TAMUCC Physics I Lab Manual
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PHYS 1401 – General Physics I (Algebra-based)
PHYS 2425 – University Physics I (Calculus-based)
Summer 2016 Edition. Prepared by Dr. Jeffery Spirko, based on material by Galina Reid and
previous instructors.
Table of Contents
Table of Contents
Lab Syllabus – Lab Policies
Lab Syllabus – Do’s and Don’ts of Lab Reports
Summer 2016 Lab Schedule
Pre-Lab 0: Mathematical Exercises
Pre-Lab 1: Measurements and Calculations
Pre-Lab 2: Kinematics in One Dimension
Pre-Lab 3: Free Fall
Pre-Lab 4: Projectile Motion
Pre-Lab 5: Vector Addition and Equilibrium
Pre-Lab 6: Atwood’s Machine
Pre-Lab 7: Friction
Pre-Lab 8: Archimedes’ Principle
Pre-Lab 9: Motion on an Incline
Pre-Lab 10: Circular Motion
Pre-Lab 11: Linear Momentum and Collisions
Pre-Lab 12: Ballistic Pendulum
Lab 0 – Introductory Excel Exercises
Lab 1 – Measurements and Error Propagation
Lab 2 – Graphing, Kinematics in One Dimension
Lab 3 – Free Fall, Choosing a Measurement Technique
Lab 4 – Projectile Motion
Lab 5 – Vectors and Force Equilibrium
Lab 6 – Atwood’s Machine
Lab 7 – Friction
Lab 8 – Archimedes’ Principle and Buoyancy
Lab 9 – Inclined Plane
Lab 10 – Circular Motion and Centripetal Acceleration
Lab 11 – Collisions, Momentum, and Energy Conservation
Lab 12 – Ballistic Pendulum
Appendix A: Math Topics
Appendix B: Sample Data Report
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Lab Syllabus – Lab Policies
Lab Safety
Lab Safety is our highest priority. Students must be informed about safety issues and must obey
any safety rules posted in the lab or given by the instructor.
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University policy, OSHA regulations, and common sense require that all students are
informed about laboratory safety before doing anything remotely dangerous in the
lab. You must be registered for SMTE 0095, a zero-credit, zero-cost course, and you
must complete that course on Blackboard.
Complete SMTE 0095 Physics Safety Seminar by the Census Date. During a long
semester, this is the 12th class day. During a 5-week summer session, this is Friday of
the 1st week.
Closed shoes are required by University policy.
You must occasionally wear impact-resistant goggles. We have several pair of these
in the lab, but you may want to bring your own.
You must have access to chemical lab goggles. These must be goggles with indirect
venting to prevent chemicals from getting in your eyes. Check the lab instructions ahead
of time to know what kind of eye protection is required for what labs. Here are some
sources of chemical goggles:
o University bookstore (~$10)
o Home Depot, Lowes (~$4)
o Amazon.com (~$5)
General Information
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All experiments will be performed in groups (maximum 3 students per group), and one report
will be submitted for each group.
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Students must participate in the lab activity to get credit. If you miss a lab, help your lab
partners prepare the report. Later, you must make up the lab activity to get credit for it.
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Plagiarism is forbidden. This means you can’t cut-and-paste from the lab instructions, take
data from another lab group without giving credit, or copy text from any book or
website. Plagiarized work will get no credit, and we may refer the case to the office of
Judicial Affairs. According to http://academicaffairs.tamucc.edu/Rules_Procedures/, even
accidental plagiarism is a violation.
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All reports will be graded electronically based on the rubric posted on the report submission
page.
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Before each Lab:
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o
Prelab Practice: The pre-lab practice must be studied before coming to lab.
o
Lab Instructions: Read them so you are somewhat familiar with the procedure.
o
Lab Reports: All reports will be due before the next lab session. The last lab is due
following the same pattern, as if there was another lab. If you're unsure of
Last Updated: 2016-06-15
something, turn in what you have and ask your instructor in lab. They may allow you
to submit a revised lab report and not count it as being late.
o
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Late reports will be accepted at the discretion of the lab instructor but with the likely
penalty of up to 10% per day.
During each Lab:
o
Prelab Practice: We’ll go over the practice in class.
o
There will be a pre/post QUIZ during first part of each lab. The Prelab portion of the
QUIZ will be similar to the pre-lab practice for each lab. The Post-lab portion of the
QUIZ will contain questions related to the prior experiment.
o
Make sure you get all of your measurements and save a copy of the
data. Suggestion: Take your data in a spreadsheet like Excel.
o
The analysis can be done in lab, but it can be completed at home.
Homework After each Lab:
o
Finish the data analysis.
o
Complete the lab report. Copy your data tables into a word processor like Word.
o
Prepare for the next lab.
The lab grade is 25% of the overall class grade and will be the weighted average of the
following items:
o
(20%) Pre-lab quizzes
o
(50%) Data reports (See "Lab Report Format" from Home Page)
o (30%) Formal Lab Reports (See "Lab Report Format" from Home Page)
There is a makeup day at the end of the semester. If you have a valid excuse, you can make
up one lab with 100% of grade. Otherwise it can be only 85%. You MUST CONTACT
THE INSTRUCTOR at least 1 week before to make up a lab during one of those days.
Lab Reports
We have slightly unusual lab report formats, but they are designed to prepare students for
writing scientific papers. Academic publishers do most of the formatting for authors, even to
the point of placing Figures and Tables where they look best. Additionally, every article to be
published requires an Abstract. So we’ve worked these elements into the lab report format.
Formal Lab Report
This format closely matches what academic publishers want for journal articles. A Formal Lab
Report contains:
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Heading – We’d like this to be at the top of every page, which all word processors can do.
o Title of Lab Report
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List of Authors
Date of Lab
Abstract – This is a one or two paragraphs that summarizes the lab report. It roughly
follows the Main Body, but is not divided into sections. See the Data Report format
below for more info.
Main Body– This is the main text of your lab report. Divide this text into sections:
o Introduction – for the reader, not from the Instructions!
o Methodology – Not a “standard operating procedure”. Just describe what you
did, with enough detail that a knowledgeable person could figure it out.
o Discussion – Describe the measurement results, analysis, and results of the
analysis. Be clear about how each value was obtained. Don’t list lots of similar
values. Give a sample value or range of values, and refer to the Tables for the
rest. Intermediate calculation results aren’t needed here.
o Conclusion – A sentence or two that summarizes the scientific results (not the
skills gained). For example, you might compare the results of your data analysis
with the expected values to say whether your results support or appear to
contradict the theory. The theories we test are well-established, so if the theory is
contradicted, you should explain where the errors may have come from. The
conclusion must depend on your numerical results! It cannot contradict your
results.
Tables and Figures – All of your results of measurements and analysis should be
presented in Tables and/or Figures. Each must be labeled (Table 1, etc.) and
captioned. Do beautify your Tables. Each data point in a Table must have:
o A label in the row or column heading. This should include the name and/or
symbol, along with the units in parentheses.
o The value with a reasonable number of digits.
o The expected accuracy of the value (when requested).
In a graph, the axes must be labeled with a name or symbol, with units in parentheses.
o
o
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Data Reports
We understand how time-consuming writing lab reports can be. So we created this format to
speed the process while still giving you valuable experience and giving us something to grade. A
Data Report contains:
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Heading – We’d like this to be at the top of every page, which all word processors can do.
o Title of Lab Report
o List of Authors
o Date of Lab
Abstract – This is a one or more paragraphs that summarizes the lab activity. It roughly
follows the Main Body (above), but is not divided into sections.
o The Introduction should be 1-3 sentences.
o The Methodology should be fairly general. Only a few sentences are needed. This
is an Abstract, so more detail would be provided in the Main Body.
o The Discussion should contain just a “taste” of the data. The final result of the
analysis should be given. If there is a list of final values, give the most important
one or a typical value.
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The Conclusion should be 1-3 sentences. In a Formal Report, it might be copied
from the Main Body.
Tables and Figures – Each must be labeled (Table 1, etc.) and captioned.
o
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Note that there is no Main Body (the longest part of most Lab Reports).
A sample report is located at: https://goo.gl/Kfl57H (lower-case ell in the middle)
The corresponding spreadsheet is also online, at https://goo.gl/apFZ5U.
Notice that the table was formatted for presentation in the report.
Lab Syllabus – Do’s and Don’ts of Lab Reports
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Introduction – Give the experiment context and a goal. Don’t copy the introduction
from the instructions. Don’t even paraphrase it.
Methodology – Describe what you did.
o Feel free to use first-person plural (“we”). Don’t use “I” because all authors are
listed and nobody knows who wrote that particular sentence. Some publishers
don’t like first person, but we don’t mind.
o Don’t use second person (i.e. “you”) in your reports. This also means
you shouldn’t use imperative sentences (which are telling the reader what to
do with an implied “you”).
o Don’t say “we recorded the values in Excel” or words to that effect.
o Don’t describe your data analysis or calculations. Stick to the experimental
activity with the equipment.
o Don’t write it as instructions. Avoid using a list format.
Discussion – Summarize the results and discuss them.
o If there is a physical quantity that is a final result, give its value.
o Discuss the uncertainty (expected error) and if appropriate, the actual error.
o Describe any sources of error in your experiment. Note: you should not use
"human error" as a source of error. The term is ambiguous.
 If you make a mistake during the experiment, just fix it by discarding
those results and repeating that part of the experiment.
 If a competent lab worker can't do better, then say something specific
about the kind of error that is introduced by the person. This is a
measurement uncertainty.
Conclusion - Make sure you provide a conclusion that is based on your quantitative
results! Simply presenting the numbers isn’t enough. You must interpret them for the
reader. For example:
o As seen in Table 5, Joe was the fastest walker, with a speed of 1.5 ± 0.3 m/s.
Within the calculated uncertainties, nobody else could have walked faster than
Joe.
Abstract – This should be a brief summary of everything in the lab activity.
o One sentence of introduction is enough for the Abstract.
o In the Abstract, your methodology can be very brief. Just say what quantities you
measured and roughly how you measured them.
o Provide a summary of the discussion, including the important results.
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Summarize the conclusion.
In a Formal Lab Report, write the Abstract last. It can only summarize what is in
the rest of the report; it cannot introduce a new idea or conclusion.
Be careful about sentence structure and wording, especially sentence fragments and runons. This means pass your report through Grammar Check and Spell Check before
submitting. Yes, this is a physics class, but a well written paper is important regardless.
o
o
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Refer to your tables and graphs in the discussion (i.e. the abstract or the main body),
especially when talking about the results. For example:
…experimentaly, π was found to be 3.23 (Table 1), which is different from the
accepted value of π by ...
In the example sentence, we state our experimental value for Pi and then add Table 1 in
parenthesis at the end; this directs the reader to refer to Table 1 if desired.
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In a sentence, units go next to the number.
In a table, units go next to the heading, in parentheses.
All tables and figures must have captions! They should be descriptive but concise. Like
the Lab Syllabus says, describe which values are measured, which values are calculated,
and roughly how calculated values were obtained. For Example:
o Table 2. The volume of a cylinder. The volume was calculated from the
measured diameter and the measured height of the can. The nominal volume
was provided on the can.
The example caption covers all the requirements without being too wordy.
Other Mistakes to Avoid
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Formatting: Adhere to our Data Report or Formal Report format.
Our rulers and meter sticks are numbered in centimeters. 1 cm = 0.01 m
Stopwatches that show 2 digits for fractions of a second are showing hundredths.
When you subtract similar values, the result is significantly less accurate than the
original values.
Don’t round until you finish all of your calculations.
Round uncertainties to 1 or 2 significant figures.
Round values to the same number of decimal places as their uncertainties. You don’t
want to throw away any important detail, but there’s no sense presenting garbage digits,
either.
Include units, in figure/table labels and when units are mentioned in the text.
Check your plots! Make sure the values on the graph match the values in the data table.
Include your trendline equation in the graph. That’s what the graph is created for.
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Makeup Policy
Lab is a vital part of the Physics experience, so it’s a required part of the course. There are only
12 or so labs, and each one is important. Generally, don’t miss lab. Adjust your work schedule
so that it doesn’t conflict with lab. If you do miss a lab, we still want you to get that experience.
Allowable Absences
Students may miss lab under the following circumstances:
 With 24-hour minimum advance notice, for school-related events, job interviews, NCAA
athletic competitions, etc.
 Without advance notice, for emergencies only. Examples of emergencies are a car
accident, court subpoena, doctor appointment you can’t reschedule, death or sickness in
the family. Documentation is required within one week of a missed lab for an
emergency. The documentation may include an accident report, citation, doctor’s
excuse.
MakeUp Methods
To make up a missed lab, a student may:
 Attend another lab section. This only works if the other lab section is doing the
same lab you missed. Use the procedures below to communicate and get
permission. After doing the lab, you should get together with your normal lab group and
help them write the report. You can use either set of data (yours or theirs) in the lab
report. Hand it in with your lab group as usual.
 Exempted from lab. This is only allowed if you cannot make up a lab for unforeseen
circumstances. Your instructor must agree that it was unavoidable. In this rare case, it
can be exempted from the lab grade computation.
 Accept a zero (0).
Lab Sections
Look at the Course Schedule posted on http://sail.tamucc.edu/ (don’t log in, it’s in the lower left
corner) to see what other lab sections exist. Remember that PHYS 1401 and PHYS 2425 have
cross-listed lab sections.
All of the lab instructors are part of our course on Blackboard. To contact one of us, use
Blackboard Email to start the conversation.
Communication and Permission
Talk to your lab instructor as soon as you know you will miss lab, especially if you are an
athlete. Also, students making up labs must:
 At least 24 hours ahead of time, send an email, copied to both your lab instructor and the
lab instructor who oversees the lab you want to attend.
 In the email, introduce yourself and say why you think you should be allowed a
makeup. Make sure to include in the message:
o Your full name,
o Your lab section (number and day/time)
o Date and title of the lab,
o Date you want to make it up
 When attending the other lab section, bring a piece of paper with the same
information. (Maybe print the email.)
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Summer 2016 Lab Schedule
Lab Schedule (Summer 2016)
 Mon May 30 – Memorial Day - No Lab
 Tue May 31 –Introduction to Physics Lab, Math Exercise, Excel Exercise
1. Wed Jun 1 – Measurements
Above are due Monday June 6, 8:00 am.
2. Mon Jun 6 – Kinematics in One Dimension
3. Tue Jun 7 – Free Fall
4. Wed Jun 8– Projectile Motion*
Above are due Monday, June 13, 8:00 am.
5. Mon Jun 13 – Vector Addition and Force Equilibrium
6. Tue Jun 14 – Atwood’s Machine
7. Wed Jun 15 – Friction*
Above are due Monday June 20, 8:00 am.
8. Mon Jun 20 – Archimedes’ Principle and Buoyancy
9. Tue Jun 21 – Inclined Plane
10. Wed Jun 22 – Circular Motion and Centripetal Acceleration*
Above are due Monday June 27, 8:00 am.
11. Mon Jun 27 – Collisions: Momentum and Energy
12. Tue Jun 28 – Ballistic Pendulum
 Wed Jun 29 – No Lab
Above are due Thu June 30, 11:59 pm.
*
Labs 4, 7, and 10 are Formal Lab Reports
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Pre-Lab 0: Mathematical Exercises
This exercise for the students’ benefit. It’s not for credit. The answers are on the page after all
of the questions.
1. Scientific Notation: Review the linked page about Scientific Notation. Make sure you
can enter numbers into your calculator in Scientific Notation, and that you can interpret
numbers that the calculator reports to you in Scientific notation. Please note:
1. The “e notation” is used in calculators and when programming computers, but it
should not be used on paper or in scientific reports. Use notation similar
to this in reports: 6.02 × 1023
2. Be careful about square roots and division. On some calculators, division will
require parentheses in the denominator.
Express the following in scientific notation:
1. 38,000 =
2. 34,598,000 =
3. 0.0702 =
4. 0.281 =
5. 0.002 =
2. Significant Figures: Review the linked page about Significant Figures. Understand
that when a value is given to you, it is only as accurate as the significant digits it contains.
In general, you should report results with only as many digits as are needed to accurately
represent your value. Technically, a non-decimal number with zeroes at the end has
fewer significant figures than digits (e.g. 120 has 2 significant figures because the 0
doesn’t count). But, if you are presented with a value in that form, take it in context. Do
not reduce your calculation to one significant figure just because one of the data points
happens to have a value of 200. If the other data points have 2 significant figures,
assume that for this point as well. Please note:
 During a calculation, you should not round your values. Use the value that
the calculator or computer gives you. The “Ans” button in the calculator is your
friend. You should also learn how to save temporary values, so that you can use
them again in a few minutes. In Excel, let the computer do the calculating. Don’t
do calculations by hand, just to type them into Excel.
How many significant figures are in the following numbers?
6. 80
7. 85.0
8. 0.1492
9. 0.000587
3. Order of Operations: Review the linked page about the Order of Operations. The
biggest out-of-order problem we encounter is accidentally splitting the denominator of a
fraction. For example, if you want to do the following calculation:
1.2+3.4
5.6×107
There are many acceptable ways of typing this:
 (Preferred): (1.2+3.4)/5.6ᴇ7
 (Optional):
(1.2+3.4)/(5.6*10^7)
 (Excel):
=(1.2+3.4)/5.6e7
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Notice that the ᴇ symbol is being used for the “EE” or “Exp” button on the calculator.
This is the symbol that usually comes up on the screen. In Microsoft Excel, you would
use just the letter “e”. Note that Excel requires you to start a formula with an equals
sign. The most important thing about this is to make sure that the 107 doesn’t “pop out”
of the fraction.
1.2+3.4/5.6*10^7
means 1.2 +
3.4
⋅ 107
5.6
There are two different things that went wrong, and each should be carefully avoided.
Also, be careful about things like (1⁄2𝑥 ). It’s hard to tell whether the writer means
1⁄(2𝑥) or if they mean (1⁄2)𝑥. Generally, in text it’s the first choice, but in a calculator
or in Excel, it’s the second.
Evaluate the following using your calculator:
10.
11.
453⋅1234
13+94
123 −4
+ 75
32 −5
−11
12. 6.67 × 10
+
700×6.59×1012
6.02×1023
13. 0.92512⁄3
14. 12 + √33
15. (6 × 103 )1⁄2
4. Algebraic Manipulations: Review the linked page if necessary. Very occasionally, we
must factor or solve a quadratic equation. Expand the following (i.e. carry out the
multiplication and write an expression without parentheses):
16. (𝑥 + 3)(𝑥 − 3)
17. (𝑥 2 + 3𝑥 − 7)(𝑥 − 8)
In the following, factorize and write an expression that is a product of two factors (i.e.
make them look like the questions above):
18. 𝑥 2 + 2𝑥 − 3
19. 𝛼 2 − 𝛽 2
5. Proportions: Much of the math we deal with is the math of proportionality, possibly
with powers involved. Do remember that you can move a multiplicative factor into the
2
2𝑥
numerator (i.e. 3 𝑥 = 3 ) and that dividing is the same as multiplying by the reciporical
2
2 1
2
(i.e. 3 ÷ 4 = 3 ⋅ 4 = 3⋅4). Solve for 𝑥 in each of the following:
20. 𝑘𝑥 = 𝑚𝑔
𝑞 𝑞
21. 𝑥1 2 2 = 𝑘𝑥
22.
𝛼
𝛽
=
5𝑥
𝛿
6. Solving Linear Equations: This is the second most common type of manipulation.
In each case, solve for the indicated variable:
23. Solve for 𝑥: 2𝑥 − 7 = 11
24. Solve for 𝑦: 𝑎𝑦 + 𝑏𝑥 = 𝑐
25. Solve for 𝑧: 3 = 12 − 7𝑧
26. Solve for 𝑘: 3𝑐 = 2𝑏 − 7𝑘
7. Solving Quadratic Equations: Very occasionally, we will have to solve a quadratic
equation. We won’t do it much, but it is an important skill for dealing with projectile
motion. In case you forget, the standard form of a quadratic equation is:
𝐴𝑥 2 + 𝐵𝑥 + 𝐶 = 0
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and if you have a quadratic equation in standard form, the solutions are:
−𝐵±√𝐵2 −4𝐴𝐶
𝑥=
2𝐴
Solve for the variable in each of these:
27. 2𝑥 2 − 8𝑥 + 8 = 0
28. 𝑎2 + 3𝑎 − 2 = 0
8. Interpreting Graphs: We will frequently use graphs to summarize or interpret data sets.
Find the requested data about this dimensionless graph:
Figure 1: Sample graph for questions 29-31.
29. Slope of the above graph:
30. y-intercept of the above graph:
31. Write the equation of the graph in standard form (𝑦 = 𝑚𝑥 + 𝑏, but filling in the
known values):
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Answer Key:
1. 3.8 × 104
2. 3.4598 × 107
3. 7.02 × 10−2
4. 2.81 × 10−1
5. 2 × 10−3
6. 1
7. 3
8. 4
9. 3
10. 5224
11. 506
12. 7.73 × 10−9
13. 0.9494
14. 18 (unless the integers are exact, then ~17.74…)
15. 80 (unless you don’t believe it’s really one sig fig, then ~77.46)
16. 𝑥 2 − 9
17. 𝑥 3 − 5𝑥 2 − 31𝑥 + 56
18. (𝑥 + 3)(𝑥 − 1)
19. (𝛼 + 𝛽)(𝛼 − 𝛽)
20. 𝑚𝑔⁄𝑘
1
𝑞1 𝑞2 ⁄3
)
𝑘
𝛼𝛿
5𝛽
21. (
22.
3
𝑞1 𝑞2
𝑘
=√
23. 𝑥 = 9
24. 𝑦 = (𝑐 − 𝑏𝑥)⁄𝑎
25. 𝑧 = 9 ⁄ 7
26. 𝑘 = (2𝑏 − 3𝑐)⁄7
27. 𝑥 = 2 (There is no second solution.)
28. 𝑎 = −3.562 or 𝑎 = 0.562
29. 5
30. 20
31. 𝑦 = 5𝑥 + 20
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Pre-Lab 1: Measurements and Calculations
Review the Textbook:
 PHYS 1401: Serway & Vuille: Appendix A and Chapter 1.
 PHYS 2425: Serway & Jewett: Appendix B and Chapter 1.
The instantaneous speed of a car measured from the speedometer reading every second for 10
seconds, in units of miles per hour are: 0, 1, 9, 11, 17, 10, 28, 40, 36, 55.
1. Average of the above readings in mph is about
(21)
2. Sample standard deviation of the above readings is about
(18)
3. Mass of a ball is measured as (60.2 ± 0.5)g. What is the same measurement, given with a
relative uncertainty?
(60.2g ± 0.83%)
4. How many significant figures does the measured value of the mass of the ball have?
(3)
5. A mass of a box is described as 52 ± 0.5 kilograms, what is the range of values, in kilograms
in which you expect to find the mass of the box?
(51.5, 52.5)
6. The volume of milk in a jug is determined by pouring it out into a container, first using a
50.0 ml jar in which a 0.5 ml error in measurement is estimated and then using a smaller 5.0
ml flask in which a 0.25 ml error in measurement is estimated. The milk was poured out 5
times using the 50.0 ml jar and 8 times using the 5.0 ml flask. What is the measured volume
of milk in the jug, in ml? State in the standard form (Total volume + estimated uncertainty in
volume)
(290.0 + 4.5)
(Alternative: 290 ± 5. It’s common to round the uncertainty to one significant figure.)
7. The dimensions of a table are measured with a meter stick. The length of the table is
measured 3.431 m and width of the table is measured 2.187 m. The uncertainty of each of
these measurements is 2 mm. What is the area of the table in m2 and its uncertainty?
(7.504+0.011)
(Alternative: 7.50 ± 0.01. It’s common to round the uncertainty to one significant figure.)
8.
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Pre-Lab 2: Kinematics in One Dimension
Review the Textbook:
 PHYS 1401: Serway & Vuille: Sec 2.1-2.5.
 PHYS 2425: Serway & Jewett: Sec 2.1-2.6.
Distance vs. Time
35
Distance (m)
30
25
20
15
10
5
0
0
1
2
3
4
5
6
7
Time (s)
Figure 1. Position of a train on a straight track, as a function of time.
1. What is the distance traveled by the train between 0.0 and 6.0 seconds?
(25.0 m)
2. What is the displacement of the train between 0.0 and 6.0 seconds?
(15.0 m)
3. What is the average speed of the train between 0 and 6.0 seconds?
(4.17 m/s)
4. What is the average velocity of the train between 0.0 and 6.0 seconds?
(2.5 m/s)
5. What is the instantaneous velocity of the train at 2.0 seconds?
(5 m/s)
6. A car has a position that is can be described by x(t) = 20 +4t2, with x in meters and t in
seconds. What is the instantaneous velocity of the car at t =4.0 s?
(32 m/s)
7. A car has a position that is can be described by x(t) = 20 +4t2, with x in meters and t in
seconds. What is the instantaneous acceleration of the car at t =4.0 s?
(8 m/s2)
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Pre-Lab 3: Free Fall
Review the Textbook:
 PHYS 1401: Serway & Vuille: Chap 2.
 PHYS 2425: Serway & Jewett: Chap 2.
Use 𝑔 = 9.8 m/s 2 . Assume no effect from air resistance.
1. For a ball released from rest (dropped), how much time will it take to fly through first 3.5
meters?
(0.85 s)
2. A ball is thrown upward with an initial speed of 5.4 m/s. How far up will it rise?
(1.5 m)
3. A ball is thrown upward with an initial speed of 5.4 m/s. How much time will it take to
reach the top point of the motion?
(0.55 s)
4. A ball is thrown downward with an initial speed of 0.40 m/s. How far will it travel during
first 2.1 seconds?
(22 m)
5. An object is constantly accelerated from rest. If it reaches a distance of 2.8 meters in 1.2
seconds, what is the acceleration?
(3.9 m/s2)
6. A ball is thrown upward with an initial speed of 5.4 m/s. How long it will take the ball to
return to the starting point?
(1.1 s)
7. A ball dropped from the roof of a building reached the ground in 3 seconds. How tall is the
building?
(44.1 m)
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Pre-Lab 4: Projectile Motion
Review the Textbook:


