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S a n A n t t o n i i o C o l l l l e g e

WELCOME TO PHYSICS!
Physics is our human attempt to explain the workings of the world. The success of that attempt is evident in the technology of our society. We are surrounded by the products resulting from the application of that understanding, technological inventions including clocks, cars, and computers. You have already developed your own physical theories to understand the world around you. Some of these ideas are consistent with the accepted theories of physics while others are not. This laboratory is designed to focus your attention on your interactions with the world so that you can recognize where your ideas agree with those accepted by physics and where they do not.
This course integrates the lecture and laboratory into a single class with all sections meeting in the
Laboratory room. Each section consists of a set of problems that ask you to make decisions about the real world. As you work through the problems in this manual, remember that the goal is not to make a lot of measurements. The goal is for you to examine your ideas about the real world.
Traditional courses have three components of the course - lecture, discussion section, and laboratory - each serve a different purpose. However, in this course all activities will be done in a laboratory format. The laboratory is where physics ideas, often expressed in mathematics, come to grips with the real world.
The amount you learn in this course will depend on the time you spend in preparation before coming to class.
Before coming to lab each week you must read the appropriate sections of your text, read the assigned problems to develop a fairly clear idea of what will be happening, and complete the prediction and method questions for the assigned problems.
Often, your group will be asked to present its predictions and data to other groups so that everyone can participate in understanding how specific measurements illustrate general concepts of physics. You should always be prepared to explain your ideas or actions to others in the class. To show your instructor that you have made the appropriate connections between your measurements and the basic physical concepts, you will be asked to write a laboratory report. Guidelines for preparing lab reports can be found in the lab manual appendices and in this introduction. An example of a good lab report is shown in Appendix E.
Please do not hesitate to discuss any difficulties with your fellow students or the lab instructor.
Relax. Explore. Make mistakes. Ask lots of questions, and have fun.
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A. What to bring to each session:
1. Bring an 8" by 10" graph-ruled lab journal, to all lab sessions. Your journal is your "extended memory" and should contain everything you do in class and all of your thoughts as you are going along. As such, your journal is a legal document; consequently you should never tear pages from it. For this reason, your lab journal must be bound, such as the Avery(R) Quadrille Laboratory Notebook, and not of the varieties that allow pages to be easily removed, for example spiral bound notebooks.
2. Bring a "scientific" calculator.
3. Bring this lab manual.
B. Prepare for each session:
Sometimes all groups will work on the same problem, other times groups will work on different problems and share results.
1. Before beginning a new problem, you should carefully read the Introduction, Objectives and
Preparation sections. Read the sections of the text specified in the Preparation section.
2. Before you come to a lab, be sure you have completed the assigned Prediction and Method
Questions. The Method Questions will help you build a prediction for the given problem.
It is usually helpful to answer the Method Questions before making the prediction. These individual predictions will be checked (graded) by your lab instructor immediately at
the beginning of each lab session.
This preparation is crucial if you are going to get anything out of your laboratory work.
There are at least two other reasons for preparing: a) There is nothing more dull or exasperating than plugging mindlessly into a procedure you do not understand. b) The work is a group activity where every individual contributes to the thinking process and activities of the group. Other members of your group will not be happy if they must consistently carry the burden of someone who isn't doing his/her share.
C. Problem Reports
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Occasionally during the semester you will be assigned to write up one of the experimental problems. Your report must present a clear and accurate account of what you and your group members did, the results you obtained, and what the results mean. A report is not to be copied or fabricated. To do so constitutes Scientific Fraud. To make sure no one gets in that habit, such behavior will be treated in the same manner as cheating on a test: A failing
grade for the course and possible expulsion from the College. Your report should describe your predictions, your experiences, your observations, your measurements, and your conclusions. A description of the lab report format is discussed at the end of this introduction. Each lab report is due, without fail, within two days of the end of that lab.
D. Grades
Satisfactory completion of the problems is required as part of your course grade. Those not completing all problem assignments by the end of the quarter at a 60% level or better will receive a
quarter grade of F for the entire course. The problem grade makes up 20% of your final course grade.
There are two parts of your grade for each problem: (a) your journal, and (b) your formal problem report. Your journal will be graded by the lab instructor during the laboratory sessions. Your problem report will be graded and returned to you in your next lab session.
If you have made a good-faith attempt but your report is unacceptable, your instructor may allow you to rewrite parts or all of the report. A rewrite must be handed in again within two days of the return of the report to you by the instructor, in order to obtain an acceptable grade.
F. The class forms a local scientific community. There are certain basic rules for conducting business in this community.
1.
In all discussions and group work, full respect for all people is required. All disagreements about work must stand or fall on reasoned arguments about physics principles, the data, or acceptable procedures, never on the basis of power, loudness, or intimidation.
2.
It is OK to make a reasoned mistake. It is in fact, one of the more efficient ways to learn.
This is an academic laboratory in which to learn things, to test your ideas and predictions by collecting data, and to determine which conclusions from the data are acceptable and reasonable to other people and which are not.
What do we mean by a "reasoned mistake"? We mean that after careful consideration and after a substantial amount of thinking has gone into your ideas you simply give your best prediction or explanation as you see it. Of course, there is always the possibility that your idea does not accord with the accepted ideas. Then someone says, "No, that's not the way I see it and here's why." Eventually persuasive evidence will be offered for one viewpoint or the other.
"Speaking out" your explanations, in writing or vocally, is one of the best ways to learn.
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3. It is perfectly okay to share information and ideas with colleagues. Many kinds of help are okay.
Since members of this class have highly diverse backgrounds, you are encouraged to help each other
and learn from each other.
However, it is never okay to copy the work of others.
Helping others is encouraged because it is one of the best ways for you to learn, but copying is completely inappropriate and unacceptable. Write out your own calculations and answer questions in your own words. It is okay to make a reasoned mistake; it is wrong to copy.
No credit will be given for copied work. It is also subject to University rules about plagiarism and cheating, and may result in dismissal from the course and the University. See the University course catalog for further information.
4. Hundreds of other students use this laboratory each week. Another class probably follows directly after you are done. Respect for the environment and the equipment in the lab is an important part
of making this experience a pleasant one.
The lab tables and floors should be clean of any paper or "garbage." Please clean up your area before you leave the lab.
The equipment must be either returned to the lab instructor or left neatly at your station, depending on the circumstances.
In summary, the key to making any community work is RESPECT.
Respect yourself and your ideas by behaving in a professional manner at all times.
Respect your colleagues (fellow students) and their ideas.
Respect your instructor and his/her effort to provide you with an environment in which you can learn.
Respect the laboratory equipment so that others coming after you in the laboratory will have an appropriate environment in which to learn.
1. Before you leave the laboratory, have the instructor assign the problem you will write up and initial your cover sheet.
2. A cover sheet for each problem must be placed on top of each problem report handed in to the instructor. A cover sheet can be found at the end of every lab. It gives you a general outline of
what to include in a report.
3. A problem report is always due within two days of the end of the lab.
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4. A problem report should be an organized, coherent display of your thoughts, work, and
accomplishments. It should be written neatly (word processor recommended) in English that is clear, concise, and correct. It may help you to imagine that hundreds of people will read your report and judge you by it. Communication is the goal of the report. In many cases, communication can be aided greatly by use of tables and graphs.
5. Sample reports are included in Appendix E. Listed below are the major headings which most laboratory reports use.
COVER SHEET
See page 11 of this introduction for a sample cover sheet.
TITLE
Write a descriptive title with your name, name of partners, date performed and Professor name.
STATEMENT OF THE PROBLEM
State the problem you were trying to solve, and how you went about it. Describe the general type of physical behavior explored and provide short summary of experiment.
PREDICTION
This is a part of the lab where you try to the outcome of the experiment based on the general knowledge of Physics. Generally, you start from fundamental laws or principles and derive the theoretical expression for the measured quantity. Later you are going to use your prediction to compare with experimental results.
EXPERIMENT AND RESULTS
Following the Prediction is a Experiment and Results section, containing a detailed description of how you made your measurements and what results you obtained. This usually involves an organized and coherent display of labeled diagrams, tables of measurements, tables of calculated quantities, and graphs. Explanations of all results must occur in correct grammatical English that would allow a reader to repeat your procedure.
Also, include all information necessary to repeat your experiment.
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Mathematical calculations connecting fundamental physics relationships to the quantities measured should be given. Any interesting behavior should be explained. Difficulties performing the experiment should be described as well as any subtleties in the analysis.
All data presented must be clearly identified and labeled. Calculated results should be clearly identified. Anybody should be able to distinguish between quantities you measured, those you calculated, and those you included from other sources. Clearly assign uncertainties to ALL measured values -- without uncertainties, the data is nearly
meaningless.
CONCLUSIONS
The Conclusions section should include your answers to the following questions: What generalized behavior did you observe? Was it different from what you expected? Why?
(E.g.: What were the possible sources of uncertainties? Did you have any major experimental difficulties?) How do your results compare with the theory presented in your textbook or during lectures? Can you think of any other ways to check your theory with your data?
REFERENCES
List books, journals or any other resources that you used to write your lab report.
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Table of Contents
Introduction
Introductory Laboratory
Introduction to Computational Modeling
Laboratory I: Description of Motion in One Dimension
Problem #1: Constant Velocity Motion
Problem #2: Soap Box Derby
Problem #3: Motion on a Level Surface With an Elastic
Cord
Laboratory II: Description of Motion in Two Dimensions
Laboratory III: Forces
Problem #1: Egg Drop
Problem #2: Bouncing
Problem #1: Forces in Equilibrium
Problem #2: Forces in Motion
Problem #3: Normal and Kinetic Frictional Force
Error and Uncertainty
Computational Modeling
Laboratory IV: Conservation of Energy and Momentum
Air Drag in Two Dimensions
Apollo Lab
Problem #1: Perfectly Inelastic Collisions
Problem #2: Kinetic Energy and Work
Problem #3: Energy and Collisions When the Objects
Bounce Apart
Laboratory VI: Rotational Kinematics
Laboratory VII: Rotational Dynamics
Laboratory VIII: Mechanical Oscillations
Problem #4: Conservation of Angular Momentum
Problem #1: Measuring Spring Constants
Problem #2: The Effective Spring Constant
Problem #3: Oscillation Frequency with Two Springs
Problem #4: Oscillation Frequency of an Extended System
Appendix A: Significant Figures
Appendix B: Accuracy, Precision, and
Uncertainty
Appendix C: Graphing
Appendix D: Sample Laboratory Reports
Problem #4: Energy and Friction
Problem #1: Angular Speed and Linear Speed
Problem #1: Moment of Inertia of a Complex System
Problem #2: Equilibrium
Problem #3: Forces, Torques, and Energy
Problem #5: Driven Oscillations
A
B
C
E
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Science is a way of thinking much more than it is a body of knowledge.
Carl Sagan - Broca's Brain
Background
Doing science is a way of observing and thinking about natural phenomena with the aim of trying to understand them. Models are human generated representations of natural phenomena. What sets science apart from other human intellectual endeavors is that scientific knowledge is based on experiments and observations with the ultimate authority being nature itself. Scientific knowledge is based on the cumulative work of scientists over the past 300 years. Science is a human activity, and each scientist has his/her own personal style, guided by his/her training, talents, biases, and personality. As a start, your common sense and intuition are highly reliable guides to scientific thinking. As you learn some of the fundamental scientific theories and laws in physics, you can hone your scientific modeling and reasoning skills. Generally, this will involve using scientific theory as well as your own common sense and intuition to interpret your experimental data.
Purpose
1) To learn how to set up and perform an experiment using a LabPRO and a Motion Detector to acquire data.
2) To learn how to use VideoAnalysis tools to measure the motion of objects.
3) To understand the basics of error analysis.
4) To model and interpret scientific data.
Outcomes:
1) Write-up in your lab notebook on measurement of acceleration due to gravity from motion detector.
2) Write-up in your lab notebook on measurement of acceleration due to gravity from Video Analysis.
3) Write-up in your lab notebook on measurement variability.
Procedure
1) Start up the Logger Pro software application on the computer by double clicking on the Logger Pro icon.
2) When you open LoggerPro you will see a Window such as the one pictured.
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Menu Bar
Tool Bar
Data Table
Graph Window
Set up your computer to run your experiment in Logger Pro by following these steps: a.
Click on "Setup Sensors"
Setup
Sensors b.
When the window appears, choose the icon labeled "Motion Detector" and then drag it to the Port labeled DIG/SONIC1. Make sure all other ports say NONE and click OK. Be sure that your Motion
Detector is plugged in to Port 1 on the LabPro.
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c.
LoggerPro automatically sets up the graph and data tables it thinks you will need for your experiment. Set up your Graph Window with a title of Position vs. Time and appropriately labeled axes. Do this by double clicking on the Position vs. Time graph. When the dialog box appears, enter the appropriate title. d.
Click on the button at the top of the screen with a small clock. This is the "Data Collection.." button and will allow you to change the settings of the motion detector. When the window appears, click on the "Sampling" tab. The sampling rate determines how often the motion detector takes a reading and also the total time data are collected. Change the sampling rate to something appropriate for the experiment planned. For example, for Experiment 1 you might select an Experimental Length of 3 seconds and a sampling speed of 30 samples/second. This means the motion detector will send out
30 pulses of sound every second, for 3 seconds.
Data
Collection
To Start Data
Collection
Settings
4) For your first measurement with Logger Pro and the Motion Detector, why not measure how high the ceiling is above you? Click the Collect button. You may hear a clicking sound from the Motion Detector, which is normal. Observe the graph that is being drawn. Logger Pro will take data for five seconds.
After the 5 seconds, you may need to click on the
Autoscale button to bring the line on scale. Check
Autoscale
Graph to see that the distance reported makes sense, and verify it with a meter stick. If the Motion Detector measurement is noisy or obviously wrong, then you've got some other object in the way of the Motion Detector. It's possible that the Motion Detector is reporting the distance to a light fixture, so try different placements on the table.
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Now hold your hand steady about a meter above the Motion Detector and click Collect. Before data collection stops, move your hand up and down a bit to see how the graph responds. Don't get closer than 0.2 m to the detector.
Can you point to the place on the graph when your hand was farthest from the
Motion Detector? How about the closest? If you click on the Autoscale button the graph will rescale and make it easier to read the values.
Logger Pro can also display the velocity and acceleration of the detected object. Click on the y-axis label, and choose Acceleration from the list of data.
5) Mount your Motion Detector on a Pole Stand and drop an object below it. Be sure that you have smooth data for your position and velocity graph. If you don’t have smooth data, adjust your motion detector and repeat your experiment.
6) Click the Curve Fit button on the toolbar. This will bring up the
Curve Fit dialog box. Notice that only the graph of this run shows up on the graph. Move your cursor to the beginning of the straight part of the graph and click and drag across to the end of the straight section. Release your mouse. The linear part of the graph is now highlighted. Click on mt+b (Linear) in the list of General Equations. Click on Try Fit and notice that the coefficients on the right side change and a dark line appears on the graph. If this fit is satisfactory, click on OK.
The slope, intercept and correlation will be displayed along with the curve fit on the graph. Drag this helper box to the lower right corner of the graph.
7) Use your data to determine the acceleration due to gravity of your falling ball.
Curve Fit
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Logger Pro can import videos of your experiment. You can then synchronize the video to your data, so when the experiment is replayed, the data and video will be viewed together. This can be a powerful tool.
In this tutorial and the next, you will learn how to
1. Capture a video into an experiment file.
3. Use the replay feature to analyze the experiment.
The experiment in this tutorial records the position, velocity and acceleration of a ball as it is dropped in an identical fashion to the previous experiment.
First, you will capture a movie. Pull down the Insert menu and choose Video Capture. It may ask you to choose a picture format in which case choose 400x300. You should see an image on the screen. If you do not, call your instructor for help.
Now click on the Options button and change the setting to take a 2 second movie. Then choose Camera
Settings -> Adjustments -> the image tab. Since we are trying to follow the motion of a fast moving object you will want to use the following settings:
 Set the Gain to Maximum
 Set the Shutter to 1
 Set the Exposure to Manual and a value of 150.
Close these settings and get ready to take your movie. You will need three people to make the movie. One person to take the movie, another person to drop the ball and a third person to hold a meter stick parallel to the path along which the ball falls.
Click Capture Video . The movie will appear in the Logger Pro window. Close the video capture dialog and under the Page Menu Choose Auto Arrange . Play your movie to be sure that it fully captures the motion.
If not, delete this movie and try again.
Note the five buttons on the bottom of the Movie Player. The button furthest on the left is the Play button.
Click it now to play the movie. The next button to the right is the Stop button. It will stop the movie during replay. The third button from the left "rewinds" the video to the beginning. The next two buttons move the video to the previous or next frames. The slider tool allows you to "slide" through the movie.
Click the far right button in the movie player. This action will open up the video analysis tools, which appear on the right side of the movie player. Under the Page Menu
Add
Point
Choose Auto Arrange again. A graph will also be displayed. By default, the graph will display both the x and y positions versus time. For this tutorial you will analyze the y position only. To set that up, click the y-axis label of the graph and then click "Y". The y axis should now show "Y" only.
Now we need to set a scale for the analysis. Click the Set Scale button, which is the fourth button from the top of the movie player on the right side. Now click and drag the mouse from one end of the meter stick to the other end. A pop-up dialog box will appear. The default of "1" for the Distance and "m" for the unit of measure is correct, so click OK.
Set
Scale
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Now click the Next Frame button and advance the movie until the ball has just leaves your lab partners hand. Now click the Add Point button (second from the top). Move your mouse over the movie and use the cross hairs to identify the middle of the ball. Click the mouse once. Notice that a mark is left on the screen, and the movie advances one frame. Mark the center of the ball again and repeat this process until the ball almost hits the floor. Now click the movie Play button to see the ball move through the air, and watch the trail of its horizontal and vertical position. The graph shows vertical position vs. time. Based on what your learned about curve fitting previously, fit a quadratic to the graph.
You may be interested in analyzing the horizontal motion. If so click the y-axis label and choose "X".
Use your data to determine the acceleration due to gravity of your falling ball.
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Installing VPython (Python 2.2 plus the Visual module):
Step 1 :
Load up your browser and visit www.vpython.org
and select Download.
Step 2:
Download Python-2.5.msi
and install it by double-clicking on the file.
Step 3:
Download VPython-Win-Py2.5-3.2.10.exe
and install it. It will automatically go into the Python 2.5 folder.
This version of VPython has "newzoom" as the default: hold down both the left and right mouse buttons to zoom.
Also, Visual no longer automatically executes "from random import random, randint, uniform". If you need these functions, add this statement to your program.
Step 4:
If you are using Windows 95, you may need to install OpenGL , the graphics library used by VPython.
To start using VPython:
Start IDLE by doubleclicking the "IDLE for VPython" shortcut on the desktop.
Open a demo program -- for example, bounce2.py.
Press F5 to run (or use the Run menu).
Every time you run, your files are automatically saved (if you have changed them).
Press F1 for documentation.
You can also start IDLE from the "Python 2.5: IDLE for VPython" entry on the Programs portion of the
Windows Start menu, then open a demo program in Programs\Demos.
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1 Overview
VPython is a programming language that is easy to learn and is well suited to creating 3D interactive models of physical systems.
VPython has three components that you will deal with directly:
 Python, a programming language invented in 1990 by Guido van Rossem, a Dutch computer scientist. Python is a modern, object-oriented language which is easy to learn.
 Visual, a 3D graphics module for Python created by David Scherer while he was a student at Carnegie Mellon University.
Visual allows you to create and animate 3D objects, and to navigate around in a 3D scene by spinning and zooming, using the mouse.
 IDLE, an interactive editing environment, written by van Rossem and modified by Scherer, which allows you to enter computer code, try your program, and get information about your program. IDLE is currently not available on pre-OSX
Macintosh.
This tutorial assumes that Python and Visual are installed on the computer you are using.
2 Spinning and Zooming
> Start IDLE by doubleclicking on the snake icon on your desktop. A window labeled “Untitled” should appear.
Pick
Open on the File menu and choose the program randombox.py
. Press F5 to run this program. (On Linux or Macintosh OSX, type visual-demos in a typescript to start IDLE in the demo folder. On pre-OSX Macintosh, drag the program randombox.py
onto PythonInterpreter)
Hold down the right mouse button (shift key on Macintosh) and drag the mouse to spin the “camera” around the scene. Hold down the middle mouse button (left+right buttons on a two-button mouse; control key on Macintosh) and drag the mouse to zoom into and out of the scene. These navigation tools are built into all VPython programs unless specifically disabled by the author of the program.
> Close all the VPython windows to start over.
3 Your First Program
> Start IDLE by double-clicking on the snake icon on your desktop. A window labeled “Untitled” should appear.
This is the window in which you will type your program. (On pre-OSX Macintosh, use an editor of your choice.) (On Linux or
Macintosh OSX, type idle in a typescript to start IDLE in the current folder.)
> As the first line of your program, type the following statement: from visual import *
This statement instructs Python to use the Visual graphics module.
> As the second line of your program, type the following statement: sphere ( )
4 Running the Program
> Now run your program by pressing F5 (or by choosing “Run program” from the
“Run” menu). (On pre -OSX Macintosh, save your program. In the Finder, drag your program file onto PythonInterpreter.)
When you run the program, two new windows appear. There is a window titled
“VPython,” in which you should see a white sphere, and a window titled “Output”. Move the Output window to the bottom of your screen where it is out of the way but you can still see it (you may make it smaller if you wish) .
> In the VPython w indow, hold down the “middle” mouse button and move the mouse. You should see that you are able to zoom into and out of the scene. (On a 2button mouse, hold down both left and right buttons; on a 1-button mouse, hold down the control key and the mouse button.)
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> Now try holding down the right mouse button (hold down shift key with a one-button mouse) . You should find that you are able to rotate around the sphere (you can tell that you are moving because the lighting changes) .
To avoid confusion about motion in the scene, it is helpful to keep in mind the idea that you are moving a camera around the object. When you zoom and rotate you are moving the camera, not the object.
5 Stopping the Program
Click the close box in the upper right of the display window (VPython window) to stop the program. Leave the Output window open, because this is where you will receive error messages.
6 Save Your Program
Save your program by pulling down the File menu and choosing Save As. Give the program a name ending in “.py,” such as
“ MyProgram.py
”. You must type the “.py” extension; IDLE will not supply it. Every time you run your program, IDLE will save your code before running. You can undo changes to the program by pressing CTRL-z.
7 Modifying Your Program
A single white sphere isn’t very interesting. Let’s change the color, size, and position of the sphere.
> Change the second line of your program to read: sphere(pos=(-5,0,0), radius=0.5, color=color.red)
What does this line of code do? To position objects in the display window we set their 3D cartesian coordinates. The origin of the coordinate system is at the center of the display window. The positive x axis runs to the right, the positive y axis runs up, and the positive z axis comes out of the screen, toward you. The assignment pos=(-5,0,0) sets the position of the sphere by assigning values to the x, y, and z coordinates of the z center of the sphere. The assignment radius=0.5 gives the sphere a radius of 0.5 of the same units. Finally, color = color.red makes the sphere red (there are 8 colors easily accessible by color.xxx: red,green, blue, yellow, magenta, cyan, black, and white) .
> Now press F5 to run your program. Note that VPython automatically makes the display window an appropriate size, so you can see the sphere in your display.
7.1 Objects and attributes
The sphere created by your program is an object. Properties like pos, radius, and color are called attributes of the object.
Now let’s create another object. We’ll make a w all, to the right of the sphere.
> Add the following line of code at the end of your program. box(pos=(6,0,0), size=(0.2,4,4), color=color.green) y x
> Before running your program, try to predict what you will see in your display. Run your program and try it.
7.2 Naming objects
In order to refer to objects such as the sphere and the box, we need to give them names.
> To name the sphere “ball”, change the statement that creates the red sphere to this: ball = sphere(pos=(-5,0,0), radius=0.5, color=color.red)
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> Similarl y, give the box the name “wallR” (for right wall) . Your program should now look like this: from visual import *
sphere(pos=(-5,0,0), radius=0.5, color=color.red) wallR = box(pos=(6,0,0),size=(0.2,4,4),color=color.green)
If you run your program you should find that it runs as it did before.
8 Order of Execution
You may already know that when a computer program runs, the computer starts at the beginning of the program and executes each statement in the order in which it is encountered. In Python, each new statement begins on a new line. Thus, in your program the computer first draws a red sphere, then draws a green box (of course, this happens fast enough that it appears to you as if everything was done simultaneously). When we add more statements to the program, we’ll add them in the order in which we want them to be executed. Occasionally we’ll go back and insert statements near the beginning of the program because we want them to be executed before statements that come later.
9 Animating the Ball
We would like to make the red ball move across the screen and bounce off of the green wall. We can think of this as displaying
“snapshots” of the position of the ball at successive times as it moves across the screen. To specify how far the ball moves, we need to specify its velocity and how much time has elapsed.
We need to specify a time interval between “snapshots.” We’ll call this very short time interval “dt”. In the context of the program, we are talking about virtual time (i.e. time in our virtual world); a virtual time interval of 1 second may take much less than one second on a fast computer.
> Type this line at the end of your program: dt = 0.05
We also need to specify the velocity of the ball. We can make the velocity of the ball an attribute of the ball, by calling it
“ball.velocity”. Since the ball will move in three dimensions, we must specify the x, y, and z components of the ball’s velocity.
We do this by making “ball.velocity” a vector.
> Type this line at the end of your program: ball.velocity = vector(2,0,0)
If you run the program, nothing will happen, because we have not yet given instructions on how to use the velocity to update the ball’s position.
Since distance = speed*time, we can calculate how far the ball moves (the displacement of the ball) in time dt by multiplying ball.velocity by dt. To find the ball’s new position, we add the displacement to its old position: ball.pos = ball.pos + ball.velocity*dt
Note that just as velocity is a vector, so is the position of the ball. 9.3
The meaning of the equal sign
If you are new to programming, the statement ball.pos = ball.pos + ball.velocity*dt may look very odd indeed. Your math teachers would surely have objected had you written a = a + 2.
In a Python program (and in many other computer languag es), the equal sign means “assign a value to this variable.” When you write: a = a + 2
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you are really giving the following instructions: Find the location in memory where the value of the variable a is stored. Read up that value, add 2 to it, and store the result in the location in memory where the value of a is stored. So the statement: ball.pos = ball.pos + ball.velocity*dt really means: “Find the location in memory where the position of the object bal is stored. Read up this value, and add to it the result of the vector expression ball. velocity*dt. Store the result back in the location in memory where the position of the object bal is stored.
9.4 Running the animation
Your program should now look like this: from visual import * ball = sphere(pos=(-5,0,0), radius=0.5, color=color.red) wallR = box(pos=(6,0,0), size=(0.2,4,4), color=color.green) dt = 0.05 ball.velocity = vector(0.2,0,0) ball.pos = ball.pos + ball.velocity*dt
> Run your program.
Not much happens! The problem is that the program only took one time step; we need to take many steps. To accomplish this, we write a while loop. A while loop instructs the computer to keep executing a series of commands over and over again, until we tell it to stop.
> Delete the last line of your program. Now type: while (1==1):
Don’t forget the colon! Notice that when you press return, the cursor appears at an indented location after the while. The indented lines following a while statement are inside the loop; that is, they will be repeated over and over. In this case, they will be repeated as long as the number 1 is equal to 1, or forever. We can stop the loop by quitting the program.
> Indented under the while statement, type this: ball.pos = ball.pos + ball.velocity*dt
(Note that if you position your cursor at the end of the while statement, IDLE will automatically indent the next lines when you press ENTER. Alternatively, you can simply press TAB to indent a line.) All indented statements after a while statement will be executed every time the loop is executed.
Your program should now look like this: from visual import * ball = sphere(pos=(-5,0,0), radius=0.5, color=color.red) wallR = box(pos=(6,0,0), size=(0.2,4,4), color=color.green) dt = 0.05 ball.velocity = vector(2,0,0) while (1==1): ball.pos = ball.pos + ball.velocity*dt
> Run your program. You should observe that the ball moves to the right, quite rapidly.
> To slow it down, insert the following statement inside the loop (after the while statement): rate(100)
This specifies that the while loop will not be executed more than 100 times per second.
> Run your program.
You should see the ball move to the right more slowly. However, it keeps on going right through the wall, off into empty space, because this is what we told it to do.
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10 Making the ball bounce: Logical tests
To make the ball bounce off the wall, we need to detect a collision between the ball and the wall. A simple approach is to compare the x coordinate of the ball to the x coordinate of the wall, and reverse the x component of the ball’s velocity if the ball has moved too far to the right. We can use a logical test to do this: if ball.x > wallR.x: ball.velocity.x = -ball.velocity.x
The indented line after the “if” statement will be executed only if the logical test in the previous line gives “true” for the comparison. If the result of the logical test is “false” (that is, if the x coordinate of the ball is not greater than the x coordinate of the wall), the indented line will be skipped.
However, since we want this logical test to be performed every time the ball is moved, we need to indent both of these lines, so they are inside the while loop. Insert tabs before the lines, or select the lines and use the “Indent region” option on the “Format” menu to indent the lines. Your program should now look like this: from visual import * ball = sphere(pos=(-5,0,0), radius=0.5,color
= color.red) wallR = box(pos=(6,0,0), size=(0.2,4,4)
, color=color.green) dt = 0.05 ball.velocity = vector(2,0,0) while (1==1): rate(100) ball.pos = ball.pos + ball.velocity*dt if ball.x > wallR.x: ball.velocity.x = -ball.velocity.x
> Run your program.
You should observe that the ball moves to the right, bounces off the wall,and then moves to the left, continuing off into space. Note that our test is not very sophisticated; because ball.x is at the center of the ball and wallR.x is at the center of the wall, the ball appears to penetrate the wall slightly.
> Add another wall at the left side of the display, and make the ball bounce off that wall also. Now modify your program so that it bounces off both walls. You will need to add another logical comparison between the ball and wallL.
>
Note that we inserted the statement to draw the left wall near the beginning of the program, before the while loop. If we had put the statement after the “while” (inside the loop), a new wall would be created every time the loop was executed. We’d end up with thousands of walls, all at the same location. While we wouldn’t be able to see them, the computer would try to draw them, and this would slow the program down considerably.
11 Making the ball move at an angle
To make the program more interesting, let’s make the ball move at an angle.
> Change the ball’s velocity to make it move in the y direction, as well as in the x direction. Before reading further, try to do this on your own.
Since the ball’s velocity is a vector, all we need to do is give it a nonzero y component. For example, we can change the statement: ball.velocity = vector(2,0,0) to ball.velocity = vector(2,1.5,0)
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Now the ball moves at an angle. Unfortunately, it now misses the wall! However, you can fix this later by extending the wall, or by adding a horizontal wall above the other walls.
12 Visualizing velocity
We will often want to visualize vector quantities, such as the ball ’s velocity. We can use an arrow to visualize the velocity of the ball. Before the while statement, but after the program statement setting the ball’s velocity, ball.velocity = vector(2,1.5,1) we can create an arrow: bv = arrow(pos=ball.pos, axis=ball.velocity, color=color.yellow)
It’s important to create the arrow before the while loop. If we put this statement in the indented code after the while, we would create a new arrow in every iteration. We would soon have thousands of arrows, all at the same location! This would make our program run very slowly.
> Run your program.
You should see a yellow arrow with its tail located at the ball’s initial position, pointing in the direction of the ball’s initial velocity. However, this arrow doesn’t change when the ball moves. We need to update the position and axis of the velocity vector every time we move the ball.
> Inside the while loop, after the last line of code, add these lines: bv.pos = ball.pos bv.axis = ball.velocity
The first of these lines moves the tail of the arrow to the location of the center of the ball. The second aligns the arrow with the current velocity of the ball.
Let’s also make the walls a bit bigger, by saying size= ( 0.2, 12,12 ), so the ball will hit them.
Your program should now look like this: from visual import * ball = sphere(pos=(-5,0,0), radius=0.5, color=color.red) wallR = box(pos=(6,0,0), size=(0.2,12,12), color=color.green) wallL = box(pos=(-6,0,0), size=(0.2,12,12), color=color.green) dt = 0.05 ball.velocity = vector(2,1.5,1) bv = arrow(pos=ball.pos, axis=ball.velocity, color=color.yellow) while (1==1): rate(100) ball.pos = ball.pos + ball.velocity*dt if ball.x > wallR.x: ball.velocity.x = -ball.velocity.x if ball.x < wallL.x:
ball.velocity.x = -ball.velocity.x bv.pos = ball.pos bv.axis = ball.velocity
> Run the program. The arrow representing the ball’s velocity should move with the ball, and should change direction every time the ball collides with a wall.
13 Leaving a trail
Sometimes we are interested in the trajectory of a moving object, and would like to have it leave a trail. We can make a trail out of a curve object. A curve is an ordered list of points, which are connected by a line (actually a thin tube) . We’ll create the curve before the loop, and add a point to it every time we move the ball.
> After creating the ball, but before the loop, add the following line: ball.trail = curve(color=ball.color)
This creates a curve object whose color is the same as the color of the ball, but without any points in the curve.
> At the end of the loop, add the following statement (indented) : ball.trail.append(pos=ball.pos)
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This statement adds a point to the trail. The position of the point is the same as the current position of the ball. Your program should now look like this: from visual import * ball = sphere(pos=(-5,0,0), radius=0.5, color=color.red) wallR = box(pos=(6,0,0), size=(0.2,12,12), color=color.green) wallL = box(pos=(-6,0,0), size=(0.2,12,12), color=color.green) dt = 0.05 ball.velocity = vector(2,1.5,1) bv = arrow(pos=ball.pos, axis=ball.velocity, color=color.yellow) ball.trail = curve(color=ball.color) while (1==1): rate(100) ball.pos = ball.pos + ball.velocity*dt if ball.x > wallR.x: ball.velocity.x = -ball.velocity.x
if ball.x < wallL.x: ball.velocity.x = -ball.velocity.x
bv.pos = ball.pos bv.axis = ball.velocity ball.trail.append(pos=ball.pos)
> Run your program. You should see a red trail behind the ball.
14 How your program works
In your while loop you continually change the position of the ball (ball.pos); you could also have changed its color or its radius. While your code is running, VPython runs another program (a “parallel thread”) which periodi cally gets information about the attributes of your objects, such as ball.pos and ball.radius. This program does the necessary computations to figure out how to draw your scene on the computer screen, and instructs the computer to draw this picture. This happens many times per second, so the animation appears continuous.
15 Making the ball bounce around inside a large box
> You are now at a point where you can try the following modifications to your program.
 Add top and bottom Walls.
 Expand all the walls so they touch, forming part of a large box.
 Add a back wall, and an “invisible” front wall. That is, do not draw a front wall, but include an “if” statement to prevent the ball from coming through the front.
 Give your ball a component of velocity in the z direction as well, and make it bounce off the back and front walls.
 Add gravity into your program so that there is an acceleration of -9.8 m/s 2 in the negative y direction.
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These problems are intended to introduce you to using a computer to model matter, interactions, and motion. You will build on these small calculations to build models of physical systems in later chapters.
Some parts of these problems can be done with almost any tool (spreadsheet, math package, etc.). Other parts are most easily done with a programming language. We recommend the free 3D programming language VPython. Your instructor will introduce you to an available computational tool and assign problems, or parts of problems, that can be addressed using the chosen tool.
Problem 1.1 Move an object across a computer screen
(a) Write a program that makes an object move from left to right across the screen at speed v. Make v a variable, so you can change it later. Let the time interval for each step of the computation be a variable dt, so that the position x increases by an amount v*dt each time. If it can be done with the tool you are using, erase the previous position of the object as you go, to simulate the actual motion of an object.
(b) Modify a copy of your program to make the object run into a wall and reverse its direction.
(c) Make a modification so that the object's speed v is no longer a constant but changes smoothly with time. Is the speed change clearly visible to an observer? Try to make one version in which the speed change is clearly noticeable, and another i n which it is not noticeable.
(d) Corresponding to part (c), make a computer graph of xvs. t, where t is the time.
(e) Corresponding to part (c), make a computer graph of v vs. t, where t is the time.
Turn in your programs for parts (c), (d), and (e).
Problem 1.2 Move an object at an angle
(a) Write a program that makes an object move at an angle.
(b) Change the component of velocity of the object in the x direction but not in the y direction, or vice versa. What do you observe?
(c) Start the object moving at an angle and make it bounce off at an appropriate angle when it hits a wall.
Turn in your answer to part (b), and the final version of your computation, part (c).
Problem 1.3 Move an object, leave a trail
Write a program that makes an object move smoothly from left to right across the screen at speed v, leaving a trail of dots on the screen at equal time intervals. If the dots are too close together, leave a dot every N steps, and adjust N to give a nice display.
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In this laboratory you will study the motion of objects that can go in only one dimension; that is, along a straight line. You will be able to measure the position of these objects by capturing video images on a computer. Through these measurements and their analysis you can investigate the relationship of quantities that are useful to describe the motion of objects. Determining these kinematic quantities (position, time, velocity, and acceleration) under different conditions allows you to improve your intuition about their quantitative relationships. In particular, you should be able to determine which relationships depend on specific situations and which apply to all situations.
There are many possibilities for the motion of an object. It might either moves at a constant speed, speed up, slow down, or have some combination of these motions. In making measurements in the real world, you need to be able to quickly understand your data so that you can tell if your results make sense to you. If your results don't make sense, then either you have not set up the situation properly to explore the physics you desire, you are making your measurements incorrectly, or your ideas about the behavior of objects in the physical world are incorrect. In any of the above cases, it is a waste of time to continue making measurements. You must stop, determine what is wrong and fix it. If it is your ideas that are wrong, this is the time to correct them by discussing the inconsistencies with your partners, rereading your text, or talking with your instructor. Remember, one of the reasons for doing physics in a laboratory setting is to help you confront and overcome your incorrect preconceptions about physics, measurements, calculations, and technical communications. Because most people are much faster at recognizing patterns in pictures than in numbers, the computer will graph your data as you go along.
BJECTIVES
After you successfully complete this laboratory, you should be able to:
 Describe completely the motion of any object moving in one dimension using the concepts of position, time, velocity, and acceleration.
 Distinguish between average quantities and instantaneous quantities when describing the motion of an object.
 Express mathematically the relationships among position, time, velocity, average velocity, acceleration, and average acceleration for different situations.
 Graphically analyze the motion of an object.
 Begin using technical communication skills such as keeping a laboratory journal and writing a laboratory report.
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REPARATION
Read YOUNG AND FREEDMAN: Chapter 2. Before coming to the lab you should be able to:
 Define and recognize the differences among these concepts:
- Position, displacement, and distance.
- Instantaneous velocity and average velocity.
- Instantaneous acceleration and average acceleration.
 Find the slope and intercept of a straight-line graph. If you need help, see Appendix C.
 Determine the slope of a curve at any point on that curve. If you need help, see Appendix C.
 Determine the derivative of a quantity from the appropriate graph.
 Use the definitions of sin  , cos  , and tan  with a right triangle.
These laboratory instructions may be unlike any you have seen before. You will not find worksheets or step-by-step instructions. Instead, each laboratory consists of a set of problems that you solve before coming to the laboratory by making an organized set of decisions (a problem solving strategy) based on your initial knowledge. The instructions are designed to help you examine your thoughts about physics.
These labs are your opportunity to compare your ideas about what "should" happen with what really happens. The labs will have little value in helping you learn physics unless you take time to predict what will happen before you do something. While in the laboratory, take your time and try to answer all the questions in this lab manual. In particular, the exploration questions are important to answer before you make measurements. Make sure you complete the laboratory problem, including all analysis and conclusions, before moving on to the next one.
Since this design may be new to you, this first problem contains both the instructions to explore constant velocity motion and an explanation of the various parts of the instructions. The explanation of the instructions is in this font and is preceded by the double, vertical lines seen to the left.
Why are we doing this lab problem? How is it related to the real world? In the lab instructions, the first paragraphs describe a possible situation that raises the problem you are about to solve. This emphasizes the application of physics in solving real-world problems.
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To earn some extra money, you have taken a job as a camera operator for the Los Angeles Grand Prix automobile race. Since the race will be simulcast on the Internet, you will be using a digital video camera that stores the images directly on a computer. You notice that the image is distorted near the edges of the picture and wonder if this affects the measurement of a car’s speed from the video image. You decide to model the situation using a toy car, which moves at a constant velocity.
?
Does the measured speed of a car moving with a constant velocity depend on the position of the car in a video picture?
The question, framed in a box and preceded by a question mark, defines the experimental problem you are trying to solve. You should keep the question in mind as you work through the problem.
E QUIPMENT
To make a prediction about what you expect to happen, you need to have a general understanding of the apparatus you will use before you begin. This section contains a brief description of the apparatus and the kind of measurements you can make to solve the laboratory problem. The details should become clear to you as you use the equipment.
For this problem, you will use a motorized toy car, which moves with a constant velocity on an aluminum track. You will also have a stopwatch, a meter stick, a video camera and a computer with a video analysis application to help you analyze the motion.
P REDICTION
Everyone has his/her own "personal theories" about the way the world works. One purpose of this lab is to help you clarify your conceptions of the physical world by testing the predictions of your personal theory against what really happens. For this reason, you will always predict what will happen before collecting and analyzing the data. Your prediction should be completed and written in your lab journal before you come to lab.
The “Method Questions” in the next section are designed to help you determine your prediction and should also be completed before you come to lab. This may seem a little backwards.
Although the prediction question is given before the method questions, you should complete the method questions before making the prediction.
The prediction question is given first so you know your goal.
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Spend the first few minutes at the beginning of the lab session comparing your prediction with those of your partners. Discuss the reasons for any differences in opinion. It is not necessary that your predictions are correct, but it is necessary that you understand the basis of your prediction.
How would each of the graphs of position-versus-time, velocity-versus-time, and acceleration-versus-time show a distortion of the position measurement? Sketch these graphs to illustrate your answer. How would you determine the speed of the car from each of the graphs? Which method would be the most sensitive technique for determining any distortions? Appendix B might help you answer this question.
Sometimes, as with this problem, your prediction is an "educated guess" based on your knowledge of the physical world. In these problems exact calculation is too complicated and is beyond this course. However, it’s possible to come up with a qualitative prediction by making some plausible simplifications. For other problems, you will be asked to use your knowledge of the concepts and principles of physics to calculate a mathematical relationship between quantities in the experimental problem.
M ETHOD Q UESTIONS
Method Questions are a series of questions intended to help you solve the experimental problem. They either help you make the prediction or help you plan how to analyze data. Method Questions should be answered and written in your lab journal before you come to lab.
To determine if the measured speed is affected by distortion, you need to think about how to measure and represent the motion of an object. The following questions should help with the analysis of your data.
1. How would you expect an instantaneous velocity-versus-time graph to look for an object moving with a constant velocity? Make a rough sketch and explain your reasoning. Write down the equation that describes this graph. If this equation has any constant quantities in it, what are the units of those constant quantities? What parts of the motion of the object does each of these constant quantities represent? For a toy car, what do you estimate should be the magnitude of those quantities? How would a distortion affect this graph? How would it affect the equation that describes the graph? How will the uncertainty of your position measurements affect this graph? How might you tell the difference between uncertainty and distortion?
2. How would you expect a position-versus-time graph to look for an object moving with a constant velocity? Make a rough sketch and explain your reasoning. What is the relationship between this graph and the instantaneous velocity versus time graph? Write down the equation that describes this graph. If this equation has any constant quantities in it, what are the units of those constant quantities? What parts of the motion of the object does each of these constant quantities represent? For a toy car, what do you estimate should be the magnitude of those quantities? How would a distortion affect this graph? How would it affect the equation that describes the graph? How will the uncertainty of your position measurements affect this graph? How might you tell the difference between uncertainty and distortion?
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3. How would you expect an instantaneous acceleration-versus-time graph to look for an object moving with a constant velocity? Make a rough sketch and explain your reasoning. Write down the equation that describes this graph. If this equation has any constant quantities in it, what are the units of those constant quantities? What parts of the motion of the object does each of these constant quantities represent? For a toy car, what do you estimate should be the magnitude of those quantities? How would a distortion affect this graph? How would it affect the equation that describes the graph? How will the uncertainty of your position measurements affect this graph? How might you tell the difference?
E XPLORATION
This section is extremely important — many instructions will not make sense, or you may be lead astray, if you do not take the time to carefully explore your experimental plan.
In this section you practice with the apparatus before you make time-consuming measurements which may not be valid. This is where you carefully observe the behavior of your physical system, before you begin making measurements. You will also need to explore the range over which your apparatus is reliable.
Remember to always treat the apparatus with care and respect . Your fellow students in the next lab section will need to use the equipment after you are finished with it. If you are unsure about how the apparatus works, ask your lab instructor.
Most apparatus has a range in which its operation is simple and straightforward. This is its range of reliability. Outside of that range, complicated corrections need to be applied. You can quickly determine the range of reliability by making qualitative observations at what you consider to be the extreme ranges of your measurements. Record your observations in your lab journal. If you observe that the apparatus does not function properly for the range of quantities you were considering measuring, you can modify your experimental plan before you have wasted time taking an invalid set of measurements.
The result of the exploration should be a plan for doing the measurements that you need. Record your measurement plan in your journal.
Place one of the metal tracks on your lab bench and place the toy car on the track. Turn on the car and observe its motion. Determine if it actually moves with a constant velocity. Use the meter stick and stopwatch to determine the speed of the car.
Turn on the video camera and look at the motion as seen by the camera on the computer screen. Go to the first lab for instructions about using the video recorder.
Do you need to focus the camera to get a clean image? How do the room lights affect the image? Which controls help sharpen the image? Record your camera adjustments in your lab journal.
Move the position of the camera closer to the car. How does this affect the video image on the screen? Try moving it farther away. Raise the height of the camera tripod. How does this affect the image? Decide where you want to place the camera to minimize the distortion.
Practice taking a video of the toy car. Looking at the video frame by frame allows you to check whether the computer has missed any frames (the motion should be smooth). The capacity of the computer to take
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in all of the data from the video camera depends on the amount of data. If your computer is dropping too many frames, you will not have enough data to analyze. You can minimize the number of frames dropped by decreasing the amount of data in the video picture by adjusting the picture size and keeping the picture as feature free as possible. Check out these effects. Write down the best situation for taking a video in your journal for future reference. You will be doing a lot of this. When you have the best movie possible, save it and open the video analysis application.
Make sure everyone in your group gets the chance to operate the camera and the computer.
M EASUREMENT
Now that you have predicted the result of your measurement and have explored how your apparatus behaves, you are ready to make careful measurements. To avoid wasting time and effort, make the minimal measurements necessary to convince yourself and others that you have solved the laboratory problem.
Measure the speed of the car using a stopwatch as it travels a known distance. How many measurements should you take to determine the car’s speed? (Too few measurements may not be convincing to others, too many and you may waste time and effort.) How much accuracy do you need from your meter stick and stopwatch to determine a speed to at least two significant figures? Make the number of measurements you need and record them in a neat and organized manner so that you can understand them a month from now if you must. Also make sure to record precisely how you make these measurements. Some future lab problems will require results from earlier ones.
Take a video of the motion of the car to determine its speed. Measure some object you can see in the video so that you can tell the analysis program the real size of the video images when it asks you to calibrate. The best object to measure is the car itself. When you digitize the video, why is it important to click on the same point on the car’s image? Estimat e your accuracy in doing so. Be sure to take measurements of the motion of the car in the distorted regions (edges) of the video.
Make sure you set the scale for the axes of your graph so that you can see the data points as you take them.
Use your measurements of total distance the car travels and total time to determine the maximum and minimum value for each axis before taking data.
Are any points missing from the position versus time graph? Missing points result from more data being transmitted from the camera than the computer can write to its memory. If too many points are missing, make sure that the size of your video frame is optimal (see Appendix D). It may also be that your background is too busy. Try positioning your apparatus so that the background has fewer visual features.
Note: Be sure to record your measurements with the appropriate number of significant figures (see Appendix A) and with your estimated uncertainty (see Appendix B).
Otherwise, the data is nearly meaningless.
A NALYSIS
Data by itself is of very limited use. Most interesting quantities are those derived from the data, not direct measurements themselves. Your predictions may be qualitatively correct but quantitatively very wrong. To see this you must process your data.
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Always complete your data processing (analysis) before you take your next set of data. If something is going wrong, you shouldn't waste time taking a lot of useless data. After analyzing the first data, you may need to modify your measurement plan and re-do the measurements. If you do, be sure to record the changes in your plan in your journal .
Calculate the average speed of the car from your stopwatch and meter stick measurements. Determine if the speed is constant within your measurement uncertainties. Can you determine the instantaneous speed of your car as a function of time?
Analyze your video to find the instantaneous speed of the car as a function of time. Determine if the speed is constant within your measurement uncertainties. See Appendix D for instructions on how to do video analysis.
Why do you have less data points for the velocity versus time graph compared to the position versus time graph? Use the data tables generated by the computer to explain how the computer generates the velocity graphs.
C ONCLUSIONS
After you have analyzed your data, you are ready to answer the experimental problem. State your result in the most general terms supported by your analysis. This should all be recorded in your journal in one place before moving on to the next problem assigned by your lab instructor. Make sure you compare your result to your prediction.
Compare the car’s speed measured with video analysis to the measurement using a stopwatch. How do they compare? Did your measurements and graphs agree with your answers to the Method Questions? If not, why? What are the limitations on the accuracy of your measurements and analysis?
Do measurements near the edges of the video give the same speed as that as found in the center of the image within the uncertainties of your measurement? What will you do for future measurements?
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You need to design a soap box derby car to compete in the grand physics 4A derby. In the soapbox derby, two cars are released from rest at the top of a ramp. The one that reaches the bottom first wins. Someone wants to make the car as heavy as possible to give it the largest acceleration. You need to test if this contention is correct, than determine the ideal distribution of mass to design the fastest soap box derby car.
?
How does the acceleration of an object down a ramp depend on its mass?
E QUIPMENT
For this problem you will have a stopwatch, a meter stick, an end stop, a wood block, a video camera and a computer with a video analysis application .You will also have a cart to roll down an inclined track and additional cart masses to add to the cart.
P REDICTION
Make a sketch of how you think the acceleration-versus-mass graph will look for carts with different mass released from rest from the top of an inclined track.
Do you think the acceleration of the cart increases, decreases, or stays the same as the mass of the cart increases? Make your best guess and explain your reasoning.
M ETHOD Q UESTIONS
The following questions should help with your prediction and the analysis of your data.
1. Make a sketch of the acceleration-versus-time graph for a cart released from rest on an inclined track. On the same graph sketch how you think the acceleration-versus-time will look for a cart with a much larger mass. Explain your reasoning? Write down the equation that best represents each of these accelerations. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph?
2. Write down the relationship between the acceleration and the velocity of the cart. Use that relationship to construct an instantaneous velocity versus time graph for each case. The connection between the derivative of a function and the slope of its graph will be useful. Write down the equation that best represents each of these velocities. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph? Can any of these constants be determined from the constants in
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the equation representing the acceleration? Which do you think best represents the velocity of the cart? Change your prediction if necessary.
3. Write down the relationship between the velocity and the position of the cart. Use that relationship to construct a position versus time graph for each case. The connection between the derivative of a function and the slope of its graph will be useful. Write down the equation that best represents each of these positions. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph? Can any of these constants be determined from the constants in the equation representing the velocity? Which do you think best represents the position of the cart?
Change your prediction if necessary.
E XPLORATION
Observe the motion of several carts of different mass when released from rest at the top of the track. BE SURE TO CATCH THE CART BEFORE IT HITS THE END STOP! From your estimate of the size of the effect, determine the range of mass that will give the best results in this problem. Determine the first two masses you should use for the measurement.
How do you determine how many different masses do you need to use to get a conclusive answer?
How will you determine the uncertainty in your measurements? How many times should you repeat these measurements? Explain.
What is the total distance through which the cart rolls? How much time does it take? These measurements will help you set up the graphs for your computer data taking.
Write down your measurement plan.
Make sure everyone in your group gets the chance to operate the camera and the computer.
M EASUREMENT
Using the plan you devised in the exploration section, make a video of the cart moving down the track at your chosen angle. Make sure you get enough points for each part of the motion to determine the behavior of the acceleration. Don't forget to measure and record the angle (with
estimated uncertainty).
Choose an object in your picture for calibration. Choose your coordinate system. Is a rotated coordinate system the easiest to use in this case?
Why is it important to click on the same point on the car’s image to record i ts position? Estimate your accuracy in doing so.
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PROBLEM #2: MASS AND MOTION DOWN AN INCLINE
Make sure you set the scale for the axes of your graph so that you can see the data points as you take them.
Use your measurements of total distance the cart travels and the total time to determine the maximum and minimum value for each axis before taking data.
Are any points missing from the position versus time graph? Missing points result from more data being transmitted from the camera than the computer can write to its memory. If too many points are missing, make sure that the size of your video frame is optimal (see Appendix D). It may also be that your background is too busy. Try positioning your apparatus so that the background has fewer visual features.
Make several videos with carts of different mass to check your qualitative prediction. If you analyze your data from the first two masses you use before you make the next video, you can determine which mass to use next. As usual you should minimize the number of measurements you need.
A NALYSIS
Choose a function to represent the position-versus-time graph. How can you estimate the values of the constants of the function from the graph? You can waste a lot of time if you just try to guess the constants.
What kinematic quantities do these constants represent?
Choose a function to represent the velocity versus-time-graph. How can you calculate the values of the constants of this function from the function representing the position-versus-time graph? Check how well this works. You can also estimate the values of the constants from the graph. Just trying to guess the constants can waste a lot of your time. What kinematic quantities do these constants represent?
From the velocity-versus-time graph determine the acceleration as the cart goes down the ramp. Use the function representing the velocity-versus-time graph to calculate the acceleration of the cart as a function of time.
Make a graph of the cart’s acceleration down the ramp as a function of the cart’s mass. Do you have enough data to convince others of your conclusion about how the acceleration of the cart depends on its mass?
As you analyze your video, make sure everyone in your group gets the chance to operate the computer.
C ONCLUSION
Did your measurements of the cart's motion agree with your initial predictions? Why or why not? What are the limitations on the accuracy of your measurements and analysis?
Does the acceleration of the car down its track depend on its total mass? State your result in the most general terms supported by your analysis.
Using your cart and masses, design the fastest possible cart for the soap box derby race. The race will be held as a timed competition with the fastest cart winning.
33

