phys181 lab manual spring 2015

Physics 181

Laboratory Manual

Course Coordinator: Dr Haar

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This Laboratory Manual was produced for the

PHYSICS DEPARTMENT

UNIVERSITY OF ARIZONA

By Brian LeRoy

2013 Based in part on previous

Physics Department Laboratory Manuals

Copyright 2013 All rights reserved

Arizona Board of Regents

Spring 2015

Sun Monday Tues Wednesday Thursday Friday Sat

Jan 11Wk 1

12 13 14 15 16 Jan 17

Jan 18Wk 2

19 MLK day 20 21 22 23 Jan 24

Jan 25Wk 3

26 27 28 29 30 Jan31

Feb 1Wk 4

2 3 4 5 6 Feb 7

Feb 8Wk 5

9 10 11 12 13 Feb 14

Feb 15Wk 6

16 17 18 19 20 Feb 21

Feb 22Wk 7

23 24 25 26 27 Feb 28

Mar 1Wk 8

2 3 4 5 6 Mar 7

Mar 8Wk 10

9 10 11 12 13 Mar 14

Mar 15 16 Spring Break 17 Spring Break 18 Spring Break 19 Spring Break 20 Spring Break Mar 21

Mar 22Wk 10

23 24 25 26 27 Mar 28

Mar 29Wk 11

30 Mar 31 Apr 1 2 3 Apr 4

Apr 5Wk 12

6 7 8 9 10 Apr 11

Apr 12Wk 13

13 14 15 16 17 Apr 18

Apr 19Easter

20 21 22 23 24 Apr 25

Apr 27Wk 15

27 28 29 Apr 30 May 1 May 2

May 3Wk 16

4 5 6 7 Reading 8 Finals May 9

May10

11 Finals 12 Finals 13 Finals 14 Finals 15 Graduation May 16Grad

Last Withdraw April 14Summer I June 8 - July 9Summer II July 13 - Aug 12 Fall 2014 begins Aug 24Labor Day Sept 7Vet Day Nov 11 Wednesday

Spring 2015

Sun Monday Tues Wednesday Thursday Friday Sat

Jan 11Wk 1

12 13 NO LABS Jan 17

Jan 18Wk 2

19 MLK day Length, Velocity, Acceleration Jan 24

Jan 25Wk 3

26 Range vs Height Write report # 1 Jan31

Feb 1Wk 4

2 Force Table Feb 7

Feb 8Wk 5

9 Acceleration Mass over a Pulley Write report # 2 Feb 14

Feb 15Wk 6

16 Friction Feb 21

Feb 22Wk 7

23 Energy Conservation Write report #3 Feb 28

Mar 1Wk 8

2 Collision Write report #4 Mar 7

Mar 8Wk 10

9 Circular Motion Mar 14

Mar 15 16 Spring Break 17 Spring Break 18 Spring Break 19 Spring Break 20 Spring Break Mar 21

Mar 22Wk 10

23 Springs and Pendula Write report #5 Mar 28

Mar 29Wk 11

30 Waves Apr 4

Apr 5Wk 12

6 Archimedes Write report #6 Apr 11

Apr 12Wk 13

13 LABORATORY PRACTICAL EXAM Apr 18

Apr 19Easter

20 Makeup Latent Heat of Liquid Nitrogen Apr 25

Apr 27Wk 15

27 Heat Engines May 2

May 3Wk 16

4 NO LABS 7 Reading 8 Finals May 9

May10

11 Finals 12 Finals 13 Finals 14 Finals 15 Graduation May 16Grad

Last Withdraw April 14Summer I June 8 - July 9Summer II July 13 - Aug 12 Fall 2014 begins Aug 24Labor Day Sept 7Vet Day Nov 11 Wednesday

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Physics 181 Laboratory Manual Contents

0. Laboratory Policies and Guidelines ......................................................... 4 1. Position, Velocity and Acceleration ........................................................ 13 2. Range Versus Height ............................................................................... 23 3. Vectors and Forces ................................................................................... 31 4. Newton’s Second Law .............................................................................. 41 5. Friction ...................................................................................................... 48 6. Energy conservation ................................................................................ 60 7. Collisions ................................................................................................... 71 8. Circular Motion ....................................................................................... 79 9. Pendula and Springs ................................................................................ 88 10. Standing Waves ........................................................................................ 97 11. Archimedes’ principle ........................................................................... 105 12. Ideal Gas Law and Heat Engine ........................................................... 114 13. Heat capacity (makeup) ......................................................................... 125 Appendices