PHYS 1401: Serway & Vuille: Sec. 3.3-3.4.
PHYS 2425: Serway & Jewett: Sec. 4.1-4.3.
1. A ball is fired horizontally from a launcher that is 2.42 m above the floor. The range of the
shot is 4.35 m. What was the initial speed of the ball?
(6.19 m/s)
2. The experimental range for shot is 1.68 m and the theoretical range for the same shot is
4.73 m. What is the percent difference between these two ranges? Use the theoretical range
as the reference value.
(64.5%)
A ball is fired from a launcher an initial speed is 7.00 m/s at a 30o angle from the
horizontal. The point of firing is 2.42 m above the floor.
3. What is the horizontal component of the ball’s initial velocity?
(6.06 m/s)
4. What is the vertical component of the ball’s initial velocity?
(3.5 m/s)
5. How much time does it take a ball to reach to the top point of the trajectory? (Hint: what is
so special about the vertical component of the velocity at that point?)
(0.3571 s)
6. What was the total time of the flight??
(1.145 s)
7. What is the predicted range of the shot?
(6.94 m)
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Pre-Lab 5: Vector Addition and Equilibrium
Review the Textbook:


PHYS 1401: Serway & Vuille: Sec. 3.2, Appendix A.5., Ex 4.2,4.6
PHYS 2425: Serway & Jewett: Sec. 3.4, Appendix B.4, Ex 5.1,5.4
1. The hypotenuse of a right triangle is 7.0 m, and the side opposite to angle θ is 5.0 m. What is
the value of cos(θ)?
(0.70)
⃗ has a magnitude of 8.0 m and is at an angle of 45° CCW from the x-axis.
Vector ⃗𝑨
2. What is 𝐴𝑥 , the 𝑥-component of vector A? (5.66 m)
3. What is 𝐴𝑦 , the 𝑦-component of vector A? (5.66 m)
⃗ has components of 𝑩𝒙 = −𝟑. 𝟎 𝐦/𝐬 and 𝑩𝒚 = −𝟕. 𝟎 𝐦/𝐬
Vector ⃗𝑩
⃗ ? (7.62 m/s)
4. What is the magnitude of vector 𝐵
5. What is the direction of vector 𝐵?
(-113.2° or 246.8°, where angles are measured CCW from the x-axis)
Given vectors r1 = (1.40m; 30o), r2 = (3.40m; 90o), and the equation of a new vector
R = − 4.00 r1 + 1.50 r2
6.
7.
8.
9.
What is 𝑅𝑥 ? (-4.85 m)
What is 𝑅𝑦 ? (2.3 m)
What is the magnitude of R? (5.37 m)
What is the direction of R? (154.6°)
10. Solve for C in the following: r1 + 2r2 + C = 0.
(C = (−1.21m) 𝑥̂ − (7.5m) 𝑦̂)
A horizontal ring is being pulled in three directions by force vectors F1, F2, and F3. The forces
balance, meaning the vectors add up to zero. Two of the forces are measured to be:
F1 = (4N)x – (8N)y and F2 = (-3N)x – (6N)y
11. Express vector 𝐹3 in component form.(𝐹3 = (−1 N)𝑥̂ + (14 N)𝑦̂
12. Express vector 𝐹3 in magnitude-direction form.
(𝐹3 = (14 N)∠94.1°)
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Pre-Lab 6: Atwood’s Machine
Review the Textbook:
 PHYS 1401: Serway & Vuille: Chap 4, Example 4.11, Problem 4.29
 PHYS 2425: Serway & Jewett: Chap 5, Example 5.7
A 2.0 kg mass and a 1.0 kg mass are tied together and placed on a horizontal frictionless
surface as shown in Figure 3. The 2.0 kg mass is pulled with a 6.0 N force.
Figure 3. A system of objects pulled by a force.
1. What is the acceleration of the system?
(2.0 m/s2)
2. If the 1.0 kg mass were pulled with a 6.0 N force instead, what would be the acceleration of
the system?
(2.0 m/s2)
3. What is the magnitude of the tension force on the 2.0 kg. object?
(2.0 N)
4. What is the magnitude of the tension force on the 1.0 kg. object?
(2.0 N)
5. Compare tension force vectors on 2.0 and 1.0 kg objects.
(They are equal in magnitude but opposite in direction)
6. Two objects are connected by a string and the string run over the pulley making both masses
suspended vertically. If masses of the objects are equal, what would be the acceleration of the
system?
(0.0 m/s2)
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Pre-Lab 7: Friction
Review the Textbook:


PHYS 1401: Serway & Vuille: Section 4.6, Examples 4.13 and 4.14.
PHYS 2425: Serway & Jewett: Section 5.8, Examples 5.12 and 5.13.
A 1200 kg car is initially going 30 m/s (about 65 MPH). It skids to a stop on an icy road,
and the coefficient of friction is only 0.1.
1. What is the normal force exerted on the car?
(11760 N)
2. What is the magnitude of a kinetic friction force acting upon the car?
(1176 N)
3. What is the magnitude of the acceleration of the car?
(0.98 m/s2)
4. How much time does it take the car to stop?
(30.6 s)
5. How far does the car go in that time?
(459 m)
Figure 4. A system of masses connected over a pulley.
6. In Figure 4, m1 = m2 = 300 g. If the surface of the table is frictionless, what is the acceleration
of the system?
(4.9 m/s2)
7. In Figure 4, m1 = m2 = 300 g. If the surface of the table has a coefficient of friction of 0.5,
what is the acceleration of the system?
(2.45 m/s2)
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Pre-Lab 8: Archimedes’ Principle
Review the Textbook:
 PHYS 1401: Serway & Vuille: Section 9.6, Example 9.8 & 9.9
 PHYS 2425: Serway & Jewett: Section 14.4, Example 14.5 & 14.6
A wooden block of 1.1cm×1.8cm×6.0cm is floating in water. The density of water
is 1000 kg/m3 and the density of the wooden block is 600 kg/m3.
1. What is the magnitude of the gravitational force acting on the block?
(0.070 N)
2. What is the magnitude of the buoyancy force acting on the block?
(0.070 N)
3. What is the specific gravity of the block?
(0.6)
4. If the block is floating such that its long side is oriented vertically, how deep is the long side
submerged in water?
(3.6 cm)
5. What force must be applied to the block to have the block be exactly half--submerged? State
both, the magnitude and the direction of the force.
(0.012 N upward)
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6. What force must be applied to the block to have the block be completely submerged? State
both, the magnitude and the direction of the force.
(0.046 N downward)
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Pre-Lab 9: Motion on an Incline
Review the Textbook:


PHYS 1401: Serway & Vuille: Chap 4, Examples 4.7 - 4.9.
PHYS 2425: Serway & Jewett: Chap 5, Example 5.6.
A suitcase of 50.0 kg is pulled with a force of 900N at an angle of 10.0o above the horizontal
over a frictionless surface.
1. What is the vertical component of the applied (pulling) force?
(156 N)
2. What is the horizontal component of the applied force?
(886 N)
3. What is the magnitude of the normal force exerted by the ground?
(334 N)
4. A box slides down a frictionless slope of 25°. What is the acceleration of the box?
(4.14 m/s2)
5. If you want a box to glide down a frictionless slope with an acceleration of 1 m/s2, what
should the slope be?
(5.9°)
6. On a 2° frictionless slope, a box is observed to have an acceleration of 0.32 m/s2. What is
the experimental gravitational acceleration in this case?
(9.2 m/s2)
7. A slope that is 0.64 meters long has one end 2cm above the other as shown in Figure 2. What
is the angle of the slope?
(1.8o)
Figure 2. Dimensions of the slope.
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Pre-Lab 10: Circular Motion
Review the Textbook:
PHYS 1401: Serway & Vuille: Section 7.4
PHYS 2425: Serway & Jewett: Section 10.3
A 350 gram mass is tied to a string and spun in a horizontal circle with a radius of 11.0 cm.
The speed of the mass is held constant and the period of rotation is 0.65s
1. What is the angular speed of this mass?
(9.67 rad/s)
2. What is the linear speed of this mass?
(1.06 m/s)
3. What is the spatial orientation of the linear velocity vector?
(horizontal and tangential to the circle )
4. What is the magnitude of acceleration of this mass?
(10.3 m/s2)
5. What is the spatial orientation of the acceleration vector?
(horizontal and radial to the circle)
6. What is the magnitude of the centripetal force on this mass?
(3.61 N)
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Pre-Lab 11: Linear Momentum and Collisions
Review the Textbook:


PHYS 1401: Serway & Vuille: Chap 5&6, Example 6.4
PHYS 2425: Serway & Jewett: Chap 7&8, Example 9.5
Two gliders moving from opposite directions collide on an air-track. The masses of the gliders
are 0.30 kg and 0.70kg. Before the collision, the speeds of gliders are 0.30m/s and 0.45m/s,
respectively. After the collision, the first glider rebounds with speed 0.40m/s. Assume the first
glider is initially moving in a positive direction.
1. What is the momentum of the first glider before the collision?
(0.09 kgm/s )
2. What is the momentum of the second glider before the collision?
(-0.32 kgm/s )
3. What is the total momentum of the system before the collision?
(-0.23 kgm/s )
4. What is the momentum of the first glider after the collision?
( -0.12 kgm/s )
5. Apply the conservation of the linear momentum to calculate the speed of the second glider
after the collision.
(0.15 m/s)
6. Did second glider rebond after collision?
(No)
7. What is the kinetic energy of the system before the collision?
(0.084 J )
8) What is the kinetic energy of the system after the collision?
(0.032J )
9. Is this an elastic collision? Justify your answer.
(Because this collision results in a loss of kinetic energy by the system, this is not an elastic
collision)
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Pre-Lab 12: Ballistic Pendulum
Review the Textbook:
PHYS 1401: Serway & Vuille: Example 5.10 & 6.5
PHYS 2425: Serway & Jewett: Example 9.6
A 3 kg ball rises up 4.0m above the ground when it is thrown. Use the ground as the reference
point (y=0) for measuring gravitational potential energy
1. What is the potential energy of the ball at ground level?
(0.0 J)
2. What is the potential energy of the ball at the highest point?
(117.6 J)
3. What is the kinetic energy of the ball at ground level?
(117.6 J)
4. What is the kinetic energy of the ball at the highest point?
(0.0 J)
5. What is the speed of the ball at ground level just after the throw?
(8.85 m/s)
6. On the way down the ball loses 40% of its mechanical energy due to air resistance though
there was no loss of mechanical energy on the way up. What is the speed of the ball as it returns
to the ground?
(6.85 m/s)
A bead moves along a frictionless circular track with a radius of 3.5 m as shown in Figure 5.
The speed of the bead at the lowest point A of the track is 5.0 m/s.
Figure 5. A bead moving along a frictionless circular track.
7. What is the maximum height (at B) attained by the bead above the lowest point of the track?
(1.27 m)
8. The center of the circular track is C. What is the angle ACB?
(50.6°)
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Lab 0 – Introductory Excel Exercises
Equipment

Computer, Calculator
Note: For completing these exercises correctly you’ll receive a free grade.
Put everything in an Excel Spreadsheet and show the instructor when you are finished.
Objectives

To familiarize students with some basic methods of data entry, calculation, and graph
analysis using a spreadsheet such as Microsoft Excel.
Exercise 1: Data Entry and Data Summaries
Typically in data tables, we place the units with the column or row heading, in parentheses. The
numerical values of the quantities don’t have units written next to them. Some methods of
summarizing data are to give the average, standard deviation, change, etc.
1. Enter the data and headings from the table into Excel. Notice that the data columns
are labeled, and that the units are placed in the label in parentheses.
o Excel Note: No units: only the numbers should be in the data cells. Because of
this, labels (with units) are always required above or at the left of a data series.
A
1
2
3
4
5
6
7
8
Averages:
9
10 Standard Devs:
11 Final - Initial:
12
13 Δd/Δt (m/s):
B
C
t (s)
16
13
6.3
1.1
-2
-7.9
d (m)
1.7
2.4
4.6
6.1
6.9
8.8
D
d/t (m/s)
Table 1. Sample data set to be entered into Excel. The labels are provided for you. Don’t type
in the grey headings; Excel shows them above and to the left of your spreadsheet.
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2. Calculate the ratios: Here you will calculate the ratios of distance divided by time.
Computers are great at repeating calculations without much effort. The “Fill handle” is a
key to getting these calculations done easily.
 Enter the first formula to calculate the ratio of d/t in the first blank cell of
column D. Click on Cell D2, press “=” to start the formula, click on the top of the
fraction (cell C2), type “/” for the division symbol, then click on the bottom of the
fraction (cell B2). You could type the whole formula, but clicking reduces
mistakes.
 Copy the formula down by dragging the “fill handle” of cell D2 downward.
The “Fill Handle” is a solid black dot at the bottom corner of the current cell. To
see it, click once (not twice!) to select the cell you want to copy. Move your
mouse cursor to the lower-right corner so that it turns into a plus sign (+). Drag
the fill handle down, up, or across the cells that you want to fill.
o Excel Note: Formulas start with an equals sign (=). Use formulas with cell
references (e.g. “B2”) as much as possible. Don’t retype data values. That only
gives you the opportunity to make a mistake. You may want to use your
calculator to check the results, but don’t type in the results from your calculator.
3. Calculate the averages of all three data sets (Column B, C, D) by using the built-in
average function. On the “Average” row, in cell B9 type “=average(B2:B7)”. Use the
fill handle to copy the formula across for series C and D.
4. Calculate the Standard Deviations: Use the standard deviation function in cells
B10 thru D10. The first one should look something like “=stdev(B2:B7)” .
5. Calculate the Change in each Quantity. In B11, calculate “Delta t”, which is just
“=B7−B2”.
o Excel Note: To get the Δ, you can use Insert-Symbol in Excel, or just Google for
“Unicode Math Symbols” and copy-and-paste what you need. Alternatively, you can just
use the word “Delta”.
6. Calculate a mediocre approximation of the slope. In B13, divide Delta d by
Delta t to produce Δ𝑑/Δ𝑡.
Exercise 2: Graphing
Using the MS Excel features, make two scatter plot graph using the above set of data points.
Note that for most scientific graphs you will need to use the scatter plot option (click “insert”
tab and look for “scatter plot” on upper bar).


Page 27
Graph 1, scatter plot of 𝒅 vs. 𝒕. This one is easy, because the data is in the right order.
Graph 2, scatter plot of 𝒕 vs. 𝒅. The data series are “backwards” for this graph. There
are two ways to deal with this:
o Method 1: Alternatively, you can copy-and-paste the data table into a blank
area of the spreadsheet and then use cut-and-paste to reverse the columns. I call
this a “helper table”.
o Method 1: Create a blank graph and use the Select Data option within the
graph and manually choose the 𝑥 and 𝑦 data series. This is cumbersome but
works well if you can figure it out.
Last Updated: 2016-06-15
Make sure to label each axis (“t (s)” or “d (m)” like the column labels) of each plot properly and
display the title of the plot on the chart. If you have difficulties, ask for help from the lab
instructor.
Important Note: We refer to any graph by the vertical axis first. For example, in a “𝒅 vs. 𝒕”
graph, 𝒅 is on the 𝒚-axis and 𝒕 is on the 𝒙-axis.
Exercise 3: Graph analysis and Linear Fitting
On each graph, obtain the equation of best fit, using Excel’s trendline feature. Go back to the
Excel graphs and fit a linear trendline to each graph (right click on one of the graph points and
choose Trendline option). To display the equation of the fitting line, select “Display Equation”
from “Options”. Use this equation to find out the slope and intercept of the graph. Excel
displays the equation using the variables 𝑥 and 𝑦. Of course your variables are different, so you
should correct the equation to show the correct variables.
Graph 1: 𝒅 vs. 𝒕
Graph 2: 𝒕 vs. 𝒅
Displayed Equation
Corrected Equation
Slope
Units of Slope
y-Intercept
Units of y-Intercept
Table 2. Data extracted from the linear fit. Use an empty area of your spreadsheet for this
information.
Once you get your results, we’d like you to notice 2 things:
 The slopes are inverses (1/𝑥) of each other.
 The slope of 𝑑 vs. 𝑡 is not equal to 𝑑/𝑡 (Cells D2-D7) or even the average 𝑑/𝑡 (cell D8).
It’s close to the rise/run = Δ𝑑/Δ𝑡 (Cell B13) between the first and last data points.
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Exercise 4: Trigonometry Practice.
Excel does all trigonometry in radians. This means you have to frequently convert between
degrees and radians. Fortunately, it provides formulas to do this.
You want
Math notation
Excel notation
Sine
sin(𝜃)
=sin(radians(c4))
Cosine
cos(𝜃)
=cos(radians(c4))
Tangent
tan(𝜃)
=tan(radians(c4))
Sine with mult.
𝑑 sin(𝜃)
=b4*sin(radians(c4))
tan−1 (𝑦/𝑥)
=degrees(atan(e5/d5))
atan2(𝑥, 𝑦)
=degrees(atan2(d5,e5))
√𝑥 2 + 𝑦 2
=sqrt(d5^2+e5^2)
Inverse Tangent
Square Root
Notice that there’s a special version of the inverse tangent that takes the 𝑥 and 𝑦 values
separately, instead of having them divided. This eliminates the need to sometimes add 180°.
Use the spreadsheet to calculate the total displacement for a two-stage trip. Many formulas are
given to help you along. Use the “Fill Handle” as much as possible. The math steps in the
calculation are:



Calculate the horizontal and vertical parts of each Leg.
Add the horizontal and vertical parts separately to get the totals.
Use the Pythagorean Theorem and inverse tangent to calculate the Distance and Angle of the
total.
A
B
𝑑, Distance
40
(m)
41 Leg 1
1
2
42 Leg 2
43
Total
√𝑥𝑇2 + 𝑦𝑇2
C
D
E
𝜃, Angle (deg)
𝑥, Horiz (m)
𝑦, Vert (m)
30
45
𝑑1 cos(𝜃1 )
𝑑1 sin(𝜃1 )
𝑑2 cos(𝜃2 )
𝑑2 sin(𝜃2 )
atan2(𝑥𝑇 , 𝑦𝑇 )
𝑥1 + 𝑥2
𝑦1 + 𝑦2
Table 3. Sample data for practice calculations. Some sample formulas are given using math
notation. The final results in B43 and C43 should be a distance of 2.977 m at an angle of 40.01°.
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Lab 1 – Measurements and Error
Propagation
Equipment







Computer, Calculator
Ruler
Calipers
Measuring Tape
Stopwatch
Soda can
Ball to drop
We’ll spend a lot of time this semester measuring things and doing calculations. If we don’t
properly determine and keep track of the uncertainty of the values, it’s impossible to assess
whether they provide physics insight at the end.
Objectives


To get familiar with simple measurements (e.g. time and length) and the use of
measured values for simple calculations (e.g. the value of π, volume, speed). (Note: this
is a “learning objective” and wouldn’t be listed as a goal or objective in a lab report.)
To determine which lab partner walks the fastest at a normal, comfortable walking pace.
Notes about Labs and Lab Reports




Page 30
Frequently, the lab activity can be divided into Procedure and Analysis stages.
o The Procedure involves recording data, so must usually be done in the lab.
o The Analysis involves calculations and interpretation. Often, the analysis can be
done at home, as the lab report is being prepared. If you have time to do the
analysis in lab, your instructor can check it.
In all lab activities, you should record your data in a table in a spreadsheet.
We’ll often provide template tables like Table 1.
In your lab report, copy-and-paste the table from the spreadsheet into the results
section of your document. Then:
o Trim values to a reasonable number of significant figures.
o Provide a caption.
o Format the table so it looks presentable. Don’t let the table get split between
pages, and don’t let it spill into the margin.
We will frequently ask questions in the lab instructions. They are intended to help you
figure out what to write for discussion and conclusions. Do not copy the questions
into your lab report. This would be plagiarism. Instead, make sure that your lab
report addresses the answer in the discussion or conclusion.
Last Updated: 2016-06-15
Exercise 1: Measuring the value of π.
The circumference of a circle is supposed to be related to the diameter of a circle using the
formula:
𝐶 =𝜋⋅𝑑
(1)
Here, π = 3.14159… is a well-known irrational number. We can test the theory by ignoring the
fact that we know π and instead measuring the circumference and diameter of a circular object.
We can then calculate an experimental value of π by the formula.
Procedure
1. Measure the circumference of a soda can. Be sure to include units with your answer. To
measure the curved distance, you could do one of the following:
 Wrap a piece of string around the can, mark one circumference, then measure the
string.
 Roll the can across a piece of paper and mark where one point on the can starts
and ends up.
 Wrap a measuring tape around the can.
2. Using a pair of calipers, measure the diameter of the can.
Analysis
.
3. Use formula (1) to solve for the experimental value of π
4. Compare your value of π to the accepted value. For similar values, a good way to
compare them is to calculate the “percent difference”. This is just the difference between
the values, expressed as a percentage of the “reference value”. In this case, it is clear that
the reference value should be the true, mathematical value of pi, as given by your
calculator. Sometimes you’ll have to carefully decide which value being compared is the
reference value.
5. What could be the cause of the error? In your lab report, specify exactly how it could
have happened. (Don’t ever just answer “human error”. That sounds like you made a
mistake. If you make a mistake, just fix it by re-measuring. If you genuinely can’t do a
better measurement, the error comes from precision of the measurement. There should
still be an explanation of what limits the precision.)
Description (with units)
Value
Circumference (cm)
Diameter (cm)
Experimental π
Accepted π
3.14159265
% Difference
Table 1: Quantities measured and/or calculated in Exercise 1.
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Exercise 2: The Volume of a cylinder.
The volume of a cylinder can be determined by using the formula:
𝑉 = 𝜋𝑟 2 ℎ
(2)
where 𝑟 is the radius and ℎ is the height of the cylinder.
Procedure
1. Make necessary measurements to calculate the volume of a soda can that resembles a
cylinder. Note, because the radius is half of the diameter, which you have above, you
don’t have to re-measure it.
Analysis
2. Calculate the radius and volume of the can. Notice that because the volume comes from
three values in cm, all multiplied together (the radius counts twice because it is squared),
its units are cubic centimeters (cm3 ).
3. Record the can volume as given on the label.
4. Calculate the % difference between your calculated volume and the volume on the can
label. Note: 1ml = 1cm3
5. What could be the cause of the error?
Description (with units)
Value
Measured Diameter (cm)
Measured Height (cm)
Calculated Radius (cm)
Calculated Volume (cm3)
Nominal Volume (ml)
% Diference
Table 2. Measurements and calculations to determine the volume of a can.
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Exercise 3: Reaction Time
When a person uses a stopwatch, there are several possible sources of error in the measurement.
One of them is “reaction time”, which is a time delay between when a person sees something
happen and when they can actually do something about it. The reaction time could add or
subtract from the value being measured, depending on how the experiment is set up. Knowing
the reaction time of a person operating a stop watch gives us an estimate of the uncertainty in
the time measurement.
A simple experiment could help to determine a reaction time of a person:
Procedure
1. Have one person hold a ruler with two fingers, letting it hang loosely with the 0 cm mark
at the bottom.
2. The person being measured (the “partner”) will place her/his fingers around 0 cm mark,
without touching the ruler, and prepare to catch the ruler.
3. Without warning the first person will drop the ruler. The partner should catch it.
4. The measurement is the distance that the ruler fell. Read the measurement on the ruler
where the partner’s fingers grabbed it.
Analysis
The time 𝑡 during which ruler was in the air and the distance ℎ are related according to:
1
ℎ = 2 (9.8)𝑡 2
Use this relationship to estimate the reaction time of each partner. Note: ℎ must be in meters.
5. Solve for 𝑡 in the equation above, and calculate the time of the ruler fall for each
partner’s catch.
6. Which Partner would be the best stopwatch operator? Why it would be wise to delegate
this duty to the partner with the smallest reaction time to work with the stop watch?
Description
(with units)
Name of
Partner
Distance of
Ruler Fall (m)
Time of Ruler
Fall (s)
Partner 1
Partner 2
Partner 3
Table 4: Measurements and calculations for the reaction time measurement. The amount of
time that the ruler was falling is the person’s reaction time.
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Exercise 4: Timing a ball drop.
Procedure and Analysis
1. Drop a ball (tennis or baseball) from three different heights ten (10) times each and time
each fall with a stopwatch. Choose three different heights: a tabletop, a top of your
reach, a hallway balcony. For each trial, be careful to drop the ball from exactly the same
height.
2. Calculate the average drop time and the uncertainty of the drop time for each height.
For the uncertainty, use the standard deviation of the trial times.
3. For the relative uncertainty (also called fractional uncertainty), divide the uncertainty by
the average value and multiply by 100%.
Discussion/Conclusions:
4. Which ball drop took the most time? Is the result certain, or are the values
indistinguishable? Does the result match your intuition?
5. Which of the three measurements is the most reliable?
Description
Height 1, 𝒕 (s) Height 2, 𝒕 (s) Height 3, 𝒕 (s)
Description of height
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Trial 6
Trial 7
Trial 8
Trial 9
Trial 10
Average
Uncertainty
Relative Uncertainty (%)
Table 4. Measurements of the times for a ball to drop from various heights.
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Exercise 5: Measuring Walking Speeds (Optional, instructor will
decide)
We don’t have a device to measure walking speed directly. But we can measure a distance
travelled and the time it took, and calculate the speed. This is called an indirect measurement
of the speed, and we still often just call it the measured speed.
Procedure
Make the necessary measurements to calculate a walking speed of each partner, and estimate
the uncertainty for each measurement. The reaction time of the person working the stopwatch
can be used as the uncertainty in time. The uncertainty in distance should be estimated based
on the method of the measurement and the measuring device.
Analysis
Calculate the speed and the uncertainty of the speed for each person’s walk.