You are helping a friend design a new ride for the State Fair. In this ride, a cart is pulled along a long straight track by a stretched elastic cord like a bungee cord. Before spending money to build it, your friend wants to you to determine if this ride will be safe. Since sudden changes in velocity can lead to whiplash, you decide to find out how the acceleration of the cart changes with time. In particular, you want to know if the greatest acceleration occurs when the sled is moving the fastest or at some other time. To solve this problem, you decide to model the situation in the laboratory with a cart pulled by an elastic cord along a level surface.
?
How does an object pulled by an elastic cord along a level surface accelerate? Is the acceleration greatest when the velocity is greatest?
E QUIPMENT
For this problem you will have a stopwatch, a meter stick, a scissor, a video camera and a computer with a video analysis application .You will also have a cart to roll on a level track. You can attach one end of an elastic cord to the cart and the other end of the elastic cord to an end stop on the track.
P REDICTION
Make a sketch of how you expect the acceleration-versus-time graph to look for a cart pulled by an elastic cord. Just below that graph make a graph of the velocity versus time on the same time scale. Identify where on your graphs the velocity is largest and the acceleration is largest.
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M ETHOD Q UESTIONS
The following questions should help with your prediction and the analysis of your data:
1. Make a sketch of how you expect an acceleration-versus-time graph to look for a cart pulled by an elastic cord. Explain your reasoning. For a comparison, make acceleration-versus-time graphs for a cart moving at constant velocity and a constant acceleration. Write down the equation that best represents each of these accelerations. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graphs?
2. Write down the relationship between the acceleration and the velocity of the cart. Use that relationship to construct a velocity-versus-time graph for each case. The connection between the derivative of a function and the slope of its graph will be useful. Write down the equation that best represent each of these velocities. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph? Can any of these constants be determined from the constants in the equation representing the acceleration? Which do you think best represents the velocity of the cart?
Change your prediction if necessary.
3. Write down the relationship between the velocity and the position of the cart. Use that relationship to construct a position-versus-time graph for each case. The connection between the derivative of a function and the slope of its graph will be useful. Write down the equation that best represents each of these positions. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph? Can any of these constants be determined from the constants in the equation representing the velocity? Which do you think best represents the position of the cart?
Change your prediction if necessary.
E XPLORATION
Test that the track is level by observing the motion of the cart. Attach an elastic cord to the cart and track. Gently move the cart along the track to stretch out the elastic. Be careful not to stretch
the elastic too tightly. Start with a small stretch and release the cart. BE SURE TO CATCH THE
CART BEFORE IT HITS THE END STOP! Slowly increase the starting stretch until the cart's motion is long enough to get enough data points on the video, but does not cause the cart to come off the track or snap the elastic. Be sure to catch the cart before it collides with the end stop.
Practice releasing the cart smoothly and capturing videos.
Write down your measurement plan.
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Make sure everyone in your group gets the chance to operate the camera and the computer.
M EASUREMENT
Using the plan you devised in the exploration section, make a video of the cart’s motion. Make sure you get enough points to determine the behavior of the acceleration.
Choose an object in your picture for calibration. Choose your coordinate system.
Why is it important to click on the same point on the car’s image to record its position? Estimate your accuracy in doing so.
Make sure you set the scale for the axes of your graph so that you can see the data points as you take them.
Use your measurements of total distance the cart travels and total time to determine the maximum and minimum value for each axis before taking data.
Are any points missing from the position versus time graph? Missing points result from more data being transmitted from the camera than the computer can write to its memory. If too many points are missing, make sure that the size of your video frame is optimal (see Appendix D). It may also be that your background is too busy. Try positioning your apparatus so that the background has fewer visual features.
A NALYSIS
Choose a function to represent the position-versus-time graph. How can you estimate the values of the constants of the function from the graph? You can waste a lot of time if you just try to guess the constants.
What kinematic quantities do these constants represent?
Choose a function to represent the velocity-versus-time graph. How can you calculate the values of the constants of this function from the function representing the position-versus-time graph? Check how well this works. You can also estimate the values of the constants from the graph. Just trying to guess the constants can waste a lot of your time. What kinematic quantities do these constants represent?
From the velocity-versus-time graph determine the acceleration of the cart. Use the function representing the velocity-versus-time graph to calculate the acceleration of the cart as a function of time.
Make a graph of the cart’s ac celeration as a function of the time. Do you have enough data to convince others of your conclusion?
As you analyze your video, make sure everyone in your group gets the chance to operate the computer.
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C ONCLUSION
How does your acceleration-versus-time graph compare with your predicted graph? Are the positionversus-time and the velocity-versus-time graphs consistent with this behavior of acceleration? What is the difference between the motion of the cart in this problem and its motion along an inclined track? What are the similarities? What are the limitations on the accuracy of your measurements and analysis?
What will you tell your friend? Is the acceleration of the cart greatest when the velocity is the greatest?
How will a cart pulled by an elastic cord accelerate along a level surface? State your result in the most general terms supported by your analysis.
How would the acceleration-versus-time graph look if you attached another piece of elastic to the other end of the cart and to the opposite end stop of the track? How about the velocity-versustime and position-versus-time graphs? If you have time, try it .
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Name and ID#: ______________________________________________________
Date performed: ________________ Day/Time section meets: ______________
Lab Partners' Names: ________________________________________________
__________________________________________________________________
__________________________________________________________________
Problem # and Title: _________________________________________________
Lab Instructor's Initials: ____________
Grading Checklist Points
LABORATORY JOURNAL:
PREDICTIONS
(individual predictions and methods questions completed in journal before each lab session)
LAB PROCEDURE
(measurement plan recorded in journal, tables and graphs made in journal as data is collected, observations written in journal)
PROBLEM REPORT:*
ORGANIZATION
(clear and readable; logical progression from problem statement through conclusions; pictures provided where necessary; correct grammar and spelling; section headings provided; physics stated correctly)
DATA AND DATA TABLES
(clear and readable; units and assigned uncertainties clearly stated)
RESULTS
(results clearly indicated; correct, logical, and well-organized calculations with uncertainties indicated; scales, labels and uncertainties on graphs; physics stated correctly)
CONCLUSIONS
(comparison to prediction & theory discussed with physics stated correctly ; possible sources of uncertainties identified; attention called to experimental problems)
TOTAL(incorrect or missing statement of physics will result in a maximum of 60% of the total points achieved; incorrect grammar or spelling will result in a maximum of 70% of the total points achieved)
BONUS POINTS FOR TEAMWORK
(as specified by course policy)
* An "R" in the points column means to rewrite that section only and return it to your lab instructor within two days of the return of the report to you.