How to use the Xplorer GLX ............................................................ 133 Example lab report ........................................................................... 141

Laboratory Policies and Guidelines

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Physics 181 Policies

n A student is graded on the basis of 12 labs (11 for students whose lab day falls on a holiday)

and associated reports, lab quizzes, worksheets, and a lab practical at the end of the semester.

n Each individual student is required to write his or her own unique and individual lab report. (See Academic Integrity Standards)

n A student will FAIL the course if there are three or more labs which are not completed. The

following are the conditions for a lab to be considered completed: • The student must attend the lab, be an active participant in the lab activities, and not be

disruptive. • The student must submit in a timely fashion, a lab report and worksheets which must

include in-lab notes signed by the TA. Reports and/or worksheets that are turned in more than 48 hours late, earn a grade of zero. In this case, the lab is considered to be not completed.

• The student must submit an electronic version of the text of their lab report to Turnitin.com via the D2L Drop Box at the same time as the paper copy and in an acceptable format.

• The student must not have violated the Code of Academic Integrity for any part of the lab. For example, a lab report that includes a violation of the Code will be considered a lab which was not completed.

• The student must refrain from reckless behavior that endangers her/himself, others or the equipment. The student must not intentionally damage or dissemble the equipment. Students are expected to straighten up their work area before they leave.

I. Work load and grading

Item Points each

Number of items

Total for Category

Quizzes 4 12 48 Weekly Worksheets On weeks when there is not a Formal Report

15 6 90

Formal Report #1 25 1 25 Formal Reports #2 - #6 50 5 250 Lab Practical 75 1 75 Total 488

A. Weekly quizzes

• Usually 2 questions worth 2 points each. • The quiz will usually consist of one question from the previous week and one from

the current week. Typical questions will be a drawing of the equipment, a Free Body Diagram, a simple graph, a simple calculation or a definition.

• Quizzes will be graded very coarsely, i.e. basically 0 or 2 points on each part.


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B. Worksheets

• No part of the Worksheets except acquiring the data is a “Team Project”! Turning in the same worksheet as your partner will result in a violation of the Code of Academic Integrity. Neither student will receive credit for completing the lab.

• For labs without a formal report, a worksheet must be completed. • Worksheets will only be spot graded, i.e. checked for blanks, graphs, drawings. Also

2 or 3 of the questions will be graded. • 3 of the 15 points will be awarded for participation, i.e. your involvement in doing the

experiment, cleaning up your work space, and your skills and care with the equipment and general lab behavior.

• 2 of the 15 points will be for the quality of the data. • Your TA must sign and date your original data before leaving the lab. This original

data must be turned in with the worksheet. • You may not use someone else’s data. This is a violation of the Code of Academic

Integrity.

C. Formal reports • No part of the formal report except acquiring the data is a “Team Project”! Turning

in a similar report to your partner will result in a violation of the Code of Academic Integrity. Neither student will receive credit for completing the lab.

• The requirements for the formal report are discussed later. • There will be six formal reports during the semester. The first one is worth half as

much to give some practice and feedback on writing the reports. • Formal reports are due at the beginning of the next lab. A paper copy must be

submitted in class and an electronic version turned into the D2L drop box. The report is not considered turned in until both tasks are completed.

• You should staple the worksheet associated with the lab to the end of the lab report. • Your TA must sign and date your original data before leaving the lab. This original

data must be turned in with the formal report. • You may not use someone else’s data. This is a violation of the Code of Academic

Integrity.

D. Lab practical • Covers common lab skills and the ability to perform common laboratory tasks.

E. Return of graded material

• Lab Reports, quizzes and worksheets should be returned to the students at the next lab meeting after they are turned in. If this is not happening, please contact the lab coordinator, Roger Haar [email protected].


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II. Late lab reports • Lab reports (both the text and D2L versions) and the worksheets are due at the

beginning of the next lab meeting. The time and date shown in the D2L dropbox is NOT the time things are due!