The speed is calculated by dividing two values. Therefore, its fractional
uncertainty is found by adding the fractional uncertainties of the original values.
Calculate the fractional uncertainty for each quantity used in the calculation of the average
∆d
d
speed (
∆v
∆t
t
and ). Then, just add them to produce the fractional uncertainty of the average speed
( v ). The uncertainty in the average speed (Δ𝑣) is calculated from this. You might need some
scratch space in your spreadsheet to hold intermediate results. Just use the space below the
table. For your lab report, you only need the values in the template table.


Which lab partner walks the fastest?
Based on the uncertainties of the measurements, is your conclusion certain?
Partner’s Name
Distance walked (m)
Uncertainty in distance (m)
Time of the walk (s)
Uncertainty in time (s)
Average speed (m/s)
Uncertainty in speed (m/s)
Table 5. Measurements and calculations to find the walking speed of three lab partners.
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Lab 2 – Graphing, Kinematics in One
Dimension
Equipment




Calculator, Computer, PASCO Interface
PASCO Motion Sensor
Beach Ball
Capstone file: Kinematics Blue/Black Motion Sensor (depends on which color you have).
Objectives

To track the position and velocity of an object under constant acceleration conditions.
Introduction
Kinematics is the description of motion. When describing the motion of an object, it is essential
to know its position relative to a reference point, how fast and in what direction it is moving, and
how it is accelerating (changing its rate of motion). A sonar ranging device such as the PASCO
Motion Sensor uses pulses of ultrasound that reflect from an object to determine the position of
the object. As the object moves, the change in its position is measured many times each second.
The change in position from moment to moment determines the velocity (in meters per second).
The change in velocity from moment to moment determines the acceleration (in meters per
second per second). The position of an object at a particular time can be plotted on a graph.
You can also graph the velocity and/or acceleration of the object versus time. A graph is a
mathematical picture of the motion of an object. For this reason, it is important to understand
how to interpret a graph of position, velocity, or acceleration versus time.
Experimental Setup
The experimental setup consists of a motion sensor to determine the distance between the
object and the sensor, a Lab Interface that communicates the data obtained by the motion
sensor to the PC, and the PC. A beach ball will be the object in motion.
Figure 1. The PASCO motion sensor and 850 Interface.
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Capstone Setup
1. Make sure that the PASCO Interface is turned on, otherwise you may have to help
Capstone find the interface.
2. There are two Capstone files in the web folder with your lab manual.
● Kinematics Black Motion Sensor.cap
● Kinematics Blue Motion Sensor.cap
Load the one that corresponds to the Motion Sensor that you have. It automatically
configures the sensor, makes a Table with Position and Velocity, and makes graphs of
Position and Velocity.
3. Connect the actual sensor to the interface by inserting the yellow plug into socket #1 and
the black plug into socket #2.
4. Place the Motion Sensor on the table orienting it such that the sensor points up.
Warning!!! You will be tossing a beach ball up and catch it on its way down. Clear the area
around the motion sensor for at least 1 meter.
Experimental Procedure
Exercise 1: Position vs. Time
Movie directors use the sequence “Lights, Camera, Action, Cut” to start filming. We’ll do almost
the same thing.
1.
2.
3.
4.
5.
o “Lights”: Get the equipment ready.
o “Camera”: Start the data acquisition with the Run button.
o “Action”: Make the equipment do its thing.
o “Cut”: Hit Stop.
When you are ready, hold the beach ball above the Motion Sensor.
The software organizes its data in “Runs”. To start a run, click “Record” button in
“Controls” (lower left corner).
Toss the beach ball a foot or two straight up and catch it before it hits the sensor.
When your Run is complete, click Stop.
A successful run should have a portion that is perfectly parabolic, with at least 12 data
points, and no spikes. Generally, save any scientific data that might be good. Delete
genuinely bad runs (red X down in “Controls”).
Analyzing the Data:
1. Highlight the parabolic portion of the Position graph with “Highlight” tool (icon that
locates above the graph and looks like a marker). Bring the sides of the Highlight box in
near the parabola, and use the bottom edge of the Highlight box to chop off the nonparabolic portion. We are trying to analyze the ball’s motion while it is in the air. That
graph should be parabolic. A parabola slopes more steeply as you get away from the
peak.
2. Use the Autozoom Tool (First icon above the graph) to make your data fill the graph.
3. In your data table, some of the data should now be highlighted. Scroll to the highlighted
data, select all of the highlighted points (all columns), right-click to copy, and then paste
it into Excel.
4. Based on the recorded time intervals determine the following times:
 When the ball started its parabolic ascent (𝑡𝑖 )
 When it reached the maximum height (𝑡max )
 When it reached the lowest point of the parabolic descent (𝑡𝑓 )
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5. Based on the recorded position of the ball find its initial (𝑥𝑖 ), maximum reached (𝑥max)
and final (𝑥𝑓 ) positions.
6. From the obtained data calculate the distance traveled by the ball (𝑑) and the
displacement (Δ𝑥) of the ball during that time.
7. Use calculated positions to determine the average speed and average velocity of the ball.
8. Back in Capstone, look at any particular point on the Position graph when the ball was
ascending. Find the time and slope of at that point.

In the graph toolbar, the “Crosshair” tool lets you see the values of a given
point.

Again in the graph toolbar, the “tangent line” tool shows you the slope at a
point. After clicking on the tool, find the dashed line, and drag it around until it
points at the data point you are interested in.
9. Use a similar method to find the time and instantaneous velocity for the time of your
choice during the ball’s descent.
10. In Excel, make a scatterplot of Distance vs. Time using the data copied from Capstone.
Obtain the equation of the best fit for this graph using the trendline feature. When
fitting, choose wisely between a linear, a power, and a polynomial function. Display the
equation on the graph. Edit the trendline equation to take out the 𝑥 and 𝑦 variables that
Excel uses, and put in 𝑡 and 𝑥(𝑡) in their place. Make sure to label the axis and display a
title on the graph.
11. Using the best fit line, determine trendline equation 𝑥(𝑡), the acceleration, the “initial
velocity” 𝑣0 at time 𝑡 = 0, the velocity function 𝑣(𝑡), the instantaneous velocity at time
𝑡 = 𝑡𝑖 (your initial time). (Note: the “initial velocity” in this case is not the velocity the
ball was moving at any time. It’s extrapolated back to 𝑡 = 0. Your actual initial velocity
would be at 𝑡 = 𝑡𝑖 . To get this, you have to actually plug 𝑡𝑖 into the velocity equation.)
Initial
Max Height
Time (s)
Position (m)
Table 1. Recorded positions and times for a single parabolic motion of the ball.
Final
Distance Traveled (m)
Average Speed (m/s)
Displacement (m)
Average Velocity (m/s)
Table 2. Calculated values related to distance and velocity for the single parabolic motion of
the ball.
During Ascent
During Descent
Time (s)
Instantaneous Velocity (m/s)
Table 3. Velocities extracted from the position vs. time graph, using the Capstone “Tangent
Line” function.
Trendline equation:
𝑥(𝑡) =
Acceleration 𝑎 (m/s2):
“Initial velocity” 𝑣0 at 𝑡 = 0 (m/s):
Equation of velocity:
𝑣(𝑡) =
Initial Velocity 𝑣𝑖 at 𝑡 = 𝑡𝑖 (m/s):
Table 4. Trendline of position graph and values extracted from the trendline. The “initial
velocity” at 𝑡 = 0 is in quotes because the ball was never actually going that velocity. The initial
velocity at time 𝑡 = 𝑡𝑖 is found using the 𝑣(𝑡) equation with the initial time from Table 1.
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Exercise 2: Velocity vs. Time
Do not repeat the experiment. Instead, we will use the same Run in DataStudio that you used
above.
Analyzing the data:
(Note: if you’re using the new Spring 2016 Capstone files, your velocity data was probably
copied to Excel with the position data above. If this is true, skip to step 4.)
1. Get Capstone to plot the Velocity of the same data Run by opening another graph
display.
2. Highlight the same portion of the Velocity graph as Position graph (i.e. from 𝑡𝑖 to 𝑡𝑓 ).
Open another table (Velocity and Time). While the ball is in the air, the velocity is linear.
Anything that’s not linear is not part of the free flight of the ball.
3. Using a Table of Velocity vs. Time, copy the highlighted data into Excel.
4. In Excel, make a scatterplot of Velocity vs. Time. Obtain the best fit function 𝑣(𝑡) and
make the graph nice, as described above in the Position vs. Time section.
5. Use this new 𝑣(𝑡) to find the same parameters as in Table 4 above.
Trendline Equation:
𝑣(𝑡) =
Acceleration 𝑎 (m/s2):
“Initial velocity” 𝑣0 at 𝑡 = 0 (m/s):
Initial velocity 𝑣𝑖 at 𝑡 = 𝑡𝑖 (m/s):
Table 5. Trendline of the velocity vs. time graph and values extracted from the trendline. The
“initial velocity” at 𝑡 = 0 is in quotes because the object was never actually going that velocity.
The initial velocity at 𝑡 = 𝑡𝑖 is found by plugging the initial time into the velocity equation.
During Ascent
Time (s)
Instantaneous Velocity (m/s)
Table 6. Velocities calculated from the velocity vs. time trend line.
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During Descent
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Exercise 3: Questions
These short-answer questions are to help you decide what to include in your report.
1. Compare the values in Tables 5 and 6 to those in Tables 4 and 3. If the quantities are not
equal, find the percent difference, using the first table as the “reference” values.
2. In your abstract, your “result of the experiment” should be few sentences that
incorporate all of the numerical quantities in Exercise 1 (though not necessarily the
fit equations).
3. With Motion Sensor measurements, a positive direction (sign) is defined as the direction
away from the sensor Based on the experience you gained predict the sign of the velocity
and acceleration for the following cases. Include this as another Table.
Sign of the Initial
Velocity
Sign of the
Acceleration
The sensor is placed on the floor and the ball is
tossed up above the sensor
The sensor is placed on the floor and the ball is
dropped down towards the sensor
The sensor is attached to the ceiling and the
ball is tossed towards the sensor
The sensor is attached to the ceiling and the
ball is dropped beneath the sensor
You can test your predictions with the actual ball and the Motion Sensor.
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Lab 3 – Free Fall, Choosing a
Measurement Technique
Equipment



Calculator, Computer, PASCO Interface
Ball, Stopwatch, Foam Pad
PASCO Photogate, “Picket Fence”
Objectives

To compare two different techniques used to determine the acceleration due to gravity,
𝑔.
Introduction
Galileo was the first to describe gravity on Earth’s surface correctly. Through experiments, he
found that, neglecting air resistance, any object falls with a constant acceleration, 𝑔 = 9.8 m/s2
downward. That means that if the object is dropped from rest,
1
2
Δ𝑦 = − 𝑔𝑡 2
(1)
In this lab, you’ll drop two different kinds of objects and measure their fall either by human or
electronic timing.
Method 1: Human Timing with a Stopwatch
Experimental Setup
In this method, you will have a ball drop from several known heights, and measure the time it
takes for it to fall.
1. Place a foam pad on the floor, and hold a ball so that it is the desired height from the
floor. The distance should be measured from the pad to the bottom of the ball.
2. Use a stopwatch to record the time of the fall. It’s best to have one lab partner drop the
ball and another do the timing.
3. Repeat the drop for at least 5 trials at each height.
4. Repeat the experiment for heights of 0.5, 1.0, 1.25, 1.5, 1.75, and 2.0 meters.
The table of measured data for your report should look something like Table 1.
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Height
(m)
Times of Falls (s)
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
0.5
1.0
…
Table 1. Data recorded for ball drops from various heights.
Method 2: Electronic Timing with a Picket Fence and Photogate
Sensor setup:
1. Mount the Photogate so that the picket fence can be dropped between the jaws.
2. Make sure the foam mat is under the Photogate.
3. Plug the Photogate into socket #1 of the PASCO Interface box and start Capstone. There
is no Capstone setup file for this lab.
4. In Capstone Hardware Setup window, click on socket #1 and choose the Photogate.
5. In Capstone Timer Set up (below Hardware Setup) window, choose “Next” in steps 1 and
2; then, choose “Picket Fence” in step 3 and “Speed” in step 4; continue choosing “Next”
till “Finish” and “Save”.
6. From “Displays” menu, choose graph and select “Speed” as the vertical axis label.
Experimental Procedure:
1. Click on “Record” in Capstone.
2. Hold the Picket Fence so it is hanging vertically just above the Photogate.
3. Drop the Picket Fence through the Photogate. It should be freely falling the entire time it
is in the Photogate.
4. Stop recording. You should see a sloped linear graph that may have a few data points at
the beginning or at the end.
5. Generate the Linear Fit line in Capstone. Look above the graph for the Fit button (a red
line surrounded by blue dots) and pick “Linear”.
6. The Slope of the 𝑣(𝑡) vs. 𝑡 graph should be the acceleration.
Repeat the Experimental Procedure for this method for at least 5 trials. Your recorded data
table should look something like Table 2.
Trial
1
2
3
4
5
Acceleration (m/s2)
Table 2. Data recorded from Capstone for the Picket Fence drop.
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Data Analysis
Method 1
For each height of the ball drop, do the following:
1. Calculate the average time and the uncertainty of the time. (For the uncertainty, you can
use the standard deviation, as recommended by NIST2.) This should be easy using Excel.
Recall that the standard deviation formula is: =stdev(range)
2. Calculate the acceleration 𝑎, using the model formula (Eq. 1) in the introduction.
3. Calculate the relative uncertainty (%Unc) of the time
4. Determine the relative uncertainty of the acceleration. For powers, the %Unc of the
input value gets multiplied by the power to produce the %Unc of the result. In this case,
the power is 2, so just multiply the %Unc of time by 2, and that gives you the %Unc of 𝑎.
5. Calculate the uncertainty of the acceleration.
Find the average accelerations, which is your value of 𝑔 for this method. For the uncertainty of
your final value, use the larger of:


The standard deviation of the individual accelerations.
The average uncertainty of the individual accelerations, divided by √𝑁 (where 𝑁 is the
number of data points.)
The table of calculated results in your report should look something like Table 3.
Height (m)
Average
Δ𝑡 (s)
Unc. of
Δ𝑡 (s)
%Unc of
Δ𝑡 (%)
Acceleration
𝑎 (m/s2)
%Unc of 𝑎
(%)
Unc. of 𝑎
(m/s2)
0.5
…
Average
Table 3. Calculated times and accelerations for the ball drop, as timed by hand.
Method 2
Find the average acceleration and its uncertainty using the same method used in Method 1.
2
http://physics.nist.gov/cgi-bin/cuu/Info/Uncertainty/index.html
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Comparing the Methods
Record the average Acceleration (with uncertainty) of each method, and find the error and %
Error for each.
Desc
Method 1
Method 2
Acceleration (m/s2)
Accepted 𝒈 (m/s2)
Error
% Error
Table 4: Summary of results.
In your abstract paragraph, you should state which method you think is more accurate. Also
give reasons why each value differs from the 𝑔. (Remember, “human error” is not a reason for
the difference. You have to point to something specific.)
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Lab 4 – Projectile Motion
Safety Requirements
Safety Glasses (not necessarily chemical goggles) are required.
Safety equipment is required based on the hazards present in the room, not the activities being
performed at the time. Whenever the projectile launchers will be out, glasses are required.
Equipment






Calculator, Computer
Projectile Launcher
Ramrod
Greenish Yellow Plastic Ball
Carbon Paper
Meter Stick, Measuring Tape, Ruler
Objectives