This laboratory allows you to continue the study of accelerated motion in more realistic situations. The carts you used in Laboratory I moved in only one dimension. Objects don't always move in a straight line. However, motion in two and three dimensions can be decomposed into a set of one-dimensional motions, so that what you learned in the first lab can be applied in this lab. You will also need to think of how air resistance could affect your results. Can this always be neglected? You will study the motion of an object in free fall, an object tossed into the air, and an object moving in a circle. As always, if you have any questions, please ask your fellow students or your instructor.
BJECTIVES
After successfully completing this laboratory, you should be able to:
• Determine the motion of an object in free-fall by considering what quantities and initial conditions affect the motion.
• Determine the motion of a projectile from its horizontal and vertical components by considering what quantities and initial conditions affect the motion.
• Determine the motion of an object moving in a circle from its horizontal and vertical components by considering what quantities and initial conditions affect the motion.
REPARATION
Read YOUNG AND FREEDMAN: Chapter 4. Review your results and procedures from
Laboratory I. Before coming to the lab you should be able to:
• Determine instantaneous velocities and accelerations from video images.
• Analyze a vector in terms of its components along a set of perpendicular axes.
• Add and subtract vectors.
After completing this laboratory see Appendix D (Sample Lab Report #1) for some suggestions on how to improve your lab reports.
39

You want to design a crash protection chassis for a person ejected from an airplane. Since human and animal testing is expensive, you decide to start you testing on an egg. Your task is to design a container out of straws that will protect an egg from breaking when it is dropped off the top of the science building
(approximately 18 m) Prior to starting your design you decide to investigate the problem by making measurements to determine if the free-fall acceleration of the egg depends on its mass.
?
How does the acceleration of a freely falling object depend on its mass?
E QUIPMENT
For this problem, you will have a collection of balls each with approximately the same diameter.
You will also have a stopwatch, a meter stick, a video camera and a computer with a video analysis application. After you have completed the experiment, you will be given one egg
(unboiled), 2 meters of scotch tape and a collection of 40 straws.
P REDICTION
Make a sketch of how you expect the average acceleration-versus-mass graph to look for falling objects with the same size and shape but different masses.
Do you think that the free-fall acceleration increases, decreases, or stays the same as the mass of the object increases? Make your best guess and explain your reasoning.
M ETHOD Q UESTIONS
1. Write down an outline of how you will determine the acceleration of the object from video data.
2. Sketch a graph of acceleration as a function of time for a constant acceleration and next to it one for half of that acceleration. Below those graphs, make graphs for velocity as a function of time and position as a function of time. Write down the equation that best represents each of these accelerations. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph?
3. Make a sketch of the acceleration-versus-time graph you expect for a heavy falling ball. Next to that graph make another one for the acceleration of a ball with one quarter of the mass.
Explain your reasoning? Write down the equation that best represents each of these accelerations. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph? How do they differ from your constant acceleration graphs?
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4.
Write down the relationship between the acceleration and the velocity of an object. How could you write an equation for the velocity in which the acceleration was not constant but changed linearly with time? What do the constant quantities in that equation represent?
5. Use the relationship between the acceleration and the velocity of the ball to construct an instantaneous velocity-versus-time graph for each case from Method Question 3. The connection between the derivative of a function and the slope of its graph will be useful. Write down the equation that best represents each of these velocities. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph? Can any of these constants be determined from the constants in the equation representing the acceleration? Change your prediction if necessary.
6. Write down the relationship between the velocity and the position of the ball. Use that relationship to construct a position-versus-time graph for each case. The connection between the derivative of a function and the slope of its graph will be useful. Write down the equation that best represents each of these positions. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph? Can any of these constants be determined from the constants in the equation representing the velocity? Change your prediction if necessary.
7. Compare your graphs to those for constant acceleration (Method Question 2). What are the differences, if any, that you might observe in your data? The similarities?
E XPLORATION
Review your lab journal from the problems in Lab 1. Position the camera and adjust it for optimal performance. Make sure everyone in your group gets the chance to operate the camera and the computer.
Practice dropping one of the balls until you can get the ball's motion to fill the least distorted part of the screen. Determine how much time it takes for the ball to fall and estimate the number of video points you will get in that time. Are there enough points to make the measurement? Adjust the camera position and screen size to give you enough data points without dropping too many. You should be able to reproduce the conditions described in the Predictions.
Although the ball is the best item to use to calibrate the video, the image quality due to its motion might make this difficult. Instead, you might hold an object of known length in the plane of motion of the ball, near the center of the ball’s trajectory, for calibration purposes. Where you place your reference object does make a difference in your results. Check your video image when you put the reference object close to the camera and then further away. What do you notice about the size of the reference object in the video image? The best place to put the reference object to determine the distance scale is at the position of the falling ball.
Step through the video and determine which part of the ball is easiest to consistently determine. When the ball moves rapidly you may see two images of the ball due to the interlaced scan of a TV camera. You should use the same image for each measurement.
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Write down your measurement plan.
M EASUREMENT
Measure the mass of a ball and make a video of its fall according to the plan you devised in the exploration section. Make sure you can see the ball in every frame of the video.
Digitize the position of the ball in enough frames of the video so that you have the sufficient data to accomplish your analysis. Make sure you set the scale for the axes of your graph so that you can see the data points as you take them. Use your measurements of total distance the ball travels and total time to determine the maximum and minimum value for each axis before taking data.
Complete your data analysis as you go along (before making the next video), so you can determine how many different videos you need to make. Don’t waste time in co llecting data you don't need or, even worse, incorrect data. Collect enough data to convince yourself and others of your conclusion.
Repeat this procedure for different balls.
A NALYSIS
Choose a function to represent the position-versus-time graph. How can you estimate the values of the constants of the function from the graph? You can waste a lot of time if you just try to guess the constants.
What kinematic quantities do these constants represent?
Choose a function to represent the velocity-versus-time graph. How can you calculate the values of the constants of this function from the function representing the position-versus-time graph? Check how well this works. You can also estimate the values of the constants from the graph. Just trying to guess the constants can waste a lot of your time. What kinematic quantities do these constants represent?
If you cannot get one function to describe your velocity graph in a consistent way, you can try using one function for the first half of the motion and another for the last half. To do this you must go through the video analysis process twice and record your results each time.
From the velocity-versus-time graph determine the acceleration of the ball. Use the function representing the velocity versus time graph to calculate the acceleration of the ball as a function of time. Is the average acceleration different for the beginning of the video (when the object is moving slowly) and the end of the video (when the object is moving fast)?
Determine the average acceleration of the object in free fall for each value of its mass and graph this result.
Do you have enough data to convince others of your conclusions about your predictions?
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C ONCLUSION
Did the data support your predicted relationship between acceleration and mass? (Make sure you carefully review Appendix C to determine if your data really supports this relationship.) If not, what assumptions did you make that were incorrect? Explain your reasoning.
What are the limitations on the accuracy of your measurements and analysis?
Do your results hold regardless of the masses of balls? Would the acceleration of a falling
Styrofoam ball be the same as the acceleration of a falling baseball? Explain your rationale. Make sure you have some data to back up your claim. Will the acceleration of a falling canister depend on its mass? State your results in the most general terms supported by your analysis.
Now prepare your egg in a container and prepare to drop it off the top of the building. You will film the egg on the way down and then prepare graphs of position, velocity and acceleration vs. time and use them to calculate the terminal velocity of your egg. The egg that survives the drop with the highest terminal velocity will win the contest.
43

You have been hired by a toy company to produce a videotape that guarantees to teach people how to juggle. Since it is easier to grab slow moving objects, you decide to determine how the velocity of a ball changes as it is thrown through the air. To catch the ball, you must also know both its horizontal and vertical position. For that reason, you decide to separately determine the horizontal motion of the ball and the vertical motion of the ball. To do this analysis, you decide to analyze a video of a ball thrown in a manner appropriate to juggling.
?
1. How does the horizontal component of the ball's velocity change with time?
2. How does the horizontal component of the ball's position change with time?
3. How does the vertical component of the ball's velocity change with time?
4. How does the vertical component of the ball’s position change with time?
E QUIPMENT
For this problem, you will have a ball, a stopwatch, a meter stick, a video camera and a computer with a video analysis application.
P REDICTION
1. Write down the equation that describes the horizontal velocity component of the ball as a function of time. Present that equation as a graph.
Do you think the horizontal component of the object's velocity changes during its flight? If so, how does it change? Or do you think it is constant (does not change)? Make your best guess and explain your reasoning.
2. Write down the equation that describes the horizontal position of the ball as a function of time.
Present that equation as a graph.
3. Write down the equation that describes the vertical velocity component of the ball as a function of time. Present that equation as a graph.
Do you think the vertical component of the object's velocity changes during its flight? If so, how does it change? Or do you
think it is constant (does not change)? Explain your reasoning.
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4. Write down the equation that describes the vertical position of the ball as a function of time.
Present that equation as a graph.
M ETHOD Q UESTIONS
The following questions may help with the prediction and the analysis of your data.
1. Make a large (about one-half page) drawing of the trajectory of the ball on a coordinate system. Be sure to label the horizontal and vertical axes of your coordinate system.
2. On your drawing, put the acceleration vectors of the ball (relative sizes and directions) for five different positions; two going up, two going down, and the maximum height. Explain your reasoning. Decompose each acceleration vector into its vertical and horizontal components.
3. On your drawing, put the velocity vectors of the ball at the same positions you chose to draw your acceleration vectors. Decompose each velocity vector into its vertical and horizontal components. Check to see that the change of the velocity vector is consistent with the acceleration vector. Explain your reasoning.
4. Looking at your drawing, how does the ball's horizontal acceleration change with time? How does the magnitude of the ball’s acceleration compare to the gr avitational acceleration? Write an equation giving the ball’s horizontal acceleration as a function of time. Make the graph that describes this equation. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph?
5. Looking at your drawing, how does the ball's horizontal velocity change with time? Is this consistent with your statements about the ball’s acceleration given for the previous question?
Write an equation givi ng the ball’s horizontal velocity as a function of time. Make the graph that describes this equation. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph?
6. Based on the equation describing the ball’s horizontal velocity, write an equation giving the ball’s horizontal position as a function of time. Make the graph that describes this equation. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph?
7. Looking at your drawing, how does the ball's vertical acceleration change with time? How does the magnitude of the ball’s acceleration compare to the gravitational acceleration? Write an equation giving the ball’s vertical acceleration as a function of time. Make the graph that describes this equation. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph?
5. Looking at your drawing, how does the ball's vertical velocity change with time? Is this consistent with your statements about the ball’s acceleration given for the previous question?
Write an equation giving the ball’s vertical velocit y as a function of time. Make the graph that
45

describes this equation. If there are constants in your equation, what kinematic quantities do they represent. How would you determine these constants from your graph?
6. Based on the equation describing the ball’s vertical velocity , write an equation giving the ball’s vertical position as a function of time. Make the graph that describes this equation. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph?
E XPLORATION
Position the camera and adjust it for optimal performance. Make sure everyone in your group gets
the chance to operate the camera and the computer.
Practice throwing the ball until you can get the ball's motion to fill the video screen (or at least the undistorted part of the video screen) after it leaves your hand. Determine how much time it takes for the ball to travel and estimate the number of video points you will get in that time. Is that enough points to make the measurement? Adjust the camera position and screen size to give you enough data points without dropping too many. You should be able to reproduce the conditions described in the predictions.
Although the ball is the best item to use to calibrate the video, the image quality due to its motion might make this difficult. Instead, you might need to place an object of known length in the plane of motion of the ball, near the center of the ball’s trajectory, for calibration purposes. Wher e you place your reference object does make a difference in your results. Check your video image when you put the reference object close to the camera and then further away. What do you notice about the size of the reference object?
Determine the best place to put the reference object for calibration.
Step through the video and determine which part of the ball is easiest to consistently determine. When the ball moves rapidly you may see two images of the ball due to the interlaced scan of a TV camera. You should use the same image for each measurement.
Write down your measurement plan.
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M EASUREMENT
Make a video of the ball being tossed. Make sure you can see the ball in every frame of the video.
Digitize the position of the ball in enough frames of the video so that you have the sufficient data to accomplish your analysis. Make sure you set the scale for the axes of your graph so that you can see the data points as you take them. Use your measurements of total distance the ball travels and total time to determine the maximum and minimum value for each axis before taking data.
A NALYSIS
Choose a function to represent the horizontal position versus time graph and another for the vertical position graph. How can you estimate the values of the constants of the functions from the graph? You can waste a lot of time if you just try to guess the constants. What kinematic quantities do these constants represent?
Choose a function to represent the velocity versus time graph for each component of the velocity. How can you calculate the values of the constants of these function from the functions representing the position versus time graphs? Check how well this works. You can also estimate the values of the constants from the graph. Just trying to guess the constants can waste a lot of your time. What kinematic quantities do these constants represent? Determine the launch velocity of the ball from this graph. Is this value reasonable? Determine the velocity of the ball at its highest point. Is this value reasonable?
From the velocity-versus-time graph determine the acceleration of the ball independently for each component of the motion. Use the functions representing the velocity-versus-time graph for each component to calculate the acceleration of the ball as a function of time for each component. Is the acceleration constant from just after launch to just before the ball is caught? Determine the magnitude of the ball’s acceleration at its highest point. Is this value reasonable?
C ONCLUSION
Did your measurements agree with your initial predictions? Why or why not? Did your measurements agree with those taken by other groups? Why or why not? What are the limitations on the accuracy of your measurements and analysis?
How do the horizontal components of a juggled ball's velocity and position depend on time?
How do the vertical components of a juggled ball's velocity and position depend on time? State your results in the most general terms supported by your analysis. At what position does the ball have the minimum velocity? Maximum velocity?
47

You have a summer job working for NASA to design a low cost landing system for a Mars mission. The payload will be surrounded by a big padded ball and dropped onto the surface. When it reaches the surface, it will simply bounce. The height and the distance of the bounce will get smaller with each bounce so that it finally comes to rest on the surface. Your task is to determine how the ratio of the horizontal distance covered by two successive bounces depends on the ratio of the heights of each bounce and the ratio of the horizontal components of the velocity of each bounce. After making the calculation you decide to check it in your laboratory on Earth.
?
How does the ratio of the horizontal distance covered by two successive bounces depend on the ratio of the heights of each bounce and the ratio of the horizontal components of the velocity of each bounce?
E QUIPMENT
For this problem, you will have a ball, a stopwatch, a meter stick, and a computer with a video camera and an analysis application .
P REDICTION
Calculate the ratio of the horizontal distance of two successive bounces if you know the ratio of the heights of the bounces and the ratio of the horizontal components of the initial velocity of each bounce.
Be sure to state your assumptions so your boss can tell if they are reasonable for the Mars mission.
M ETHOD Q UESTIONS
The following questions should help you make the prediction.
1. Draw a good picture of the situation including the velocity and acceleration vectors at all relevant times. Decide on a coordinate system to use. Make sure you define the positive and negative directions. During what time interval does the ball have a motion that is easiest to calculate? Is the acceleration of the ball during that time interval constant or is it changing?
Why? What is the relationship between the acceleration of the ball before and after a bounce?
Are the time intervals for two successive bounces equal? Why or why not? Clearly label the horizontal distances and the heights for each of those time intervals. What reasonable assumptions will you probably need to make to solve this problem? How will you check these assumptions with your data?
2. Write down all of the kinematic equations that apply to the time intervals you selected under the assumptions you have made. Make sure you clearly distinguish the equations describing the horizontal motion and those describing the vertical motion. These equations are the tools
48

you will use to solve the problem. Start the problem by calculating the quantity you wish to find: the horizontal distance covered between two successive bounces.
3. Write down an equation that gives the horizontal distance that the ball travels during the time between the first and second bounce. For that time interval, select an equation that gives the horizontal distance that the ball travels between bounces as a function of the initial horizontal velocity of the ball, its horizontal acceleration, and the time between bounces.
4. The only additional unknown in your equation is the time interval between bounces. You can determine it from the vertical motion of the ball. Select an equation that gives the change of vertical position of the ball between the first and second bounce as a function of the initial vertical velocity of the ball, its vertical acceleration, and the time between bounces.
5. An additional unknown, the ball’s initial vertical velocity, has been introduced. Determine it from another equation giving the height that the ball bounces as a function of the initial vertical velocity of the ball, its vertical acceleration, and the time from the bounce until it reaches that position. How is that time interval related to the time between bounces? Show that this is true.
6. Combining the previous steps gives you an equation for the horizontal distance of a bounce in terms of the ball’s horizontal velocity, the height of the bounce, and the vertical acceleration of the ball.
7. Repeat the above process for the next bounce and take the ratio of horizontal distances to get your prediction.
E XPLORATION
Review your lab journal from any previous problem requiring analyzing a video of a falling ball.
Position the camera and adjust it for optimal performance. Make sure everyone in your group gets
the chance to operate the camera and the computer.
Practice bouncing the ball without spin until you can get at least two full bounces to fill the video screen
(or at least the undistorted part of the video screen). Three is better so you can check your results. It will take practice and skill to get a good set of bounces. Everyone in the group should try to determine who is best at throwing the ball.
Determine how much time it takes for the ball to have the number of bounces you will video and estimate the number of video points you will get in that time. Is that enough points to make the measurement?
Adjust the camera position and screen size to give you enough data points without dropping too many.
Although the ball is the best item to use to calibrate the video, the image quality due to its motion might make this difficult. Instead, you might need to place an object of known length in the plane of motion of the ball, near the center of the ba ll’s trajectory, for calibration purposes. Where you place your reference object does make a difference to your results. Determine the best place to put the reference object for calibration.
49

Step through the video and determine which part of the ball is easiest to consistently determine. When the ball moves rapidly you may see two images of the ball due to the interlaced scan of a TV camera. You should only use one of the images.
Write down your measurement plan.
M EASUREMENT
Make a video of the ball being tossed. Make sure you can see the ball in every frame of the video.
Digitize the position of the ball in enough frames of the video so that you have the sufficient data to accomplish your analysis. Make sure you set the scale for the axes of your graph so that you can see the data points as you take them. Use your measurements of total distance the ball travels and total time to determine the maximum and minimum value for each axis before taking data.
A NALYSIS
Analyze the video to get the horizontal distance of two successive bounces, the height of the two bounces, and the horizontal components of the ball’s velocity for each bounce. You will probably want to calibrate the video independently for each bounce so you can begin your time as close as possible to when the ball leaves the ground. The point where the bounce occurs will usually not correspond to a video frame taken by the camera so some estimation is necessary to determine this position.
Choose a function to represent the horizontal position-versus -time graph and another for the vertical position graph for the first bounce. How can you estimate the values of the constants of the functions?
You can waste a lot of time if you just try to guess the constants. What kinematic quantities do these constants represent? How can you tell where the bounce occurred from each graph? Determine the height and horizontal distance for the first bounce.
Choose a function to represent the velocity-versus-time graph for each component of the velocity for the first bounce. How can you calculate the values of the constants of these functions from the functions representing the position-versus-time graphs? Check how well this works. You can also estimate the values of the constants from the graph. Just trying to guess the constants can waste a lot of your time.
What kinematic quantities do these constants represent? How can you tell where the bounce occurred from each graph? Determine the initial horizontal velocity of the ball for the first bounce. What is the horizontal and vertical acceleration of the ball between bounces? Does this agree with your expectations?
Repeat this analysis for the second bounce.
What kinematic quantities are approximately the same for each bounce? How does that simplify your prediction equation?
50

C ONCLUSION
How do your graphs compare to your predictions and method questions? What are the limitations on the accuracy of your measurements and analysis?
Will the ratio you calculated be the same on Mars as on Earth? Why?
What additional kinematic quantity, whose value you know, can be determined with the data you have taken to give you some indication of the precision of your measurement? How close is this quantity to its known value?
51

Name and ID#: ______________________________________________________
Date performed: ________________ Day/Time section meets: ______________
Lab Partners' Names: ________________________________________________
__________________________________________________________________
__________________________________________________________________
Problem # and Title: _________________________________________________
Lab Instructor's Initials: ____________
Grading Checklist Points
LABORATORY JOURNAL:
PREDICTIONS
(individual predictions and methods questions completed in journal before each lab session)
LAB PROCEDURE
(measurement plan recorded in journal, tables and graphs made in journal as data is collected, observations written in journal)
PROBLEM REPORT:*
ORGANIZATION
(clear and readable; logical progression from problem statement through conclusions; pictures provided where necessary; correct grammar and spelling; section headings provided; physics stated correctly)
DATA AND DATA TABLES
(clear and readable; units and assigned uncertainties clearly stated)
RESULTS
(results clearly indicated; correct, logical, and well-organized calculations with uncertainties indicated; scales, labels and uncertainties on graphs; physics stated correctly)
CONCLUSIONS
(comparison to prediction & theory discussed with physics stated correctly ; possible sources of uncertainties identified; attention called to experimental problems)
TOTAL(incorrect or missing statement of physics will result in a maximum of
60% of the total points achieved; incorrect grammar or spelling will result in a maximum of 70% of the total points achieved)
BONUS POINTS FOR TEAMWORK
(as specified by course policy)
* An "R" in the points column means to rewrite that section only and return it to your lab instructor within two days of the return of the report to you.
52

This laboratory will allow you to investigate the effect of specific interactions (forces) on the motion of objects. In the first problem, you will investigate the effect of forces on a sliding object. The second problem illustrates the application of the force concept and, in particular, the vector nature of forces, to a situation in which nothing is moving. The last three problems investigate the behavior of a specific interaction (in this case, the frictional force).
BJECTIVES
After successfully completing this laboratory, you should be able to:
• Make and test quantitative predictions about the relationship of forces on objects and the motion of those objects for real systems.
• Use forces as vector quantities.
• Characterize the behavior of the friction force.
• Improve your problem solving skills.
REPARATION
Read YOUNG AND FREEDMAN: Chapter 3 and Chapter 5. Review the motion of a cart moving down a ramp.
Before coming to lab you should be able to:
• Define and use sine, cosine and tangent for a right triangle.
• Recognize the difference between mass and weight.
• Determine the net force on an object from its acceleration.
• Draw and use force diagrams.
• Resolve force vectors into components and determine the total force from the components.
• Explain what is meant by saying a system is in "equilibrium."
• Write down the force law for a frictional force.
53

You have a summer job with a research group studying the ecology of a rain forest in South America.
To avoid walking on the delicate rain forest floor, the team members walk along a rope walkway that the local inhabitants have strung from tree to tree through the forest canopy. Your supervisor is concerned about the maximum amount of equipment each team member should carry to safely walk from tree to tree. If the walkway sags too much, the team member could be in danger, not to mention possible damage to the rain forest floor. You are assigned to set the load standards.
Each end of the rope supporting the walkway goes over a branch and then is attached to a large weight hanging down. You need to determine how the sag of the walkway is related to the mass of a team member plus equipment when they are at the center of the walkway between two trees. To check your calculation, you decide to model the situation using the equipment shown below.
?
How does the vertical displacement of an object suspended on a string halfway between two branches, depend on the mass of that object?
E QUIPMENT
The system consists of a central object, B, suspended halfway between two pulleys by a string. The whole system is in equilibrium. The picture below is similar to the situation with which you will work. The objects A and C, which have the same mass (m), allow you to determine the force exerted on the central object by the string.
You do need to make some assumptions about what you can neglect. For this investigation, you will also need a meter stick, two pulley clamps, three mass hangers and a mass set to vary the mass of objects.
P REDICTION
Calculate the change in the vertical displacement of the central object B as you increase its mass (M). You should obtain an equation that predicts how the vertical displacement of central object B depends on its mass (M), the mass (m) of objects A and C, and the horizontal distance (L) between the two pulleys.
Use your equation to make a graph of the vertical displacement of object B as a function of its mass (M).
54

M ETHOD Q UESTIONS
To solve this problem it is useful to have an organized problem-solving strategy such as the one outlined in the following questions. You should use a technique similar to that used in Problem 1 (where a more detailed set of Method Questions is given) to solve this problem. You might also find the Problem Solving section 4-6 of your textbook is useful, see also Problems 4-6, 4-7.
1. Draw a sketch similar to the one in the Equipment section. Draw vectors that represent the forces on objects A, B, C, and point P. Use trigonometry to show how the vertical displacement of object B is related to the horizontal distance between the two pulleys and the angle that the string between the two pulleys sags below the horizontal.
2. The "known" (measurable) quantities in this problem are L, m and M; the unknown quantity is the vertical displacement of object B.
3. Use Newton's laws to solve this problem. Write down the acceleration for each object. Draw separate force diagrams for objects A, B, C and for point P (if you need help, see your text).
What assumptions are you making?
Which angles between your force vectors and your horizontal coordinate axis are the same as the angle between the strings and the horizontal?
4.
For each force diagram, write Newton's second law along each coordinate axis.
5.
Solve your equations to predict how the vertical displacement of object B depends on its mass (M), the mass (m) of objects A and C, and the horizontal distance between the two pulleys (L). Use this resulting equation to make a graph of how the vertical displacement changes as a function of the mass of object B.
6.
From your resulting equation, analyze what is the limit of mass (M) of object B corresponding to the fixed mass (m) of object A and C. What will happen if M>2m?
E XPLORATION
Start with just the string suspended between the pulleys (no central object), so that the string looks horizontal. Attach a central object and observe how the string sags. Decide on the origin from which you will measure the vertical position of the object.
Try changing the mass of objects A and C (keep them equal for the measurements but you will want to explore the case where they are not equal).
Do the pulleys behave in a frictionless way for the entire range of weights you will use? How can you determine if the assumption of frictionless pulleys is a good one?
Add mass to the central object to decide what increments of mass will give a good range of values for the measurement. Decide how measurements you will need to make.
55

M EASUREMENT
Measure the vertical position of the central object as you increase its mass. Make a table and record your measurements.
A NALYSIS
Make a graph of the vertical displacement of the central object as a function of its mass based on your measurements. On the same graph, plot your predicted equation.
Where do the two curves match? Where do the two curves start to diverge from one another? What does this tell you about the system?
What are the limitations on the accuracy of your measurements and analysis?
C ONCLUSION
What will you report to your supervisor? How does the vertical displacement of an object suspended on a string between two pulleys depend on the mass of that object? Did your measurements of the vertical displacement of object B agree with your initial predictions? If not, why? State your result in the most general terms supported by your analysis.
What information would you need to apply your calculation to the walkway through the rain forest?
Estimate reasonable values for the information you need, and solve the problem for the walkway over the rain forest.
56

You are a volunteer in the city’s children’s summer program. One suggested activity is for the children to build and race model cars along a level surface. To ensure that each car has a fair start, your co-worker recommends a special launcher be built. The launcher uses a string attached to the car at one end and, after passing over a pulley, the other end of the string is tied to a block hanging straight down. The car starts from rest and the block is allowed to fall launching the car along the track. After the block hits the ground, the string no longer exerts a force on the car and the car continues moving along the track. You want to know how the launch speed of the car depends on the parameters of the system that you can adjust. You decide to calculate how the launch velocity of the car depends on the mass of the car, the mass of the block, and the distance the block falls.
?
What is the velocity of the car after being pulled for a known distance?
E QUIPMENT
Released from rest, a cart is pulled along a level track by a hanging object as shown below:
Car t Tr ack
A
You will be able to vary the mass of Object A (the “mass hanger”) and the Cart. They are connected together using a light string. Object A falls through a distance which is significantly shorter than the length of track. You will have a meter stick, a stopwatch, a mass hanger, a mass set, cart masses, a pulley, a pulley clamp, one piece of string and the video analysis equipment. In this experiment we ignore the friction between the cart and the track.
P REDICTION
Calc ulate the cart’s velocity after object A has hit the floor, as a function of the mass of the object A, the mass of the cart, and the distance object A falls.
Make a graph of the cart’s velocity after object A has hit the floor, as a function of the mass of object
A for the same cart mass and height through which object A falls.
57

Make a graph of the cart’s velocity after object A has hit the floor, as a function of the mass of the cart for the same mass of object A and height through which object A falls.
Make a graph of the cart’s velocity after object A has hit the floor, as a function of the distance object A falls for the same cart mass and mass of object A.
Calculate the frictional force on the cart as a function of the mass of object A, the mass of the cart, and the acceleration of the cart while it is being pulled by the string. In the same way calculate the frictional force on the cart after object A hits the floor and the string is no longer pulling on the cart.
Make an educated guess about the relationship between the frictional forces in the two situations.
M ETHOD Q UESTIONS
To test your prediction, you must determine how to calculate the velocity from the quantities you can measure in this problem. It is useful to have an organized problem-solving strategy such as the one outlined in the following questions. You might also find the Problem Solving techniques in the Competent Problem Solver useful.
1. Make a sketch of the problem situation similar to the one in the Equipment section. Draw vectors that show the direction and relative magnitude of the motion of the objects (velocity and acceleration) at the interesting times in the problem: when the cart starts from rest, just before the object A hits the floor, and just after object A hits the floor. Draw vectors to show all of the forces on object A and the cart. Assign appropriate symbols to all of the quantities describing the motion and the forces. If two quantities have the same magnitude, use the same symbol but write down your justification for doing so. For example, the cart and object A have the same magnitude of velocity when the cart is pulled by the string. Explain why. Decide on your coordinate system and draw it.
2. The "known" quantities in this problem are the mass of object A, the mass of the cart, and the height above the floor where object A is released. Assign these quantities symbols so that you can use them in algebra. The unknown quantities are the velocity of the cart and of object A just before object A hits the floor. There are other unknowns as well. List them. What is the relationship between what you really want to know (the velocity of the cart after object A hits the floor) and what you can calculate (the velocity of the cart just before object A hits the floor)?
3. Write down what principles of Physics you will use to solve the problem. Because forces determine the motion of the cart, using Newton's 2nd Law to relate the sum of the forces on each object to its motion is a good bet. Since you need t o determine forces, Newton’s 3rd Law is probably also useful. Will you need any of the principles of kinematics? Write down any assumptions you have made which are necessary to solve the problem and justified by the physical situation.
4. Draw separate free-body diagrams for object A and for the cart after they start accelerating (if you need help, see section 4-6 and section 4-7 in chapter 4 of your text book). Check to see if any of these forces are related by Newton’s 3rd Law (Third Law Pairs). For easy reference, it is
58

useful to draw the acceleration vector for the object next to its free-body diagram. It is also useful to put the force vectors on a separate coordinate system for each object (force diagram).
Remember the origin (tail) of all vectors is the origin of the coordinate system. For each force diagram (one for the cart and one for object A), write down Newton's 2nd law along each axis of the coordinate system. It is important to make sure that all of your signs are correct. For example, if the acceleration of the cart is in the + direction, is the acceleration of object A + or -?
Your answer will depend on how you define your coordinate system.
5. Since you are interested in the velocity of the cart and the distance object A falls but Newton’s
2nd Law gives you an acceleration, write down any kinematic equations which are appropriate to this situation. You will have to decide if the acceleration of each object is constant or varies during the time interval for which your calculation is valid. What is that time interval?
6. You are now ready to plan the mathematics of your solution. Write down an equation, from those you have collected in steps 4 and 5 above, which relates what you want to know (the velocity of the cart just before object A hits the ground) to a quantity you either know or can find out (the acceleration of the cart and the time from the start until just before object A hits the floor). Now you have two new unknowns (acceleration and time). Choose one of these unknowns (for example, time) and write down a new equation (again from those collected in steps 4 and 5) which relates it to another quantity you either know or can find out (distance object A falls). If you have generated no additional unknowns, go back to determine the other original unknown (acceleration). Write down a new equation that relates the acceleration of the cart to other quantities you either know or can find (forces on the cart). Continue this process until you generate no new unknowns. At that time you should have as many equations as unknowns.
You can now solve your mathematics to give the prediction.
E XPLORATION
Adjust the length of the string such that object A hits the floor well before the cart runs out of track.
You will be analyzing a video of the cart after object A has hit the floor. Adjust the string length to give you a video that is long enough to allow you to analyze several frames of motion.
Choose a mass for the cart and find a range of masses for object A that allows the cart to achieve a reliably measurable velocity before object A hits the floor. Practice catching the cart before it hits the end stop on the track. Make sure that the assumptions for your prediction are good for the situation in which you are making the measurement. Use your prediction to determine if your choice of masses will allow you to measure the effect that you are looking for. If not, choose different masses.
Choose a mass for object A and find a range of masses for the cart that allows the cart to achieve a reliably measurable velocity before object A hits the floor. Practice catching the cart before it hits the end stop on the track. Make sure that the assumptions for your prediction are good for the situation in which you are making the measurement. Use your prediction to determine if your
59

choice of masses will allow you to measure the effect that you are looking for. If not, choose different masses.
Now choose a mass for object A and one for the cart and find a range of distances for object A to fall that allows the cart to achieve a reliably measurable velocity before object A hits the floor.
Practice catching the cart before it hits the end stop on the track. Make sure that the assumptions for your prediction are good for the situation in which you are making the measurement. Use your prediction to determine if your choice of distances will allow you to measure the effect that you are looking for. If not, choose different distances.
Write down your measurement plan.
M EASUREMENT
Carry out the measurement plan you determined in the Exploration section.
Complete the entire analysis of one case before making videos and measurements of the next case.
A different person should operate the computer for each case.
Make sure you measure and record the mass of the cart and object A (with uncertainties). Record the height through which object A (the mass hanger) falls and the time it takes.
Take a video that will allow you to analyze the data after the mass hanger has hit the floor, but before the cart has run out of track.
A NALYSIS
Ensure a different person in your group operates the computer for each case you are analyzing.
Determine the cart's velocity just after object A hits the floor from your video.
What are the limitations on the accuracy of your measurements and analysis?
C ONCLUSION
How do the predicted velocity and the measured velocity compare in each case? Did your measurements agree with your initial prediction? If not, why?
Does the launch velocity of the car depend on its mass? The mass of the block? The distance the block falls? Is there a choice of distance and block mass for which the mass of the car does not make much difference to its launch velocity?
If the same mass block falls through the same distance, but you change the mass of the cart, does the force that the string exerts on the cart change? In other words, is the force of the string on object A always equal to the weight of object A? Is it ever equal to the weight of object A?
60