• Lab reports (paper and electronic) and worksheets will be considered late ten minutes after the start of class.

• If the lab report or worksheet is late by less than 4 hours, its grade is reduced 10%. • If a lab report or worksheet is late by more than 4 hours but less than 24 hours, its

grade is reduced 25%. • If a lab report or worksheet is late by more than 24 hours and less than 48 hours, its

grade is reduced 50%. • If a lab report or worksheet is late by more than 48 hours, its grade is ZERO. The

student will also not receive credit for completing the associated lab. Notes: The clock is only running on class days (Monday through Friday) when classes

are in session. Failure to submit reports in a timely fashion may result in a student being administratively dropped.

III. Attendance

• Attendance is required and students must attend their own scheduled lab section. • Missing a lab without an acceptable documented excuse, will reduce your final lab

grade. • Additional points may be lost if the lab report is late because you missed the lab at

which the report was due! • Being late, leaving early or leaving in the middle of the lab (for example to chat on a

phone) may result in lost participation points or even being ruled absent. • Lab quizzes are given at the start of the lab. Quizzes by students who arrive late are

due when the rest of the class’ quizzes are collected. Students arriving after the quizzes are collected get a ZERO on that quiz.

• If you leave without getting your TA to sign and date your notes, you will be considered absent.

IV. Makeup Lab • There is a makeup lab, but it is reserved for those with excuses like documented

medical or family emergencies. The student must email his/her TA and Roger Haar as soon as possible after the absence. This email should include course, PHYS181 and section numbers, the date of the missed lab and the reason for the absence.

• If an absence is going to be caused by a religious holiday or an activity for which the student has a “Dean’s Excuse” the student should email the lab manager and his/her TA BEFORE the absence.

• Only one lab may be made up except in extreme cases. • If the missed lab had a formal report, the student must:

o Take the quiz, submit the worksheet and formal report on the makeup lab. o Otherwise, the student must take the quiz and submit the worksheet.

• Makeup lab reports are due at the hour the makeup lab began two academic days later.

• There will be NO makeup for the lab practical except for extreme situations.


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V. Incompletes

As per University policies, an incomplete is available only to those students who have completed the bulk of the course (75%) and have a passing grade on the completed material. A written agreement as to how the incomplete will be removed must be filed with the Academic Support Office of the Physics Department. VI. Help and Computers

The first place to go for help is your TA’s office hours. If you cannot make those office hours, try making an appointment. On most weekdays help is available in the Physics consultation room, PAS 372. Graduate students who will try to help you staff this room, but they will not do your homework for you.

Students enrolled in a Physics class, have access to the computers in PAS 272 during

weekdays except when a class is using the room. The Physics Dept provides these computers. VII. Physics 181 Course Coordinator

Dr. Roger Haar PAS 324 621-6773 or 520-282-0400 [email protected]

Contacting Dr. Haar: • Dr. Haar is usually in the PAS building 8:30-5:00. • He may be in his office with the door closed so knock. • Most of the time when he is in the PAS building, he is available for a few minutes.

§ Try calling him § Try emailing him § When emailing or leaving a message

a. Include your name, class, and section. b. Include the general nature of your problem. c. Give your availability.

• Dr. Haar almost never checks his University e-mail or phone messages on weekends or after 5:00 PM.

• If you are emailing Dr. Haar about any administrative question you must include: Your name, course, section and TA’s name. Unless your message is about your TA, you should cc your message to your TA.

VIII. General Lab Rules

1. No food or beverage, other than water, is allowed in the laboratory room 2. Shirt and shoes must be worn in the laboratory. 3. Roller skates and roller blades may not be worn in the laboratory. 4. Bicycles and skateboards are not allowed in the laboratory. 5. Cell phones must be off except for use as calculators or cameras to record setups. 6. No phone calls, no texting.


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IX. D2L • It will be assumed that all students will be familiar with D2L. D2L will be used as an

official channel of communication from the instructors to the students as well as a way students can track their grades.

• If you take this course, you are agreeing to submit your papers online to a plagiarism-prevention program called TurnItIn.com. This is done through a dropbox on D2L.

• You should note that TurnItIn.com – always without your name and any personal information – will retain your paper as part of their database so that students who plagiarize from it can be detected.

