m is the mass of each cylindrical mass, R is the radius of the pulley that
the rod is attached to, and I0 is the moment of inertia of all of the rotating system except the two
cylindrical masses. Ask your TA for assistance if needed.
3. Complete the last two columns of the Data Table. Disconnect the Rotary Sensor and open a
MgR
fresh LoggerPro file. Plot the magnitude of
(y axis) vs. r2 (x axis). Perform a linear fit.

4. What is the slope of the linear fit? How is the slope related to the mass, m, of each
cylindrical mass? Determine m from the slope and compare it with the value obtained by
weighing. Calculate the percentage difference.
59
DATA TABLE
Average mass, m, of short cylinder (kg) =
Mass, M, of mass hanger (kg)
=
Radius, R, of largest pulley (m)
=
Run #
Angular
acceleration, 
(rad/s2)
Distance, r
MgR / 
r2
(m)
(kg.m2)
(m2)
1
2
3
4
5
6
7
Slope of graph (kg)
=
Percentage difference =
60
Lab 11. Simple Harmonic Motion
Lots of things vibrate or oscillate. A vibrating tuning fork, a moving child’s playground swing,
and the speaker in a headphone are all examples of physical vibrations. There are also electrical
and acoustical vibrations, such as radio signals and the sound you get when blowing across the
top of an open bottle. Adding heat to a solid increases the vibration of atoms and molecules.
One simple system that vibrates is a mass hanging from a spring. The force applied by an ideal
spring is proportional to how much it is stretched or compressed. Given this force behavior, the
up and down motion of the mass is called simple harmonic and the position can be modeled with
y = Asin (2ft +  )
In this equation, y is the vertical displacement from the equilibrium position, A is the amplitude
of the motion, f is the frequency (number of oscillations per second), t is the time, and  is a
phase constant that tells us the value of y at t = 0. This experiment will clarify each of these
1 k
terms. The frequency f =
where k is the spring constant and m is the mass.
2 m
Figure 1
OBJECTIVES
Measure the position and velocity as a function of time for an oscillating mass and spring
system.
Determine the amplitude, period, and phase constant of the observed simple harmonic
motion.
Compare the observed motion of a mass and spring system to a mathematical model of
simple harmonic motion.
MATERIALS
computer
Labquest Mini
Logger Pro
Vernier Motion Detector
200 g and 300 g masses
ring stand, rod, and right-angle clamp
spring, with a spring constant of
approximately 15 N/m
twist ties
wire basket
61
PRELIMINARY QUESTIONS
1. Attach the 200 g mass to the spring and hold the free end of the spring in your hand, so the
mass and spring hang down with the mass at rest. Lift the mass about 5 cm and release.
Observe the motion. Sketch a graph of position vs. time for the mass.
2. Just below the graph of position vs. time, and using the same length time scale, sketch a
graph of velocity vs. time for the mass.
3. Measure the spring extension when you hang the 200 g mass and calculate the spring
constant, k. Then calculate the theoretical frequency with which the system will oscillate.
PROCEDURE
1. Attach the spring to a horizontal rod connected to the ring stand and hang the mass from the
spring, as shown in Figure 1. Securely fasten the 200 g mass to the spring and the spring to
the rod, using twist ties so the mass cannot fall. Adjust the height of the mass so that the
bottom of the mass is about 33–55 cm from the table top or floor.
2. Set the Motion Detector sensitivity switch to Ball/Walk. Connect the
Motion Detector to a digital (DIG) port of the interface.
3. Place the Motion Detector below the mass. No objects should be near the path between the
detector and mass, such as a table edge. Place the wire basket over the Motion Detector to
protect it.
4. Open the file “Simple Harmonic Motion” from the Lab 11 folder.
5. Make a preliminary run to verify things are set up correctly. Lift the mass upward a few
centimeters and release. The mass should oscillate along a vertical line only. Click
begin data collection.
to
6. When data collection is complete, the position graph should show a clean sinusoidal curve. If
it has flat regions or spikes, reposition the Motion Detector and try again.
7. Compare the position graph to your sketched prediction in the Preliminary Questions. How
are the graphs similar? How are they different? Also, compare the velocity graph to your
prediction.