To determine how the angle at which a ball is fired affects its horizontal range.
Introduction
In this experiment, you will study how the range of a projectile changes as the angle at which it
is fired is changed. A plastic ball fired from a spring gun is the projectile. The gun is placed on a
floor, and the ball is fired into the air at different angles, striking the floor. To help locate the
point of impact on the floor, use a carbon paper and a white paper. Ask your instructor for any
other simple devices and instruments. Vary the angles by as great a range as possible.
The experiment will proceed in two stages. First, we will use a horizontal shot to establish the
launch speed of the ball. Then, we will launch the ball at angles above horizontal to observe the
effect on the range.
Our basic assumption is that a flying object has an acceleration of 9.8 m/s2 downward. The
basic formulas of kinematics apply. When we put them together with the initial velocity and
height, we can derive Eq. 1. (This comes from the basic Projectile Motion equations, in Sec 4.3
of Serway/Jewett 9th Edition, or Sec 3.4 of Serway/Vuille 10th Edition. You can feel free to solve
the basic equations instead of using Eq. 1.)
We will use a comparison between experimentally measured ranges 𝑥 and the same ranges
calculated using Eq. 1 as the criteria for the success of the experiment.
𝑥 = 𝑣0 𝑐𝑜𝑠𝜃
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𝑣0 𝑠𝑖𝑛𝜃 + √2𝑔ℎ + (𝑣0 𝑠𝑖𝑛𝜃)2
𝑔
Last Updated: 2016-06-15
(1)
Experimental Setup
The experimental setup consists of a spring-loaded launcher (Figure 1) and a plastic ball used as
a projectile. A sheet of carbon paper placed on the top of a sheet of white paper will record the
landing position of the projectile. Tape the white paper to the floor first, then place the carbon
paper on top of it, shiny side down. When you want to see where the ball landed, just look at the
white paper for a black mark from the carbon.
Figure 1. Projectile launcher set to an angle above horizontal. (Image credit: PASCO)
Initial Velocity of the Projectile:
Each launcher has a unique spring which results in different initial velocity of the projectiles.
You first need to determine the initial velocity of your launcher. Note: you have to use the same
“range” setting on the launcher for every shot. That is an important experimental parameter to
make note of and communicate to your readers.
1.
2.
3.
4.
Use the plumb line of the launcher to position the barrel horizontally.
Measure the height, ℎ, from which the projectile is launched.
Fire several shots horizontally and measure the resulting ranges, 𝑥𝑖
Calculate the initial speed of the projectile.
Height
(m)
Range (m)
Trial 1
Trial 2
Trial 3
Trial 4
Trial 5
Average
Speed
(m/s)
Table 1. Recorded ranges of the horizontally-launched projectile.
Data Recording:
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For the angled launches, use the hanging plumb bob to measure the angle of the launcher.
1. Fire the plastic ball at increasing angles, ranging from 10° to 80°, in increments of 10°. You
might want to add an extra data point at 45°.
2. At each angle, do 3 trials. You may need practice shots to figure out where to place the
carbon paper. Practice shots don’t count as trials unless you can definitely say where the ball
landed.
3. For each trial, measure the range from the launcher.
A sample table for recording you table is below.
Angle (°)
Calculated
Range (m)
Measured Range (m)
Trial 1
Trial 2
Trial 3
Average
0
10
20
30
40
45
50
60
70
80
Table 2. Recorded data for angled launches of the projectile.
Data Analysis
1. For the horizontal launch, calculate the launch speed in Table 1. (Hint: The time it takes
the projectile to fall from the launch height to the ground is a good middle step.
Alternatively, just set θ = 0° in Eq. 1.)
2. Calculate the expected Range for each angle in Table 2.
3. Plot the two series of values (experimental and calculated) on the same graph. Use
scatter plot of Range vs. Angle. (Remember, the *FIRST* variable is the *VERTICAL*
axis.)
4. Fit the two series into polynomial function and display equations on the graph.
5. Examine both trends and conclude how the launch angle affects the range; make sure to
mention at what launch angle the projectile reaches its maximum range.
6. Comment on the agreement between the experimental and calculated values of the
range.
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Lab 5 – Vectors and Force Equilibrium
Equipment




Calculator, Computer
Circular force table, 3 pulleys
String, Ring for anchoring string
Mass hangers, mass hanger set
Objectives

To predict and verify the conditions necessary for static equilibrium.
Introduction
Scalars are quantities completely specified by a magnitude. Examples include temperature,
mass, and time. Vectors are quantities represented by both a magnitude and a direction.
Examples include displacement, velocity, and force.
While working with scalar quantities one can just algebraically add, subtract, multiply or divide
numbers. However, when working with vectors one need to account for the direction of the
vector and therefore a simple addition or subtraction would be incorrect.
A force (typically represented by the symbol 𝐹 ) is an example of a vector quantity. An object is in
equilibrium when the sum of all forces acting on the object is zero. In equilibrium, a
stationary object will remain stationary. In order to predict equilibrium, it is necessary to add
all the forces acting on the object following the rules of vector addition. If the vector sum of all
forces is zero, the condition of equilibrium is satisfied, and the object should remain still (if it
wasn’t moving to begin with).
Before starting this experiment, refresh your knowledge of the rules of vector manipulations.
(Chap 3 of Serway/Vuille or Serway/Jewett) Your review should include the addition of vectors
and two ways of representing vectors: vector components and magnitude/direction.
Experimental Setup
The Force Table is a round platform with a protractor printed on it. In the center is a peg, and
pulleys are clamped around the edge. A ring serves as our object of study. Several strings are
tied to the ring, and it is centered over the peg. The other ends of the strings go over the pulleys
and masses hang from them. A typical setup is depicted in Figure 1.
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Figure 1: Typical Force Table setup, in this case with 3 equally spaced pulleys that have equal
masses dangling from them. (Image credit: PASCO)
General Instructions









The setup will have 2 or 3 strings exerting forces on the ring. When only 2 strings are in
use, the third string can just lay on the force table.
The ring must be centered around the pin:
The masses added to the holder should be between 25 g and 150 g.
In this set up, the magnitude of the force equals the magnitude of the gravitational force
on the mass attached to the string. Fg = mg
Vector Addition in a Spreadsheet: https://www.youtube.com/watch?v=KqjZd6yyslA
When calculating components in Excel, the angle should be converted to radians.
Example: =B2*cos(radians(C2)).
DO NOT add up the Magnitude or Direction columns. Instead, calculate the Magnitude
and Direction of the Net Force using the components.
According to theory, the Net Force should be zero. In the experiment, it should be found
that the Net Force magnitude is less than 5% of any of the individual force magnitudes.
See if there is a symmetry in the magnitudes of the forces or in the angles between the
forces.
Experiment 1: Object submitted to two forces
1. If only one force is applied to the object, the object will not be in an equilibrium state.
Try it. Set up the force table as shown in Figure 1, including the plastic ring (the object),
but with only one string pulling on the ring. Apply 1 force to the object by suspending a
mass holder with some masses from one of the strings attached to the ring with the
string pulled over a pulley on the side of the table. Notice that the ring only stays in
place if it is resting against the pin.
2. Prepare a second mass holder and hang it over another pulley so the object reaches
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equilibrium (the ring must be stationary at the center of the force table). Adjust the
mass and move the pulley until the ring is in equilibrium while it is centered on the pin.
3. Record you data in the table below.
Force
Magnitude
Direction (°)
x-component
y-component
Force 𝐹1 (N)
Force 𝐹2 (N)
Net Force (N)
Table 1. Measurements and calculations for two forces.
Experiment 2: Object submitted to three forces
1.
2.
3.
4.
Part A:
Set up Force 1 and Force 2 by hanging two equal masses at ±60° (i.e. 60° and 300°).
Calculate the resultant force of Force 1 and Force 2. Predict what Force 3 will be
needed to balance the resultant and achieve equilibrium. Hint: Force 3 should be the
opposite of the resultant.
Set up Force 3 and Move the Pulley 3 until the ring is in equilibrium in the middle.
Record the forces in the table similar into Table 2.
Part B:
Even if the magnitude of the forces are not changing, any change in direction of any force will
affect the resultant and the neutralizer.
1. Don’t change Force 1 (i.e. don’t change Mass 1 or move Pulley 1).
2. Don’t change Mass 2.
3. Redirect Force 2 by moving Pulley 2 towards or away from (your choice) Pulley 1 by
about 10°.
4. Calculate the resultant of Force 1 and Force 2. Predict what Force 3 will be needed to
neutralize the resultant.
5. Set up Mass 3 and Pulley 3 to re-establish equilibrium.
6. Record the forces and directions in a new table. The table should be a copy of Table 2,
but with a fresh caption, because it’s for a new situation.
In abstract, mention how the adjustment of the angle affected both, the magnitude and the
direction of the resultant and neutralizer.
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Force
Magnitude
Direction (°)
x-component
y-component
Force 𝐹1 (N)
Force 𝐹2 (N)
Resultant (N)
Neutralizer ⃗⃗⃗⃗
𝐹3 (N)
Net Force (N)
Table 2. Measurements and calculations for three forces.
Experiment 3: Object submitted to three forces, each of different
magnitude.
1. Set up the force table with three different strength forces (i.e. three different masses).
2. Adjust the positions of the pulleys to achieve equilibrium with the ring centered on the
pin.
3. Record the forces and directions in a new table. The table should be a copy of Table 2,
but with a fresh caption, because it’s for a new situation.
Extra credit (10 points)
Consider four equal-strength forces placed at 72° intervals. (Perhaps set them at 72°, 144°, 216°,
and 288°.) Calculate the magnitude and direction of the Net Force.
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Lab 6 – Atwood’s Machine
Equipment









Computer, PASCO Interface
Table Clamp
Double pulley apparatus (one smart pulley)
Smart Pulley Data Cable
String
Two Mass Hangers
One Mass Set (1×500, 2×200, 1×100, 1×50, 2×20, 1×10, 1×5, 2×2, 1×1)
Styrofoam Pad
Capstone setup file: Smart Pulley with Photogate.cap
Objectives

To verify Newton’s Second Law using the Atwood’s Machine
Introduction
According to Newton’s 2nd Law the acceleration of an object depends on the net applied force
and the object’s mass.
(1)
∑𝐹 = 𝑚𝑎
In an Atwood's Machine, there are two objects, each one is a mass hanging from one end of a
string. The string is hung over a pulley so that both masses are hanging. The weight of one mass
pulls the system in one direction, while the weight of the other mass pulls in the other direction.
Figure 1. Individual free body diagrams of each of the hanging masses.
In the above free body diagrams, 𝑇 is the tension in the string, Assuming that the pulley is
massless and frictionless, and the string has no mass and doesn’t stretch, the tension T will be
the same throughout the string. The weights are 𝐹𝑔1 = 𝑚1 𝑔 and 𝐹𝑔2 = 𝑚2 𝑔 where 𝑔 is the
acceleration due to gravity. Taking the convention that up is positive and down is negative, the
net force equations for 𝑚1 and 𝑚2 are:
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∑𝐹1𝑦 = 𝑇 − 𝑚1 𝑔 = 𝑚1 𝑎1
(2)
∑𝐹2𝑦 = 𝑇 − 𝑚2 𝑔 = 𝑚2 𝑎2
(3)
The key to solving this is to realize that in the system of 𝑚1 and 𝑚2 , one mass accelerates
upward while the other accelerates downward. Let’s say 𝑚1 (the lighter mass) accelerates
upward and call its acceleration plain 𝑎. Then the acceleration of 𝑚2 is just 𝑎2 = −𝑎. We can
subtract equations (2) and (3) to obtain:
𝑚2 𝑔 − 𝑚1 𝑔 = 𝑚1 𝑎 + 𝑚2 𝑎
Δ𝑚𝑔 = 𝑚tot 𝑎
(4)
(5)
Solving for a, the acceleration of the system of both masses, the theoretical acceleration is g
times the difference in mass divided by the total mass:
Δ𝑚
𝑎 = 𝑔(
)
𝑚tot
(6)
If 𝑚2 > 𝑚1 , then 𝐹𝑔2 is stronger, and the net force on the system accelerates 𝑚2 downward and
𝑚1 upward.
Treating the Atwood’s Machine as a System.
Notice that equation (5) looks a lot like Newton’s 2nd Law (1). The “object” is the pair of masses
attached to either end of a string. On the right side of (5), the mass of the system is the the total
mass of the two hangers. The acceleration of the system is directly proportional to the exerted
force and inversely proportional to the mass of the system.
On the left of (5), we have the net force on the system 𝐹net = Δ𝑚𝑔. The net force acting on the
system of both masses is just the difference in weight between two hanging masses. This net
force accelerates both of the hanging masses. The heavier mass is accelerated downward, and
the lighter mass is accelerated upward.
Experimental Plan
Newton’s 2nd Law can be verified by setting up the Atwood’s Machine under various conditions.
By carefully selecting the masses, we can graph:
(1) 𝐹net vs. 𝑎 while keeping the system mass 𝑚tot constant.
(2) 𝑎 vs. 1/𝑚tot while keeping the net force Δ𝑚𝑔 constant. Our trendline equation is:
1
𝑎 = (𝑔Δ𝑚) (𝑚 )
So the slope of the graph will be (𝑔Δ𝑚).
tot
In graph (1), the result should be linear with the slope of the graph equal to the total mass and in
graph (2), the result should be linear with the slope of the graph equal to the exerted net force
Δ𝑚𝑔. If this is true, then Newton’s 2nd Law is verified.
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Experimental Setup
We will use the Photogate/pulley system to measure the motion of both masses as one moves
upward and the other moves downward. Capstone software will record the changing speed of the
masses as they move. The slope of the velocity vs. time graph is the acceleration of the system.
There is a saved Capstone setup file, called “Smart Pulley with Photogate”, posted with the
lab instructions. Please use it to avoid Capstone setup headaches.
Hardware Setup
Figure 2a: (For Instructors Only) Disassembled smart pulley. Caution! If assembly or
adjustment is necessary, do not “bottom out” the threads on the long threaded side of the
rod. It can break the pulley and/or dislodge the bearings. (1) Snug the smart pulley on the
short side, with the photogate acting as a washer. (2) Thread the other pulley on the other
side of the rod, but don’t “bottom out” the threads. It will be loose. Hold it so the pulleys are
planar, then (3) tighten the nut against the pulley to hold it in place.
1. Mount the universal clamp to the edge of the table.
2. Insert the connecting rod of the pulley into the clamp.
3. Plug the photogate into digital channel 1of the interface. You can tell it’s working by
spinning the pulley. A little red light near the pulley should blink.
4. Use a piece of string just long enough so that when one mass is on the floor/table, the other
is a few centimeters below its pulley.
5. Hang mass hangers from each end of the string, and hang the string over the pulleys, as
shown in Figure 2.
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Figure 2b. Schematic representation of the Atwood’s Machine. (Image source: PASCO)
6. Make sure the string isn’t rubbing against anything, such as the data cable.
Data Recording
Experiment 1: Constant Total Mass
Start with 𝑚1 being a total of around 340 g. When recording masses, don’t forget to include the
50 g for the mass hanger. Start with 𝑚2 being a total of 350 g.
In each Run, change the amount of mass on 𝑚1 by moving some masses to 𝑚2 . This will keep
the total mass constant. Shift 10 g or 20 g at a time, and you don’t have to have evenly spaced
points.
With the masses listed above, you can definitely make m1 have values of 340 g, 320 g, 300 g, 290
g, 270 g, and 250 g. Remember, in this experiment, the total 𝑚1 + 𝑚2 should always be 690 g.
Set up the desired amounts of mass on 𝑚1 and 𝑚2 .
Move the masses so that 𝑚2 is hanging near the top and 𝑚1 is on the table/floor.
When you are ready to start, press Start in Capstone.
When 𝑚2 reaches the bottom, press Stop.
For each data run, fill in the columns of the data table.
 Run #: Record the run number from Capstone, in case you have to come back to it.
 𝑚1 , and 𝑚2 : The individual masses.
 𝐹net = Δ𝑚 𝑔 as described in the introduction.
 𝑚tot: the total mass.
 𝑎exp : the experimental acceleration, measured directly in Capstone:
 Use the Highlighter to select the “good” points.
 Use the Linear Fit feature just above the graph to draw the fit and see the slope. This
is the experimental acceleration.
 𝑎theory = 𝐹net /𝑚tot is the acceleration as calculated from theory.
 Calculate the % difference using the theoretical value as your “reference”.
6. Move masses around as needed and repeat.
1.
2.
3.
4.
5.
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Run #
𝒎𝟏 (kg)
𝒎𝟐 (kg)
𝑭net (N)
𝒎tot (kg)
𝒂exp
(m/s2)
𝒂theory
(m/s2)
Percent
Difference
…
Table 1. Example table for recording run parameters, results, and calculations.
Experiment 2: Constant 𝑭net
The procedural steps are the same as in Experiment 1; however, the difference between m1 and
m2 should be kept constant of 100 g. In between each run, add or subtract the same amount of
mass from each hanger. Record the data in another Excel spreadsheet table built from an
example of Table 1.
With the masses listed above, you can definitely make m1 have values of 300 g, 270 g, 250 g, 200
g, 170 g, and 150 g. Remember, in this experiment 𝑚2 should always be 100 g more.
Make sure to record your mass difference, probably in the caption of your data table.
Data Analysis
Once you have all of the Runs calculated, make two summarizing graphs, one for each
Experiment.
1. Expt 1: Each data run has the same total mass. Make a graph of 𝐹net vs. 𝑎exp . The slope of
the linear trendline should be the total mass of the system. Record your results in something
like Table 2.
Total Mass (kg)
Slope of 𝐹net vs. 𝑎exp (kg)
% Error
Table 2. Measured and calculated total mass for Experiment 1.
2. Expt 2: Each data run has the same 𝐹net . Make a graph of 𝑎exp vs. 1/𝑚tot. You’ll need to
make a helper column to hold the values of 1/𝑚tot. Since Newton’s 2nd Law says 𝐹net =
𝑚tot 𝑎, the slope of the graph should be 𝐹net . Make a Table 3, similar to Table 2, recording
your 𝐹net , slope of the graph, and the % Error. (See also the intro.)
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Lab 7 – Friction
Equipment