• If you have questions or problems with TurnItIn.com, contact the D2L help site. • Students are required to submit only the text portion of their lab reports. • This text must be submitted in a format that TurnItIn.com can read: Plain Text (*.txt) or

Word formats (*.doc or *.docx). • If a report is submitted in a format that cannot be read by TurnItIn.com, the report will be

considered not submitted. • The end time on the D2L dropbox is NOT the time the report is due. Reports are due at

the beginning of your lab section.


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ACADEMIC INTEGRITY STANDARDS

Students who are retaking this class may NOT submit reports containing material from their own previously written lab reports or worksheets. This is considered a violation of the “Code of Academic Integrity.”

Each individual student is required to write his or her own unique and individual lab

report. This means that two students should not work from a common draft. Duplication and or plagiarism will be treated as a violation of the “Code of Academic Integrity” and reported to the Dean of Students.

See the Dean of Student’s website for the complete Code of Academic Integrity.

EACH STUDENT MUST INDIVIDUALLY (NOT A TEAM EFFORT):

• Write the text, including the procedure section. • Plot the graphs including information as to how the slope was extracted. • Do the analysis of data. • Draw the figures of the setup, free body diagrams, etc.

“BORROWED” IMAGES

Students must make their own diagrams. Even with attribution, images from other sources will be marked as wrong. “BORROWED” TEXT

Any text that is not the student’s own must be shown in quotes and a valid attribution given to avoid charges of plagiarism. Put quotation marks around the quoted material and place the attribution within parentheses, immediately after the quote. Even with attribution, the student may be marked down for not using his or her own words. If you quote something then explicitly explain the quote in your own words. This applies to quoting the lab manual. SANCTIONS

General list of possible sanctions: • In all cases, notice will be sent to the Dean of Students. • Attendance required at a plagiarism workshop. • Loss of credit for the entire report. This will then result in you being considered

absent on that day. Students may not makeup this lab. • Loss of credit for the report and a further reduction of the final course grade. • Failing grade for the course. • Permanent notation on the student’s transcript. • Suspension or expulsion from the University.

Anyone helping another student cheat will receive the same sanctions.


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Grading of a “Typical” Lab Report

The actual division of points will vary from experiment to experiment. Sometimes there are many graphs and drawings but little theory. This is not an English or an Art class, but the TA must be able to quickly decipher what is written or drawn, otherwise it is wrong.

(Points shown are for a typical 50 point report. For a 25 point report divide the number of points by two.)

Each section of the report is to be titled, for example the abstract should be titled “Abstract.” TITLE PAGE 1 Point

Experiment’s title plus student’s name, Lab partner’s name, TA’s name. Course and section numbers, date and time of paper and electronic submission.

ABSTRACT 3 Points

What were you trying to do? How did you try? What were your results (numbers)? This section should be 3-6 sentences.

INTRODUCTION 4 Points

Explicitly state the goal of the experiment, what you are trying to do and show. Relate the experiment to previous measurements and physical concepts.

THEORY & DERIVATIONS 8 Points

What needs to be measured? Derive the equation(s) that convert your raw data into the final results. Derivations of equations should include text helping to describe and motivate the math operations. Give the name of the law associated with equations as appropriate. Equations throughout the report should be numbered.

DESCRIPTION of the PROCEDURE 4 Points

What did you physically do and why? This should not be a copy of the lab manual. This should include what went right or wrong, tricks, hints, and limitations. How you overcame any problems. How measurements were made. This section almost always requires one or two drawings of the equipment setup.

RESULTS and SAMPLE CALCULATIONS 4 Points

Final numbers and an indication of the calculations needed to get them. The sample calculation should show the entire process of starting with the original measured data and ending with the final results.

CONCLUSION and DISCUSSION 10 Points

• Develop a logical case for your conclusions. • Were the goals of the experiment met? • Was it a “good” experiment? Given the equipment are the final numbers reliable? • What went right or wrong? And how do these things affect the measurements? • What were the uncertainties and how did they affect your final values? • Do NOT say “human error, be explicit! • If you claim a terrible measurement was good you should expect to be marked down.