8. Estimate the equilibrium position of the 200 g mass. Do this by allowing the mass to hang
free and at rest. Click
to begin data collection. After collection stops, click
Statistics, , to determine the average distance from the detector. Record this value as
position (y0) for Run 1 in your data table.
9. Now lift the mass upward about 5 cm and release it. The mass should oscillate along a
vertical line only. Click
to collect data. Examine the graphs. The pattern you are
observing is characteristic of simple harmonic motion.
10. Use the position graph to measure the time interval between maximum positions. This is the
period, T, of the motion. For improved accuracy, measure the time interval between, say, the
first and eleventh maxima and divide this by 10. (There are 10 periods between the first and
eleventh maximum.) The frequency, f, is the reciprocal of the period, f = 1/T. Based on your
62
period measurement, calculate the frequency. Record the period and frequency of this motion
in your data table.
11. The amplitude, A, of simple harmonic motion is the maximum distance from the equilibrium
position. Estimate values for the amplitude from your position graph. Enter the values in your
data table. If you drag the mouse from a peak to an adjacent trough, Logger Pro will report
the change in position over that region.
12. Repeat Steps 9–11 with the same 200 g mass, but with a larger amplitude than in the first run.
13. Change the mass to 300 g and repeat Steps 8–11. Use an amplitude of about 5 cm. Keep a
good run made with this 300 g mass on the screen. You will use it for several of the Analysis
questions.
ANALYSIS
1. View the graphs of the last run. Compare the position vs. time and the velocity vs. time
graphs. How are they the same? How are they different?
2. Click Examine, , to use the Examine tool. Move the mouse cursor back and forth across the
graph to view the data values for the last run on the screen. In your data table, record time
and position values for when v = 0. Also record time and position values for a point when the
velocity is greatest. Relative to the equilibrium position, where is the mass when the velocity
is zero? Where is the mass when the velocity is greatest?
3. Does the frequency, f, appear to depend on the amplitude of the motion? Do you have
enough data to draw a firm conclusion?
4. Does the frequency, f, appear to depend on the mass used? Did it change much in your tests?
5. How does the frequency with the 200 g mass compare with the theoretical value that you
obtained in Preliminary Questions?
6. You can compare your experimental data to the sinusoidal function model using the Model
feature of Logger Pro. Try it with your 300 g data. The model equation in the introduction,
which is similar to the one in many textbooks, gives the displacement from equilibrium.
However, your Motion Detector reports the distance from the detector. To compare the
model to your data, add the equilibrium distance to the model; that is, use
y = A sin (2ft +  ) + y0
where y0 represents the equilibrium distance. The phase parameter, , is called the phase
constant and is used to adjust the y value reported by the model at t = 0 so that it matches
your data.
d.
e.
f.
g.
Click once on the position graph to select it.
Choose Model from the Analyze menu and select Latest.
Select the Sine function from the General Equation list.
The Sine equation is of the form y=A*sin(Bt +C) + D. Compare this to the form of the
equation above to match variables; e.g.,  corresponds to C, and 2f corresponds to B.
h. Adjust the values for A, B and D to reflect your values for A, f and y0. You can either enter
the values directly in the dialog box or you can use the up and down arrows to adjust the
values.
63
i. The optimum value for  will be between 0 and 2. Find a value for  that makes the model
come as close as possible to the data of your 300 g experiment. You may also want to
adjust y0, A, and f to improve the fit. Write down the equation that best matches your data.
7. Does the model fit the data well? How can you tell?
8. Predict what would happen to the plot of the model if you doubled the parameter for A by
sketching both the current model and the new model with doubled A. Now double the
parameter for A in the model dialog box to compare to your prediction.
9. Similarly, predict how the model plot would change if you doubled f, and then check by
modifying the model definition.
EXTENSION
1. Investigate the effect of "damping" by attaching a damper (a flat sheet of 8.5"11" paper ) to
the mass. Set the motion detector to zero with the mass/paper at the equilibrium position.
2. Make measurements of position vs time for a 10 s period. The amplitude will exhibit a
typical "exponential decay," so the previous sinusoidal function will be modified by a
multiplying term, e-Bt, where B is called the "damping coefficient."