Explain your reasoning. Devise a test of your prediction using the same equipment used in this problem. Try it if you have time.
Was the frictional force the same whether or not the string exerted a force on it? Does this agree with your initial prediction? If not, why?
Do you need to repeat the measurement to get a definitive result? Determine how many times and do it.
Do you need to change the conditions of the measurement to get a definitive result? Do it.
61

?
You are working for a company that contracts to test the mechanical properties of different materials of systems. One of the customers wants your group to determine the coefficient of kinetic friction for wood on aluminum.
You decide to measure the coefficient of kinetic friction by graphing the frictional force as a function of the normal force using a wooden block sliding on an aluminum track. The coefficient of kinetic friction is the slope of that graph. Of course, there is measurement uncertainty no matter how you do the measurement. For this reason, you decide to vary the normal force in two different ways to see if you get consistent results.
What is the coefficient of kinetic friction for wood on aluminum?
E QUIPMENT
A wooden block slides down an aluminum track, as shown below.
The tilt of the aluminum track with respect to the horizontal can be adjusted. You can change the mass of the wooden block by attaching additional mass on it. Video analysis equipment will allow you to determine the acceleration of the wooden block sliding down the aluminum track. Two wooden blocks, a meter stick, a stopwatch and a mass set are available for this experiment.
62

P REDICTIONS
The coefficient of kinetic friction is the slope of the graph of the kinetic frictional force as a function of normal force. Using the Table of Coefficients of Friction on page 25, sketch a graph of the kinetic frictional force on the block against the normal force on the block.
Explain your reasoning.
M ETHOD Q UESTIONS
To test your prediction you must determine how to calculate the normal force and the kinetic frictional force from the quantities you can measure in this problem. You do not need to know much about friction to make this prediction. It is useful to have an organized problem-solving strategy such as the one outlined in the following questions. You should use a technique similar to that used in Problem 1 (where a more detailed set of Method Questions are given) to solve this problem.
1. Make a drawing of the problem situation similar to the one in the Equipment section. Draw vectors to represent all quantities that describe the motion of the block and the forces on it.
What measurements can you make with a meter stick to determine the angle of incline?
Choose a coordinate system. What is the reason for using the coordinate system you picked?
2. What measurements can you make to enable you to calculate the kinetic frictional force on the block? What measurements can you make to enable you to calculate the normal force on the block? Do you expect the kinetic frictional force on the wooden block to increase, decrease, or
stay the same as the normal force on the wooden block increases? Explain your reasoning.
3. Draw a free-body diagram of the wooden block as it slides down the aluminum track. Draw the acceleration vector for the block near the free-body diagram. Transfer the force vectors to your coordinate system. What angles between your force vectors and your coordinate axes are the same as the angle between the aluminum track and the table? Determine all of the angles between the force vectors and the coordinate axes.
4. Write down Newton’s 2nd Law along each coordinate axis.
5. Using the equations in step 4, determine an equation for the kinetic frictional force in terms of quantities you can measure. Next determine an equation for the normal force in terms of quantities you can measure. In our experiment, the measurable quantities include the mass of the block, the angle of incline and the acceleration of the cart.
E XPLORATION
Find an angle at which the wooden block accelerates smoothly down the aluminum track. Try this when the wooden block has different masses on top of it.
63

Select an angle and series of masses that will make your measurements most reliable.
Find a mass for which the wooden block accelerates smoothly down the aluminum track. Try this several different angles of the aluminum track. Try different block masses.
Select a block mass that gives you the greatest range of track angles for reliable measurements.
M EASUREMENT
Keeping the aluminum track at the same angle, take a video of the wooden block's motion. Keep the track fixed when block is sliding down. Make sure you measure and record that angle. You will
need it later.
Repeat this procedure for different block masses to change the normal force. Make sure the block moves smoothly down the incline for each new mass. Make sure every time you use the same surface of the block to contact the track.
Collect enough data to convince yourself and others of your conclusion about how the kinetic frictional force on the wooden block depends on the normal force on the wooden block.
Keeping the wooden block's mass fixed, take a video of its motion. Keep the track fixed when block is sliding down. Make sure you measure and record each angle.
Repeat this procedure for different track angles. Make sure the block moves smoothly down the incline for each new angle. Make sure every time you use the same surface of the block to contact the track.
Collect enough data to convince yourself and others of your conclusion about how the kinetic frictional force on the wooden block depends on the normal force on the wooden block.
A NALYSIS
For each new block mass and video, calculate the magnitude of the kinetic frictional force from the acceleration. Also determine the normal force on the block.
Graph the magnitude of the kinetic frictional force against the magnitude of the normal force, for a constant angle of incline. On the same graph, show your predicted relationship.
For each angle and video, calculate the magnitude of the kinetic frictional force from the acceleration. Also determine the normal force on the block.
Graph the magnitude of the kinetic frictional force against the magnitude of the normal force for a constant block mass. On the same graph, show your predicted relationship.
C ONCLUSION
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What is the coefficient of kinetic friction for wood on aluminum? How does this compare to your prediction based on the table?
What are the limitations on the accuracy of your measurements and analysis? Over what range of values does the measured graph match the predicted graph best? Where do the two curves start to diverge from one another? What does this tell you?
If available, compare your value of the coefficient of kinetic friction (with uncertainty) with the value obtained by the different procedure given in the next problem. Are the values consistent?
Which way of varying the normal force to measure the coefficient of friction do you think is better? Why?
65

Surfaces
Steel on steel
Aluminum on steel
Copper on steel
Steel on lead
Copper on cast iron
Copper on glass
Wood on wood
Glass on glass
Metal on metal (lubricated)
Teflon on Teflon
Rubber on concrete
Ice on ice
Wood on Aluminum
* All values are approximate.
 s
0.74
0.61
0.53
0.9
1.1
0.7
0.25 - 0.5
0.94
0.15
0.04
1.0
0.1
 k
0.57
0.47
0.36
0.4
0.07
0.04
0.8
0.9
0.3
0.5
0.2
0.03
0.25-0.3
66

Name and ID#: ______________________________________________________
Date performed: ________________ Day/Time section meets: ______________
Lab Partners' Names: ________________________________________________
__________________________________________________________________
__________________________________________________________________
Problem # and Title: _________________________________________________
Lab Instructor's Initials: ____________
Grading Checklist Points
LABORATORY JOURNAL:
PREDICTIONS
(individual predictions and methods questions completed in journal before each lab session)
LAB PROCEDURE
(measurement plan recorded in journal, tables and graphs made in journal as data is collected, observations written in journal)
PROBLEM REPORT:*
ORGANIZATION
(clear and readable; logical progression from problem statement through conclusions; pictures provided where necessary; correct grammar and spelling; section headings provided; physics stated correctly)
DATA AND DATA TABLES
(clear and readable; units and assigned uncertainties clearly stated)
RESULTS
(results clearly indicated; correct, logical, and well-organized calculations with uncertainties indicated; scales, labels and uncertainties on graphs; physics stated correctly)
CONCLUSIONS
(comparison to prediction & theory discussed with physics stated correctly ; possible sources of uncertainties identified; attention called to experimental problems)
TOTAL(incorrect or missing statement of physics will result in a maximum of
60% of the total points achieved; incorrect grammar or spelling will result in a maximum of 70% of the total points achieved)
BONUS POINTS FOR TEAMWORK
(as specified by course policy)
* An "R" in the points column means to rewrite that section only and return it to your lab instructor within two days of the return of the report to you.
68

In this lab you will analyze the motion of objects in two dimensions. It is important that every member of your group learn how to use the computer and software. Be sure that you all take turns and that, when you are the operator, your partners understand what you are doing.
Stop and Think – What is a digital image?
Before trying to capture a video in the computer, let’s be sure that we understand what the moving image on a television screen actually consists of. It is a series of individual stationary, or still, pictures, appearing at a rate of 30 pictures, or frames , a second. Standard movie films consist of strips of such still pictures, recorded photographically at a slightly lower frame rate.
Digital computers, and hence our electronic imaging systems, work only in quantized units. Your camera divides the area that it sees into over 76,000 elements, arranged into a grid 320 cells wide by 240 cells high. You will see later that LoggerPro different movies have different grids. In either case, a 1-step by
1-step cell is called a pixel
, for “picture element.”
To determine the coordinates of an object in an image frame using LoggerPro, we just position the cursor over the object, and the system will count the pixel rows and columns to get to the location of the cursor, starting at the lower left of the image on the screen. Then, if we know how many pixels correspond to a meter long object, we can calculate real distances from the pixel counts. This is called “scaling the image.” We will return to this shortly.
With real distances known from scaling the images, and with the time interval between two chosen frames also known (a multiple of 1/10 sec, say), we can easily calculate such things as velocities and accelerations of moving objects in our videos.
Note that the system reports locations rounded to the nearest integer pixel count. As a result, no position measurement can be known more accurately than within ± 0.5 pixel spacing, or to within ± 1/2 of whatever distance is equivalent to one pixel in real units
(mm, meters, etc.).
In this lab you will analyze the trajectory of a bouncing ball. The ball will be accelerated by gravity in the – y direction, but not in the x direction.
Assuming that only gravitational forces are significant, the ball's motion should obey the equations x = x
0
+ v
0x t (Eqn. 1)
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y = y
0
+ v
0y t + 1/2 gt
2
. (Eqn. 2)
In this experiment, a ball is launched from a spring powered projectile launcher. After it leaves the launcher it falls freely, The camera takes a series of pictures of the trajectory of the ball when the ball is moving freely. By measuring the position of the ball in each picture, you can test Equations 1 and 2.
Open up LoggerPro and select the Insert Menu and choose Movie. Open up the folder 2D Movies and insert the movie: Plastic Ball at 45 degrees.
Collect data for the position of the ball through the entire film. Now stop and look at LoggerPro's Table
Window . It shows the time of each frame of your movie, and records the x and y pixel locations of each position that you identified by clicking on a frame. The three data columns are similar to those in a spread sheet program, but don't allow any manipulation of their contents. (Later on, we will copy and paste the data to and Excel spread sheet and do some calculations.)
If you make a mistake in positioning your mouse in a particular frame, you may want to correct the position of the point you chose. You can move the old point to a new location by selecting the select point button and dragging the point you want to correct.
The Initial Physics Discussion
Print out and think about the following graphs.
X position vs time, Y position vs. Time, X Velocity vs. Time, Y velocity vs. Time.
Each of you should have your own copy of the group's printouts, and note your own conclusions on your own plots .
1. Do the plots of X position, velocity, and acceleration all confirm that the ball is subject to no horizontal force?
2. Does a plot of Y position vs. time look like you expected? (What is that?)
3. What does a plot of Y velocity vs. time look like? What does that show?
4. What about a plot of Y acceleration vs. time?
You should know that we expect to see a constant, negative, acceleration of the ball in the y direction.
Numerically, we expect that the acceleration of the golf ball will be g, which is approximately equal to 9.80 m/sec
2
. But we have yet to scale our movie, in order to find the conversion factor between pixels and meters.
Before we scale the movie, let's look at our results in terms of pixels, and take a look at how accurate those results seem to be.
Now for some more quantitative analysis. First, let's fit a quadratic curve to y as a function of time. The computer can show us the quadratic function that "comes closest" to passing through all of the data points on our y vs. t plot.
70

After you make the fit you will notice At the top of the graph is the algebraic function resulting from the plotted fit. ( For now you should ignore the computer's RMSE (Root mean squared error) parameter. It is simply one of several statistical measure of the goodness of the fit, and LoggerPro often calculates it incorrectly.)
The number multiplying the squared term in the fit corresponds to the " 1/2 g " factor in Equation 2. If we multiply our number by 2, that gives our experimental value for g , in units of pixels / sec
2
.
Before moving on to scale the movie, and get g in m/sec
2
, let's look at how closely our data follows the computer's "best fit" curve.
Zooming in LoggerPro; Looking at Data "Jitter"
Maximize your plot of y vs. t . Probably all of the points lie close to the curve, but how close? "Magnify" a region of your LoggerPro plot, by "zooming in" on it with the following steps:
1. Hold down on the mouse button and use the cursor to draw a rectangular box around a region of interest containing one of your data points. The graph then shows only that region, in a magnified view. Now click on the Zoom In button which looks like a magnifying glass with a + inside it.
2. If you want to magnify some more, do it again. (But if you magnify too much, the fit line may disappear, and you'll have to start over by returning to the unmagnified graph.)
3. When you want to return to your original unmagnified view, just click onthe Autoscale button .
Use the zooming feature to look at several points of your graph, and to judge how far the points are vertically away from the best-fit curve. For now, it is easier to use the cursor readout at the lower left corner of the graph to estimate the differences between your data point and the nearby curve.]
After picking 5 or so points, type your data into excel and use it to calculate the Root Mean Square Error.
How does this compare to the value given by LoggerPro? How does it compare to other lab groups?
The average deviation which you have calculated is an estimate for how much an individual measurement varies from the "true" value indicated by the curve. It includes the random ± 0.5 pixel "quantization error," plus any inaccuracies you introduced in picking off your points.
Before we convert pixels to meters and get our final physical result for the value of g and its uncertainty, let's review some of the concepts we have used.
1. Things are accelerated by gravity in the vertical direction, but not horizontally.
2. Constant acceleration leads to a quadratic variation in position.
3. An electronic camera can measure positions in pixels, which have later to be converted to meters. The integer-pixel position measurement leads to an uncertainty of ± 0.5 pixel in our knowledge of the true position.
4. Every individual measurement is inexact, and we need to understand what a typical uncertainty in a measured quantity might be.
5. Our result for the value of g (in pixels/sec
2
) is inexact, because the measurements are inexact.
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Our focus on the deviations of measured points from the fitted line assumes that the fit itself is our best measure of the true function, and that the deviations are due to random jitter in the individual measurements. Our manual study of the typical deviations, and the computer's calculation of RMSE, are attempts to characterize the range of random variations (including pixel quantization, mouse vibration, etc.).
These are examples of random errors, which are presumed to exist independently at each measurement.
Other errors affect all of your measurements at once. They are called systematic errors. An example occurs when you convert pixels to meters in your video. The conversion factor clearly affects all of your points at once.
We now have to determine how many pixels in our picture correspond to one meter in the real world.. We will do this by measuring how long a meter stick was in our video, as measured in pixels. The resulting value for our conversion factor (pixels per meter, or the inverse, meters per pixel) will allow us to convert any measurement from pixels to meters (or pixels/secs to meters/sec 2 ).
Our conversion factor will have some error, which we can estimate based on the pixel quantization error, plus a "guesstimate" as to how reproducibly we can position our cursor on the ends of the meter stick.
Once we know the conversion factor, and its estimated error, we can use it to convert all of our measured x and y positions to meters from the origin, rather than pixels. But the conversions will all involve the factor, which has some error associated. This is a systematic error, since it affects all measurements equally.
First, maximize the size of your movie, so that it fills the entire screen. Now click the Set Active
Point Button. This will create another data set. Choose any frame in which the meter stick is clearly visible.
Move the cursor manually to either end of the meter stick, and click to create a point. Note the pixel coordinates at the two positions, from the table. Calculate the distance between the two points, in pixels.
Using the distance in pixels, calculate your movie’s scale factor, in units of pixels per meter. Use this to convert your value for the acceleration of the ball from pixels/sec 2 to m/sec 2 . Note the result in your lab book .
Is your answer close to the expected value of g ? What effects might systematically bias your answer away from 9.8 m/s
2
?
Now let's consider what is the accuracy with which your measurements determine a value for g . If the uncertainty in g is called ∆g, then the fractional, or relative, uncertainty in g is defined as ∆g/ g .
When we want to estimate the uncertainty in our determination of g , we will have to use both the computer's estimate of the effects of random error, RMSE(a
2
), and our estimate of the uncertainty in the conversion factor from pixels to meters. For the moment, you should just add the relative uncertainties to get the relative uncertainty in g . That is,
∆g / g = RMSE(a
2
) / a
2
+ ∆F / F, where F is your scale factor. It is up to you to estimate a value for ∆F, perhaps by repeated trials. Taking responsibility for estimating the accuracy with which you make a measurement is not easy, but you have to do it. Take time to think it through!
Record your final result for g and for ∆g . Note any comments you might have on the reliability of your conclusion, and on a comparison of your result with the accepted value for g , 9.80 m/sec
2
.
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You will want to store your LoggerPro analysis file. As before, put it in your lab section’s folder.
Now we want to analyze the trajectory of a Styrofoam ball launched from the same launcher. Your analysis last week should have informed you as to the type of behavior to expect from the object in the vertical direction. .
Repeat the analysis of the Hard Plastic Ball for the Styrofoam ball using the Movie:
Styrofoam ball at 45 degrees. This time use the Set Scale function to scale your film instead of doing it manually. For this analysis you should have:
Graphs:

X position vs. time

Y position vs. time

X velocity vs. time

Y velocity vs. time
Answer the questions:
1. Do the plots of X position, velocity, and acceleration all confirm that the ball is subject to no horizontal force? What do they show?
2. Does a plot of Y position vs. time look like you expected? (What is that?)
3. What does a plot of Y velocity vs. time look like? What does that show? Does it fit with what you saw last week?
4. What about a plot of Y acceleration vs. time? What does this show?
Calculate your value for the acceleration of gravity from the quadratic fit of the y position vs. time. Now calculate the acceleration due to gravity from the slope of the y velocity vs time. What do you see?
Find the RMSE for the y position graph as you did previously and for the y velocity graph. How do these values compare. What does this mean?
Now report your final value for g with uncertainty from these two graphs. Include the random error from the
RMSE calculation and the systematic error from the Scaling calculation.
∆g / g = RMSE(a
2
) / a
2
+ ∆F / F,
Record your final result for g and for ∆g from the Styrofoam ball . Note any comments you might have on the reliability of your conclusion, and on a comparison of your result with the accepted value for g , 9.80 m/sec
2
.
Be sure that everything is entered in you lab notebook before you leave. You should plan to meet with your group sometime between now and next week to discuss any issues that might come up during the analysis.
73

Computational science involves using mathematical models and computer programs to improve our understanding of the way the world works. There are three parts to the process:
 Find a complete mathematical description of the system that will include all the behavior you are interested in.
 Construct an efficient computer code that solves these equations
 Extract results and predictions that can be compared with real world, with experiments or with other scientific theories.
These steps are repeated to refine both the computer program and our understanding of the problem.
You now should have the skills to create the more complex program that will simulate an object under the influence of general gravitation.
This laboratory introduces gravitational motion and explains with the help of examples, motion under the gravitational force. You should have completed the Vpython tutorial
Start IDLE by clicking on the snake icon on the panel at the bottom of your desktop. A window labeled
'Untitled' should appear. This is the window in which you will type your program.
Before we start the simulation we need to understand the concept of gravitation force given by Sir Isaac
Newton in 1665.
Newton was the first to show that the force that holds the Moon in its orbit is the same force that makes an apple fall. Newton concluded that not only does the Earth attract an apple and the Moon but every body in universe attracts every other body, this phenomena of bodies to move towards each other is called gravitation . Quantitatively, Newton proposed a law, which is famously called as Newton's Law of
Gravitation . This law states that every particle attracts any other particle with a gravitational force whose magnitude is given by
F=
Gm
1 m
2 r
12 r 2 here m
1
and m
2
are the mass of the particles, r = r
2
 r
1
is the distance between them, r
12
is a vector of unit length pointing from r
1
to r
2
and G is the gravitational constant, whose value is known to be
(1)
G = 6.67 ×10  11 N m 2 /kg 2
The force F acting on the two particles is opposite in direction and equal in magnitude.
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To help us visualize and make this tutorial more interesting we will simulate the flight path of Apollo 13 when subjected to the Earth's gravitational force. We will need to calculate the parking orbit (orbit of the space craft to revolve around the Earth) of Apollo 13.
For successfully implementing the simulation we will need to know the acceleration, position and velocity of Apollo 13, which can be respectively found by equation ( 1 ) and the equations of motion learnt in the projectile tutorial.
In every computer we need to specify some basic variables without which a computer program fails to run.
In this and the next section we will fill in these attributes, from visual import *
G=6.67e-11 dt=10
Here
G is the gravitational constant and dt is the time interval.
In VPython we can display various windows, but initially there is one Visual display window named scene.
Any objects we begin creating will by default go into the scene. In this program we embellish the display window by modifying the attributes with the help of the display function. scene = display(title = 'Earth and Apollo 13',width = 800, height = 600, background=(0,1,1))
Here we create a visual display window 800 by 600, with 'Earth and Apollo 13' in the title bar and with a background color of cyan filling the window. Various other attributes can also be specified. For more information visit the documentation of VPython in www.vpython.org
Now we specify the attributes concerning the earth and apollo 13 needed for the simulation. This is done as:
###################################################
# Declaration of Variables #
###################################################
#Declaring variables for earth earth= sphere(pos=(0,0,0), color=color.green) earth.velocity = vector(0,0,0) earth.mass=5.97e24 earth.radius = 6378.5e3
#Declaring variables for apollo apollo=sphere(pos=(6551e3,0,0),color=color.red) apollo.velocity=vector(0,7801,0) apollo.radius= 100e3
#Creating the trails for graphing the orbital path of Apollo apollo.trail=curve(pos=[apollo.pos],color=apollo.color)
VPython allows us to label objects in the display window with the label() command. In our program we label Earth and Apollo in the following manner.
#Labelling Earth and Apollo earthlabel = label(pos = earth.pos,text='Earth',xoffset = 10, yoffset = 10,
space = earth.radius, height = 10, border = 6) apollolabel = label(pos = apollo.pos,text='Apollo13',xoffset = 10, yoffset = 10,
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space = apollo.radius, height = 10, border = 6)
A unique feature of the label object is that several attributes are given in terms of screen pixels instead of the usual "world-space" coordinates. For example the height of the text is given in pixels, with the result the text remains readable even when the object is moved far away.
To calculate the position and velocity of Apollo 13 as it moves around the earth we will need to know the acceleration of the space ship. We can then apply this acceleration to the equations of motion learnt from the projectile tutorial.
The acceleration of an object is determined by the forces on it, which in turn depends on its position compared to other objects. For example the force of gravity between two objects in space is given by equation ( 1 ). The acceleration of the two bodies can be calculated by m apollo
a apollo
=
 a apollo
=
Gm apollo
M earth r 2
GM earth r 2
In Vpython we can calculate the acceleration of the two planets can be calculated as:
####################################################
# Loop for moving the Apollo 13 around earth #
#################################################### while (1==1):
rate(1000)
# Calculating the acceleration of the Apollo
R12 = mag(earth.pos-apollo.pos)
direction12=norm(earth.pos-apollo.pos)
apollo.acceleration=(G*earth.mass/R12**2)*direction12
Here mag gives the magnitude of the distance between the two planets and norm gives the normalised unit vector.
In our program we will need to insert this in the while loop part of the program because the acceleration of
Apollo 13 will constantly change with respect to time. The condition while (1==1): specifies the simulation to continue forever, as the condition 1==1 can never become false. We can exit the program by just closing down the display window.
We will use the equations of motion used in the projectile tutorial to determine the new velocity and position of Apollo 13 at some later point in time. This is done in the form of
#Calculating the updated position of the Apollo
apollo.velocity=apollo.velocity+apollo.acceleration*dt
apollo.pos=apollo.pos+apollo.velocity*dt
apollo.trail.append(pos=apollo.pos)
Now we have managed to complete our simulation of Apollo 13, around the earth, due to earth's gravitational force.
NOTE: We have ignored the force exerted by Apollo 13 on the earth because of its small magnitude.
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The situation for with gravitational interactions between more than two objects/planets is more complicated than that for two planets. We need to use the principle of superposition of gravitational forces
F
1
= F
12
+F
13
+F
14
+  + F
1n
(2)
Here F
1
is the net force on planet 1 and F
1n
is the force exerted on particle 1 by particle n, which is calculated by the gravitational force formula given in equation ( 1 ).
Now we will take our simulation a step further and will now try to simulate how Apollo 13 escaped from it's parking orbit and went to Moon.
After Apollo 13 was launched, it orbited the earth 1.5 times, which is roughly 2.5hrs. after being launched from earth. At this time the translunar injection(TLI) took place. The translunar injection (ignition of engines designed to escape earth) accelerates the spacecraft such that its velocity changes from 7801 m/s to
10920 m/s. The direction of velocity will be determined by the current orientation of the spacecraft. It is therefore important to initiate the TLI in precisely the correct moment to put Apollo 13 on the correct path to the moon. In the following simulation we aim to determine this moment for the TLI.
In this exercise we will simulate this TLI and Apollo 13 path to the moon and make it hit the moon.
The circular motion of the spacecraft in the parking orbit, can be parameterized as follows r(  ) = r park
( cos(  ), sin(  ),0) and v(  ) = v park
(cos(  ), sin(  ), 0) where r park
= 6551km and v park
= 7801m/s. The angle  for circular motion is given by  (t) = [(2  t)/T] where T is the period of one orbit.
(3)
Question: What is the time taken by Apollo 13 to complete one orbit around the earth?
To compute the path for the passage from the earth's parking orbit to the moon's orbit, we use the initial conditions for the position r as given in equation ( 3 ). However, for the velocity use the same direction as in equation ( 3 ) but with a higher magnitude v trans
= 10920m/s that the spacecraft has after burning the TLI engine. r(  ) = r park
(cos(  ), sin(  ),0) and v
0
(  ) = v trans
(  sin(  ), cos(  ),0) (4)
Now we will briefly touch the logical reasoning steps to the simulation:
1.
We will need to know the parameters of the moon, time taken complete one orbit and angle at which the TLI takes place.
2.
We will need to check Apollo 13 acceleration due to gravitational force from the earth and moon.
3.
We will need to keep a check on angle for the TLI, as at that specified angle the velocity and radius of the spacecraft take the form of equation ( 3 ). But please note this method of increasing the velocity of the spacecraft from 7801m/s to 10920m/s is used to make the simulation easy to understand. In reality the spacecraft gradually increases it's velocity to the specified velocity.
77