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TABLES, DRAWINGS, and GRAPHS 10 Points n TABLES

• Tables should be numbered and labeled, for example as “Table 4.” • In your text, you should refer to your tables by these labels. • Tables should have column labels with units. • Each table should have a caption which explains what the table is and what is

important, and if there is a graph of the data in the table. • In your text, you should refer to each table at least once.

n FIGURES

• For most experiments, there should be a schematic drawing of the setup. • This is not the same as a photograph, but rather is a picture showing the relationship

of the equipment, often with dimensions and notes. • Figures are not to be “borrowed” from anywhere! • As needed there should be free body diagrams. • Figures (drawings) should be numbered and labeled, for example as “Figure 2." • In your text, you should refer to your figures by these labels • Each figure should have a caption that explains what the figure is and what is

important in the figure. • In your text, you should refer to each figure at least once. Often this would be an

expansion of the caption.

n GRAPHS • Each graph should use at least half of the sheet of graph paper. • Graphs should have labeled axes with units, and as appropriate: the errorbars, best fit

line, range of fit lines, and indications of what values were used to extract the slope. • Data points should not be connected dot-to-dot. Instead show a best-fit line. • Graphs should be numbered and labeled, for example as “Graph 3.” • In your text, you should refer to your graphs by these labels. • Each graph should have a caption that explains what the graph shows about the data.

Also the caption should identify the table that contains the plotted data. • In your text, you should refer to each graph at least once. Often this would be an

expansion of the caption. You should use your graphs to support your conclusions.

If computer graphs are allowed: • The student is responsible for seeing that all of the above requirements are met. • Each graph should cover most of a half page of paper. • Unnecessary items like R2 should not be displayed. • The background should be white. • There should be reasonably spaced vertical and horizontal gridlines. • Slopes should still be determined manually using the rise-over-run method.

WORKSHEET 1 Point PARTICIPATION 3 Points QUALITY OF DATA 2 Points


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Grading Chart Course_________ Section_______ Student’s Name ________________________________ Experiment ________________________________________________________ ITEM out of score Comment

TITLE 1

ABSTRACT 3

INTRODUCTION 4

THEORY & DERIVATIONS

8

PROCEDURE 4

RESULTS & SAMPLE CALCULATIONS

4

CONCLUSION and DISCUSSION

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TABLES, DRAWINGS, and GRAPHS

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WORKSHEET 1

PARTICIPATION 3

QUALITY OF DATA 2

TOTAL 50

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Position, Velocity and Acceleration

The physical quantities of position, velocity and acceleration allow us to study the physics of objects in motion. With these three quantities the motion of objects of all sizes from bacteria to galaxies can be understood. These three quantities are all related to each other. The average velocity 𝑣 is defined as

𝑣 =𝛥𝑥𝛥𝑡 (1)

where Δ𝑥 is the change in position and 𝛥𝑡 is the change in time. Similarly the average acceleration 𝑎 is defined as

𝑎 =𝛥𝑣𝛥𝑡 (2)

where Δ𝑣 is the change in velocity. Therefore, if we make a plot of the position of an object as a function of time, the slope of this graph will give the average velocity. The slope of a graph of the velocity as a function of time gives the average acceleration. Furthermore, if the acceleration is constant, we can find the position of an object as a function of time using

𝛥𝑥 = 𝑥! − 𝑥! = 𝑣!𝑡 +!!𝑎𝑡! (3)

where 𝑥! is the initial position, 𝑥! is the final position and 𝑣! is the initial velocity. In this lab we will be measuring the position of objects as a function of time using a tape timer and photogates. From these measurements, we will find the velocity and acceleration of the objects.

A photogate consists of an infrared beam that is interrupted by objects passing through it. By measuring the time that an object takes to travel from one gate to another or through one gate, its velocity can be determined. A tape timer works by making a mark on a piece of tape at well-defined intervals of time. By measuring the distance between these marks, the position of the object as a function of time is determined. Using this position data, we will calculate and graph the velocity and acceleration of the object using the above equations.

Goals of the this lab: • Measure the size of various objects • Learn how to determine velocity using photogates • Analyze the motion of a cart using a tape timer • Understand the relationships between position, velocity and acceleration

Position, velocity and acceleration

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Lab equipment: Rulers and Calipers These will be used to measure distances in the room and the size of various objects.