Calculator, Computer, PASCO Interface
Force Sensor (pick one):
o PASCO Force Sensor
o Spring scale
PASCO Photogate with Pulley and clamp for edge of table.
Dual-surface block.
String (~50-100 cm)
Mass Hanger and mass set.
Capstone Setup file: Force Sensor - Graph.cap
Capstone Setup file: Smart Pulley with Photogate.cap
Objectives
The purpose of the following experiments is to verify the statement that in general 𝜇𝑠 > 𝜇𝑘 .
Introduction
Friction is a resisting force that acts along the tangent to two surfaces in contact with each other
when one body slides or attempts to slide across another. The direction of the frictional force is
opposite the body's motion or attempted motion. Our simple model is that the available
frictional force (𝑓𝑠 or 𝑓𝑘 ) is directly proportional to the normal force (𝐹𝑁 ). (We’re referring to the
force magnitudes, so no arrow or boldface font is necessary.)
There are two different kinds of friction: the static friction force 𝑓𝑠 and the kinetic friction force
𝑓𝑘
When two surfaces are still at rest, but attempt to slide one over another, static friction is
present. The static frictional force 𝑓𝑠 is has a magnitude that can have any value, up to a limit:
𝑓𝑠 ≤ 𝜇𝑠 𝐹𝑁
(1)
Here, 𝐹𝑁 is the magnitude of the normal force between the surfaces.
When two surfaces are moving with respect to each other, the other kind of friction occurs. It is
the kinetic friction. The kinetic frictional force 𝑓𝑘 has a specific magnitude:
𝑓𝑘 = 𝜇𝑘 𝐹𝑁
(2)
In the expressions above 𝜇𝑠 is the static friction coefficient and 𝜇𝑘 is the kinetic friction
coefficient. In general 𝜇𝑠 > 𝜇𝑘 . This means that when enough force is exerted on an object to
overcome the static frictional force and the object starts moving, the kinetic frictional force will
be less than the applied force and the object will accelerate. Another way of thinking of this is
that it is harder to get something moving than it is to keep it moving.
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Experiment 1: Static Friction Coefficient
In this experiment, you will determine the static friction coefficient of a block sitting on the lab
bench. To do this, first measure the limit of the static friction force with a force sensor by
pulling very gently on the force sensor till block moves. Second, calculate the normal force by
assuming that the table surface is perfectly horizontal so that the normal force and gravitational
forces on the block are equal in magnitude. Since fsmax = μsn, the slope of the graph friction vs.
normal force should be the coefficient of static friction.
PASCO Force Sensor setup



Plug the Force Sensor into the PASCO Interface box. Be VERY CAREFUL about the
orientation of the pins. Make sure they line up correctly with the socket.
Open “Force Sensor – Graph” Capstone setup file from the lab web folder.
Whenever you are going to take a measurement, you must “tare” the sensor. To do this,
get the sensor into the correct position and orientation, but exert no force on the hook.
(This means the string should be slack.) Then, press the “tare” or “zero” button.
Data Recording
1. Measure and record the mass of the dual-surface block.
2. Your block should have two different surfaces. They may be wood and metal, or they
may be wood and felt. For each run, make note of what surface is in contact with the
table!
3. Tie a string between the block and the force sensor. Place the block near the edge of the
table. See Figure 1 below.
Figure 1. Schematic setup for experiment 1.
4. Place the block, making note of which side is down, on the table, and record the pull
needed to make it start sliding. Start very gently so that the object doesn’t move at
first. Gradually increase the pull until the object finally moves. Once the object moves,
you’re already done taking data for that pull. Don’t drag the block across the desk. This
is a measurement of static friction, not sliding friction.
5. Record the peak magnitude of the force.
 PASCO Force Sensor: Stop the recording once the block starts to move. In
Capstone, use a cross-wire tool to determine the coordinates of the point at the
positive or negative peak on the graph. Ignore the negative sign.
 Spring scale: Simply watch the scale as you pull, and record the maximum force
just as the block starts to slide.
6. Repeat the measurement a few times to see if you are getting a consistent value. If not,
try pulling more gently and increase the force more gradually. Use the average of the few
most consistent values for your force measurement.
7. Repeat the measurements with different parameters: for the next sample, increase the
mass on top of the block by at least 50 g. Do a total of 5 samples gradually increasing the
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mass.
8. Flip the block over and repeat the experiment with the other side down.
Side
touching
Table
Mass of the
block (kg)
Added
mass(kg)
Total mass
(kg)
Normal force (N)
Maximum Static
Friction Force (N)
Table 1. Measured and Calculated values from experiment 1. (Note: There should be 10
different samples, 5 each for the two sides of the block.)
Data Analysis
9. For each data run, calculate the normal force 𝐹𝑁 assuming it equals the gravitational
force on the total mass.
10. Make 2 graphs, one for each side that was sliding against the table. Graph the friction
force 𝑓𝑠 vs. the normal force 𝐹𝑁 . Determine the linear trendline. The slope is your static
friction coefficient. If you want, you can put both sets of data on the same graph.
Experiment 2: Kinetic Friction Coefficient
Measuring the kinetic friction coefficient is trickier because we can only measure the pulling
force, not the friction force directly. They are only equal if the block is moving with zero
acceleration. It is difficult for a person to pull on the force sensor with a zero acceleration.
Fortunately, we can set up a situation with a constant acceleration. A block of mass 𝑚block is
placed on the table top, and a weight hanger of mass 𝑚hang is tied to a string, which is passed
over a pulley paired with a photogate and tied to the block, see Figure 2. When the weight
hanger is released, the entire system will accelerate and we will get to measure that acceleration.
Figure 2. Schematic setup for Experiment 2
The external forces affecting the acceleration of the system are:


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𝐹𝑔 pulling down on the mass hanger.
𝑓𝑘 resisting the motion
Last Updated: 2016-06-15
Since these forces are in opposite directions, the sum of the forces is the difference of their
magnitudes. This must equal the total mass times the acceleration.
𝐹net,ext = 𝑚hang 𝑔 − 𝑓𝑘 = 𝑚total 𝑎
(3)
The vertical forces on the block are just gravity and the normal force and assuming a perfectly
horizontal table top, they are equal and opposite.
𝐹𝑔 = 𝑚block 𝑔
(4)
So here’s the basic idea: Place a mass of around 250 g on a hanger and add 200 g of extra mass
on the block. Set the system in motion and measure the acceleration. Record 𝑚block, 𝑚hang , and
acceleration 𝑎. Calculate the friction force. Since fk = μkn, the slope of the graph friction vs.
normal force should be the coefficient of kinetic friction.
Experimental Setup and Data Recording
1. Tie a string between the block and a mass hanger, and pass the string over a smart
pulley. Make sure the string isn’t rubbing against anything.
2. Place the desired amount of mass on the block and hold it so it doesn’t slide on the table.
3. Close Capstone. Open the “Smart Pulley with Photogate” Capstone file. Recording the
data in this part of the experiment is just like Lab 6 – Atwood’s Machine. Review those
instructions if you need a reminder. Remember to get the slope of the upward sloping
part of the velocity graph.
4. Let go of the block and let the computer record the velocity graph.
5. For each run, record 𝑚block , 𝑚hang , and the acceleration 𝑎.
6. Repeat the measurement for 5 different 𝑚extra values and for each side of the block. Keep
𝑚hang the same throughout Expt 2.
Data Analysis
7. Calculate 𝑓𝑘 from the experiment by solving Equation (3), and calculate 𝐹𝑁 using the
mass of the block. You can either:
 Solve 𝑓𝑘 = 𝜇𝑘 𝐹𝑁 directly for each run, and average the results to obtain a final value
of 𝜇𝑘 . If the values are consistent, this method works well.
 Graph 𝑓𝑘 vs. 𝐹𝑁 , and use the slope as your final value of 𝜇𝑘 . If the points form a
straight line graph, this method works well.
Final Comparison
Organize the results in a summary table such that it would be easy to compare the experimental
values of coefficients of static and kinetic friction for each surface.
For friction coefficients of the same surfaces (such as metal on table or wood on table), we
expect that 𝜇𝑠 is larger than 𝜇𝑘 . Discuss if this holds.
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Lab 8 – Archimedes’ Principle and
Buoyancy
Equipment












Chemical splash goggles (Students bring their own)
Distilled/Deionized Water, Isopropyl alcohol
Calculator, Computer
Set of Digital Calipers
Spring Scale or Force Sensor
Plastic bins to catch overflow.
Graduated cylinder
Aluminum Container with spout
Vertical stand, perpendicular clamp, horizontal rod (between 20 cm and 50 cm)
Brass Ball with a string attached to it
Wooden cylinder with pencil lines marking off equal lengths
Capstone setup file: Force Sensor – Digits Display.cap
Objectives
Verify Archimedes’ principle and use the Archimedes’ principle to determine the density of a
given liquid.
Introduction
The famous legend tells us that Archimedes was the person who discovered that the volume of
displaced water equals the volume of a submerged object. He came up with that idea as he was
trying to measure the volume of a crown of unusual shape. Puzzled he had filled his bathtub
flush with water and water overflowed when he got inside of the tub. The idea that the amount
of water splashed out of the tub is exactly the volume of his own body struck him and he ran
outside of his house crying “Eureka!” This means, “I have found it”.
Archimedes’ Principle itself isn’t directly about volume, it’s about buoyancy. It states that the
buoyant upward force acting on an object entirely or partially submerged in a fluid is equal to
the weight of the fluid displaced by the object.
For a given object, the weight can be directly calculated from the mass or from the density and
volume:
𝐹𝑔 = 𝑚𝑔 = 𝜌𝑉𝑔
The buoyant force is found by applying the same idea to the fluid instead of the object:
𝐹𝐵 = 𝑚fluid 𝑔 = 𝜌fluid 𝑉displaced 𝑔
(1)
Here, 𝑚fluid is the mass of the displaced fluid, which is broken down as the density of the fluid
𝜌fluid multiplied by the submerged volume of the object 𝑉displaced .
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For a prism-shaped object like a cylinder, the submerged volume is equal to the cross-sectional
area, 𝐴, multiplied by the submerged depth, 𝑑. So the buoyant force can be written as:
𝐹𝐵 = 𝜌fluid 𝐴𝑑𝑔
(2)
If the object is lowered into the fluid while the buoyant force is measured, the slope of the graph
of 𝐹𝐵 versus 𝑑 is proportional to the density of the fluid.
Figure 1: Schematic picture for the experiment
Part I. Verifying Archimedes Principle
Figure 2. Ready to perform Experiment 1. Read the instructions below; this picture is just so
you have an idea of what is going on. The upper container (with the spout) was set in place, and
then it was “topped off” with water. (The extra water fell into the plastic bin.) The graduated
cylinder is ready to catch the overflow that will come out when the brass ball is lowered into the
water. (Do not copy our picture into your lab report!)
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Experiment 1. Volume of the Displaced Liquid
The purpose of this experiment is to verify Archimedes’ “finding” that the volume of the
displaced liquid is the same as the volume of the object immersed. A metal ball will be used as
the solid object.
1. Measure the volume of the ball by measuring its diameter and using that to calculate the
volume of the ball, assuming that it’s a perfect sphere.
2. Submerge the ball in water and determine the volume of the water displaced. (See figure
2.)
a. To do so place the aluminum container (the one with a spout) in position where
you can catch any overflow with the graduated cylinder.
b. Fill the container with water so it just overflows (don’t catch this water), then
allow it to stop dripping. Note that if you move the container after it’s full, you’ll
slosh some water out, so get the container fixed in position first, then fill it.
c. Prepare to catch any additional water that comes out of the spout with the
graduated cylinder.
d. Lower the ball in the water while catching the overflow in the graduated cylinder.
3. Measure the displaced water volume using the graduated cylinder. Keep this cylinder
of water for Experiment 2!
4. Compare the results. If you’re far off, try it again, being more careful not to lose water
before submersing the ball. (Note: 1 cm3 = 1 ml)
Ball diameter (cm)
Vball (cm3)
Vwater (ml)
% Error
Table 1: Replace this text with an appropriate caption.
Experiment 2. Weight of the Displaced Liquid
The purpose of this experiment is to verify Archimedes' Principle for a sunken object. In other
words, you will attempt to verify that the buoyant force is equal to the weight of displaced liquid.
To do this you will need a force sensor. We have two options for the force sensor:


Spring scale: These show the tension in newtons. (The equivalent mass in grams is
also shown; ignore this.)
PASCO Force Sensor: These require the PASCO 850 Universal Interface and setup in
PASCO Capstone. The file “Force Sensor – Digits Display.cap” is configured for the force
sensor.
Rememner, the value we record is actually the tension FT in the sensor.
DATA RECORDING:
1. Suspend the ball from the hook of the Force Sensor and record its weight in air.
2. Bring a cup of water up from underneath the ball. (Figure 3)
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3. Make sure the ball isn’t touching the side or bottom of the cup. While the ball is
submerged, record the “weight” of the ball in water.
4. Using one of the digital balances, find the mass of the graduated cylinder from
Experiment 1 with and without the water.
ANALYZING THE DATA:
1. Calculate the buoyancy force by finding the difference of the weight of the ball in air and
in water.
2. Calculate the mass of the displaced water by subtracting the mass measurements from
above.
3. Calculate the weight of the displaced water. Note that 𝑔 = 0.0098 N/g.
4. Compare the experimental value of the buoyancy force with the weight of the displaced
water by calculating % Error.
𝐹𝑇 in air (N)
𝐹𝑇 in water (N)
𝐹𝐵 (N)
𝑚1 (g)
𝑚2 (g)
𝑚water (g)
𝐹𝑔,water (N)
% Error
Table 2: Replace this text with an appropriate caption.
Part II. Measuring Density
Experiment 3. Density of water/alcohol
The purpose of this experiment is to apply Archimedes’ principle to determine the density of a
given liquid. For objects with constant cross sectional areas (𝐴) such as cylinders or cubes one
can use Equation 2 above.
From the equation you can see that the magnitude of the buoyant force is directly proportional
to the portion of the object submerged in the liquid 𝑑. The graph 𝐹𝐵 vs. 𝑑 is a linear function
with slope equal to 𝜌water 𝐴𝑔.
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Figure 3. Taking data for Experiment 3. The cup of water has been brought up under the
mass until it reached the desired line. The tension recorded by the spring scale is not the
weight of the block, as it has been reduced by the buoyancy force. (Do not copy our picture
into your lab report!)
DATA RECORDING:
1. Hang the wooden block from the Force Sensor and display the reading as with the
sphere.
2. Take a reading of the block’s weight when measured in air (𝐹𝑇 = 𝐹𝑔 ). Note that 𝐹𝑔 doesn’t
change throughout the experiment.
3. Submerge the wooden block suspended from the force sensor down in water to the first
pencil mark (1cm depth) and record its “weight in water” (𝐹𝑇 , which is not 𝐹𝑔 ). You can
lower the block into the water or raise the water under the block; it doesn’t matter which.
4. Repeat the same procedure for each mark until the wooden cylinder starts floating.
5. Once the block starts actually floating, or if the block leans against the side of the
container, don’t any more data.
6. Record your experimental data in a table similar to Table 3.
ANALYZING THE DATA:
1. Calculate the buoyant force (𝐹𝐵 = 𝐹𝑔 − 𝐹𝑇 ) for each data point.
2. Plot the buoyant force 𝐹𝐵 versus the submerged depth, 𝑑.
3. Determine the slope of the graph and compute the density of the liquid. Record this in
something similar to Table 4.
7. Calculate the % Difference between the accepted value (1,000 kg/m3) for the density of
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water and the value you found from the slope.
Depth 𝒅 (m)
𝑭𝒈 (N)
𝑭𝑻 (N)
𝑭𝑩 (N)
Table 3: Replace this text with an appropriate caption.
Slope of 𝐹𝐵 vs. 𝑑 (N/m)
Density of liquid, 𝜌 (kg/m³)
Expected 𝜌 (kg/m³)
% Error
Table 4. Replace this text with an appropriate caption.
Experiment 4. Density of isopropyl alcohol





Wear chemical safety goggles. If you get isopropyl alcohol in your eye, go to the eyewash
station and flush it out.
Don’t drink the isopropyl alcohol; it’s poisonous not drinkable.
If you get isopropyl alcohol on your skin, dry it off. Isopropyl alcohol is also known as
rubbing alcohol and getting a small amount on your skin shouldn’t harm you.
Don’t spill the alcohol.
Ask the instructor about waste collection.
Repeat Experiment 3, but using isopropyl alcohol as the fluid instead of water. Make data tables
and a graph.
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Lab 9 – Inclined Plane
Equipment