4.
We simulate the TLI with initial conditions given by equation ( 3 ) and check when to end simulation.
Let's try incorporate step 1 from the previous section i.e. to include the parameters of the moon, time taken to complete one orbit and angle at which the TLI takes place. In VPython we define these variables in the
Declaration of Variables section,
###################################################
# Declaration of Variables #
###################################################
#Time(in secs) to complete 1 revolution around earth
T = 5040
#Time(in secs) for calculating simulation time t = 0
#Angle at which Translunar Injection occurs
TLI_theta = pi
#Angle to check for Translunar Injection theta = 0
Now we include the coordinates of the moon
#Declaring variables for earth earth= sphere(pos=(0,0,0), color=color.green) earth.velocity = vector(0,0,0) earth.mass=5.97e24 earth.radius = 6378.5e3
#Declaring variables for Apollo apollo=sphere(pos=(6551e3,0,0),color=color.red) apollo.velocity=vector(0,7801,0) apollo.radius= 100e3
#Declaring variables for Moon moon=sphere(pos=(384e6,0,0),color=color.black) moon.velocity=vector(0,0,0) moon.mass=7.3480e22 moon.radius=2.238e6
Now we create the trail for Apollo 13 and label the earth and moon.
#Creating the trails for graphing the orbital path of Apollo apollo.trail=curve(pos=[apollo.pos],color=apollo.color)
#Labelling Earth and Moon earthlabel = label(pos = earth.pos,text='Earth',xoffset = 10, yoffset = 10,
space = earth.radius, height = 10, border = 6) moonlabel = label(pos = moon.pos,text='Moon',xoffset = 10, yoffset = 10,
space = moon.radius, height = 10, border = 6)
Note: We had labeled the Apollo previously, but for simplicity we will not label it anymore.
With the introduction of the moon into the simulation, the Apollo 13 experiences gravitational from earth and moon. So the net force on Apollo 13 is,
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This makes the acceleration of Apollo 13,
F apollo
= F moon
+F earth m apollo
a apollo
=
Gm apollo
M earth r
12
2
+
Gm apollo
M moon r
13
2
(5)
(6)
 a apollo
=
GM earth r
12
2
+
GM moon r
13
2
(7) where r
12
is distance between the Earth and Apollo 13 and r
13
is distance between the Moon and Apollo 13.
This part of the simulation is similar to the previous simulation where we only had the Earth and Apollo 13.
The only difference is we keep a check on the angle Apollo 13 is making and when this angle equals the angle for TLI, we switch to the TLI part of the simulation. We perform this check by using an if statement of
VPython and checking when the angle of Apollo becomes equal to angle of TLI.
Note: We have defined Apollo as object 1, the Earth as object 2 and the Moon as object 3. R
12
is the radius from Apollo to the Earth. R
13
is the radius from Apollo to the moon and so on.
####################################################
# Loop for moving the Apollo 13 around earth #
####################################################
TLI = true while (apollo.pos.x<1.2*moon.pos.x):
rate(1000)
#Updating time
t = t+dt
#Checking angle for TLI
theta = (2*pi*t)/T
# At point TLI_theta increase velocity of craft
if( (theta > TLI_theta) & TLI):
TLI = false
TLI_velocity = 10920
R_parking = 6551e3
apollo.velocity = TLI_velocity*vector(-sin(TLI_theta),cos(TLI_theta),0)
apollo.pos = (6551e3*cos(TLI_theta), 6551e3*sin(TLI_theta), 0)
# Calculating the acceleration of Apollo before TLI
R12 = mag(earth.pos-apollo.pos)
direction12=norm(earth.pos-apollo.pos)
a12 = (G*earth.mass/R12**2)*direction12
R13 = mag(moon.pos-apollo.pos)
direction13 = norm(moon.pos - apollo.pos)
a13 = (G*moon.mass/R13**2)*direction13
# Note now we calculate the acceleration of the Apollo as the sum of the acceleration from the earth and the moon.
apollo.acceleration = a12 + a13
#Calulating the updated position of the Apollo
apollo.velocity=apollo.velocity+apollo.acceleration*dt
apollo.pos=apollo.pos+apollo.velocity*dt
apollo.trail.append(pos=apollo.pos)
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The program will enter the if loop, only if the statement if((theta > TLI_theta) & TLI comes true. If this statement comes true then all the instructions under it are executed. Here we are using a logical variable
TLI to flag whether the TLI has been performed. We only want to jump into the statements in the if statement once.
The simulation is stopped (in the while statement) if the x position of the craft ever gets much larger that the x position of the moon. We intend to make the Apollo 13 loop around the moon and return to earth.
Try to calculate the perfect angle to enable the spacecraft to travel towards the moon and return, the so called free return trajectory.
We have simulated the motion of Apollo 13 around the Earth and performing a Translunar Injection to escape the Earth's force and travel towards the Moon. The equations of motion have been used to calculate the updated velocity and position of Apollo 13.
More sophisticated programs solving gravitational interactions are used in recent research to:
 test the long term stability of the solar system
 study the evolution of the solar system and
 find what range of planetary orbits were possible
 explain the distribution of orbits within the asteroid belt
 study the evolution of galaxies (Python example stars.py)
 study motion near black holes (visualisation BlackHoles.java)
For more information on the history of research into solar system motion try Newton's Clock: Chaos in the
Solar System, by Ivars Peterson
Now you should be prepared to complete the following problems
80

2 Problems
Problem 1 Planetary orbits
In this problem you will study the motion of a planet around a star. To start with a somewhat familiar situation, you will begin by modeling the motion of our Earth around our Sun. Write answers to questions either as comments in your program or on paper, as specified by your instructor.
Planning
(a) The Earth goes around the Sun in a nearly circular orbit, taking one year to go around. Using data on page 50, what initial speed should you give the Earth in a computer model, so that a circular orbit should result? (If your computer model does produce a circular orbit, you have a strong indication that your program is working properly.)
(b) Estimate an appropriate value for I'1t to use in your computer model. your
Remember that t is in seconds, and consider your answer to part (a) in making this estimate. If
 is too small, your calculation will require many tedious steps. Explain briefly how you decided on an appropriate step size.
Circular orbit
(c) Starting with speed calculated in (a), and with the initial velocity perpendicular to a line connecting the Sun and the Earth, calculate and display the trajectory of the Earth. Display the whole trail, so you can see whether you have a closed orbit (that is, whether the Earth returns to its starting point each time around). Include the Sun's position rs =
<x s
' Ys' zs> in your computations, even if you set its coordinates to zero. This will make it easier to modify the computation to let the Sun move (Problem 2
Binary stars).
Computational accuracy
(d) As a check on your computation, is the orbit a circle as expected? Run the calculation until the accumulated time t is the length of a year: does this take the Earth around the orbit once, as expected?
What is the largest step size you can use that still gives a circular orbit and the correct length of a year?
What happens if you use a much larger step size?
Noncircular trajectories
(e) Set the initial speed to 1.2 times the Earth's actual speed. Make sure the step size is small enough that the orbit is nearly unaffected if you cut the step size. What kind of orbit do you get? What step size do you need to use to get results you can trust? What happens if you use a much larger step size?
Produce at least one other qualitatively different noncircular trajectory for the Earth. "What difficulties would humans have in surviving on the Earth if it had a highly noncircular orbit?
Force and momentum
(f) Choose initial conditions that give a noncircular orbit. Continuously display a vector showing the momentum of the Earth (with its tail at the Earth's position), and a different colored vector showing the force on the Earth by the Sun (with its tail at the Earth's position). You must scale the vectors appropriately to fit into the scene. A way to figure out what scale factor to use is to print the numerical values of momentum and force, and compare with the scale of the scene. For example, if the width of the scene is W meters, and the force vector has a typical magnitude F (in newtons), you might scale the force vector by a factor 0.1 W/F, which would make the length of the vector be one-tenth the width of the scene. Is the force in the same direction as the momentum? How does the momentum depend on the distance from the star?
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Problem 2 The effect of the Moon and Venus on the Earth
In Problem 2.1 you analyzed a simple model of the Earth orbiting the Sun, in which there were no other planets or moons. Venus and the Earth have similar size and mass. At its closest approach to the Earth, Venus is about 40 million kilometers away (4x10 10 m). The Moon's mass is about 7xl0 22 kg, and the distance from Earth to Moon is about 4xl0 8 m (400,000 km, center to center).
(a) Calculate the ratio of the gravitational forces on the Earth exerted by Venus and the Sun. Is it a good approximation to ignore the effect of Venus when modeling the motion of the Earth around the Sun?
(b) Calculate the ratio of the gravitational forces on the Earth exerted by the Moon and the Sun. Is it a good approximation to ignore the effect of the Moon when modeling the motion of the Earth around the Sun?
Problem 3 Other force laws
Modity your orbit computation to use a different force law, such as a force that is proportional to 1/ r or 1/ r3 , or a constant force, or a force proportional to r (this represents the force of a spring whose relaxed length is nearly zero). How do orbits with these force laws differ from the circles and ellipses that result from a
1/r2 law? If you want to keep the magnitude of the force roughly the same as before, you will need to adjust the force constant G.
Problem 4 The three-body problem
Carry out a numerical integration of the motion of a three-body gravitational system and plot the trajectory, leaving trails behind the objects. Calculate all of the forces before using these forces to update the momenta and positions of the objects. Otherwise the calculations of gravitational forces would mix positions corresponding to different times.
Try different initial positions and initial momenta. Find at least one set of initial conditions that produces a long-lasting orbit, one set of initial conditions that results in a collision with a massive object, and one set of initial conditions that allows one of the objects to wander off without returning. Report the masses and initial conditions that you used.
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Problem 5 The Ranger 7 mission to the Moon
The first U.S. spacecraft to photograph the Moon close up was the unmanned "Ranger 7" photographic mission in 1964. The spacecraft, shown in the illustration at the right (NASA photograph), contained television cameras that transmitted close-up pictures of the Moon back to Earth as the spacecraft approached the Moon. The spacecraft did not have retrorockets to slow itself down, and it eventually simply crashed onto the Moon's surface, transmitting its last photos immediately before impact.
Figure 2.31 is the first image of the Moon taken by a U.S. spacecraft, Ranger 7, on July
31, 1964, about 17 minutes before impacting the lunar surface. The large crater at center right is Alphonsus (108 km diameter); above it (and to the right) is Ptolemaeus and below it is Arzachel. The Ranger 7 impact site is off the frame, to the left of the upper left corner.
You can find out more about the actual Ranger lunar missions at http://nssdc.gsfc.nasa.gov /planetary /lunar / ranger.html
To send a spacecraft to the Moon, we put it on top of a large rocket containing lots of rocket fuel and fire it upward. At first the huge ship moves quite slowly, but the speed increases rapidly. When the "first-stage" portion of the rocket has exhausted its fuel and is empty, it is discarded and falls back to Earth. By discarding an empty rocket stage we decrease the amount of mass that must be accelerated to even higher speeds. There may be several stages that operate for a while and then are discarded before the spacecraft has risen above most of Earth's atmosphere (about 50 km, say, above the Earth), and has acquired a high speed. At that point all the fuel available for this mission has been used up, and the spacecraft simply coasts toward the Moon through the vacuum of space.
We will model the Ranger 7 mission. Starting 50 km above the Earth's surface (5X10 4 m), a spacecraft coasts toward the Moon with an initial speed of about 10 4 m/s. Here are data we will need: mass of spacecraft = 173 kg mass of Earth"" 6xlO 24 kg mass of Moon "" 7x10 22 kg radius of Moon = 1.75x10
6 m distance from Earth to Moon"" 4x10 8 m (400,000 km, center to center)
We're going to ignore the Sun in a simplified model even though it exerts a sizable gravitational force. We're expecting the Moon mission to take only a few days, during which time the Earth (and Moon) move in a nearly straight line with respect to the Sun, because it takes 365 days to go all the way around the Sun. We take a reference frame fixed to the Earth as representing
(approximately) an inertial frame of reference with respect to which we can use the momentum principle.
Figure 2.31 The first Ranger 7 photo of the Moon (NASA photograph).
For a simple model, make the Earth and Moon be fixed in space during the mission. Factors that would certainly influence the path of the spacecraft include the motion of the Moon around the Earth, and the motion of the Earth around the Sun. In addition, the Sun and other planets exert gravitational forces on the spacecraft. As a separate project you might like to include some of these additional factors.
(a) Compute the path of the spacecraft, and display it either with a graph or with an animated image. Here, and in remaining parts of the problem, report the step size t-.t that gives accurate results. (that is, cutting this step size has little effect on the results).
(b) By trying various initial speeds, determine the approximate Minimum launch speed needed to reach the Moon, to two significant figures (this is the speed that the spacecraft obtains from the multistage rocket, at the time of release above the Earth's atmosphere). What happens if the launch speed is less than this minimum value? (Be sure to check the step size issue.)
(c) Use a launch speed 10% larger than the approximate minimum value found in part (b). How long does it take to go to the Moon, in hours or days? (Be sure to check the step size issue.)
(d) What is the "impact speed" of the spacecraft (its speed just before it hits the Moon's surface)? Make sure that your spacecraft crashes on the surface of the Moon, not at the Moon's center! (Be sure to check the step size issue.)
You may have noticed that you don't actually need to know the mass m of the spacecraft in order to carry out the computation. The gravitational force is proportional to m, and the momentum is also proportional to m, so m cancels.
However, nongravitational forces such as electric forces are not proportional to the mass, and there is no cancellation in that case. We kept the mass m in the analysis in order to illustrate a general technique for predicting motion, no matter what kind of force, gravitational or not.
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Problem 6 The effect of Venus on the Moon voyage
In the Moon voyage analysis, you used a simplified model in which you neglected among other things the effect of
Figure 2.32 The relative positions of
Venus. An important aspect of physical modeling is making estimates of how large the neglected effects
Venus, Earth, and Moon in Problem 6. might be. If we take Venus into account, make a rough estimate of whether the spacecraft will miss the
Moon entirely. How large a sideways deflection of the crash site will there be? Explain your reasoning and approximations.
Venus and the Earth have similar size and mass. At its-closest approach to the Earth, Venus is about
40 million kilometers away (4X10 10 m). In the real world, Venus would attract the Earth and the Moon as well as the spacecraft, but to get an idea of the size of the effects, pretend that the Earth, the Moon, and Venus are all fixed in position (Figure 2.32) and just investigate Venus's effect on the spacecraft.
Problem 7 Determining the mass of an asteroid
In June 1997 the NEAR spacecraft ("Near Earth Asteroid Rendezvous"; see http://nearJhuapl.edu/), on its way to photograph the asteroid Eros, passed within 1200 km of asteroid Mathilde at a speed of 10 km/s relative to the asteroid (Figure 2.34). From photos transmitted by the 805 kg spacecraft,
Mathilde's size was known to be about 70 km by 50 km by 50 km. It is presumably made of rock. Rocks on Earth have a density of about 3000 kg/m 3 (3 grams/cm 3 ).
(a) Make a rough diagram to show qualitatively the effect on the spacecraft of this encounter with Mathilde. Explain your reasoning. Figure 2.34 The NEAR spacecraft passes the asteroid Mathilde (Problem 7).
(b) Make a very rough estimate of the change in momentum of the spacecraft that would result from encountering Mathilde. Explain how you made your estimate.
(c) Using your result from part (b), make a rough estimate of how far off course the spacecraft would be, one day after the encounter.
(d) From actual observations of the location of the spacecraft one day after encountering Mathilde, scientists concluded that Mathilde is a loose arrangement of rocks, with lots of empty space inside.
What was it about the observations that must have led them to this conclusion?
Experimental background: The position was tracked by very accurate measurements of the time that it takes for a radio signal to go from Earth to the spacecraft followed immediately by a radio response from the spacecraft being sent back to Earth. Radio signals, like light, travel at a speed of 3x10 8 mis, so the time measurements had to be accurate to a few nanoseconds (1 ns =: 109 S ).
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Problem 8 Dark matter in the Universe
In the 1970's the astronomer Vera Rubin made observations of distant galaxies that she interpreted as indicating that perhaps 90% of the mass in a galaxy is invisible to us ("dark matter"). She measured the speed with which stars orbit the center of a galaxy, as a function of the distance of the stars from the center. The orbital speed was determined by measuring the "Doppler shift" of the light from the stars, an effect which makes light shift toward the red end of the spectrum ("red shift") if the star has a velocity component away from us, and makes light shift toward the blue end of the spectrum if the star has a velocity component toward us. She found that for stars farther out from the center of the galaxy, the orbital speed of the star hardly changes with distance from the center of the galaxy, as is indicated in the diagram below. The visible components of the galaxy (stars, and illuminated clouds of dust) are most dense at the center of the galaxy and thin out rapidly as you move away from the center, so most of the visible mass is near the center.
About the same speed at a different radius
(a) Predict the speed v of a star going around the center of a galaxy in circular orbit, as a function of the star's distance r from the center of the galaxy, assuming that almost all of the galaxy's mass M is concentrated at the center.
(b) Construct a logical argument as to why Rubin concluded that much of the mass of a galaxy is not visible to us. Reason from principles discussed in this chapter, and your analysis of part (a). Explain your reasoning. You need to address the following issues:
•• Rubin's observations are not consistent with your prediction in (a) .
•• Most of the visible matter is in the center of the galaxy .
•• Your prediction in (a) assumed that most of the mass is at the center.
This issue has not yet been resolved, and is still a current topic of astrophysics research. Here is a reference to the original work: "Dark Matter in Spiral Galaxies" by Vera C. Rubin,
Scientific American, June 1983 (96-108). You can find several graphs of the rotation curves for spiral galaxies on page 101 ( this article.
Problem 9 Orbital periods
(a) Many communication satellites are placed in a circular orbit around the Earth at a radius where the period (the time to go around the Earth once) is 24 hours. If the satellite is above some point on the equator, it stays above that point as the Earth rotates, so that as viewed from the rotating Earth the satellite appears to be motionless. That is why you see dish antennas pointing at a "fixed" point in space. Calculate the radius of the orbit of such a "synchronous" satellite. Explain your calculation in detail.
(b) Electromagnetic radiation including light and radio waves travels at a speed of 3x10 8 m/ s. If a phone call is routed through a synchronous satellite, what is the minimum delay between saying something and getting a response? Explain. Include in your explanation a diagram of the situation.
(c) Some human-made satellites are placed in "near-Earth" orbit, just high enough to be above almost all of the atmosphere. Calculate how long it takes for such a satellite to go around the
Earth once, and explain any approximations you make.
(d) Calculate the orbital speed for a near-Earth orbit, which must be provided by the launch rocket. (Recently large numbers of near-Earth communications satellites have been launched.
Their advantages include making the signal delay unnoticeable, but with the disadvantage of having to track the satellites actively and having to use many satellites to ensure that at least one is always visible over a particular region.)
(e) When the first two astronauts landed on the Moon, a third astronaut remained in an orbiter in circular orbit near the Moon's surface. During half of every complete orbit, the orbiter was
85 when radio contact was lost.

Figure 2.35 Three stars in a circular orbit
(Problem 10).
Observed positions of a star from 1995 to
1999 near the center of our Milky Way
Galaxy, and an orbit fit to the data (Problem
11).
Problem 10 Analytical3-body orbit
There is no general analytical solution for the motion of a 3body gravitational system. However, there do exist analytical solutions for very special initial conditions. Figure
2.35 shows three stars, each of mass m, which move in the plane of the page along a circle of radius r. Calculate how long this system takes to make one complete revolution. (In many cases 3-body orbits are not stable: any slight perturbation leads to a break-up of the orbit.)
Problem 11 The center of our galaxy
Remarkable data indicate the presence of a massive black hole at the center of our Milky Way galaxy. One of the two
W. M. Keck 10 meter diameter telescopes in Hawaii was used by Andrea Chez and her colleagues to observe infrared light coming directly through the dust surrounding the central region of our galaxy (visible light is multiply scattered by the dust, blocking a direct view). Stars were observed for several consecutive years to determine their orbits near a motionless center that is completely invisible
("black") in the infrared but whose precise location is known due to its strong output of radio waves, which are observed by radio telescopes. The data were used to show that the object at the center must have a mass that is huge compared to the mass of our own Sun, whose mass is a "mere" 2xlO kg.
(a) The speeds of the stars as a function of distance from the center were consistent with Newtonian predictions for orbits around a massive central object. How should the speed of each star depend on its distance from the center, and the mass of the central object?
(b) The star nearest the center of our galaxy was observed to be about 1014 m from the center, with a period of about
15 years. What is the speed of this star in m/ s? Also express the speed as a fraction of the speed of light. (The actual orbit is an ellipse as reproduced in Figure 2.36, but for purposes of this analysis you may approximate it as being circular.)
(c) This is an extraordinarily high speed for a star. Is it reasonable to approximate the star's momentum as mv?
(d) Calculate the mass of the massive black hole about which this star is orbiting.
(e) How many of our Suns does this represent?
It is thought that all galaxies may have such a black hole at their centers, as a result of long periods of mass accumulation. When many bodies orbit each other, sometimes an object happens in an interaction to acquire enough speed to escape from the group, and the remaining objects are left closer together. Some simulations show that over time, as much as half the mass may be ejected, with agglomeration of the remaining mass. This could be part of the mechanism for forming massive black holes.
For more information, see the home page of Andrea Ghez, http:/ /www.astro.ucla.edu/faculty/ghez.htm. The papers available there refer to "arcseconds," which is an angular measure of how far apart objects appear on the sky, and to
"parsecs," which is a distance equal to 3.3 lightyears (a lightyear is the distance light goes in one year).
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In this lab you will use conservation of momentum to predict the motion of objects resulting from interactions that are difficult to analyze with force concepts or conservation of energy. In these problems, you will explore the applicability of conservation of momentum in predicting the motion resulting from collisions. Fortunately, conservation principles can be used to relate the motion of an object before a collision to the motion after a collision, without knowledge of the complicated details of the collision process itself. To fully analyze an interaction, one must often use both conservation of energy and conservation of momentum.
•
•
•
•
BJECTIVES
Successfully completing this laboratory should enable you to:
• Use conservation of momentum to predict the outcome of interactions between objects.
•
•
•
Choose a useful system when using conservation of momentum.
Identify momentum transfer (impulse) when applying energy conservation to real systems.
Use the principles of conservation of energy and of momentum together as a means of describing the behavior of systems.
REPARATION
Read YOUNG AND FREEDMAN: Chapter 6. You should also be able to:
Analyze the motion of an object using video analysis.
Calculate the work transferred to or from a system by an external force.
Calculate the total energy of a system of objects.
Calculate the total momentum of a system of objects.
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You have a summer job at NASA with a group designing a docking mechanism that would allow two space shuttles to connect with each other. The mechanism is designed for one shuttle to move carefully into position and dock with a stationary shuttle. Since the shuttles may be carrying different payloads and have consumed different amounts of fuel, their masses may be different when they dock. Your supervisor wants you to calculate the velocity of the docked shuttles as a function of the initial velocity of the moving shuttle and the masses of the shuttles. You decide to calculate the resulting velocity for a system of two objects sticking together after colliding. You then build a laboratory model using carts to check your calculation.
?
When two objects stick together after a collision, what is the final velocity of the objects as a function of the initial velocity of the moving object and their masses?
E QUIPMENT
You will use the track and carts with which you are familiar. For this problem, cart A is given an initial velocity towards a stationary cart B. Pads at the end of each cart are used to get the carts to stick together after the collision. The video analysis equipment is available. You also need a meter stick, a stopwatch, two end stops and cart masses. This is exactly the same situation as Problem #3 in Lab IV. If you did that
Problem, you can use that data to check your prediction. moving stationary
A B
P REDICTION
Calculate the final velocities of the carts as a function of the initial velocity of cart A and the masses of the two carts.
You will investigate the following three cases in which the total mass of the two carts is constant
(m
A
+ m
B
= constant): (a) the masses of the carts are equal (m
A
= m
B
); (b) the moving cart has a larger mass than the stationary cart (m
A
> m
B
); (c) the moving cart has a smaller mass than the stationary cart (m
A
< m
B
). How do the final velocities of the stuck-together carts compare to one another and to the initial velocity of cart A? Discuss both magnitude and direction.
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M ETHOD Q UESTIONS
It is useful to have an organized problem-solving strategy such as the one outlined in the following questions.
1. Make two drawings that show the situation before the collision and after the collision. Show and label velocity vectors for every object in your drawings. Define your system.
2. Write down the momentum conservation equation for this situation and identify all of the terms in the equation. Are there any of these terms that you cannot measure with the equipment at hand?
Write down the energy conservation equation for this situation and identify all the terms in the equation. Are there any of these terms that you cannot measure with the equipment at hand?
Which conservation principle should you use to predict the final velocity of the stucktogether carts, or do you need both equations? Why?
E XPLORATION
If you have done Problem #2 in Lab IV, you should be able to skip this part.
Practice setting the cart into motion so the carts stick together after the collision. Also, after the collision carefully observe the carts to determine whether or not either cart leaves the grooves in the track. Adjust your procedure to minimize this effect so that your results are reliable.
Try giving the moving cart various initial velocities over the range that will give reliable results. Note qualitatively the outcomes. Keep in mind also that you want to choose an initial velocity that gives you a good video.
Try varying the masses of the carts so that the mass of the initially moving cart covers a range from greater than the mass of the stationary cart to less than the mass the stationary cart while keeping the total mass of the carts the same. Be sure the carts still move freely over the track. What masses will you use in your final measurement?
M EASUREMENT
Record the masses of the two carts. Make a video of their collision. Examine your video and decide if you have enough frames to determine the velocities you need. Are there any peculiarities that you notice in the data that might suggest that the data is unreliable?
Analyze your data as you go along (before making the next video), so you can determine how many different videos you need to make, and what the carts' masses should be for each video. Collect enough
89

data to convince yourself and others of your conclusion about how the final velocity of both carts in this type of collision depends on velocity of the initially moving cart and the masses of the carts.
A NALYSIS
Determine the velocities of the carts (with uncertainty) before and after each collision from your video.
Calculate the momentum of the carts before and after the collision.
Now use your Prediction equation to calculate each final velocity (with uncertainty) of the stuck-together carts.
C ONCLUSION
How do your measured and predicted values of the final velocity compare? Compare both magnitude and direction. What are the limitations on the accuracy of your measurements and analysis?
When a moving shuttle collides with a stationary shuttle and they dock (stick together), how does the final velocity depend on the initial velocity of the moving shuttle and the masses of the shuttles? State your results in the most general terms supported by the data.
Use your results and conservation of energy to calculate the energy dissipation you will want to same this data for Lab V.
90

In this lab you will begin to use conservation of energy to determine the motion resulting from interactions that are difficult to analyze using force concepts. You will explore how conservation of energy is applied to real interactions. Although energy is always conserved, it is sometimes difficult to calculate the value of all of the energy terms for an interaction. Not all of the initial energy of a system ends up as visible energy of motion. Some energy is transferred into or out of the system and some becomes internal energy of the system. Since this energy is not observable in the macroscopic motion of objects, we sometimes say that the energy is "dissipated" in the collision. The first four problems explore the application of conservation of energy using the carts. The fifth problem deals specifically with the very real complication of friction.
BJECTIVES
Successfully completing this laboratory should enable you to:
• Use conservation of energy to predict the outcome of interactions between objects.
• Choose a useful system when using conservation of energy.
• Identify different types of energy when applying energy conservation to real systems.
• Decide when conservation of energy is not useful to predict the outcome of interactions between objects.
REPARATION
Read YOUNG AND FREEDMAN : Chapter 11, Chapter 12 and Chapter 13. You should also be able to:
• Analyze the motion of an object using video analysis.
• Calculate the kinetic energy of a moving object.
• Calculate the work transferred to or from a system by an external force.
• Calculate the total energy of a system of objects.
• Calculate the gravitational potential energy of an object with respect to the earth.
92

You are working at a company that designs pinball machines and have been asked to devise a test to determine the efficiency of some new magnetic bumpers. You know that when a normal pinball rebounds off traditional bumpers, some of the initial energy of motion is "dissipated" in the deformation of the ball and bumper, thus slowing the ball down. The lead engineer on the project assigns you to determine if the new magnetic bumpers are more efficient. The engineer tells you that the efficiency of a collision is the ratio of the final kinetic energy to the initial kinetic energy of the system.
Since you want to measure the efficiency of the bumper (not the ball), you decide to use a cart with a magnet colliding with a magnetic bumper. You will use a level track and give the cart an initial velocity measuring its velocity before and after the collision. You begin to gather your camera and data acquisition system when your colleague suggests that you don’t need all that equipment. Just start the cart from rest on an inclined track and make the necessary measurements with a meter stick. You are not sure you believe it and besides you like using fancy technology. Instead of arguing, you decide to work together, measure the energy efficiency both ways, and determine the extent to which you get consistent results.
You divide your group into two teams. One team will use the level track and the video analysis equipment while the other will use the incline and meterstick.
?
Determine the efficiency of the bumper in the situation of an inclined track and a level track?
E QUIPMENT
You will have a meter stick, a stopwatch, cart masses and wooden blocks to create the incline. You may also use the video analysis equipment to estimate the effect of friction for measuring the efficiency.
P REDICTIONS
Calculate the energy efficiency of the bumper (with friction and without) in terms of the least number of quantities that you can easily measure in the situation of an inclined track.
Calculate the energy efficiency of the bumper in terms of the least number of quantities that you can easily measure in the situation of a level track.
M ETHOD Q UESTIONS
It is useful to have an organized problem-solving strategy. The following procedures should help with the analysis of your data:
93

1.
Make a drawing of the cart on the level track before and after the impact with the bumper.
Define your system. Label the velocity and kinetic energy of all objects in your system before and after the impact.
2.
Write an expression for the efficiency of the bumper in terms of the final and initial kinetic energy of the cart.
3.
Write an expression for the energy dissipated during the impact with the bumper in terms of the kinetic energy before the impact and the kinetic energy after the impact.
4.
Make a drawing of the cart on the inclined track at its initial position (before you release the cart) and just before the cart hits the bumper. Define the system. Label the kinetic energy of all objects in your system at these two points, the distance the cart traveled, the angle of incline, and the initial height of the cart above the bumper.
5.
Draw a force diagram of the cart as it moves down the track. Which force component does work on the cart (i.e., causes a transfer of energy into the cart system)? Write an expression for the work done on the cart. How is the angle of the ramp related to the distance the cart traveled and the initial height of the cart above the bumper? How does the kinetic energy of
the cart just before impact compare with the work done on the cart?
6.
Now make another drawing of the cart on the inclined track just after the collision with the bumper and at its maximum rebound height. Label kinetic energy of the cart at these two points, the distance the cart traveled, the angle of the ramp, and the rebound height of the cart above the bumper.
7.
Draw a force diagram of the cart as it moves up the track. Which force component does work on the cart (i.e., causes a transfer of energy out of the cart system)? Write an expression for the work done on the cart. How does the kinetic energy of the cart just after
impact compare with the work done on the cart?
8.
Write an expression for the efficiency of the bumper in terms of the kinetic energy of the cart just before the impact and the kinetic energy of the cart just after the impact.
9.
Write an expression for the energy dissipated during the impact with the bumper in terms of the kinetic energy of the cart just before the impact and the kinetic energy of the cart after the impact.
10.
Repeat the same procedure and consider the effect of friction.
E XPLORATION
Review your exploration notes for measuring a velocity using video analysis. Practice pushing the cart with different velocities. Make sure that the cart will never contact the bumper (end stop) during the impact. Find the proper rang of velocity for your measurement. Set up the camera and tripod to give you best video (undistorted, in focus, minimal missing frames) of the collision immediately before and after the cart collides with the bumper.
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Find a useful range of heights and inclined angles that will not cause damage to the carts or bumpers. Make sure that the cart will never contact bumper (end stop) during the impact. Decide how you are going to measure the height of the cart.
You may want to estimate the effect of friction. Make a schedule to test the effect of friction by the video analysis equipment. How can you find the average frictional force when the cart moves on the inclined track? How much energy is dissipated by friction?
M EASUREMENT
Take the measurements necessary to determine the kinetic energy of the cart just before and after the impact with the bumper. Take data for several different initial heights.
Take the measurements necessary to determine the kinetic energy before and after the impact with the bumper. What is the most efficient way to measure the velocities with the video equipment? Take data for several different initial velocities.
A NALYSIS
Calculate the efficiency of the bumper for the inclined track. Does your result depend on the velocity of the cart before it hits the bumper?
Calculate the efficiency of the bumper for the level track. Does your result depend on the velocity of the cart before it hits the bumper?
C ONCLUSION
What is the efficiency of the magnetic bumpers? How much energy is dissipated in an impact? State your results in the most general terms supported by your analysis. Is effect of friction significant?
If available, compare your value of the efficiency (with uncertainty) with the value obtained by the different procedure given in the preceding problem. Are the values consistent? Which way to measure the efficiency of the magnetic bumper do you think is better? Why?
95