Photogates The photogates have an infrared sensor in them that detects the motion of objects through them. They will be used to measure time.

Tape timer The tape timer produces marks on a piece of paper every 0.1 or 0.025 seconds. They will be used to record the motion of carts on the track.

Track These are 1.2 meter long tracks for the carts to travel along.

Cart These carts move along the track and have a mass of 500 grams.


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Lab Procedures

I. Measuring lengths When measuring the size of an object it is important to have the correct tools. We would not

want to measure the width of your hair with a meter stick or the length of a football field with a caliper. Matching the tool with the measurement that needs to be done is critical for obtaining accurate results. You have several different measuring tools at your workstation ranging from a 2-meter stick to a 12 inch ruler to a caliper. Use them to measure the following lengths and record them in the worksheet at the end of this lab. Remember that all measurements must include the units to be meaningful. § Measure the length and width of the room. § Measure your height. § Measure the distance from the tip of your outstretched hand to your elbow (This is traditionally

called a cubit). § Find a part of your hand that has a dimension of about 0.1 meters (10 cm). § Find a part of your hand that has a dimension of about 0.01 meters (1 cm). § Find an object that you carry around that has a dimension of about 0.001 meters (1 mm).

II. Determining velocity using photogates

We do not have any sensors that can directly measure the velocity of an object. In order to determine the velocity, we need to measure the change in position in some interval of time. For the remainder of the lab, we will be measuring the velocity of objects using photogates and tape timers. In this first section we will use a set of photogates to measure the velocity of your finger. • Set the photogates 1 meter apart with the gap pointed upward so that they form a U-shape. Both

photogates need to be connected to the timing unit. The setup should look like the picture below.

• Set the timer to the Pulse mode and the 1 ms resolution. Turn the memory ON. In this configuration the timer starts running when an object (your finger in this case) passes through the first gate and stops when an object passes through the second gate. The timer then reads the interval of time required for the object to pass between the first and second gates. Below is a picture of the timer. The distance between the gates is known, so you can calculate the average velocity of the object.

Photogates

Timing unit

1 meter


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• Move your finger at 1 m/s from the first gate to

the second one. What should be the reading on the timer?

• Move your finger at 2 m/s. What should be the reading on the timer?

• Move your finger at ½ m/s. What should be the reading on the timer?

Now we are going to see how fast you can move

an object through the photogates. In this measurement, we are only going to use one photogate. • Set the timer to the Gate mode and the 0.1 ms resolution. Turn the memory ON. In this

configuration the timer starts running when on object (the meter stick in this case) blocks the infrared beam. The timer continues to run as long as the beam is blocked. After the object leaves the gate, the timer reads the interval of time that the beam was blocked. If you know the size of the object, you can calculate its velocity when passing through the photogate. § Measure the width of the meter stick. § See how fast you can move the width of the meter stick through the photogate. § Repeat these measurements for the thickness of the meter stick. § Calculate the maximum velocities for the width and thickness on the worksheet.

III. Analyzing motion using tape timers

In the rest of the lab, we will use tape timers to analyze the motion of carts on level and inclined tracks. The tape timer works by having a paper tape slide through the timer and marks are recorded every 0.1 seconds in the 10 Hz mode (every 0.025 seconds in the 40 Hz mode). You will be measuring the distance between these marks for a level and inclined track. • Place the track in the location you want to use it. Make sure it is level using the level at your

workstation. The feet at the bottom of the track are adjustable to level it. • Place one photogate at each end of the track. They should be more than 1 meter apart as they will

hold the tape for the tape timer. • Cut a piece of tape that is long enough to go from one

photogate to the other. • Thread the tape through the tape timer as shown in the

picture. You should make sure that the tape goes through the slotted tabs on the sides of the timer and under the carbon disk.

• Place the tape timer on top of the cart on the track. • Adjust the heights of the photogates so that the tape runs

parallel to the track. Use masking tape to hold the paper tape to the two photogates. It should not sag or be pulled upwards by the photogates or cart. The setup should look like the picture below. Everything is setup correctly if the cart can move freely from one end of the track to the other.