3. Click on Analysis, then Curve Fit, then select "damped oscillator with offset".
4. Report the damping coefficient, B, and the frequency in the Data Table.
5. Design an experiment in which you remove the damping but without changing the mass.
Then repeat steps 2 to 4.
6. Did the introduction of damping have a significant effect on the frequency? On the damping
coefficient?
64
DATA TABLE
Run
Mass
(g)
y0
(m)
A
(m)
T
(s)
f
(Hz)
1
2
3
Time
(s)
when v = 0
Position
(m)
____________
when v is maximum ____________
Fitted equation with parameters
Effect of damping
Damping coeff. (s-1)
Frequency (Hz)
With damping
Without damping
65
Lab 12. Sound Waves and Beats
Sound waves consist of a series of air pressure variations. A Microphone diaphragm records
these variations by moving in response to the pressure changes. The diaphragm motion is then
converted to an electrical signal. Using a Microphone and an interface, you can explore the
properties of common sounds.
The first property you will measure is the period, or the time for one complete cycle of
repetition. Since period is a time measurement, it is usually written as T. The reciprocal of the
period (1/T) is called the frequency, f, the number of complete cycles per second. Frequency is
measured in hertz (Hz). 1 Hz = 1 s–1.
A second property of sound is the amplitude. As the pressure varies, it goes above and below the
average pressure in the room. The maximum variation above or below the average pressure is
called the amplitude. The amplitude of a sound is closely related to its loudness.
In analyzing your data, you will see how well a sine function model fits the data. The pressure in
the medium carrying a periodic wave can be modeled with a sinusoidal function. Your textbook
may have an expression resembling this one:
y = Asin ( 2π f t )
In the case of sound, which is a longitudinal wave, y refers to the variation in air pressure as a
function of time, t (See Fig. 1). A is the amplitude of the wave and f is the frequency. The factor
of 2 ensures that when t = T, the sine function will have gone through one complete cycle.
When two sound waves overlap, air pressure variations will combine. For sound waves, this
combination is additive. We say that sound follows the principle of linear superposition. Beats
are an example of superposition. Two sounds of nearly the same frequency will create a
distinctive, cycling variation of sound amplitude, which we call beats.
Figure 1
OBJECTIVES
•
Measure the frequency and period of sound waves from a tuning fork.
• Measure the amplitude of sound waves from a tuning fork.
66
•
Observe beats between the sounds of two tuning forks.
MATERIALS
computer
Labquest Mini
Logger Pro
Vernier Microphone
tuning forks, rubber stopper
PRELIMINARY QUESTIONS
1. How can a musician use beats to tune his or her instrument?
2. Given that sound waves consist of a series of air pressure increases and decreases, what
would happen if an air pressure increase from one sound wave was located at the same place
and time as a pressure decrease from another of the same amplitude?
PROCEDURE
Part I Simple Waveforms
1. Connect the Microphone to the computer interface.
2. Practice producing a pure tone with a tuning fork. To do so, sharply rap one of the prongs
against the rubber stopper. Then hold the tuning fork firmly on a solid surface, such as the
table top, which will act as a "sounding board.".
3. Open the file “32 Sound Waves” in the Physics with Vernier folder. Adjust the setting to
collect 0.05 s of data. (For Part II, the setting will need to be re-adjusted appropriately.) Data
are collected for only 0.05 s in order to be able to display the rapid pressure variations of
sound waves. The vertical axis corresponds to the variation in air pressure and the units are
arbitrary.
4. To center the waveform on zero, it is necessary to zero the Microphone channel. With the
room quiet, click
to center waveforms on the time axis.
5. Sound one of the tuning forks. Hold it close to the Microphone and click
should be sinusoidal in form, similar to Figure 1.
. The data
6. Note the appearance of the graph. Count and record the number of complete cycles shown
after the first peak in your data.
7. Click Examine, . Click and drag the mouse between the first and last peaks of the
waveform. Read the time interval t, and divide it by the number of cycles to determine the
period of the waveform.
8. Calculate the frequency of the note in Hz and record it in your data table.
9. In a similar manner, determine amplitude of the waveform. Click and drag the mouse across
the graph from top to bottom for an adjacent peak and trough. Read the difference in y
values, shown on the graph as y.