Calculator, Computer
Air Track, Air Pump
Air Track Cart
Stopwatch
Objectives

To measure the acceleration of an object moving on a gentle, frictionless incline and to
use that measurement to determine the gravitational acceleration 𝑔.
Introduction
In an earlier experiment, you attempted to use a stopwatch to measure the gravitational
acceleration 𝑔 by timing the fall of a dropped object. It was almost impossible to get a good
result by this method. In the same lab, you used a computer to time the fall of another dropped
object (the “picket fence”) and found that by improving the timing method, a better
measurement of 𝑔 was obtained. In this lab, we’ll go back to using the stopwatch, but we’ll slow
down the action by having our object float down a gentle incline instead of dropping straight
down.
You already know the basics of kinematics. When an object undergoes one-dimensional
uniformly accelerated motion, its velocity increases linearly with time. If the initial velocity of
the object is zero at time 𝑡 = 0, then the velocity 𝑣 at any time 𝑡 is given by the following
expression:
𝑣 = 𝑎𝑡
(1)
Here, 𝑎 is the acceleration, which is constant in magnitude and direction.
Consider a time interval between 𝑡 = 0 and any later time 𝑡. The velocity is zero at 𝑡 = 0, and the
velocity at time 𝑡 is 𝑣. With uniform acceleration, the average velocity 𝑣avg during the time
interval is:
𝑣avg =
0+𝑣
= 𝑣/2
2
(2)
The displacement 𝑥 of the object during the time interval 𝑡 is given by:
1
1
𝑥 = 𝑣avg 𝑡 = 2𝑣𝑡 = 2𝑎𝑡 2
(3)
Thus, Equation 3 states that if an object is released from rest, its displacement is directly
proportional to the square of the elapsed time.
When an object is dropped, we assume free fall, which means its acceleration is 𝑔 in a
downward direction. A closely related situation is an object gliding down a slope. In this case,
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the acceleration is reduced by a factor determined by the angle of the slope. The magnitude of
the acceleration is then:
𝑎 = 𝑔 sin 𝜃
(4)
Here, 𝜃 is the angle of the slope, measured with respect to a horizontal surface. So if 𝜃 = 0, the
object doesn’t accelerate. If 𝜃 = 90°, the slope is vertical and the object falls freely. If 𝜃 is some
small angle, then the acceleration will be small. (The origin of this formula is that the force of
gravity 𝐹𝑔 is a vector which can be resolved into components that are perpendicular to the plane
and parallel to the plane. The perpendicular component is offset by the support force from the
plane. The parallel component (𝐹𝑔 sin 𝜃) pulls the cart down the plane.)
Experimental Setup
A cart is placed on an air track raised at one end to form an inclined plane whose angle of
inclination is 𝜃. Since measuring small angles is difficult, we will instead measure some
distances. The difference in height Δℎ between two points along the slope can serve as the
opposite side to our 𝜃, and the distance between these points (along the slope) can serve as the
hypotenuse.
Figure 1. Slope showing the measured parameters needed to calculate the angle.
Then, we know that our sin 𝜃 = Δℎ/𝑑.
When released, the cart glides down the inclined plane with acceleration 𝑎 = 𝑔 sin 𝜃.
Data Recording
Using the stopwatch, you will measure the time it takes for the cart to slide various distances.
Later, during the analysis, you will determine the acceleration 𝑎 and then the gravitational
acceleration 𝑔.
Please do not draw marks on the air track. If you want reference marks, use a Post-it.
Make sure the Post-it is low enough that the cart doesn’t touch it as it glides by.
1. Hook up the air track to the pump, place the cart on it, and test it out. Make sure the cart
can glide freely back and forth.
2. Place a book or a small block under one end of the air track.
3. Choose two points along the track and measure their heights above the table and the
distance 𝑑 between them. These values are vital to the success of the experiment, so you
probably want each partner to verify the measurements.
4. Have one partner hold the cart at rest near the top of the incline. Find the point 0.25 m
(25 cm) down the track. Have another partner get ready with the stopwatch.
5. Release the cart from rest and time how long it takes for the cart to glide the first 0.25 m
down the track.
6. Repeat the measurement for five (5) trials at this distance.
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7. Repeat the experiment for distances of 0.50 m, 0.75 m, 1.00 m, and 1.40 m.
Δ𝑥 (m)
𝑡1 (s)
𝑡2 (s)
𝑡3 (s)
𝑡4 (s)
𝑡5 (s)
𝑡avg (s)
2
(𝑡avg )
(s2)
0.25
…
Table 1. Measurements of the time for an air cart to slide various distances down a track, when
released from rest.
Data Analysis
When we expect a proportional relationship like that between Δ𝑥 and 𝑡 2 , one way of finding the
proportionality constant (in this case, 12𝑎) is to calculate its value for each data point (i.e. each
row in the table). The problem is that some kinds of experimental errors will keep us from ever
getting a good value of 𝑎. Even if we take the average of all of our 𝑎 values (one from each row),
the error can still linger. One way of eliminating some kinds of errors is to use the linear best
fit to measure of the proportionality constant. The slope of the trend line works great for this
purpose.
1. Make a scatter plot of Δ𝑥 vs. 𝑡 2 . (Hint: this means Δ𝑥 is on the vertical axis, and 𝑡 2 is on
the horizontal axis.
2. Add the linear trend line and its equation. Record the slope.
3. Calculate the acceleration 𝑎 from the slope. (Hint: Δ𝑥 = 12𝑎𝑡 2 = (Slope)𝑡 2 .)
4. Find the acceleration due to gravity from your cart’s acceleration.
5. In your conclusion, comment on the accuracy of your experiment and discuss possible
causes of the error.
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Description
Value
Slope (m/s 2)
Acceleration 𝑎 (m/s 2)
Height Difference Δℎ (m)
Distance between points 𝑑 (m)
sin 𝜃
𝑔experimental (m/s 2)
% Error
Table 2. Various values calculated from the measurements, ultimately producing an
experimental value of the gravitational acceleration 𝑔.
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Lab 10 – Circular Motion and
Centripetal Acceleration
Equipment









Safety Glasses required for possible flying springs.
Calculator, Computer, (optional) PASCO Interface
Ruler
Centripetal Apparatus
Clamp to secure Centripetal Apparatus to table.
Timing Option 1
o Stopwatch
Timing Option 2
o Horizontal Clamp
o PASCO Photogate, Stand, and Cable
Force Option 1
o Mass Hanger, Mass Set
Force Option 2
o PASCO Force Sensor
o Capstone Setup file: Force Sensor – Digits Display.cap
Objectives
Verify Newton’s Second Law, as applied to an object in circular motion.
Introduction
We expect that the acceleration of an object that is undergoing circular motion should be:
𝑎𝑐 =
𝑣 2 (2𝜋𝑅/𝑇)2 4𝜋 2 𝑅
=
=
𝑅
𝑅
𝑇2
(1)
Here, 𝑅 is the radius of the path, and 𝑇 is the period, which is the time it takes the object to
complete one revolution about the circle. The distance traveled by the bob in one revolution is
the circumference the circle, 2𝜋𝑅, and its average speed 𝑣 can be calculated by dividing the
circumference by the period or 𝑣 = 2𝜋𝑅/𝑇. This is the substitution made after the second
equals sign. The above acceleration is called the centripetal acceleration.
The net force that causes centripetal acceleration is called centripetal force. It’s impossible
to measure a net force, but we can measure a single tension force. If we can somehow arrange it
so that the centripetal force is equal to the force we measured, we have effectively measured the
centripetal force.
Our object will be a steel bob hanging from a string. We’ll make sure the string is vertical, so
that it doesn’t have any horizontal component. The only sideways force on the bob will be from
a spring. Each measurement will take place in two phases:
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

With the bob hanging at rest, use the Force Sensor to exert an equal and opposite
sideways force so that the bob is hanging vertically.
With the Force Sensor removed, spin the Circular Motion Apparatus so that the bob’s
own inertia makes the string vertical. In this situation, the spring is providing the
centripetal force.
Experimental Setup:






Place the centripetal apparatus on the tabletop and secure it (clamp it down). The
apparatus consists of a rotating vertical metal shaft with an adjustable crossbar and a
heavy bob suspended from the longer end of the crossbar. A reference pole is used to tell
when the bob is hanging vertically.
The bob should almost reach the little vertical reference rod, but it shouldn’t be so low
that it touches.
Timing Option 2 – if using the phogate for timing:
o The horizontal clamp should be attached to the rotating pole very near the
bottom.
o Lay the photogate stand on its side so that the horizontal clamp blocks the beam
once per rotation of the pole.
WARNING: The counterbalance must be secure.
WARNING: Secure the crossbar whenever the equipment is used.
IMPORTANT: If using the PASCO Force Sensor, make sure the TOP label is facing up
when it connecting the Force Sensor to the PASCO interface. The connector should plug
in fairly easily.
DATA RECORDING:
Preparation:
1. Detach the spring from the bob.
2. Set the position of the crossbar as desired. Start with small radii.
 To adjust it, hold it securely, loosen the screw, slide it, and tighten the screw.
3. Position of the vertical reference pole directly under the bottom of the bob.
 To move it, loosen the screws and slide the adjustable metal plate so the reference
pole is where it needs to be. Tighten the metal plate screws.
Measure the Radius:
4. With the bob hanging freely (no spring, no motion), the bob should be directly over the
vertical reference pole. (Figure 1)
5. Measure the distance between the center of the rotational shaft and the center of the
reference pole. Record this as your radius in Data Table 1.
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Figure 1. To measure the radius, the spring is detached, the bob hangs down vertically
from its strings, and the vertical reference rod is below the bob.
Measure the Spring Force at that Radius:
6. Attach the spring to the bob. Do not spin the bob yet.
7. Measure the force required to pull the bob back to the reference pole.
 Use the piece of string attached to the outside edge of the bob.
 Option 1: PASCO Force Sensor:
i. Load one of the PASCO Capstone files from the lab files folder.
 Force Sensor – HiRes Digits Display: Only does force sensor
 Circular Motion – Force Sensor and Photogate: Allows for
computer-based timing measurements.
ii. Have a lab partner with a steady hand hold the Force Sensor and pull the
bob outward, away from the shaft. Make the bob hang directly above the
vertical positioning pole.
iii. Use the Digits in Capstone to get a measurement of the tension force in
the string. As long as the bob hangs vertically downward, this should
equal the spring force.
 Option 2: Hanging Mass (Figure 2):
i. Pass the string over the pulley off to the outside of the apparatus, and
hang a mass from the end of the string.
ii. Adjust the amount of mass so that the bob hangs directly above the
vertical positioning pole. (Hint: It takes between 500 g and 2000 g of
mass.)
 Record this as your spring force in Data Table 1.
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Figure 2. To measure the spring force with a hanging mass, the spring is attached, and the
amount of mass is adjusted until the bob is directly above the vertical reference rod again.
Measure the period of Rotation:
8. Detach the Force Sensor. You can detach the string from the bob, or just loop it up
over the crossbar a few times so it doesn’t come loose.
9. Spin the bob so that it hangs vertically.
 One team member should to spin the shaft.
 Carefully build up speed until the spinning bob reaches directly above the
reference pole. If it is a little bit outside the reference pole, that’s okay, because it
will slow down a little during the timing.
10. Measure the period of the rotation.
 Option 1 (Stopwatch): Time 10 revolutions while keeping the rotation rate
consistent (the spinning bob should always pass just above the positioning pole).
If everything is working right, it shouldn’t slow down too much. The period is the
time of only one rotation, so divide by 10. Record the period in Data Table 1.
 Option 2 (Photogate):
i. Place a clamp near the bottom of the pole so that one of the thumb screws
sticks out and swings around in circles. Lay a photogate on its side so that
the thumb screw passes through the photogate once per rotation.
ii. Load the Circular Motion – Force Sensor and Photogate file into
Capstone.
iii. Rotate the mechanism so that the bob hangs vertically down from the end
of the crossbar. It should be passing directly over the vertical reference
pole.
iv. Make sure Capstone is displaying live data and/or recording.
v. Watch to make sure the bob is still passing over the pole. Read the period
from PASCO Capstone.
Repeat the above steps (Radius, Force, Period) for different crossbar positions. Move it in 1cm
increments.
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Radius of
Rotation
(m)
Period of
Rotation
(s)
Speed
(m/s)
Centripetal
Acceleration
(m/s2)
Spring Force
(N)
Table 1: Experimentally measured values and results calculated from the experiment.
(Students, make a better caption. The acceleration should be calculated from 𝑣 2 /𝑅, not from
𝐹 = 𝑚𝑎.)
ANALYZING THE DATA:
According to Newton’s 2nd Law 𝐹 = 𝑚𝑎. In this experiment the force and the acceleration were
measured independently. If Newton’s 2nd Law is valid, the slope of the graph 𝐹 vs 𝑎𝑐 should be
equal to the mass of the rotating object (the bob).
1. Calculate the period of the rotation for every radius. Hint: the period of the rotation is
the time of one revolution.
2. Calculate the centripetal acceleration using Equation (1).
3. Make a graph of 𝐹 vs. 𝑎, using experimentally determined values of 𝐹spring and 𝑎𝑐 .
4. Obtain the slope of this graph. Since 𝐹 = 𝑚𝑎, we expect that the slope should be the
mass of the bob.
5. Use the digital balance to determine the actual mass of the bob.
6. Calculate the percentage difference between the slope and the actual weight of the bob.
7. Discuss whether your experimental data validates Newton’s 2nd Law.
Slope of 𝐹 vs. 𝑎:
Actual mass of bob:
% Difference
Table 2: Final Results (Students make a better caption.)
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Lab 11 – Collisions, Momentum, and
Energy Conservation
Equipment





Calculator, Computer, PASCO Interface
Air Track, Air pump
Air Track Carts (2 of one kind, 1 of the other kind)
Photogate and Stand (2 of each)
Capstone setup file: Photogates for Collisions.cap
Objectives
The objective of this experiment is to verify the Law of Conservation of Energy and the Law of
Conservation of Momentum for collisions.
Introduction
Collisions and Laws of Conservation of Momentum and Energy
In this experiment we will measure the motion of two carts before and after a collision. The
carts will glide on an air track to minimize the effects of friction. Based on the observed motion
we will compute the linear momentum and kinetic energy of each object, both before and after
the collision. The carts are equipped with springy metal “bumpers” on each end, so that
collisions between the carts are elastic (very little energy is transformed into other forms, such
as material deformation or heat). Based on our observations we will then verify that the total
linear momentum and the total energy of the system are conserved for this type of collision.
This introduction gives some information on Momentum and Kinetic Energy, but it is certainly
not comprehensive. For more information:



Serway/Vuille, College Physics, 9th Ed, Chap 5&6, especially Ex 6.4.
Serway/Jewett, Physics for Sci. & Engr., 9th Ed, Chap 7&9, especially Ex 9.5.
http://www.physicsclassroom.com/class/momentum
Momentum
The linear momentum 𝑝 of an object is defined as the product of the object’s mass 𝑚 and
velocity 𝑣 . This definition is Equation 1 below. Linear momentum is a vector quantity that
points in the same direction as the velocity vector. The units of momentum are kg·m/s.
Definition:
𝑝𝑖 = 𝑚𝑣𝑖
(1)
Note: As a vector, momentum on an air track can be positive or negative!
Linear momentum is changed by forces exerted over a period of time, similar to how the amount
of water in a bucket is changed by the flow rate applied over a period of time. When two objects
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touch, their forces are equal and opposite, and each experiences the force for the exact same
amount of time. So any momentum lost by one object is gained by the other.
The principle of conservation of linear momentum applies to the collision between two
objects along a 1-dimensional track. The principle states:
The total linear momentum of an isolated system remains constant (conserved). An
isolated system is one for which the vector sum of the external forces acting on the system
is zero. (The forces between objects that interact within our system are internal.)
𝑝𝑓 = 𝑝𝑖
∑(𝑚𝑣 )𝑖 = ∑(𝑚𝑣 )𝑓
(2)
𝑚1 𝑣1𝑖 + 𝑚2 𝑣2𝑖 = 𝑚1 𝑣1𝑓 + 𝑚2 𝑣2𝑓
Note: in the last form of the equation, the 𝑣’s are components, not magnitudes, so they can be +
or -.
Kinetic Energy
The Kinetic Energy of an object, KE, is a positive, scalar (not vector) quantity. Energy is important
because it can be transformed back into different kinds of energy. Kinetic energy is only one kind.
1
KE = 2𝑚𝑣 2
(3)
Kinetic energy is changed by work, which is force exerted over a distance. If two objects rub
against each other, they experience the force of contact for the same time, but for different
distances. So momentum is conserved (as described above), but some kinetic energy can be lost
or gained.
If you slide to a stop in your car, your car loses kinetic energy (you exert a force and move a
distance in that direction, so you do work), but the road does not gain kinetic energy, because it
doesn’t move. If you come to a stop, you must supply new energy to get going again. Springy
objects, like springs and rubber balls, allow you to push on them to slow down, but then they are
ready to keep exerting the force right back at you, to give the energy back.
An elastic collision is one in which all of the kinetic energy that existed before the collision is
still kinetic energy after the collision. (No energy transformed by deformation of the objects or
by friction) So in an elastic collision, not only is the momentum conserved, but the kinetic
energy is conserved also:
KE𝑓
1
∑ (2𝑚𝑣 2 )
𝑖
1
1
2
2
𝑚
𝑣
+
𝑚
𝑣
2 1 1𝑖
2 2 2𝑖
= KE𝑖
1
= ∑ (2𝑚𝑣 2 )
1
𝑚 𝑣2
2 1 1𝑓
𝑓
1
2
+ 2𝑚2 𝑣2𝑓
(4)
=
Note: In the last form of the equation, the 𝑣’s are technically magnitudes, but components can
be used because they are squared and any negative sign gets cancelled out.
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Experimental Setup:
Two photogates are set up at about 1/3 and 2/3 of the length of an air track. The PASCO software will
record the time it takes for the length of a cart to blocks a photogate while the car is passing through the
photogate. The software will then calculate the speeds of the carts using these measured gate-blockage
times. You will use this information to compute the linear momentum and the kinetic energy of each cart
before and after the collision.
This experiment will investigate three different conditions before the collision.
Condition 1: Carts with Different Masses, the light cart is originally at rest. Heavy Cart is
moving to the right and collides with Light Cart, which is at rest.
Photogate 1
Heavy Cart
𝑣1𝑖 > 0
Photogate 2
Light Cart
𝑣2𝑖 = 0
Figure 1. Rough experimental setup for Condition 1.
Condition 2: Carts with Different Masses, the heavy cart is originally at rest. Light Cart is
moving to the right and collides with Heavy Cart, which is at rest.
Photogate 1
Light Cart
Photogate 2
Heavy Cart
𝑣1𝑖 > 0
𝑣2𝑖 = 0
Figure 2. Rough experimental setup for Condition 2.
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Condition 3: Two Identical Carts, initially moving toward each other. One cart is
moving to the right and collides with the other cart, which is moving to the left. Either two light
carts or two heavy carts can be used.
Photogate 1
Cart 1
Photogate 2
Cart 2
𝑣1𝑖 > 0
𝑣2𝑖 < 0
Figure 3. Rough experimental setup for Condition 3.
Initial Setup:
1. Adjust the photogates height so that only the flag (the strip of aluminum on the top of a cart)
blocks the photogate infrared beam.
2. Connect one Photogate to Channel 1 and another to Channel 2.
3. In the same location as your lab instructions, open “Photogates for Collisions.cap”.
4. Level the track. Turn on the air pump, start the Data Recording and push a cart down the track
at around 0.3 to 0.5 m/s. As the cart passes down the track, its speed should be the same in each
gate. If it slows down or speeds up, adjust the height of one end of the track. Stop the Data
Recording.
Data Recording:
5. Turn on the Air Source to lift the carts slightly above the air track and minimize the effect of
friction. Practice pushing the cart(s) gently from the ends of the air track so that they collide in
the middle of the air track. Make sure that the collision happens in between the photogates!
Adjust the photogates or your pushing technique.
6. Once you have practiced with collisions between the two carts, proceed to replicate four
collisions for each one of the conditions described in pictures 1, 2 and 3. For each collision, place
the carts according to the corresponding condition/picture and push one or both carts (depending
on the condition) such that they collide with each other.
7. After each collision, allow the carts to pass back through the photogates (if appropriate) but grab
them from the track before they strike the end bumpers.
8. Important: keep track of which cart passed which photogate, and in which direction. The
computer won’t do this for you.
9. When you record your data, carefully use the measured speeds and your knowledge of the
collision to determine each cart’s velocity (+ or −) before and after the collision.
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ANALYZING THE DATA:
1. Again, the computer only records speed. You must choose the sign (+ or −) of the velocities from
your notes about the direction of the motions.
2. Calculate the total momentum before the collision (Equation 2, left side).
3. Calculate the total kinetic energy before the collision (Equation 4, left side).
4. Calculate the total momentum after the collision (Equation 2, right side).
5. Calculate the total kinetic energy after the collision (Equation 4, right side).
According to the law of conservation of energy the total kinetic energy before the elastic collision
should be equal to the total kinetic energy after the collision it the collision is elastic. For each
collision, compare the kinetic energy before and the kinetic energy after the collision (% diff) for
each collision, and then average the calculated values.
Avg. % Diff =____________________________________
According to the law of conservation of momentum the total momentum before the collision
should be equal to the total momentum after the collision. For each collision, compare the
momentum before and the momentum after the collision (% diff) for each collision, and then
average the calculated values.
Avg. % Diff =____________________________________
Note: The lines above are provided for you to record your results for yourself. Your lab report
should include the information either in sentences in the Abstract or in a separate Data Table.
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Trial m1(kg) m2(kg) Vi1(m/s) Vi2(m/s) pitotal(kgm/s) KEitotal(J)
1
2
3
4
Table 1a: Condition 1 initial values (the light cart is originally at rest) (Make a better caption.)
Trial m1(kg) m2(kg) Vf1(m/s) Vf2(m/s) pftotal(kgm/s) KEftotal(J) p %diff
KE %diff
1
2
3
4
Table 1b: Condition 1 final values (the light cart is originally at rest) (Make a better caption.)
Trial m1(kg) m2(kg) Vi1(m/s) Vi2(m/s) pitotal(kgm/s) KEitotal(J)
1
2
3
4
Table 2a: Condition 2 initial values (the heavy cart is originally at rest)
Trial m1(kg)
m2(kg) Vf1(m/s)
Vf2(m/s)
pftotal(kgm/s
)
KEftotal(J)
p %diff
KE %diff
p %diff
KE %diff
1
2
3
4
Table 2b: Condition 2 final values (the heavy cart is originally at rest)
Trial m1(kg) m2(kg) Vi1(m/s) Vi2(m/s) pitotal(kgm/s) KEitotal(J)
1
2
3
4
Table 3a: Condition 3 initial values (two carts of same mass are moving)
Trial m1(kg)
1
2
m2(kg) Vf1(m/s)
Vf2(m/s)
pftotal(kgm/s) KEftotal(J)
3
4
Table 3b: Condition 3 final values (two carts of same mass are moving)
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Lab 12 – Ballistic Pendulum
Equipment




Safety Glasses (not necessarily chemical goggles)
Calculator, Computer
Ballistic Pendulum Apparatus (Stand, Launcher, Pendulum, Ball)
Ruler
Objectives
To determine the initial velocity of a projectile through proper application of the principles of
conservation of momentum and energy.
Introduction
The ballistic pendulum is a classic method of determining the velocity of a projectile. It is also a
good demonstration of many of the basic principles of physics. The ball is fired into the ballistic
pendulum, which then swings up a measured amount. From the height reached by the
pendulum, you can calculate its gravitational potential energy. The gravitational potential
energy is equal to the kinetic energy of the pendulum at the bottom of the swing, just after the
collision with the ball. You cannot equate the kinetic energy of the pendulum after the collision
with the kinetic energy of the ball before the swing since the collision between ball and
pendulum is inelastic, and kinetic energy is not conversed in inelastic collisions. Momentum is
conserved in all forms of collisions, so you know that the momentum of the ball before the
collision is equal to the momentum of the pendulum after the collision. Once you know the
momentum of the ball and the ball’s mass, you can determine the initial velocity.
Momentum
As discussed in Lab 11, the momentum of an object depends on its mass and velocity, and the
units of momentum are kg·m/s. Remember: As a vector, momentum in a direction can be
positive or negative!
𝑝𝑖 = 𝑚𝑣𝑖
Definition:
(1)
In a collision, the total momentum of the two objects is conserved.
𝑝𝑖 = 𝑝𝑓
𝑚1 𝑣1𝑖 + 𝑚2 𝑣2𝑖 = 𝑚1 𝑣1𝑓 + 𝑚2 𝑣2𝑓
(2)
Kinetic and Potential Energy
Again, as discussed in Lab 11, the kinetic energy of an object is a positive, scalar (not vector) quantity
that also depends on its mass and velocity. Energy is important because it can be transformed back into
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different kinds of energy. The units of energy are
joules (1 J = 1 kg ⋅ m2 /𝑠 2).
Definition:
1
KE = 2𝑚𝑣 2
(3)
Potential energy is another form of energy. For some forces (called conservative forces), there is
a corresponding potential energy. For gravity, the potential energy is:
Definition:
PE𝑔 = 𝑚𝑔𝑦
(4)
When only conservative forces do work, conservation of energy applies. This means the total of
the kinetic and potential energies must be constant. An important example is when the initial
potential energy is initially zero (because 𝑦𝑖 = 0) and the final kinetic energy is zero (because
𝑣𝑓 = 0).
KE𝑖 + PE𝑔𝑖 = KE𝑓 + PE𝑔𝑓
1
𝑚𝑣𝑖2 = 𝑚𝑔Δ𝑦
2
(5)
Experimental Setup
You should be wearing eye protection for the entire duration of the lab. The ballistic pendulum
should already be set up.
1. Set the mass of the pendulum as desired. In the picture, there are two extra masses
attached to the bottom of the pendulum.
2. Latch the pendulum at the top so it is out of the way. Set the spring to the desired launch
strength, and load the launcher. Then lower the pendulum. Use the same launch
strength every time, or the ball velocity will keep changing.
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3. Set the angle indicator to 0°.
4. Firmly hold the base of the apparatus against the table. Fire the pendulum and record
the maximum angle.
5. Repeat the angle measurement for 5 trials.
Once your trials are complete:
6. Measure the mass of the ball and record this as 𝑚ball .
7. Measure the mass of the pendulum (with the ball in it) and record this as 𝑚tot .
8. Find the center of gravity of the pendulum arm. To do this, balance it by hanging it from
piece of string or by balancing it on the launcher ramrod. Record the distance from the
pivot point to the balance point as 𝑅.
Change an experimental parameter.
9. Repeat the experiment for 3 different sets of conditions. For example, you could do the
pendulum with no extra masses on the bottom, then one extra mass, then two extra
masses. If you use the same launcher setting every time, you can expect the ball
velocities to come out the same.
Data Analysis
1. Average the angles for your trials.
2. Calculate the change in height of the center of mass of the pendulum:
Hint:
Δℎ = 𝑅(1 − cos 𝜃)
(6)
3. Use the law of conservation of energy to calculate the velocity of the pendulum (with the
ball inside) just after the collision. (Equation 5 above.)
4. Use the law of conservation of momentum to calculate the velocity of the ball just before
the collision. (Equation 2 above. Note that 𝑣2𝑖 = 0, and 𝑣1𝑓 = 𝑣2𝑓 are both the result of
the previous step.)
Condition
A
B
C
Description
No extra mass
1 extra mass
2 extra masses
Theta 1 (deg)
Theta 2 (deg)
Theta 3 (deg)
Theta 4 (deg)
Theta 5 (deg)
Theta Avg (deg)
m_ball (kg)
… (more rows for
more data and
calculations)
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Appendix A: Math Topics
Standard Deviation
What does the standard deviation of a set of values mean? Sometimes it’s called the spread of
the values. It represents approximately how far, on average, any given value is from the overall
mean.
Figure 1. A sample set of measurements demonstrating standard deviation. This could be the
length and width of a sponge. The length appears to be 15 ± 0.5 cm, while the width appears to
be 5 ± 1 cm. The uncertainties here are the standard deviations of the measurements. In each
case, most of the measurements are “within 2 standard deviations” of the measurement value.
The standard deviation is the square root of the “variance” of the values. By agreement among
measurement scientists, this is the preferred form of the uncertainty of a measurement.
[http://physics.nist.gov/cuu/Uncertainty/basic.html].
Note that this is sort of an average deviation. Some points are further away than the average,
some are closer. On average, 68% of the points will be within one standard deviation of the
average. On the graph above, about that many points have an x value between 14.5 cm and 15.5
cm. This is sometimes called the 68% confidence interval
[http://mathworld.wolfram.com/ConfidenceInterval.html]. If you want more confidence of
“catching” more the measurements in the interval, “cast a wider net”. Two standard deviations
provides about 95% confidence, while ~2.5 standard deviations provides 99% confidence.
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Propagation of Uncertainty
Description
Operations
Numerical Uncertainty
(Same units as value)
Quantity = Val ± Unc
Relative Uncertainty
(Numerically dimensionless, but written as %, ppm, or
ppb)
Quantity = Val ± %Unc
Addition and Subtraction:
Add Numerical Unc
Value = x + x + ...
Unc = Unc + Unc + ...
Multiplication and Division:
Add %Unc (or RelUnc)
Value = x x
%Unc = %Unc + %Unc + ...
Powers and Roots:
Scale %Unc (or RelUnc)
Value = x /x
%Unc = %Unc ×Power + %Unc × Power +
...
or
1
2
1
2
1
2
1
2
Power1
1
1
1
Power2
2
2
2
Quick Examples of Uncertainty
Quantities can be expressed with absolute or relative uncertainty.
 x = 50 ± 1 m = 50 m ± 2%
 y = 20 ± 1 m = 20 s ± 5%
 t = 25 ± .5 s = 25 s ± 2%
When adding or subtracting, add the uncertainties.
 x – y = 30 ± 2 m
When multiplying or dividing, add the relative uncertainties.
 xy = (50 m ± 2%) (20 m ± 5%) = (1000 m ± 7%) = (1000 ± 70 m )
 x / t = (50 m ± 2%) / (25 s ± 2%) = (2 m/s ± 4%) = (2 ± .08 m/s)
With powers, scale the relative uncertainty by the power. (Square root is the one-half power.)
 t = (25 s ± 2%) = (15625 s ± 6%) = (15625 ± 937 s ) ≈ (15600 ± 900 s )
 √(xy) = √(1000 m ± 7%) = (31.62 m ± 3.5%) = (31.62 ± 1.1 m) ≈ (31.6 ± 1 m)
Remember that when you add or scale a relative uncertainty, the result is still a relative uncertainty.
2
3
3
3
2
3
3
2
Alternate Method: Max/Min Method
An alternative way of calculating the uncertainty of a result is to try to find the maximum and minimum
possible values of the result, given that the input values may vary by their uncertainties. As an example
calculation, we know that the displacement of an accelerated object that starts from rest is related to the
time: Δx = ½ a t . Solving for the acceleration, we get a = 2 Δx / t .
2
2
Note that the maximum a comes from considering the maximum Δx and the minimum t (because t is in
the denominator). You can often get away with just calculating the maximum, because the minimum is
about as far below the average as the maximum is above the average.
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Here is a sample calculation with the max/min method:
Value w/ Unc
Max result
Min result
Δx
3.0 m ± 0.05 m
3.05 m
2.95 m
t
1.2 s ± 0.02 s
1.18 s
1.22 s
4.17 m/s
4.38 m/s
2
3.96 m/s
2
0.21 m/s
2
0.21 m/s
2
Calculations of
a = 2 Δx / t
2
2
Error in a
a
4.17 ± 0.21 m/s
2
Graphing and Trendlines
(Under construction)
We always refer to graphs by the y-axis first. So a graph of “Distance vs. Time” has Distance on
the vertical y-axis, and Time on the horizontal x-axis. The easiest way to achieve this is to have
your data series side-by-side (or in consecutive rows) with the x data first.
When a graph shows a roughly straight-line data trend, a linear trendline gives a great way to get
a single quantity that summarizes all of the data. Remember, the slope is the “rise over run” of
the trendline.
(Slope) =
Δ𝑦
Δ𝑥
Because of this, the units of the slope are the units of y divided by the units of x. If you’re
unsure of whether the graph is correct, check the units of the slope.
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Appendix B: Sample Data Report3
Meter Stick Balance - Data Report Example
Dr. Jeffery Spirko
2015-03-13
Abstract
In this experiment, we measured the mass of a meter stick using only the meter stick itself, a
200 g known mass, and a wooden dowel rod. When the meter stick is balanced on top of the
dowel rod, it is because the center of mass is lined up with the dowel rod. The center of mass is
defined in any introductory physics textbook as 𝑀𝑇 𝑥𝐶𝑀 = 𝑚1 𝑥1 + 𝑚2 𝑥2 + ⋯. Here, mi is each
mass value, xi is the position of each mass, MT is the total mass, and xcm is the position of the
center of mass. Initially, the stick was balanced by itself to establish the location of its center of
mass. Then, the 200 g mass was placed at a known location on the meter stick, and the system
was rebalanced.
All data and calculations are recorded in Table 1. During the balancing, the 200 g mass was
located at 90.0 cm, the meter stick was centered at 49.7 cm, and the balance point was
74.2 cm. Calculating the mass of the meter stick is assisted by a change of origin, setting it at the
balance point. This gives the equation 𝑚1 𝑟1 = 𝑚2 𝑟2, where r is the absolute value of the lever
arm of each mass. The experimentally measured mass of the meter stick was 128.98 g. This is
very close to the value obtained by a digital balance, which was 129.66 g. The difference is only
0.68 g, which is 0.5%, which is fairly small considering the equipment used. Since the final
equation is a proportionality, the percentage error of the result comes from the percent error of
the input values. An error of 0.5 mm in one of the positions would cause a 0.3% error in the
corresponding lever arm, so the error in 3 different positions can easily explain the overall
experimental error.
Figures and Tables
Balance Point of Stick x1 (cm) 49.7 M_Stick, Expt m1 (g) 128.98
Extra Mass m2 (g)
200 M_Stick, Direct m1 (g) 129.66
Location of Mass x2 (cm)
90.0
Error in m1 (g)
0.68
Balance Point xcm (cm)
74.2
% Error
0.5%
Lever of Stick r1 (cm)
24.5
Lever of Mass r2 (cm)
15.8
Table 1. Data and calculation results for the Meter Stick Balance experiment. The balance
point of the stick was determined by balancing the stick on a rod by itself. After m2 was placed,
the system was rebalanced at xcm. The experimental mass was determined using 𝑚1 𝑟1 = 𝑚2 𝑟2
and compared with a directly measured value.
3
Note: A version of this data report is online at https://goo.gl/Kfl57H (lower-case “ell” in the middle).
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