Paper Tape Carbon Paper

Tape Timer


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• Move the cart to one end of the track and set the tape timer to the 10 Hz setting. • Give the cart a gentle push so that it moves to the other end of the track. • Examine the paper tape. There should be a series of dots on it. If there are no dots, make sure

that everything has been setup correctly. See if you can identify and fix the problem. If not, ask your TA for help.

• Make sure that you have at least 15 dots on the paper to analyze. If you have too few points, repeat the above procedure with a new piece of paper but give the cart a gentler push this time. It should take the cart about 2 seconds to travel the length of the track.

IV. Analysis of level track data

Now we are going to analyze the data that you have acquired to find the position and velocity of the cart as a function of time. Label each dot starting with 1 for the starting location of the cart. On TABLE #1 of the worksheet, fill out the following quantities.

§ Write the dot number. § Calculate the time since the cart started. Remember that the dots are equally spaced in time. § Measure the distance from the starting position to the current dot. § Calculate the separation between the current dot and the previous one. This is the change in

position of the cart, Δxn. § Calculate the velocity of the cart at the current dot using the change in position and change in

time from the previous dot. Recall that the velocity at position n is calculated as

𝑣! =𝛥𝑥!𝛥𝑡!

=𝑥! − 𝑥!!!𝑡! − 𝑡!!!

(4)

§ Make a plot of the position of the cart as a function of time. This plot should be hand drawn. This should be labeled as GRAPH #1 and attached to your worksheet. From this plot, you can measure the slope to obtain the velocity of the cart.

§ Make a plot of the velocity of the cart as a function of time. This plot should be hand drawn. This should be labeled as GRAPH #2 and attached to your worksheet. How does the velocity on this graph compare to the slope you extracted from GRAPH #1?

The last part of the lab is to measure the motion of the cart on an incline. We will use a wooden

block at one end of the track to create an incline for the cart to roll down. • Place the 4 cm high wooden block under the foot of the track at the far end from the stop. • Setup the tape timer using the same method as for the level track. However, now the photogates

will need to be at different heights. Once again, the tape should be parallel to the track.


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• Start the tape timer on the 10 Hz setting and release the cart from the raised end of the track. Make sure that the cart starts from rest.

• Catch the cart at the end. • Make sure that your tape has at least 15 dots on it.

V. Analysis of inclined track data

Now we are going to analyze the data that you have acquired to find the position, velocity and acceleration of the cart as a function of time. Label each dot starting with 1 for the starting location of the cart. On TABLE #2 on the worksheet, fill out the following quantities. § Fill out all of the quantities that you did for TABLE #1, dot number, time, distance, Δxn and vn. § Calculate the time squared, 𝑡!, since the cart started moving. § Using the calculated velocities at each point, find the change in velocity at each point. § Calculate the acceleration of the cart at the current dot using the change in velocity and change

in time from the previous dot. Recall that the acceleration at position n is calculated as

𝑎! =𝛥𝑣!𝛥𝑡!

=𝑣! − 𝑣!!!𝑡! − 𝑡!!!

(5)

Now we are going to make a series of plots of the motion of the cart as a function of time. All of the graphs should be hand drawn. § Make a plot of the position of the cart as a function of time. This should be labeled as GRAPH

#3 and attached to your worksheet. § Make a plot of the position of the cart as a function of time squared. This should be labeled as

GRAPH #4 and attached to your worksheet. From this plot, extract the slope which is half the acceleration.

§ Make a plot of the velocity of the cart as a function of time. This should be labeled as GRAPH #5 and attached to your worksheet. From this plot, extract the slope which is the acceleration.

§ Make a plot of the acceleration of the cart as a function of time. This should be labeled as GRAPH #6 and attached to your worksheet.

Name:_________________________

Date:__________________________

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Position, velocity and acceleration worksheet

I. Measuring lengths Put your measurements in the table below. Do not forget to include units will all the values.

Length of room Width of room Your height “Cubit”

Identify the objects measured and their size in the table below.

Object that is ~0.1 m Object that is ~0.01 m Object that is ~0.001 m

II. Determining velocity using photogates Write the readings on the timer for the following velocities of your finger.

1 m/s 2 m/s ½ m/s

Put your measurements for the velocity of the meter stick in the following table. Do not forget to include units with all the values.