You still have your summer job with the Traffic Safety Board investigating the damage done to vehicles in different kinds of traffic accidents. Your boss now wants you to concentrate on the damage done in low speed collisions when a moving vehicle hits a stationary vehicle and they bounce apart. Even with new improved bumpers, your boss believes that, given the same initial energy, the damage to the vehicles in a collision when cars bounce apart will be less when the moving vehicle has a smaller mass than the stationary vehicle (e.g., a compact car hits a van) than for other situations.
To determine if your boss is correct, you model the collision with carts on the track and measure the energy efficiency of the collision, the ratio of the final kinetic energy of the system to the initial kinetic energy, as a function of mass and velocity. For your model you use the most efficient bumper you can think of, a magnetic bumper.
?
When a moving object collides with a stationary object and they bounce apart, how does the energy dissipated depend on the relative masses of the objects?
E QUIPMENT
Magnets at the end of each cart are used as bumpers to ensure that the carts bounce apart after the collision. The video analysis equipment is available to determine the velocities of the carts. You also have a meter stick, a stopwatch, two end stops and cart masses. moving stationary
A B
P REDICTION
Calculate the energy dissipated in a collision in which the carts bounce apart as a function of the mass of each cart, the initial kinetic energy of the system, and the energy efficiency of the collision.
Consider the following three cases in which the total mass of the carts is the same (m
A
+ m
B
= constant): (a) the masses of the carts are equal (m
A
= m
B
); (b) the moving cart has a larger mass than the stationary cart (m cart (m
A
< m
B
A
> m
B
); and (c) the moving cart has a smaller mass than the stationary
). Do you think the energy efficiency will be the same or different for these three cases? If different, in which case do you think the efficiency will be the largest? The smallest?
Rank each of these collisions from most efficient to least efficient. Make your best educated guess and explain your reasoning. If the kinetic energy of an incoming vehicle is the same in the three cases, use your calculation to determine which one will cause the most damage.
96

M ETHOD Q UESTIONS
The following procedure and questions may help with the prediction and analysis of your data.
1. Draw two pictures that show the situation before the collision and after the collision. In this experiment we ignore the friction between carts and the track. Draw velocity vectors on your sketch. Define your system. Write down the energy of the system before the collision and also after the collision. Is there any energy transferred into or out of the system?
2. Write down the energy conservation equation for this collision.
3. Write an equation for the efficiency of the collision in terms of the final and initial kinetic energy of the carts.
4. Solve your equations for the energy dissipated.
E XPLORATION
Practice setting the cart into motion so that the carts don’t actually touch. Also, after the collision carefully observe the carts to determine whether or not either cart leaves the grooves in the track. Adjust your procedure to minimize this effect so that your results are reliable.
Try giving the moving cart various initial velocities over the range that will give reliable results. Note qualitatively the outcomes. Keep in mind also that you want to choose an initial velocity that gives you a good video.
Try varying the masses of the carts so that the mass of the initially moving cart covers a range from greater than the mass of the stationary cart to less than the mass of the stationary cart while keeping the total mass of the carts the same. Be sure the carts still move freely over the track. What masses will you use in your final measurement?
M EASUREMENT
Record the masses of the two carts. Make a video of their collision. Examine your video and decide if you have enough positions to determine the velocities that you need. Are there any peculiarities that you notice in the data that might suggest that the data is unreliable?
Analyze your data as you go along (before making the next video), so you can determine how many different videos you need to make, and what the carts' masses should be for each video. Collect enough data to convince yourself and others of your conclusion about how the energy efficiency of this type of collision depends on the relative masses of the carts.
A NALYSIS
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Determine the velocity of the carts before and after the collision using video analysis. For each video, calculate the kinetic energy of the carts before and after the collision.
Calculate the energy efficiency of each collision. Into what other forms of energy do you think the cart's initial kinetic energy is most likely to transform?
Graph how the energy efficiency varies with mass of the initially moving cart (keeping the total mass of both carts constant). What is the function that describes this graph? Repeat for energy efficiency as a function of initial velocity.
C ONCLUSION
Which case (m
A
= m
B
, m
A
> m
B
, or m
A
< m
B
) is the energy efficiency the largest? The smallest?
Was a significant portion of the energy dissipated? How does it compare to the case where the carts stick together after the collision? Into what other forms of energy do you think the cart's initial kinetic energy is most likely to transform?
Could the collisions you measured be considered essentially elastic collisions? Why or why not? The energy efficiency for a perfectly elastic collision is 1.
Can you approximate the results of this type of collision by assuming that the energy dissipated is small?
Was your boss right? Is the same amount of damage done to the vehicles when a car hits a stationary truck and they stick together as when the truck hits the stationary car (given the same initial kinetic energies)? State your results that support this conclusion.
98

You still have your summer job with the Traffic Safety Board and your team has been assigned the task of determining the safety of trucks hauling large crates through the city. In particular your team is investigating what happens to the crate if the truck hits a stopped vehicle. If that happens, is it possible for the crate to slam into the cab of the truck? You decide to calculate the distance a crate moves on a level truck bed after the truck has come to a sudden stop as a function of the speed of the truck before the collision. You assume the worst case scenario that the crate is not tied down, as the law requires.
Since it is too expensive to use actual trucks to check your calculation, you model the situation with a cart on the track.
?
If a crate is riding on a truck, how does the distance it travels after the truck comes to a sudden stop depend on the speed of the truck?
E QUIPMENT
You will have a cart with cart masses, a track, an end stop, a wood/cloth block, a mass set, a meter stick, a stopwatch and the video analysis equipment to determine the velocity of the cart before the collision. You need to put the cart masses on the top of the cart so that the wood/cloth block can slide on the cart. You can suddenly stop the cart by colliding it with the end stop on the track. In our experiment we ignore the friction between the cart and the track. v e nd sto p b lo c k c a rt
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PROBLEM #5: ENERGY AND FRICTION
P REDICTIONS
Calculate the distance the block slides after the cart is suddenly stopped as a function of the cart’s speed before the collision. Show this result graphically.
M ETHOD Q UESTIONS
The following procedure and questions should help with the prediction and analysis of your data.
1. Draw two pictures one of which shows the situation before the collision and one after the collision. Draw velocity vectors on your sketch. Define your system. Write down the energy of the system before the collision. Write down the energy of the system after the collision. Is there any energy transferred into or out of the system? Before the collision, what is the relationship between the cart’s velocity and the wood/cloth block’s velocity? They are same or not?
2. Write down the energy conservation equation for this situation.
3. Draw a force diagram for the wood/cloth block as it slides across the cart. Identify the forces that do work on the block (i.e., result in the transfer of energy in or out of the system).
E XPLORATION
Practice setting the cart with masses into motion so the cart sticks to the end stop. What adjustments are necessary to make this happen consistently?
Place the wood/cloth block on the cart. Try giving the cart various initial velocities. Choose a range of initial velocities that give you good video data. Make sure that the initial velocity should not so fast that the wood/cloth block begins to slide on the cart before the collision. Try several masses for the cart and the block. Note qualitatively the outcomes when the cart sticks to the end stop.
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PROBLEM #5: ENERGY AND FRICTION
M EASUREMENT
Make the measurements that you need to check the prediction. Because you are dealing with friction, it is especially important that you repeat each measurement several times under the same conditions to see if it is reproducible.
A NALYSIS
Make a graph of the distance the block travels as a function of the c art’s initial speed. Determine if this result depends on the mass of the block or the mass of the cart. If the graph is not linear, make another graph of distance as a function of speed to some power such that it is (see Appendix C). What determines the slope of that line?
C ONCLUSION
Do your results agree with your predictions? What are the limitations on the accuracy of your measurements and analysis?
As a check, determine the coefficient of kinetic friction between the block and the cart from your results. Is it reasonable?
Does the distance that the crate slides depend on the mass of the truck? On the mass of the crate?
101

LAB VI: INTRODUCTION
Name and ID#: ______________________________________________________
Date performed: ________________ Day/Time section meets: ______________
Lab Partners' Names: ________________________________________________
__________________________________________________________________
__________________________________________________________________
Problem # and Title: _________________________________________________
Lab Instructor's Initials: ____________
Grading Checklist Points
LABORATORY JOURNAL:
PREDICTIONS
(individual predictions and methods questions completed in journal before each lab session)
LAB PROCEDURE
(measurement plan recorded in journal, tables and graphs made in journal as data is collected, observations written in journal)
PROBLEM REPORT:*
ORGANIZATION
(clear and readable; logical progression from problem statement through conclusions; pictures provided where necessary; correct grammar and spelling; section headings provided; physics stated correctly)
DATA AND DATA TABLES
(clear and readable; units and assigned uncertainties clearly stated)
RESULTS
(results clearly indicated; correct, logical, and well-organized calculations with uncertainties indicated; scales, labels and uncertainties on graphs; physics stated correctly)
CONCLUSIONS
(comparison to prediction & theory discussed with physics stated correctly ; possible sources of uncertainties identified; attention called to experimental problems)
TOTAL(incorrect or missing statement of physics will result in a maximum of
60% of the total points achieved; incorrect grammar or spelling will result in a maximum of 70% of the total points achieved)
BONUS POINTS FOR TEAMWORK
(as specified by course policy)
* An "R" in the points column means to rewrite that section only and return it to your lab instructor within two days of the return of the report to you.
102

So far this semester, you have been asked to think of objects as point particles rather than extended bodies. This assumption is useful and sometimes good enough. However, the approximation of extended bodies as point particles gives an incomplete picture of the real world.
Now we begin a more realistic description of the motion and interactions of objects. Real objects usually rotate as well as move along trajectories. You already have a lot of experience with rotating objects from your everyday life. Every time you open or close a door, something rotates. Rotating wheels are everywhere. Balls spin when they are thrown. The earth rotates about its axis. Rotations are important whether you are discussing galaxies or subatomic particles.
An object’s rotational motion can be described with the kinematic quantities you have already used: position, velocity, acceleration, and time. In these problems, you will explore the connection of these familiar linear kinematic quantities to a more convenient set of quantities for describing rotational kinematics: angle, angular velocity, angular acceleration, and time. Often, the analysis of an interaction requires the use of both linear and rotational kinematics.
BJECTIVES
Successfully completing this laboratory should enable you to:
• Use linear kinematics to predict the outcome of a rotational system.
• Choose a useful system when using rotational kinematics.
• Identify links between linear and rotational motion.
• Decide when rotational kinematics is more useful and when it is a not.
• Use both linear and rotational kinematics as a means of describing the behavior of systems.
REPARATION
Read YOUNG AND FREEDMAN: Chapter 8. You should also be able to:
• Analyze the motion of an object using linear kinematics and video analysis.
• Identify the connection between the linear description of motion and rotational description of motion
• Calculate the angle of rotation in radians.
• Calculate the angular speed of a rotating object.
• Calculate the angular acceleration of a rotating object.
103

You have a summer job with a group testing equipment that might be used on a satellite. To equalize the heat load from the sun, the satellite will spin about its center. Your task is to determine the forces on delicate measuring equipment when it spins at a constant angular speed. Since any object requires a net force on it to travel in a circular path at constant speed, that object must be accelerating. You decide to first find the linear speed of any object in the satellite as a function of the distance of that object from the axis of rotation. From the linear speed of an object in circular motion, you can calculate its acceleration. You know the angular speed of the satellite’s rotation.
?
What is the relationship between the linear speed of an object rotating at a constant angular speed and its distance from the axis of rotation?
What is the object’s acceleration?
E QUIPMENT
You will be using a set of rotational apparatus that spins a horizontal beam on a stand. A top view of the device is shown to the right.
You will have a meter stick, a stopwatch, and the video analysis equipment.
+
+
+
+
+
P REDICTION
Make a graph of the linear speed of a point on the beam as a function of its distance from the axis of rotation when the beam has a constant angular speed. Calculate the acceleration of a point on the beam based on this graph.
104

M ETHOD Q UESTIONS
1. Draw the trajectory of a point on the beam. Choose a coordinate system. Choose a point on that trajectory that is not on a coordinate axis. Draw vectors representing the position, velocity, and acceleration of that point.
2. Write equations giving the perpendicular components of the position vector as a function of the distance of the point from the axis of rotation and the angle the vector makes with one axis of your coordinate system. Calculate how that angle depends on time and the constant angular speed of the beam. Graph each of these equations as a function of time.
3. Using your equations for components of the position of the point, calculate the equations for the components of the velocity of the point. Graph these equations as a function of time. Compare these graphs to those representing the components of the position of the object. Put these components on your drawing and verify that their vector sum gives the correct direction for the velocity of the point.
4. Use your equations for the components of the velocity of the point to calculate its speed.
5. Write down the acceleration of the point as a function of its speed and its distance from the axis of rotation. Using step 4, calculate the acceleration of the point as a function of the beam’s angular speed and its distance from the axis of rotation.
E XPLORATION
Practice spinning the beam at different angular speeds. How many rotations does the beam make before it slows down appreciably? Select a range of angular speeds to use in your measurements.
Move the apparatus to the floor and adjust the camera tripod so that the camera is directly above the middle of the spinning beam. Make sure the beam is level. Practice taking some videos. Find the best distance and angle such that the motion has the least amount of distortion. How will you make sure that you always measure the same position on the beam?
Plan how you will measure the perpendicular components of the velocity to calculate the speed of the point. How will you also use your video to measure the angular speed of the beam?
M EASUREMENT
Take a video of the spinning beam. Be sure you have more than two complete revolution of the beam. For best results, use the beam itself when calibrating your video.
105

PROBLEM #1: ANGULAR SPEED AND LINEAR SPEED
Determine the time it takes for the beam to make two complete revolutions and the distance between the point of interest and the axis of rotation. Set the scale of your axes appropriately so you can see the data as it is digitized.
Decide how many different points you will measure to test your prediction. How will you ensure that the angular speed is the same for all of these measurements?
How many times will you repeat these measurements using different angular speeds?
A NALYSIS
Analyze your video by following a single point on the beam for at least two complete revolutions.
Use the velocity components to determine the direction of the velocity vector. Is it in the expected direction?
Analyze enough different points in the same video to make a graph of speed of a point as a function of distance from the axis of rotation. What quantity does the slope of this graph represent?
Calculate the acceleration of each point and graph the acceleration as a function of the distance from the axis of rotation. What quantity does the slope of this graph represent?
C ONCLUSIONS
How do your results compare to your predictions and the answers to the method questions?
106

Name and ID#: ______________________________________________________
Date performed: ________________ Day/Time section meets: ______________
Lab Partners' Names: ________________________________________________
__________________________________________________________________
__________________________________________________________________
Problem # and Title: _________________________________________________
Lab Instructor's Initials: ____________
Grading Checklist Points
LABORATORY JOURNAL:
PREDICTIONS
(individual predictions and methods questions completed in journal before each lab session)
LAB PROCEDURE
(measurement plan recorded in journal, tables and graphs made in journal as data is collected, observations written in journal)
PROBLEM REPORT:*
ORGANIZATION
(clear and readable; logical progression from problem statement through conclusions; pictures provided where necessary; correct grammar and spelling; section headings provided; physics stated correctly)
DATA AND DATA TABLES
(clear and readable; units and assigned uncertainties clearly stated)
RESULTS
(results clearly indicated; correct, logical, and well-organized calculations with uncertainties indicated; scales, labels and uncertainties on graphs; physics stated correctly)
CONCLUSIONS
(comparison to prediction & theory discussed with physics stated correctly ; possible sources of uncertainties identified; attention called to experimental problems)
TOTAL(incorrect or missing statement of physics will result in a maximum of
60% of the total points achieved; incorrect grammar or spelling will result in a maximum of 70% of the total points achieved)
BONUS POINTS FOR TEAMWORK
(as specified by course policy)
* An "R" in the points column means to rewrite that section only and return it to your lab instructor within two days of the return of the report to you.
108

Describing rotations requires applying the physics concepts you have already been studying – position, velocity, acceleration, force, mass, kinetic energy, and momentum to objects that can rotate. However, as we have seen, a modified set of kinematic quantities is sometimes easier to apply to objects with a definite shape – angle, angular velocity, and angular acceleration. To more closely investigate the interactions of these objects, it is also useful to define a modified set of dynamic quantities – torque, moment of inertia (or rotational inertia), rotational kinetic energy, and angular momentum.
In this laboratory, you will apply the concept of torque alongside the concept of force to design structures that are stationary as well as accelerating. You will use the concept of moment of inertia to predict the motion of objects when torques are applied. Also, you will be using a new conservation theory: the conservation of angular momentum.
BJECTIVES
After successfully completing this laboratory, you should be able to:
• Use the concept of torque for a system that is in static equilibrium.
• Relate the concepts of torque, angular acceleration and moment of inertia for rigid bodies.
• Use the conservation principles of energy, momentum, and angular momentum for rigid body motion.
REPARATION
Read YOUNG AND FREEDMAN: Chapter 9; Chapter 10.
Before coming to lab you should be able to:
• Determine the net force on an object from its acceleration.
• Know when to use mass and when to use moment of inertia to determine the motion of objects.
• Determine the net torque on an object from its angular acceleration.
• Draw and use force and torque diagrams.
• Explain what is meant by a system in "equilibrium."
• Know how to determine the period of rotation of an object.
109

While examining the engine of your friend’s snowblower you notice that the starter cord wraps around a cylindrical ring of metal. This ring is fastened to the top of a heavy, solid disk, "a flywheel," and that disk is attached to a shaft. You are intrigued by this configuration and wonder how to determine its moment of inertia. Your friend thinks that you just add them up to get the moment of inertia of the system. You are also curious if it is better to have the ring centered on the disk or off the axis. To test this idea you decide to build a laboratory model described below to determine the moment of inertia of a similar system from its motion. You think you can do it by just measuring the acceleration of the hanging weight.
?
Determine the moment of inertia from the acceleration of the hanging weight in the equipment described below. Is the moment of inertia of a system just the sum of the moments of inertia of the components?
E QUIPMENT
For this problem you will have a disk, which is mounted on a sturdy stand by a metal shaft. Below the disk there is a metal spool on the shaft to wind string around. A ring sits on the disk so both ring and disk share the same rotational axis. Alternately the ring can be moved of the central axis. A length of string is wrapped around the spool and then passes over a pulley lined up with the tangent to the spool. A weight is hung from the other end of the string so that the weight can fall past the edge of the table.
As the hanging weight falls, the string pulls on the spool, causing the ring/disk/shaft/spool system to rotate. You will also have a meter stick, a stopwatch, a pulley clamp, a mass hanger, a mass set and the video analysis equipment in this experiment.
P REDICTION
Calculate that the moment of inertia of the centered disk system as a function of the mass of the hanging weight, the acceleration of the hanging weight and the radius of the spool.
Using the table of moments of inertia in your text, write an expression for the total rotational inertia of the system as the sum of the rotational inertia of components of the system in terms of their masses and radii.
110

Will these two methods agree? If not which one, if any, is correct?
Write an expression that gives the moment of inertia of the off-centered ring/disk/shaft/spool system in terms of the distance from the center of the ring to the center of the disk and the characteristics of the components of the system such as the radius of the spool and mass. (Hint: If you need help, see the parallel axis theorem in your text.)
Compare the moment of inertia calculated from the acceleration of the equipment’s hanging weight.
Does the moment of inertia of the ring/disk/shaft/spool system increase, decrease, or stay the same when the ring is moved off-axis? Explain your reasoning.
M ETHOD Q UESTIONS
To test your predicted equation, you need to determine how to calculate the moment of inertia of the system from the quantities you can measure in this problem. It is helpful to use a problem-solving strategy such as the one outlined below:
1. Draw a side view of the equipment. Draw the velocity and acceleration vectors of the weight.
Add the tangential velocity and tangential acceleration vectors of the outer edge of the spool.
Also, show the angular acceleration of the spool. What is the relationship between the acceleration of the string and the acceleration of the weight if the string is taut? What is the relationship between the acceleration of the string and the tangential acceleration of the outer edge of the spool if the string is taut?
2. Since you want to relate the moment of inertia of the system to the acceleration of the weight, you probably want to consider a dynamics approach (Newton’s second law) especially considering the torques exerted on the system. It is likely that the relationships between rotational and linear kinematics will also be involved.
3. To use torques, first draw vectors representing all of the forces that could exert torques on the ring/disk/shaft/spool system. Identify the objects that exert those forces. Draw pictures of those objects as well showing the forces exerted on them.
4. Draw a free-body diagram of the ring/disk/shaft/spool system. Show the locations of the forces acting on that system. Label all the forces. Does this system accelerate? Is there an angular acceleration? Check to see if you have all the forces on your diagram. Which of these forces can exert a torque on the system? Identify the distance from the axis of rotation to the point where each force is exerted on the system. Write down an equation that gives the torque in terms of the distance and the force that causes it. Write down Newton's second law in its rotational form for this system. Remember that the moment of inertia includes everything in the system.
5. Use Newton’s third law to relate the force of the string on the spool to the force of the spool on the string. Following the string to its other end, what force does the string exert on the weight? Make sure all the forces on the hanging weight are included in your drawing.
111

6. Draw a free-body diagram of the hanging weight. Label all the forces acting on it. Does this weight accelerate? Is there an angular acceleration? Check to see if you have all the forces on your diagram. Write down Newton's second law for the hanging weight. How do you know that the force of the string on the hanging weight is not equal to the weight of the hanging weight?
7. Is there a relationship between the two kinematic quantities that have appeared so far: the angular acceleration of the ring/disk/shaft/spool system and the acceleration of the hanging weight? To decide, examine the accelerations that you labeled in your drawing of the equipment.
8. Solve your equations for the moment of inertia of the ring/disk/shaft/spool system as a function of the mass of the hanging weight, the acceleration of the hanging weight, and the radius of the spool. Start with the equation containing the quantity you want to know, the moment of inertial of the ring/disk/shaft/spool system. Identify the unknowns in that equation and select equations for each of them from those you have collected. If those equations generate additional unknowns, search your collection for equations that contain them. Continue this process until all unknowns are accounted for. Now solve those equations for your target unknown.
Carry out this procedure for both the centered disk and the off axis disk.
E XPLORATION
THE OFF-AXIS RING IS NOT STABLE BY ITSELF! Be sure to secure the ring to the disk, and be sure that the system is on a stable base.
Practice gently spinning the ring/disk/shaft/spool system by hand. How long does it take the disk to stop rotating about its central axis? What is the average angular acceleration caused by this friction?
Make sure the angular acceleration you use in your measurements is much larger than the one caused by friction so that it has a negligible effect on your results.
Find the best way to attach the string to the spool. How much string should you wrap around the spool? How should the pulley be adjusted to allow the string to unwind smoothly from the spool and pass over the pulley? Practice releasing the hanging weight and the ring/disk/shaft/spool system.
Determine the best mass to use for the hanging weight. Try a large range. What mass will give you the smoothest motion?
Decide what measurements you need to make to determine the moment of inertia of the system from your
Prediction equation. If any major assumptions are involved in connecting your measurements to the acceleration of the weight, decide on the additional measurements that you need to make to justify them.
Outline your measurement plan.
Make some rough measurements to be sure your plan will work.
112

M EASUREMENT
Follow your measurement plan. What are the uncertainties in your measurements? (See Appendices A and B if you need to review how to determine significant figures and uncertainties.)
Don’t forge t to make the additional measurements required to determine the moment of inertia of the ring/disk/shaft/spool system from the sum of the moments of inertia of its components. What is the uncertainty in each of the measurements? What effects do the hole, the ball bearings, the groove, and the holes in the edges of the disk have on its moment of inertia? Explain your reasoning.
A NALYSIS
Determine the acceleration of the hanging weight. How does this acceleration compare to what its acceleration would be if you just dropped the weight without attaching it to the string? Explain whether or not this makes sense. Be sure to use an analysis technique that makes the most efficient use of your data and your time.
Using your Prediction equation and your measured acceleration, the radius of the spool and the mass of the hanging weight, calculate the moment of inertia (with uncertainty) of the disk/shaft/spool system.
Adding the moments of inertia of the components of the ring/disk/shaft/spool system, calculate the value (with uncertainty) of the moment of inertia of the system. What fraction of the moment of inertia of the system is due to the shaft? The disk? The ring? Explain whether or not this makes sense.
Compare the value of moment of inertia of the each of the systems from these two methods.
C ONCLUSION
Did your measurements agree with your initial prediction? Why or why not? What are the limitations on the accuracy of your measurements and analysis?
Compare the two values for the moment of inertia of the system when the ring is off-axis. Did your measurement agree with your predicted value? Why or why not?
Compare the moments of inertia of the system when the ring is centered on the disk, and when the ring is off-axis.
What effect does the off-center ring have on the moment of inertia of the ring/disk/shaft/spool system? Does the rotational inertia increase, decrease, or stay the same when the ring is moved off-axis?
113

State your result in the most general terms supported by your analysis. Did your measurements agree with your initial prediction? Why or why not? What are the limitations on the accuracy of your measurements and analysis?
114

You have been hired as part of a team to design new port facilities for Long Beach Harbor. Your assignment is to evaluate a new crane for lifting containers from the hold of a ship. The crane is a boom (a steel bar of uniform thickness) with one end attached to the ground by a hinge that allows it to rotate in the vertical plane. Near the other end of the boom is a motor driven cable that lifts a container straight up at a constant speed. The boom is supported at an angle to the horizontal by another cable. One end of the support cable is attached to the boom and the other end goes over a pulley. That other end is attached to a counterweight that hangs straight down. The pulley is supported by a mechanism that adjusts its height so the support cable is always horizontal. Your task is to determine how the angle of the boom to the horizontal changes as a function of the weight of the container being lifted. The mass of the boom, the mass of the counterweight, the attachment point of the support cable and the attachment point of the lifting cable have all been specified by the engineers.
You will test your calculations with a laboratory model of the crane.
?
Calculate the how the angle of the boom to the horizontal depends on the weight to be lifted.
E QUIPMENT
You will have a bar, a pulley, a pulley clamp, two mass hangers, a mass set and some strings.
BAR

A B
P REDICTION
Calculate the angle of the bar from the horizontal as a function of the weight of object A, with everything else fixed.
M ETHOD Q UESTIONS
1. Draw a crane similar to the one in the Equipment section. Select your coordinate system. Identify and label the masses and lengths relevant to this problem. Draw and label all the relevant forces.
2. Draw a free-body diagram for the bar showing the location of the forces acting on it. Label these forces. Choose the axis of rotation. Identify any torques on the rod.
3. Write the equation expressing Newton's second law for forces and another equation for torques.
Remember that the bar is in equilibrium.
115

4. Identify the target quantities you wish to determine. Use the equations collected in step 3 to plan a solution for the target. If there are more unknowns than equations, reexamine the previous steps to see if there is additional information about the situation that can be expressed in an additional equation. If not, see if one of the unknowns will cancel out.
E XPLORATION
Collect the necessary parts of your crane. Find a convenient place to build it.
Decide on the easiest way to determine where the center of mass is located on the bar.
Determine where to attach the lifting cable and the support cable so that the crane is in equilibrium for the weights you want to hang. Try several possibilities. If your crane tends to lean to one side or the other, try putting a vertical rod near the end of the crane to keep your crane from moving in that direction. If you do this, what effect will this vertical rod have on your calculations?
Do you think that the length of the strings for the hanging weights will affect the balance of the crane?
Why or why not?
Outline your measurement plan.
M EASUREMENT
Build your crane.
Make all necessary measurements of the configuration. Every time only change the mass of object A and determine the angle of the bar when the system is in equilibrium. Remember to adjust the height of the pulley to keep the support string horizontal that hangs the object B for each case.
Is there another configuration of the three objects that also results in a stable configuration?
A NALYSIS
Make a graph of the bar’s angle as a function of the weight of object A.
What happens to that graph if you change the mass of object B? The position of the attachment of the support cable to the bar?
C ONCLUSION
Did your crane balance as designed? What corrections did you need to make to get it to balance? Were these corrections a result of some systematic error, or was there a mistake in your prediction? In your opinion, what is the best way to construct a crane that will allow you to quickly adjust the setup so as to meet the demands of carrying various loads? Justify your answer.
116

While examining the manual starter on a snowblower, you wonder why the manufacturer chose to wrap the starter cord around a smaller diameter ring that is fastened to the disk of a flywheel instead of around the flywheel itself. When starting a snowblower, you know you need the starter system to spin as fast as possible when you pull the starter cord. Your friend suggests that the flywheel might spin faster for the same pulling effort if the cord is wrapped around a smaller diameter. To see whether this idea is correct, you decide to calculate the final angular speed of the flywheel after pulling a fixed length of cord with a constant force as a function of the radius of the object the cord is wrapped around. To test your calculation, you set up a laboratory model of the flywheel starter assembly. Unfortunately, it is difficult to keep a constant force so you make the model in the lab where the cord is pulled by the same hanging weight for different diameter.
?
Calculate the final angular speed of the flywheel after it is pulled a fixed length by the same hanging weight along a cord wrapped around different diameters.
E QUIPMENT
You will use the same equipment as in Problem #1. A length of string can be fastened and wrapped around the ring, the disk or the spool.
As the hanging weight falls, the string causes the ring/disk/shaft/spool system to rotate.
You will also have a stopwatch, a meter stick, a mass hanger, a mass set, a pulley clamp and the video analysis equipment.
P REDICTION
Calculate the final angular speed of the ring/disk/shaft/spool system caused by a cord wrapped around different diameters that is pulled by the same hanging weight through a fixed distance.
M ETHOD Q UESTIONS
117

To complete your prediction, it is useful to use a problem-solving strategy such as the one outlined below:
1. Make two side view drawings of the situation (similar to the diagram in the Equipment section), one just as the hanging mass is released, and one just as the hanging mass reaches the ground (but before it hits). Label all relevant kinematic quantities and write down the relationships that exist between them. What is the relationship between the velocity of the hanging weight and the angular velocity of the ring/disk/shaft/spool system? Label all the relevant forces.
2.
Determine the basic principles of physics that you will use and how you will use them.
Determine your system. Are any objects from outside your system interacting with your system? Write down your assumptions and check to see if they are reasonable. In particular, how will you ensure that your equipment always pulls the cord through the same length when it is wrapped around different diameters?
3.
Use dynamics to determine what you must do to the hanging weight to get the force for each diameter around which the cord is wrapped. Draw a free-body diagram of all relevant objects.
Note the acceleration of the object in the free-body diagram as a check to see if you have drawn all the forces. Write down Newton's second law for each free-body diagram either in its linear form or its rotational form or both as necessary. Use Newton’s third law to relate the forces between two free-body diagrams. If forces are equal, give them the same symbol. Solve your equations for the force that the string exerts.
4. Use the conservation of energy to determine the final angular speed of the rotating objects.
Define your system and write the conservation of energy equation for this situation:
What is the energy of the system as the hanging weight is released? What is its energy just before the hanging weight hits the floor? Is any significant energy transferred to or from the system? If so, can you determine it or redefine your system so that there is no transfer? Is any significant energy changed into internal energy of the system? If so, can you determine it or redefine your system so that there is no internal energy change?
5. Identify the target quantity you wish to determine. Use the equations collected in steps 1, 3, and 4 to plan a solution for the target. If there are more unknowns than equations, re-examine the previous steps to see if there is additional information about the situation that can be expressed in an addition equation. If not, see if one of the unknowns will cancel out.
E XPLORATION
Practice gently spinning the ring/disk/shaft/spool system by hand. How will friction affect your measurements?
Find the best way to attach the string to the spool, disk, or ring. How much string should you wrap around each? How should the pulley be adjusted to allow the string to unwind smoothly and pass over the pulley in each case? You may need to reposition the pulley when changing the position where the cord wraps. Practice releasing the weight and the ring/disk/shaft/spool system for each case.
118