10. Calculate the amplitude of the wave by taking half of the difference, y. Record the value in
your data table.
67
11. Make a sketch of your graph or print the graph.
12. Save your data by choosing Store Latest Run from the Experiment menu.
13. Fit the function, y = A * sin(B*t + C) + D, to your data. A, B, C, and D are parameters
(numbers) that Logger Pro reports after a fit. This function is more complicated than the
textbook model, but the basic sinusoidal form is the same. Comparing terms, listing the
textbook model’s terms first, the amplitude A corresponds to the fit term A, and 2 f
corresponds to the parameter B. The time is represented by t, Logger Pro’s horizontal axis.
The new parameters C and D shift the fitted function left-right and up-down, respectively,
and are necessary to obtain a good fit. Only the values of parameters A and B are important
to this experiment. In particular, the numeric value of B allows you to find the frequency f
using B = 2 f.
a. Choose Model from the Analyze menu.
b. In the dialog box, choose Run 1|Sound Pressure and click
.
c. Select “A*sin(B*t +C) + D” from the list of equations.
d. Enter your estimate for the value of A, the amplitude.
e. Enter your estimate for the value of B (start with 2f).
f. Initially use zero for C and D.
g. Click
to view the model with your data.
h. The model and its parameters appear in a box on the graph. Adjust the values until you
have a good fit. Then, record the parameters A, B, C, and D in your data table.
14. Hide the run by choosing Hide Data Set from the Data menu and selecting Run 1 to hide.
Then, repeat Steps 5–13 for the other tuning fork. When repeating Step 13(b), choose
Run 2|Sound Pressure. When you are finished analyzing the second frequency, hide the Run
2 data.
15. Answer the Analysis questions for Part I before proceeding to Part II.
Part II Beats
16. Two pure tones with different frequencies sounded at once will create the phenomenon
known as beats. Sometimes the waves will reinforce one another and other times they will
combine to a reduced intensity. This happens on a regular basis because of the fixed
frequency of each tone. To observe beats, simultaneously sound both tuning forks and listen
for the combined sound. If the beats are slow enough, you should be able to hear a variation
in intensity. When the beats are too rapid to be audible as intensity variations, a single roughsounding tone is heard. At even greater frequency differences, two separate tones may be
heard, as well as various difference tones.
17. Collect data while the two tones are sounding. You should see a time variation of the sound
amplitude. When you get a clear waveform, choose Store Latest Run from the Experiment
menu. The beat waveform will be stored as Run 3.
18. The pattern will be complex, with a slower variation of amplitude on top of a more rapid
variation. Ignoring the more rapid variation and concentrating in the overall pattern, count
the number of amplitude maxima after the first maximum and record it in the data table.
19. Click Examine, . As you did before, find the time interval for several complete beats.
Divide the difference, t, by the number of cycles to determine the period of beats (in
68
seconds). Calculate the beat frequency in Hz from the beat period. Record these values in
your data table.
20. Proceed to the Analysis questions for Part II.
ANALYSIS
Part I Simple Waveforms
1. Did your model fit the waveform well? In what ways was the model similar to the data and in
what ways was it different?
2. Since the model parameter B corresponds to 2 f (i.e., f = B/(2)), use your fitted model to
determine the frequency. Enter the value in your data table. Compare this frequency to the
frequency calculated earlier. Which would you expect to be more accurate? Why?
3. Compare the parameter A to the amplitude of the waveform.
Part II Beats
4. How is the beat frequency that you measured related to the two individual frequencies?
Compare your conclusion with information given in your textbook.
69
DATA TABLE
Part I Simple Waveforms
Tuning fork
frequency
Number of
cycles
Tuning fork
frequency
Tuning fork
frequency
Parameter A
(arbitrary
units)
Parameter
B
(s–1)
t
(s)
Period
(s)
Calculated
frequency
(Hz)
Amplitude
(arbitrary units)
Parameter C
Parameter D
(arbitrary
units)
f = B/2
(Hz)
Part II Beats
Number of
cycles
t
(s)
Period
(s)
Calculated
beat frequency
(Hz)
70