Determine the best mass to use for the hanging weight. Remember this mass will be applied in every case.
Try a large range. What mass range will give you the smoothest motion?
Is the time it takes the hanging weight to fall different for the different situations? How will you determine the time taken for it to fall? Determine a good setup for each case (string wrapped around the ring, the disk, or the spool).
Decide what measurements you need to make to check your prediction. If any major assumptions are used in your calculations, decide on the additional measurements that you need to make to justify them. If you already have this data in your lab journal you don’t need to redo it , just copy it.
Outline your measurement plan.
Make some rough measurements to be sure your plan will work.
M EASUREMENT
Follow your measurement plan. What are the uncertainties in your measurements?
A NALYSIS
Determine the final angular velocity of the ring/disk/shaft/spool system for each case after the weight hits the ground. How is this angular velocity related to the final velocity of the hanging weight? Be sure to use an analysis technique that makes the most efficient use of your data and your time. If your calculation incorporates any assumptions, make sure you justify these assumptions based on data that you have analyzed.
C ONCLUSION
In each case, how do your measured and predicted values for the final angular velocity of the system compare?
Of the three places you attached the string, which produced the highest final angular velocity? Did your measurements agree with your initial prediction? Why or why not? What are the limitations on the accuracy of your measurements?
Given your results, how much does it matter where the starter cord is attached? Why do you think the manufacturer chose to wrap the cord around the ring? Explain your answers.
119

While driving around the city, your car is constantly shifting gears. You wonder how the gear shifting process works. Your friend tells you that there are gears in the transmission of your car that are rotating about the same axis. When the car shifts, one of these gear assemblies is brought into connection with another one that drives the car’s wheels. Thinking about a car starting up, you decide to calculate how the angular speed of a spinning object changes when it is brought into contact with another object at rest. To keep your calculation simple, you decide to use a disk for the initially spinning object and a ring for the object initially at rest. Both objects will be able to rotate freely about the same axis that is centered on both objects. To test your calculation you decide to build a laboratory model of the situation.
?
What is the final angular velocity of a disk with an initial angular speed after being connected with a ring initially at rest?
E QUIPMENT
You will use the same basic equipment in the previous problems.
P REDICTION
Calculate, in terms of the initial angular velocity of the disk before the ring is dropped onto the spinning disk and the characteristics of the system, the angular velocity of the disk after the ring is dropped onto the initially spinning disk.
120

PROBLEM #7: EQUILIBRIUM
M ETHOD Q UESTIONS
To complete your prediction, it is useful to use a problem solving strategy such as the one outlined below:
1. Make two side view drawings of the situation (similar to the diagram in the Equipment section), one just as the ring is released, and one after the ring lands on the disk. Label all relevant kinematic quantities and write down the relationships that exist between them. Label all relevant forces.
2. Determine the basic principles of physics that you will use and how you will use them.
Determine your system. Are any objects from outside your system interacting with your system? Write down your assumptions and check to see if they are reasonable.
3. Use conservation of angular momentum to determine the final angular speed of the rotating objects. Why not use conservation of energy or conservation of momentum? Define your system and write the conservation of angular momentum equation for this situation:
What is the angular momentum of the system as the ring is released? What is its angular momentum after the ring connects with the disk? Is any significant angular momentum transferred to or from the system? If so, can you determine it or redefine your system so that there is no transfer?
4. Identify the target quantity you wish to determine. Use the equations collected in steps 1 and 3 to plan a solution for the target. If there are more unknowns than equations, reexamine the previous steps to see if there is additional information about the situation that can be expressed in an addition equation. If not, see if one of the unknowns will cancel out.
E XPLORATION
Practice dropping the ring into the groove on the disk as gently as possible to ensure the best data. What happens if the ring is dropped off-center? What happens if the disk does not fall smoothly into the groove? Explain your answers.
Decide what measurements you need to make to check your prediction. If any major assumptions are used in your calculations, decide on the additional measurements that you need to make to justify them.
Outline your measurement plan.
Make some rough measurements to be sure your plan will work.
M EASUREMENTS
Lab VII - 121

Follow your measurement plan. What are the uncertainties in your measurements?
A NALYSIS
Determine the initial and final angular velocity of the disk from the data you collected. Be sure to use an analysis technique that makes the most efficient use of your data and your time. Using your prediction equation and your measured initial angular velocity, calculate the final angular velocity of the disk. If your calculation incorporates any assumptions, make sure you justify these assumptions based on data that you have analyzed.
C ONCLUSION
Did your measurement of the final angular velocity agree with your calculated value by prediction? Why or why not? What are the limitations on the accuracy of your measurements and analysis?
Could you have used conservation of energy to solve this problem? Why or why not? Use your data to check your answer.
122

Name and ID#: ______________________________________________________
Date performed: ________________ Day/Time section meets: ______________
Lab Partners' Names: ________________________________________________
__________________________________________________________________
__________________________________________________________________
Problem # and Title: _________________________________________________
Lab Instructor's Initials: ____________
Grading Checklist Points
LABORATORY JOURNAL:
PREDICTIONS
(individual predictions and methods questions completed in journal before each lab session)
LAB PROCEDURE
(measurement plan recorded in journal, tables and graphs made in journal as data is collected, observations written in journal)
PROBLEM REPORT:*
ORGANIZATION
(clear and readable; logical progression from problem statement through conclusions; pictures provided where necessary; correct grammar and spelling; section headings provided; physics stated correctly)
DATA AND DATA TABLES
(clear and readable; units and assigned uncertainties clearly stated)
RESULTS
(results clearly indicated; correct, logical, and well-organized calculations with uncertainties indicated; scales, labels and uncertainties on graphs; physics stated correctly)
CONCLUSIONS
(comparison to prediction & theory discussed with physics stated correctly ; possible sources of uncertainties identified; attention called to experimental problems)
TOTAL(incorrect or missing statement of physics will result in a maximum of
60% of the total points achieved; incorrect grammar or spelling will result in a maximum of 70% of the total points achieved)
BONUS POINTS FOR TEAMWORK
(as specified by course policy)
* An "R" in the points column means to rewrite that section only and return it to your lab instructor within two days of the return of the report to you.
123

In most of the laboratory problems so far objects have been moving with constant acceleration because the total force acting on that object was constant. In this set of laboratory problems the force on an object, and thus its acceleration, changes with position.
You are familiar with many objects that oscillate -- a tuning fork, the balance wheel of a mechanical watch, a pendulum, or the strings of a guitar. At the atomic level, atoms oscillate within molecules, and molecules within solids. All of these objects are subjected to forces which change with position. Springs are a common example of objects that exert this type of force.
In this lab you will study oscillatory motion caused by springs exerting a force on an object. You will use different methods to determine the strength of the force exerted by different spring configurations, and investigate what quantities determine the oscillation frequency of systems.
BJECTIVES
After successfully completing this laboratory, you should be able to:
• provide a qualitative explanation of the behavior of oscillating systems using the concepts of restoring force and equilibrium position.
• identify the physical quantities which influence the period (or frequency) of the oscillatory motion and describe this influence quantitatively.
• demonstrate a working knowledge of the mathematical description of an oscillator's motion.
• describe qualitatively the effect of additional forces on an oscillator's motion.
REPARATION
Read YOUNG AND FREEDMAN: Chapter 17.
Before coming to lab you should be able to:
• Describe the similarities and differences in the behavior of the sine and cosine functions.
• Recognized the difference between amplitude, frequency, angular frequency, and period for repetitive motion.
• Determine the force on an object exerted by a spring using the concept of a spring constant.
124

You are selecting springs for a large antique clock, and to determine the force they exert, you need to know their spring constants. One book recommends a static approach: hang objects of different weights on the spring and measure the displacement from equilibrium. Another book suggests a dynamic approach: hang an object on the end of a spring and measure its oscillation frequency.
To determine if these two approaches give you the same quantity, you calculate the "spring constant" from both sets of measurements, make the measurements, and compare.
?
Does measuring the oscillation frequency give a different spring constant than measuring a displacement from equilibrium?
E QUIPMENT
You can hang a spring from a metal rod that is fastened on a table clamp. Then you can attach an object to the hanging end of the spring. You can vary the mass of the object. A meter stick, a triple-beam balance, a stopwatch, a mass hanger, a mass set and the video analysis equipment are available for this experiment.
P REDICTIONS
1. Calculate the spring constant from the displacement of an object hung on the spring. How do you check this prediction?
2. Calculate the spring constant from the period of oscillation of an object hung on the spring.
How do you check this prediction?
M ETHOD Q UESTIONS
To complete your predictions, it is useful to apply a problem-solving strategy such as the one outlined below:
Method #1: Suppose you hang objects of several different masses on a spring and measure the vertical displacement of each object.
125

1. Make two sketches of the situation, one before you attach a mass to a spring, and one after a mass is suspended from the spring and is at rest. Draw a coordinate system and label the position where the spring is unstretched, the stretched position, the mass of the object, and the spring constant. Assume the springs are massless.
2. Draw a force diagram of a hanging object at rest. Label the forces. Newton's second law gives the equation of motion for the hanging object. Solve this equation for the spring constant.
3. Use your equation to sketch the expected shape of the graph of displacement (from the unstretched position) versus weight for the object-spring system. How is the slope of this graph related to the spring constant?
Method #2: Suppose you hang an object from the spring, start it oscillating, and measure the
period of oscillation.
1. Make a sketch of the oscillating system at a time when the object is below its equilibrium position. Draw this sketch to the side of the two sketches drawn for method #1. Identify and label this new position on the same coordinate axis.
2. Draw an force diagram of the object at this new position. Label the forces.
3. Apply Newton's second law as the equation of motion for the object at each of the above positions.
When the object is below its equilibrium position, how is the stretch of the spring from its unstretched position related to the position of the system's (spring & object) equilibrium position and its additional displacement from that equilibrium position to its new position.
4. Solve your equations for acceleration of the object as a function of the mass of the suspended object, the spring constant, and the displacement of the spring/object system from its equilibrium position. Keep in mind that acceleration is second derivative of position with respect to time.
5.
Try a periodic solution ( sin(
  t ) or law). Find the frequency cos(
  t )
) to your equation of motion (Newton's second
 that satisfies equation of motion for all times. How is the frequency of the system related to its period of oscillation?
E XPLORATION
Method #1: Select a series of masses that give a usable range of displacements. The smallest mass must be much greater than the mass of the spring to fulfill the massless spring assumption. The largest mass should not pull the spring past its elastic limit (about 60 cm). Beyond that point
you will damage the spring. Decide on a procedure that allows you to measure the displacement of the spring-object system in a consistent manner. Decide how many measurements you will need to make a reliable determination of the spring constant.
Method #2: Secure one end of the spring safely to the metal rod and select a mass that gives a regular oscillation without excessive wobbling to the hanging end of the spring. Again, the largest mass should not pull the spring past its elastic limit and the smallest mass should be much
126

greater than the mass of the spring. Practice starting the mass in vertical motion smoothly and consistently.
Practice making a video to record the motion of the spring-object system. Decide how to measure the period of oscillation of the spring-object system by video and stopwatch. How can you minimize the uncertainty introduced by your reaction time in starting and stopping the stopwatch? How many times should you measure the period to get a reliable value? How will you determine the uncertainty in the period?
M EASUREMENT
Method #1: Record the masses of different hanging objects and the corresponding displacements.
Method #2: For each hanging object, record the mass of the object. Use a stopwatch to roughly determine the period of the oscillation and then make a video of the motion of the hanging object.
Repeat the same procedure for objects with different masses.
Analyze your data as you go along so you can decide how many measurements you need to make to determine the spring constant accurately and reliably.
A NALYSIS
Method #1: Make a graph of displacement versus weight for the object-spring system. From the slope of this graph, calculate the value of the spring constant, including the uncertainty.
Method #2: Digitize your videos to determine the period of each oscillation. You may use the period by stopwatch as a predicted parameter in your fit equations. Make a graph of period (or frequency) versus mass for the object-spring system. If this graph is not a straight line, make another graph with the period as a function of mass to a power that is a straight line. From the slope of the straight-line graph, calculate the value of the spring constant, including the uncertainty.
C ONCLUSION
How do the two values of the spring constant compare? Which method is faster? Which method do you feel is the most reliable? Justify your answers.
127

Your company has bought the prototype for a new flow regulator from a local inventor. Your job is to prepare the prototype for mass production. While studying the prototype, you notice the inventor used some rather innovative spring configurations to supply the tension needed for the regulator valve. In one location the inventor had fastened two different springs side-by-side, as in
Figure A below. In another location the inventor attached two different springs end-to-end, as in
Figure B below.
To decrease the cost and increase the reliability of the flow regulator for mass production, you need to replace each spring configuration with a single spring. These replacement springs must exert the same forces when stretched the same amount as the original spring configurations.
?
What are the spring constants of two single springs that would replace the side-by-side and end-to-end spring configurations?
The spring constant for a single spring that replaces a configuration of springs is called its effective
spring constant.
E QUIPMENT
You have two different springs that have the same unstretched length, but different spring constants k
1
and k
2
. These springs can be hung vertically side-by-side (Figure A) or end-to-end
(Figure B). You will also have a meter stick, a stopwatch, a metal rod, a wooden rod, a table clamp, a mass hanger, a mass set and the video analysis equipment.
128

P REDICTIONS
1. Calculate the effective spring constant for the side-by-side configuration (Figure A) in terms of the two spring constants k
1
and k
2
.
2. Calculate the effective spring constant for the end-to-end configuration (Figure B) in terms of the two spring constants k
1
and k
2
.
Is the effective spring constant larger when the two springs are connected side-by-side or end-to-end?
Explain.
M ETHOD Q UESTIONS
To complete your predictions, it is useful to apply a problem-solving strategy such as the one outlined below. Apply the strategy first to the side-by-side configuration, then repeat for the end-to-end
configuration:
1. Make a sketch of the spring configuration similar to one of the drawings in the Equipment section (Figure A or Figure B). Draw a coordinate system and label the positions of each unstretched spring, the final stretched position of each spring, the two spring constants, and the mass of the object suspended. Assume that the springs are massless.
For the side-by-side configuration, assume that the light bar attached to the springs remains horizontal (i.e. it does not twist).
Now make a second sketch of a single (massless) spring with spring constant k' that has the same object suspended from it and the same total stretch as the combined springs. Be sure to label this second sketch.
2. Draw force diagrams of the object suspended from the combined springs and the same object suspended from the single replacement spring. Label the forces.
For the end-to-end configuration, draw an additional force diagram of a point at the connection of the two springs.
3. Apply Newton's laws as the equations of motion for the object suspended from the combined springs and the object suspended from the single replacement spring.
How is the total stretch of the combined springs related to the stretch of each of the springs?
Write equation of motion in terms of the displacements of the springs.
For the end-to-end configuration: At the connection point of the two springs, what is the force of the top spring on the bottom spring? What is the force of the bottom spring on the top spring?
4.
Solve your equations of motion for the effective spring constant (k') for the single replacement spring in terms of the two spring constants.
129

E XPLORATION
To test your predictions, you must decide how to measure each spring constant of the two springs and the effective spring constants of the side-by-side and end-to-end configurations.
From your results of Problem #1, select the best method for measuring spring constants. Justify your choice. DO NOT STRETCH THE SPRINGS PAST THEIR ELASTIC LIMIT (ABOUT 60
CM) OR YOU WILL DAMAGE THEM.
Perform an exploration consistent with your selected method. If necessary, refer back to the appropriate Exploration section of Problem #1.
Remember, the smallest mass must be much greater than the mass of the spring to fulfill the massless spring assumption. The largest mass should not pull the spring past its elastic limit.
Outline your measurement plan.
M EASUREMENT
Make the measurements that are consistent with your selected method. If necessary, refer back to the appropriate Measurement section of Problem #1. What are the uncertainties in your measurements?
A NALYSIS
Determine the effective spring constants (with uncertainties) of the side-by-side spring configuration and the end-to-end spring configuration. If necessary, refer back to Problem #1 for the analysis technique consistent with your selected method.
Determine the spring constants of the two springs. Calculate the effective spring constants (with uncertainties) of the two configurations using your Prediction equations.
How do the measured and predicted values of the effective spring constants for the two configurations compare?
C ONCLUSION
What are the effective spring constants of a side-by-side spring configuration and an end-to-end spring configuration? Which is larger? Did your measured values agree with your initial
130

predictions? Why or why not? What are the limitations on the accuracy of your measurements and analysis?
131

You have a summer job with a research group at the University. Because of your physics background, your supervisor asks you to design equipment to measure earthquake aftershocks.
The calibration sensor needs to be isolated from the earth movements, yet free to move. You decide to place the sensor on a track cart and attach a spring to both sides of the cart. You should now be able to measure the component of the aftershocks along the axis defined by the track. To make any quantitative measurements with the sensor you need to know the frequency of oscillation for the cart as a function of the spring constants and the mass of the cart.
?
What is the frequency of the oscillating cart?
E QUIPMENT
You will have an aluminum track, two adjustable end stops, two oscillation springs, a meter stick, a stopwatch, a cart and the video analysis equipment .
Cart
P REDICTION
Calculate the frequency of the cart in terms of its mass and the two spring constants.
M ETHOD Q UESTIONS
To complete your prediction, it is useful to use a problem-solving strategy such as the one outlined below:
1. Make two sketches of the oscillating cart, one at its equilibrium position, and one at some other position and time while it is oscillating. On your sketches, show the direction of the velocity and acceleration of the cart. Identify and label the known (measurable) and unknown quantities.
132

2. Draw a force diagram of the oscillating cart away from its equilibrium position. Label the forces.
3. Apply Newton's laws as the equation of motion for the cart. Consider both cases when the cart is in the equilibrium and displaced from the equilibrium position.
4.
Solve your equation for the acceleration, simplifying the equation until it is similar to the equation in YOUNG AND FREEDMAN
5.
Try a periodic solution ( sin(
  t ) or cos(
  t ) ) to your equation of motion (Newton's second law). Find the frequency

that satisfies equation of motion for all times. How is the frequency of the system related to its period of oscillation? Calculate frequency of the system as a function of the mass of the cart and the two spring constants.
E XPLORATION
Decide the best method to determine the spring constants based on your results of Problem #1.
DO NOT STRETCH THE SPRINGS PAST THEIR ELASTIC LIMIT (ABOUT 60 CM) OR YOU
WILL DAMAGE THEM.
Find the best place for the adjustable end stop on the track. Do not stretch the springs past 60 cm, but stretch them enough so they oscillate the cart smoothly.
Practice releasing the cart smoothly. You may notice the amplitude of oscillation decreases.
What’s the reason for it? Does this affect the period of oscillation ?
M EASUREMENT
Determine the spring constants. Record these values. What is the uncertainty in these measurements?
Record the mass of the cart. Use a stopwatch to roughly determine the period of oscillation and then make a video of the motion of the oscillating cart. You should record at least 3 cycles.
A NALYSIS
Analyze your video to find the period of oscillation. Calculate the frequency (with uncertainty) of the oscillations from your measured period.
Calculate the frequency (with uncertainty) using your Prediction equation.
How does your measured frequency compare with your predicted frequency?
133

C ONCLUSION
What is the frequency of the oscillating cart? Did your measured frequency agree with your predicted frequency? Why or why not? What are the limitations on the accuracy of your measurements and analysis? What is the effect of friction?
If you completed Problem #2: What is the effective spring constant of this configuration? How does it compare with the effective spring constants of the side-by-side and end-to-end configurations?
134

You are the technical advisor for the next Bruce Willis action adventure movie, Die Even Harder, which is filmed on Mt. Sac. The script calls for a spectacular stunt. Bruce Willis is dangling over a cliff from a long rope that is tied to the Bad Guy, who is on the ice-covered ledge of the cliff. The
Bad Guy's elastic parachute line is tangled in a tree located several feet from the edge of the cliff.
Bruce and the Bad Guy are in simple harmonic motion, and at the top of his motion, Bruce unsuccessfully tries to grab for the safety of the cliff edge while the Bad Guy reaches for his discarded knife. The script calls for Bad Guy to cut the rope just as Bruce reaches the top of his motion again.
The problem is that it is expensive to have Bruce hanging from the rope while the crew films close-ups of the Bad Guy. However, the stunt double weighs more than Bruce. The director wants to know if the stunt double will have a different motion than Bruce.
?
Does the frequency of oscillation depend on the mass of the hanging object?
You decide to solve this problem by modeling the situation with the equipment described below.
Filming stops while you are working on this problem. If you are wrong, you are sure to lose your job.
E QUIPMENT
You have an aluminum track with an adjustable end stop, a pulley, a pulley clamp, an oscillation spring, a cart, some strings, a mass hanger, a mass set, a meter stick, a stopwatch and the video analysis equipment.
The track represents the ice-covered ledge of the cliff, the adjustable end-stop represents the tree, the spring represents the elastic cord, the cart represents the Bad Guy, the string represents the rope and the hanging object represents Bruce Willis or his stunt double in this problem.
Cart
A
135

P REDICTION
Calculate how the frequency of oscillation of the system depends on the mass of the object hanging over the table.
Use your equation to sketch the expected shape of a graph of the oscillation frequency versus hanging mass. Will the frequency increase, decrease or stay the same as the hanging mass increases?
M ETHOD Q UESTIONS
To complete your prediction, it is useful to use a problem-solving strategy such as the one outlined below:
1. Make sketches of the situation when the cart and hanging object are at their equilibrium position and at some other time while the system is oscillating. On your sketches, show the direction of the acceleration of the cart and hanging object. Identify and label the known
(measurable) and unknown quantities.
2. Draw separate force diagrams of the oscillating cart and hanging object. Label the forces.
3. Independently apply Newton's laws to the cart and to the hanging object.
4.
Solve your equations for the acceleration, simplifying the equation until it is similar to the equation in YOUNG AND FREEDMAN
5.
Try a periodic solution ( sin(
  t ) or law). Find the frequency cos(
  t )
) to your equation of motion (Newton's second
 that satisfies equation of motion for all times. How is the frequency of the system related to its period of oscillation? Calculate frequency of the system as a function of the mass of the cart, the mass of the hanging object, and the spring constant.
Now you can complete your prediction. Use your equation to sketch the expected shape of the graph of oscillation frequency versus the hanging object's mass.
E XPLORATION
If you do not know the spring constant of your spring, you should decide the best way to determine the spring constant based on your results of Problem #1.
Find the best place for the adjustable end stop on the track. DO NOT STRETCH THE SPRING
PAST 60 CM OR YOU WILL DAMAGE IT, but stretch it enough so the cart and hanging mass oscillate smoothly.
Determine the best range of hanging masses to use.
136

Practice releasing the cart and hanging mass smoothly and consistently. You may notice the amplitude of oscillation decreases. What’s the reason for it? Does this affect the period of oscillation?
M EASUREMENT
If necessary, determine the spring constant of your spring. What is the uncertainty in your measurement?
For each hanging object, record the masses of the cart and the hanging object. Use a stopwatch to roughly determine the period of oscillation and then make a video of the oscillating cart for each hanging object. You should record at least 3 cycles for each video.
Analyze your data as you go along, so you can determine the size and number of different hanging masses you need to use.
Collect enough data to convince yourself and others of your conclusion about how the oscillation frequency depends on the hanging mass.
A NALYSIS
For each hanging object, digitize the video to get the period of oscillation and then calculate the oscillation frequency (with uncertainty) from your measured period.
Graph the frequency versus the hanging object's mass. On the same graph, show your predicted relationship.
What are the limitations on the accuracy of your measurements and analysis? Over what range of values does the measured graph match the predicted graph best? Do the two curves start to diverge from one another? If so, where? What does this tell you about the system?
C ONCLUSION
Does the oscillation frequency increase, decrease or stay the same as the hanging object's mass increases? State your result in the most general terms supported by your analysis.
What will you tell the director? Do you think the motion of the actors in the stunt will change if the heavier stunt man is used instead of Bruce Willis? How much heavier than Bruce would the stunt man have to be to produce a noticeable difference in the oscillation frequency of the actors?
Explain your reasoning in terms the director would understand so you can collect your paycheck.
137

You are now prepared to calibrate your seismic detector (from Problem #3). You need to determine how the amplitude of the oscillations of the detector will vary with the frequency of the earthquake aftershocks. So you replace the end stop on the track with a device that moves the end of the spring back and forth, simulating the earth moving beneath the track. The device, called a mechanical oscillator, is designed so you can change its frequency of oscillation.
?
How will the amplitude of the oscillations vary with frequency?
E QUIPMENT
You will use your seismic detector apparatus and a mechanical oscillator.
The mechanical oscillator is somewhat like a loudspeaker with one end of a metal rod attached to the center, and the other end of the rod is attached to one of the springs. The oscillator is connected to a function generator which causes the rod to oscillate back and forth with adjustable frequencies which can be read from the display on the function generator.
You will also have a meter stick, a stopwatch, a metal rod, a table clamp, a stop end, two oscillation springs and two banana cables.
P REDICTION
Make your best-guess sketch of what you think a graph of the amplitude of the cart versus the frequency of the mechanical driver will look like. Assume the mechanical oscillator has a constant amplitude of a few millimeters.
138

PROBLEM #5: DRIVEN OSCILLATIONS
E XPLORATION
Examine the mechanical oscillator. Mount it at the end of the aluminum track, using the clamp and metal rod so its shaft is aligned with the cart's motion. Connect it to the function generator, using the output marked Lo (for ``low impedance''). Use middle or maximum amplitude to observe the oscillation of the cart at the lowest frequency possible.
Determine the accuracy of the digital display on the frequency generator by timing one of the lower frequencies.
Devise a scheme to accurately determine the amplitude of a cart on the track, and practice the technique. For each new frequency, should you restart the cart at rest?
When the mechanical oscillator is at or near the undriven frequency (natural frequency) of the cart-spring system, try to simultaneously observe the motion of the cart and the shaft of the mechanical oscillator. What is the relationship? What happens when the oscillator’s frequency is twice as large as the natural frequency?
M EASUREMENT
If you do not know the natural frequency of your system when it is not driven, determine it using the technique of Problem #3.
Collect enough cart amplitude and oscillator frequency data to test your prediction. Be sure to collect several data points near the natural frequency of the system.
A NALYSIS
Make a graph the oscillation amplitude of the cart versus oscillator frequency. Is this the graph you had anticipated? Where is it different? Why? What is the limitation on the accuracy of your measurements and analysis?
C ONCLUSION
Can you explain your results? Is energy conserved? What will you tell your boss about your design for a seismic detector?
139

Name and ID#: ______________________________________________________
Date performed: ________________ Day/Time section meets: ______________
Lab Partners' Names: ________________________________________________
__________________________________________________________________
__________________________________________________________________
Problem # and Title: _________________________________________________
Lab Instructor's Initials: ____________
Grading Checklist Points
LABORATORY JOURNAL:
PREDICTIONS
(individual predictions and methods questions completed in journal before each lab session)
LAB PROCEDURE
(measurement plan recorded in journal, tables and graphs made in journal as data is collected, observations written in journal)
PROBLEM REPORT:*
ORGANIZATION
(clear and readable; logical progression from problem statement through conclusions; pictures provided where necessary; correct grammar and spelling; section headings provided; physics stated correctly)
DATA AND DATA TABLES
(clear and readable; units and assigned uncertainties clearly stated)
RESULTS
(results clearly indicated; correct, logical, and well-organized calculations with uncertainties indicated; scales, labels and uncertainties on graphs; physics stated correctly)
CONCLUSIONS
(comparison to prediction & theory discussed with physics stated correctly ; possible sources of uncertainties identified; attention called to experimental problems)
TOTAL(incorrect or missing statement of physics will result in a maximum of
60% of the total points achieved; incorrect grammar or spelling will result in a maximum of 70% of the total points achieved)
BONUS POINTS FOR TEAMWORK
(as specified by course policy)
* An "R" in the points column means to rewrite that section only and return it to your lab instructor within two days of the return of the report to you.
140

Calculators make it possible to get an answer with a huge number of figures. Unfortunately many of them are meaningless. For instance if you needed to split $1.00 among three people, you could never give them each exactly
$0.333333… The same is true for measurements. If you use a meter stick with millimeter markings to measure the length of a key, as in figure A-1, you could not measure more precisely than a quarter or half or a third of a mm. Reporting a number like 5.8142712 cm would not only be meaningless, it would be misleading.
Figure A-1
In your measurement, you can precisely determine the distance down to the nearest millimeter and then improve your precision by estimating the next figure. It is always assumed that the last figure in the number recorded is uncertain. So, you would report the length of the key as 5.81 cm. Since you estimated the 1, it is the uncertain figure. If you don't like estimating, you might be tempted to just give the number that you know best, namely 5.8 cm, but it is clear that
5.81 cm is a better report of the measurement.
An estimate is always necessary to report the most precise measurement. When you quote a measurement, the reader will always assume that the last figure is an estimate. Quantifying that estimate is known as estimating
uncertainties. Appendix B will illustrate how you might use those estimates to determine the uncertainties in your measurements.
141
What are significant figures?
The number of significant figures tells the reader the precision of a measurement. Table
A-1 gives some examples.
Table A-1
Length
(centimeters)
12.74
11.5
1.50
1.5
12.25345
0.8
0.05
Number of
Significant
Figures
4
3
3
2
7
1
1
One of the things that this table illustrates is that not all zeros are significant. For example, the zero in 0.8 is not significant, while the zero in 1.50 is significant. Only the zeros that appear after the first non-zero digit are significant.
A good rule is to always express your values in scientific notation. If you say that your friend lives 143 m from you, you are saying that you are sure of that distance to within a few meters (3 significant figures). What if you really only know the distance to a few tens of meters (2 significant figures)? Then you need to express the distance in scientific notation 1.4 x 10 2 m.
Is it always better to have more figures?
Consider the measurement of the length of the key shown in Figure A-1. If we have a scale with ten etchings to every millimeter, we could use a microscope to measure the spacing to the nearest tenth of a millimeter and guess at the one hundredth millimeter. Our measurement could be 5.814 cm with the

APPENDIX A: SIGNIFICANT FIGURES uncertainty in the last figure, four significant figures instead of three. This is because our improved scale allowed our estimate to be more precise. This added precision is shown by more significant figures. The more significant figures a number has, the more precise it is.
How do I use significant figures in calculations?
When using significant figures in calculations, you need to keep track of how the uncertainty propagates. There are mathematical procedures for doing this estimate in the most precise manner. This type of estimate depends on knowing the statistical distribution of your measurements. With a lot less effort, you can do a cruder estimate of the uncertainties in a calculated result. This crude method gives an overestimate of the uncertainty but it is a good place to start. For this course this simplified uncertainty estimate (described in Appendix B and below) will be good enough.
Addition and subtraction
When adding or subtracting numbers, the number of decimal places must be taken into account.
The result should be given to as many decimal places as the term in the sum that is given to the
smallest number of decimal places.
Examples:
Addition
6.242
+4.23
+0.013
10.485
Subtraction
5.875
-3.34
2.535
10.49 2.54
The uncertain figures in each number are shown in bold-faced type.
Multiplication and division
When multiplying or dividing numbers, the number of significant figures must be taken into account.
The result should be given to as many significant figures as the term in the product that is given to the smallest number of significant figures.
The basis behind this rule is that the least accurately known term in the product will dominate the accuracy of the answer.
As shown in the examples, this does not always work, though it is the quickest and best rule to use. When in doubt, you can keep track of the significant figures in the calculation as is done in the examples.
Examples:
Multiplication
15.84 x 2.5
7920
3168
39.600
40 69
Division
17.27 x 4.0
69.080
117
23)2691
23
39
23
161
161
1.2 x 10 2
25
75)1875
150
375
375
2.5 x 10 1
A - 142

APPENDIX A: SIGNIFICANT FIGURES
PRACTICE EXERCISES
1. Determine the number of significant figures of the quantities in the following table:
Length
(centimeters)
17.87
Number of
Significant
Figures
0.4730
17.9
0.473
18
0.47
1.34 x 10 2
2.567x 10 5
2.0 x 10 10
1.001
1.000
1
1000
1001
2.
Add: 121.3 to 6.7 x 10 2 :
[Answer: 121.3 + 6.7 x 10 2 = 7.9 x 10 2 ]
3. Multiply: 34.2 and 1.5 x 10 4
[Answer: 34.2 x 1.5 x 10 4 = 5.1 x 10 5 ]
144

APPENDIX B: SIGNIFICANT FIGURES
How tall are you? How old are you? When you answered these everyday questions, you probably did it in round numbers such as "five foot, six inches" or "nineteen years, three months." But how true are these answers?
Are you exactly 5' 6" tall? Probably not. You esti mated your height at 5’ 6" and just reported two significant figures. Typically, you round your height to the nearest inch, so that your actual height falls somewhere between 5' 5½" and 5' 6½" tall, or 5' 6" ± ½". This ± ½" is the
uncertainty, and it informs the reader of the precision of the value 5' 6".
What is uncertainty?
Whenever you measure something, there is always some uncertainty. There are two categories of uncertainty: systematic and
random.
(1) Systematic uncertainties are those which consistently cause the value to be too large or too small. Systematic uncertainties include such things as reaction time, inaccurate meter sticks, optical parallax and miscalibrated balances. In principle, systematic uncertainties can be eliminated if you know they exist.
(2) Random uncertainties are variations in the measurements that occur without a predictable pattern. If you make precise measurements, these uncertainties arise from the estimated part of the measurement.
Random uncertainty can be reduced, but never eliminated. We need a technique to report the contribution of this uncertainty to the measured value.
How do I determine the uncertainty?
This Appendix will discuss two basic techniques for determining the uncertainty:
145
estimating the uncertainty and measuring the
average deviation. Which one you choose will depend on your need for precision. If you need a precise determination of some value, the best technique is to measure that value several times and use the average deviation as the uncertainty. Examples of finding the average deviation are given below.
How do I estimate uncertainties?
If time or experimental constraints make repeated measurements impossible, then you will need to estimate the uncertainty. When you estimate uncertainties you are trying to account for anything that might cause the measured value to be different if you were to take the measurement again. For example, suppose you were trying to measure the length of a key, as in Figure B-1.
Figure B-1
If the true value was not as important as the magnitude of the value, you could say that the key’s length was 6cm, give or take 1cm. This is a crude estimate, but it may be acceptable. A better estimate of the key’s length, as you saw in Appendix A, would be 5.81cm. This tells us that the worst our measurement could be off is a fraction of a mm. To be more precise, we can estimate that fraction to be about a third of a mm, so we can say the length of the key is 5.81
± 0.03 cm.

APPENDIX B: ACCURACY, PRECISION AND UNCERTAINTY
Another time you may need to estimate uncertainty is when you analyze video data.
Figures B-2 and B-3 show a ball rolling off the edge of a table. These are two consecutive frames, separated in time by 1/30 of a second.
Figure B-2
Figure B-3
The exact moment the ball left the table lies somewhere between these frames. We can estimate that this moment occurs midway between them ( t

10 1
60 s
). Since it must occur at some point between them, the worst our estimate could be off by is 1
60 s
. We can therefore say the time the ball leaves the table is t

10 1
60

1
60 s .
How do I find the average deviation?
If estimating the uncertainty is not good enough for your situation, you can experimentally determine the un-certainty by making several measure-ments and calculating the average deviation of those measurements.
To find the average deviation: (1) Find the average of all your measurements; (2) Find the absolute value of the difference of each measurement from the average (its deviation);
(3) Find the average of all the deviations by adding them up and dividing by the number of measurements. Of course you need to take enough measure-ments to get a distribution for which the average has some meaning.
In example 1, a class of six students was asked to find the mass of the same penny using the same balance. In example 2, another class measured a different penny using six different balances. Their results are listed below:
Class 1
Penny A massed by six different students on the same balance.
Mass (grams)
3.110
3.125
3.120
3.126
3.122
3.120
3.121 average.
The deviations are: 0.011g, 0.004g,
0.001g, 0.005g, 0.001g, 0.001g
Sum of deviations: 0.023g
Average deviation:
(0.023g)/6 = 0.004g
Mass of penny A: 3.121 ± 0.004g
Class 2
Penny B massed by six different students on six different balances
Mass (grams)
3.140
3.133
3.144
3.118
3.126
3.125
3.131 average
146

APPENDIX B: ACCURACY, PRECISION AND UNCERTAINTY
The deviations are: 0.009g, 0.002g,
0.013g, 0.013g, 0.005g, 0.006g
Sum of deviations: 0.048g
Average deviation:
(0.048g)/6= 0.008g
Mass of penny B: 3.131 ± 0.008g
However you choose to determine the uncertainty, you should always state your method clearly in your report. For the remainder of this appendix, we will use the results of these two examples.
How do I know if two values are the same?
If we compare only the average masses of the two pennies we see that they are different. But now include the uncertainty in the masses.
For penny A, the most likely mass is somewhere between 3.117g and 3.125g. For penny B, the most likely mass is somewhere between 3.123g and 3.139g. If you compare the ranges of the masses for the two pennies, as shown in Figure B-4, they just overlap.
Given the uncertainty in the masses, we are able to conclude that the masses of the two pennies could be the same. If the range of the masses did not overlap, then we ought to conclude that the masses are probably different.
Figure B-4
Which result is more precise?
Mass of pennies (in grams) with uncertainties
147
Suppose you use a meter stick to measure the length of a table and the width of a hair, each with an uncertainty of 1 mm. Clearly you know more about the length of the table than the width of the hair. Your measurement of the table is very precise but your measurement of the width of the hair is rather crude. To express this sense of precision, you need to calculate the percentage uncertainty. To do this, divide the uncertainty in the measurement by the value of the measurement itself, and then multiply by 100%. For example, we can calculate the precision in the measurements made by class 1 and class 2 as follows:
Precision of Class 1's value:
(0.004 g ÷ 3.121 g) x 100% = 0.1 %
Precision of Class 2's value:
(0.008 g ÷ 3.131 g) x 100% = 0.3 %
Class 1's results are more precise. This should not be surprising since class 2 introduced more uncertainty in their results by using six different balances instead of only one.
Which result is more accurate?
Accuracy is a measure of how your measured value compares with the real value. Imagine that class 2 made the measurement again using only one balance. Unfortunately, they chose a balance that was poorly calibrated. They analyzed their results and found the mass of penny B to be 3.556 ± 0.004 g. This number is more precise than their previous result since the uncertainty is smaller, but the new measured value of mass is very different from their previous value. We might conclude that this new value for the mass of penny B is different, since the range of the new value does not overlap the range of the previous value. However, that conclusion would be
wrong since our uncertainty has not taken into account the inaccuracy of the balance. To

APPENDIX B: ACCURACY, PRECISION AND UNCERTAINTY determine the accuracy of the measurement, we should check by measuring something that is known. This procedure is called calibration, and it is absolutely necessary for making accurate measurements.
Be cautious! It is possible to make measurements that are extremely precise and, at the same time, grossly inaccurate.
How can I do calculations with values that have uncertainty?
When you do calculations with values that have uncertainties, you will need to estimate
(by calculation) the uncertainty in the result.
There are mathematical techniques for doing this, which depend on the statistical properties of your measurements. A very simple way to estimate uncertainties is to find the largest
possible uncertainty the calculation could yield.
This will always overestimate the uncertainty
of your calculation, but an overestimate is better than no estimate. The method for performing arithmetic operations on quantities with uncertainties is illustrated in the following examples:
Addition:
(3.131 ± 0.008 g) + (3.121 ± 0.004 g)
= ?
Multiplication:
(3.131 ± 0.013 g) x (6.1 ± 0.2 cm) =
?
First find the sum of the values:
3.131 g + 3.121 g = 6.252 g
Next find the largest possible value:
3.139 g + 3.125 g = 6.264 g
The uncertainty is the difference between the two:
6.264 g – 6.252 g = 0.012 g
0.004 g + 0.008 g = 0.012 g
Answer: 6.252 ± 0.012 g.
Note: This uncertainty can be found by simply adding the individual uncertainties:
First find the product of the values:
3.131 g x 6.1 cm = 19.1 g-cm
Next find the largest possible value:
3.144 g x 6.3 cm = 19.8 g-cm
The uncertainty is the difference between the two:
19.8 g-cm - 19.1 g-cm = 0.7 g-cm
Answer: 19.1 ± 0.7g
-cm.
Note: The percentage uncertainty in the answer is the sum of the individual uncertainties: percentage
0 .
013
3 .
131

100 %

0 .
2
6 .
1

100 %

0 .
7
19 .
1

100 %
148

APPENDIX B: ACCURACY, PRECISION AND UNCERTAINTY
Subtraction: Division:
(3.131 ± 0.008 g) – (3.121 ± 0.004 g)
= ?
First find the difference of the values:
(3.131 ± 0.008 g) ÷ (3.121 ± 0.004 g) = ?
First divide the values:
3.131 g  3.121 g = 1.0032
Next find the largest possible value:
3.131 g - 3.121 g = 0.010 g
3.139 g ÷ 3.117 g = 1.0071
Next find the largest possible value:
3.139 g – 3.117 g = 0.022 g
The uncertainty is the difference between the two:
1.0071 - 1.0032 = 0.0039
The uncertainty is the difference between the two:
0.022 g – 0.010 g = 0.012 g
Answer: 0.010±0.012 g.
0.004 g + 0.008 g = 0.012 g
Note: This uncertainty can be found by simply adding the individual uncertainties:
Answer: 1.003 ± 0.004
Note: The percentage uncertainty in the answer is the sum of the individual percentage uncertainties:
Notice also, that zero is included in this range, so it is possible that there is no difference in the masses of the pennies, as we saw before.
The same ideas can be carried out with more complicated calculations. Remember this will always give you an overestimate of your uncertainty. There are other calculational techniques, which give better estimates for uncertainties. If you wish to use them, please discuss it with your instructor to see if they are appropriate.
0 .
008

100 %

0 .
004

100 %

0 .
0039

3 .
131 3 .
121 1 .
0032
Notice also, the largest possible value for the numerator and the smallest possible value for the denominator gives the largest result.
100 %
These techniques help you estimate the random uncertainty that always occurs in measurements. They will not help account for mistakes or poor measurement procedures.
There is no substitute for taking data with the utmost of care. A little forethought about the possible sources of uncertainty can go a long way in ensuring precise and accurate data
PRACTICE EXERCISES:
B-1. Consider the following results for different experiments. Determine if they agree with the accepted result listed to the right. Also calculate the precision for each result. a) g = 10.4 ± 1.1 m/s 2 g = 9.8 m/s 2 b) T = 1.5 ± 0.1 sec c) k = 1368 ± 45 N/m
T = 1.1 sec k = 1300 ± 50 N/m
Answers: a) Yes, 11%; b) No, 7%; c) Yes, 3.3%
149

APPENDIX B: ACCURACY, PRECISION AND UNCERTAINTY
B-2. The area of a rectangular metal plate was found by measuring its length and its width. The length was found to be 5.37 ± 0.05 cm. The width was found to be 3.42 ± 0.02 cm. What is the area and the average deviation?
Answer: 18.4 ± 0.3 cm 2
B-3. Each member of your lab group weighs the cart and two mass sets twice. The following table shows this data. Calculate the total mass of the cart with each set of masses and for the two sets of masses combined.
Cart
(grams)
Mass set 1
(grams)
Mass set 2
(grams)
201.3
201.5
202.3
202.1
199.8
200.0
98.7
98.8
96.9
97.1
98.4
98.6
95.6
95.3
96.4
96.2
95.8
95.6
Answers:
Cart and set 1: 299.3±1.6 g.
Cart and set 2: 297.0±1.2 g.
Cart and both sets: 395.1±1.9 g.
150

APPENDIX C: GRAPHING
One of the most powerful tools used for data presentation and analysis is the graph. Used properly, graphs are an important guide to understanding the results of an experiment.
They are an easy way to make sense out of your data. Therefore, it is important to graph your data as you take it.
Loggerpro  will often provide you with a graph, but on occasion you will have to make your own. Good graphing practice, as outlined below, is a way to save time and effort while solving a problem in the laboratory.
How do I make a graph?
1. Accurate graphs are drawn on graph paper. Even if you are just making a quick sketch for yourself, it will save you time and effort to use graph paper. That is why every page of your lab journal is a piece of graph paper. Make sure to graph your data as you take it. Never put off drawing a graph until the end.
2. Every graph should have a title to indicate the data it represents. In a large collection of graphs, it is difficult to keep one graph distinct from another without clear, concise titles.
3. The axes of the graph should take up at least half a page. Give yourself plenty of space so that you can see the pattern of the data as it is developing. Both axes should be labeled to show the values being graphed and their appropriate units.
4. The scales on the graphs should be chosen so that the data occupies most of the space of your graph. You do not need to include zero on your scales unless it is important to the interpretation of the graph.
152
5. If you have more than one set of data on a set of axes, be sure to label each set to avoid confusion later.
How do I plot data and uncertainties?
Another technique that makes data analysis easier is to record all your data in a table.
A useful data table will always include a title and column headings, so that you will not forget what all the numbers mean. The column headings should include the units of the quantities listed, and usually serve as the labels for the axes of your graph. For example, look at Table C-1 and Graph C-1 on the next page. This is a position-versus-time graph drawn for a hypothetical situation.
The uncertainty for each data point is shown on the graph as a line representing a range of possible values with the principal value at the center. The lines are called error bars, and they are useful in determining if your data agrees with your prediction. Any curve
(function) that represents your data should pass through your error bars.
Table C-1:
Position vs. Time: Exercise 3, Run 1
Time/sec Position/cm
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
10 ± 5
20 ± 5
30 ± 5
39 ± 5
49 ± 5
59 ± 5
69 ± 5
79 ± 5
89 ± 5
98 ± 5

APPENDIX C: GRAPHING
Graph C-1:
What is a best fit straight line?
If your data points appear to lie on a straight line, then use a clear straight edge and draw a line through all the error bars, making sure the line has as many principal values beneath it as above it. The line does not need to touch any of the principle values. Do not connect the dots.
When done correctly, this straight line represents the function that best fits your data.
You can read the slope and intercept from your graph. These quantities usually have important physical interpretations. Some computer programs, such as Excel or Cricket Graph, will determine the best straight line for your data and compute its slope, intercept, and their uncertainties. You should check with your lab instructor, before you use a graphing program to see if it is appropriate. As an example, look at
Graph C-1. The dotted lines are possible linear fits, but the solid line is the best fit. Once you have found the best-fit line, you should determine the slope from the graph and record its value.
153
How do I find the slope of a line?
The slope of a line is defined as the ratio of the change in a line's ordinate (vertical axis) to the change in the abscissa (horizontal axis), or the
"rise" divided by "run." Your text explains slope. For Graph C-1, the slope of the best-fit line will be the change in the position of the object divided by the time interval for that change in position.
To find the value of the slope, look carefully at your best-fit line to find points along the line that have coordinates that you can identify. It is usually not a good idea to use your data points for these values, since your line might not pass through them exactly. For example, the best-fit line on Graph C-1 passes through the points
(0.15 sec, 15 cm) and (1.10 sec, 110 cm). This means the slope of the line is:
Slope =
110
1 .
10 cm
 sec

15 cm
0 .
15 sec
= Ошибка !
= 100 Ошибка !
Note that the slope of a position-versus-time graph has the units of velocity.
How do I find the uncertainty in the slope of a line?
Look at the dotted lines in Graph C-1. These lines are the largest and smallest values of the slopes that can realistically fit the data. The lines run through the extremes in the uncertainties and they represent the largest and smallest possible slopes for lines that fit the data.
You can extend these lines and compute their slopes. These are your uncertainties in the determination of the slope. In this case, it would be the uncertainty in the velocity.
How do I get the slope of a curve that is not a straight line?

APPENDIX C: GRAPHING
The tangent to a point on a smooth curve is just the slope of the curve at that point. If the curve is not a straight line, the slope will change from point to point along the curve.
To draw a tangent line at any point on a smooth curve, draw a straight line that only touches the curve at the point of interest, without going inside the curve. Try to get an equal amount of space between the curve and the tangent line on both sides of the point of interest.
The tangent line that you draw needs to be long enough to allow you to easily determine its slope. You will also need to determine the uncertainty in the slope of the tangent line by considering all other possible tangent lines and selecting the ones with the largest and smallest slopes. The slopes of these lines will give the uncertainty in the slope of the tangent line.
Notice that this is exactly like finding the uncertainty of the slope of a straight line.
How do I "linearize" my data?
A straight-line graph is the easiest graph to interpret. By seeing if the slope is positive, negative, or zero you can quickly determine the relationship between two measurements. But not all the relationships in nature produce straight-line graphs. However, if we have a theory that predicts how one measured quantity
(e.g., position) depends on another (e.g., time) for the experiment, we can make the graph be a straight line. To do this, you make a graph with the appropriate function of one quantity on one axis (e.g., time squared) and the other quantity
(e.g., position) on the other axis. This is called
"linearizing" the data.
For example, if a rolling cart undergoes constant acceleration, the position-versus-time graph is curved. In fact, our theory tells us that the curve should be a parabola. To be concrete we will
154 assume that your data starts at a time when the initial velocity of the cart was zero. The theory predicts that the motion is described by x =
½at 2 . To linearize this data, you square the time and plot position versus time squared. This graph should be a straight line with a slope of
0.5a. Notice that you can only linearize data if you know, or can guess, the relationship between the measured quantities involved.
How do I interpret graphs from Loggerpro  ?
Graphs C-2 through C-4 were produced with
Loggerpro  . They give the horizontal position as a function of time. Even though Loggerpro  draws the graphs for you, it usually does not label the axes and never puts in the uncertainty.
You must add these to the graphs yourself.
One way to estimate the uncertainty is to observe how much the data points are scattered from a smooth behavior. By estimating the average scatter, you have a fair estimate of the data’s uncertainty. Graph C -2 shows position versus time for a moving cart. The vertical axis is position in cm and the horizontal axis is time in seconds. See if you can label the axes appropriately and estimate the uncertainty for the data.
Graph C-2

Finding the slope of any given curve with
Loggerpro  should be easy after you have chosen the mathematical equation of the best-fit curve. Graphs C-3 and C-4 each show one possible line to describe the same data. From looking at these graphs, the estimate of the data’s uncertainty is less than the diameter of the circles representing that data.
Graph C-3
APPENDIX C: GRAPHING
Graph C-4
The line touching the data points in Graph C-3 has a slope of 0.38 m/sec. This is the minimum possible slope for a line that fits the given data.
The line touching the data points in Graph C-4 shows a slope of 0.44 m/sec. This is the
maximum possible slope for a line that fits the given data.
You can estimate the uncertainty by calculating the average difference of these two slopes from that of the best-fit line. Therefore the uncertainty determined by graphs C-3 and C-4 is + 0.03 m/sec.
155

PRACTICE EXERCISES:
Explain what is wrong with each of the graphs below:
156

APPENDIX C: GRAPHING
There is no set length for a problem report but experience shows good reports are typically four pages long. Graphs and photocopies of your lab journal make up additional pages. Complete reports will include the terminology and the mathematics relevant to the problem at hand. Your report should be a clear, concise, logical, and honest interpretation of your experience. You will be graded based on how well you demonstrate your understanding of the physics. Because technical communication is so important, neatness, and correct grammar and spelling are required and will be reflected on your grade.
Note: As with Problem 1 of Lab 1, the double vertical bars indicate an explanation of that part of the report. These comments are not part of the actual report.
Usually a lab report consists of the following parts:
Title,
-
Statement of the Problem,
Prediction,
-
Experiment and Results,
Conclusions,
-
References.
In this appendix there are three sample lab reports, which are based on the actual lab reports written by students who took this course before. Our main goal of putting them here is to give you some information about writing lab reports. Basically, any lab report is a simplified version of a research paper or technical report. Therefore, it is written in a scientific style and it has the same main parts as a research paper/technical report. As you go from #1 to #3, you’ll notice the i mprovement in the quality of the samples.
This is typical level of improvement of the students taking this course. Your goal is to be able to write such lab reports as sample #3 after completing this course.
It’s useful to read the sample lab reports afte r completing the following laboratories in class:
-
Sample # 1 after Projectile Motion;
-
Sample #2 after Conservation of Energy;
-
Sample #3 after Rotational Motion.
157

Write a descriptive title with your name, names of partners, date performed and Professor name.
Performed by: Roman Ivanov , Tom Johnson and Mun Tiang
In a sentence or two, state the problem you are trying to solve. List the equipment you will use and the reasons for selecting such equipment.
The problem was to determine the dependence of the time of flight of a projectile on its initial horizontal velocity. We rolled an aluminum ball down a ramp and off the edge of a table starting from rest at two different positions along the ramp. Starting from the greater height up the ramp meant the ball had a larger horizontal velocity when it rolled along the table. Since the table was horizontal, that was the horizontal velocity when it entered the air. See Figure 1 from my lab journal for a picture of the set-up.
We made two movies with the video equipment provided, one for a fast rolling ball and one for a slower one. These movies were analyzed with Loggerpro  to study the projectile’s motion in the hori zontal and vertical directions.
This is the part of the lab where you try to predict the outcome of the experiment based on your general knowledge of Physics. If your predictions were wrong and you understood it during the lab, write correct ones in your lab report. Also, attach initial predictions to your lab report and explain what was wrong. If you still have problems with the prediction, ask your Professor or go to your Professors office hours for some help. Generally, predictions are based on the fundamental laws or principles. Therefore, refer to these laws as a starting point of your predictions.
Our group predicted that the time the ball took to hit the ground once it left the table would be greater if the horizontal velocity were greater. We have observed that the faster a projectile goes initially, the longer its trajectory. Since the gravitational acceleration is constant, we reasoned that the ball would take more time to travel a larger distance. However, after completing the lab we understood that horizontal and vertical motion is independent. Therefore, for two given cases where the object had different initial horizontal velocity, the time of flight was the same. So, our initial predictions were wrong.
Presentation of your lab report is an important part of the evaluation process. It has to be written in a clear and understandable way for your peers, and Professors . We recommend using Microsoft Word for writing lab reports, especially for its handy tool -
Equation Editor. Equation Editor allows you to type complicated mathematical expressions in a compact and elegant way.
In order to use this tool go to Insert Menu  Object  Microsoft Equation 3.0.
Mathematically, we start from the definition of acceleration: a  d dt
 
158

APPENDIX C: GRAPHING and integrate twice with respect to time to see how a change in time might be related to initial velocity.
We found that: y - y o =
v o
 t + 0.5a
 t 2 (1)
In this example, the student is explaining incorrect predictions . Although motion is two- dimensional, he/she is using

0
while correct notation is

0 y
as it is a projection of velocity onto y-axis
.
With the y-axis vertical and the positive direction up, we know the acceleration is -g. We also know that v o is the initial velocity, and y o
- y is h, the height of the table. Solving for  t one finds:
 t  vo  (vo
2
 2hg) g
(2)
Faced with a choice in sign, our group chose the solution with the positive sign, deciding that a possible negative value for elapsed time does not correspond with our physical situation. From equation (2), we deduced that if v o
increased, then the time of fall also increases. This coincided with our prediction that a projectile with fastest horizontal velocity would take the most time to fall to the ground. For a graph of our predicted time of flight versus initial horizontal velocity, see Graph A from the lab journal.
Loggerpro  generated graphs of x and y positions as functions of time. Our prediction for the vertical direction was equation (1). Since the ball only has one acceleration, we predicted that equation (1) would also be true for the horizontal motion: x - x o =
v o
 t + 0.5a
 t 2
The dotted lines on the printed graphs represent these predictions.
**The Example of Two Bullets**
Our Professor asked us to compare a bullet fired horizontally from a gun to a bullet dropped vertically.
Our group decided the bullet that is fired horizontally will take longer to hit the ground than the one that is simply dropped from the same height.
This section describes your experimental method, the data that you collected, any problems in gathering the data, and any crucial decisions you made. Your actual results should show you if your prediction was correct or not.
To ensure the ball’s velocity was completely horizontal, we attached a flat plank at the end of the ramp.
The ball rolls down the ramp and then goes onto the horizontal plank. After going a distance (75 cm) along the plank, the ball leaves the edge of the table and enters projectile motion.
We measured the time of flight by simply counting the number of video frames that the ball was in the air.
The time between frames is 1/30 of a second since this is the rate a video camera takes data. This also
159

corresponds to the time scale on the Loggerpro  graphs. We decided to compare the times of flight between a ball with a fast initial velocity and one with a slow initial velocity. To get a fast velocity we started the ball at the top of the ramp. A slower velocity was achieved by starting the ball almost at the bottom of the ramp.
During the time the ball was in the air, the horizontal velocity was a constant, as shown by the velocity in the x-direction graphs for slow and fast rolling balls. From these graphs, the slowest velocity we used was
1.30 m/s, and the fastest was 2.51 m/s.
After making four measurements of the time of flight for these two situations, we could not see any correspondence between time of flight and initial horizontal velocity (see table 1 from lab journal). As a final check, we measured the time of flight for a ball that was started approximately halfway up the ramp and found it was similar to the times of flight for both the fast and the slow horizontal velocities (see table
2 from lab journal).
A discussion of uncertainty should follow all measurements. No measurement is exact. Uncertainty must be included to indicate the reliability of your data.
Most of the uncertainty in recording time of flight came from deciding the time for the first data point when the ball is in the air and the last data point before it hit the ground. We estimated that we could be off by one frame, which is 1/30 of a second. To get a better estimate of this uncertainty, we repeated each measurement four times. The average deviation served as our experimental uncertainty (see Table 1 from lab journal). This uncertainty matched our estimate of how well we could determine the first and the last frame of the projectile trajectory.
This section summarizes your results. In the most concise manner possible, it answers the original question of the lab.
Our graph indicates that the time of flight i s independent of the ball’s initial horizontal velocity (see lab journal, Graph A). We conclude that there is no relationship between these two quantities.
A good conclusion will always compare actual results with the predictions. If your prediction was incorrect, then you must discuss where your reasoning went wrong. If your prediction was correct, then you should review your reasoning and discuss how this lab served to confirm your knowledge of the basic physical concepts.
Our prediction is contradicted by the apparent independence of the time of flight and initial horizontal velocity. We thought that the ball would take longer to fall to the floor if it had a greater initial horizontal velocity. After some discussion, we determined the uncertainty in our prediction. We did not understand that the vertical motion is completely independent of the horizontal motion. Thus, in the vertical direction the equation y - y o =
v o
 t + 0.5a
 t 2 means that the v o
is the only the y-component of initial velocity. Since the ball rolls horizontally at the start of its flight, v o in this equation always equals zero.
The correct equation for the time of flight, with no initial vertical component of velocity, is actually: y - y o =
0.5a
 t 2
In this equation, there is no relationship between time of flight and initial horizontal velocity.
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Furthermore, the graphs we generated with Loggerpro  showed us that velocity in the y-direction did not change when the initial horizontal velocity changed. Velocity in the y-direction is always approximately zero at the beginning of the trajectory. It is not exactly zero because of the difficulty our camera had determining the position when the projectile motion begins. We observed that the y-velocity changed at the same rate (slope of v y
plots, graphs 1 and 2) regardless of the horizontal velocity. In other words, the acceleration in the y-direction is constant, a fact that confirms the independence of vertical and horizontal motion.
After you have compared your predictions to your measured results, it is helpful to use an alternative measurement to check your theory with the actual data. This should be a short exercise demonstrating to yourself and to your Professor that you understand the basic physics behind the problem. Most of the problems in lab are written to include alternative measurements. In this case, using the time of fall and the gravitational constant, you can calculate the height of the table.
The correct equation for the horizontal motion is x - x o
= v o
 t
The horizontal acceleration is always zero, but the horizontal distance that the ball covers before striking the ground does depend on initial velocity.
**Alternative Analysis**
Since y o
- y = h and a = -g we can check to see if our measured time of flight gives us the height of the table.
From our graph, we see that the data overlaps in a region of about 0.41 sec. With this as our time of flight, the height of the table is calculated to be 82.3 cm. Using a meter stick, we found the height of the table to be
80.25 cm. This helped convince us that our final reasoning was correct.
The example of the two bullets discussed in the Prediction section was interpreted incorrectly by our group. Actually, both bullets hit the ground at the same time. One bullet travels at a greater speed, but both have the same time of flight. Although this seems to violate "common sense" it is an example of the independence of the horizontal and vertical components of motion.
List books, journals or any other resources that you used to write your lab report..
1. Tipler, Paul A. Physics for Scientists and Engineers. 4 th , W. H. Freeman: 1999.
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The following are pages photocopied from my lab journal:
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