Width of meter stick

Shortest time for width

Greatest velocity for width

Thickness of meter stick

Shortest time for thickness

Greatest velocity for thickness

Worksheet: Position, velocity and acceleration

20


IV. Level track data Fill out TABLE #1 below with the data from the level track. The units of each quantity can be given just once in the label of the column as shown. All of your graphs must be hand drawn.

Dot

number

tn

(s)

xn

(m)

Δxn

(m)

vn

(m/s)

Attach a plot of the position of the cart as a function of time. This should be labeled as GRAPH #1. Extract the slope of this plot.____________________________ Attach a plot of the velocity of the cart as a function of time. This should be labeled as GRAPH #2.


21

V. Inclined track data Fill out TABLE #2 below with the data from the inclined track. Give the units of each quantity in the label of the column as done in TABLE #1. All of your graphs must be hand drawn.

Dot

number

tn

tn2 xn

Δxn

vn

Δvn

an

Attach a plot of the position of the cart as a function of time. This should be labeled as GRAPH #3. Attach a plot of the position of the cart as a function of time squared. This should be labeled as GRAPH #4. Extract the slope of this plot.___________________________ Attach a plot of the velocity of the cart as a function of time. This should be labeled as GRAPH #5. Extract the slope of this plot.______________________________ Attach a plot of the acceleration of the cart as a function of time. This should be labeled as GRAPH #6.


22

Questions 1. How does the velocity plotted on GRAPH #2 compare to the slope you extracted from GRAPH

#1?

2. From the graphs that you made for the inclined track, how would you find the initial velocity of the cart?

3. Compare the slopes extracted from GRAPH #4 and GRAPH #5 with the acceleration measured in GRAPH #6. Are they similar? Is this expected?

4. The acceleration of a block on an incline in the absence of friction is related to the angle of the incline by the equation, 𝑎 = 𝑔 sin𝜃 where 𝜃 is the incline angle and 𝑔 is the acceleration due to gravity. Assuming this equation is correct, use your measured acceleration to determine the value of 𝑔? Recall that you used a 4 cm block under one end of a 1.2 meter long track. How does your value compare to the actual value of 𝑔 = 9.8 m/s2?

5. Looking at a set of dots on a paper tape, how can you tell if the track is level or inclined? What happens to the spacing of the dots if the angle of the incline is increased?

23

Range Versus Height

We say that an object undergoes free fall when the acceleration is only due to gravity. An object that is in free fall experiences a constant gravitational acceleration 𝑔 that points towards the ground. The object may travel along a complicated two-dimensional path but its motion can be decomposed into two independent one-dimensional motions. Gravity acts only in the vertical (y) direction and therefore there is no acceleration in the horizontal (x) direction. Using this information, we can write expressions for the position and velocity of the object as a function of time.

𝑥 = 𝑥! + 𝑣!!𝑡 𝑣! = 𝑣!!

𝑦 = 𝑦! + 𝑣!!𝑡 −!!𝑔𝑡! 𝑣! = 𝑣!! − 𝑔𝑡

In this lab, we will determine the horizontal distance, known as the range, that a ball will travel when it is launched with different speeds.

Throughout the semester you will be measuring different variables such as length, time and mass. We need to develop an understanding of how accurate these measurements are and learn to report the uncertainty in our measurements. No measurement is perfectly accurate, there is always some experimental uncertainty associated with it. For example, in the previous lab you measured the spacing between the dots created by the tape timers. Using a ruler you measured the spacing between two dots. However, the ruler only has millimeter divisions so this means that it is difficult to measure a spacing of 5.8 mm. The best that you can do is estimate that the spacing is a little bit less than 6 mm. Another source of errors arises when a measurement is repeated multiple times. Suppose you let the cart roll down the ramp several times. The dots from the tape timer will be at slightly different locations each time. Once again, this variation represents an uncertainty in your measurement. When presenting your data, it is critical to not only report the value that was measured but also the uncertainty associated with it.

In this lab, you will repeatedly drop a ball and determine its landing location using carbon paper.

By looking at the distribution of landing spots, you will learn how to estimate the uncertainty and determine ways to reduce it to give more accurate results.

Goals of the this lab: • Graphically measure the error in a series of measurements • Determine the range of a ball as a function of its height • Understand projectile motion

Range versus height

24

Lab equipment: Rulers These will be used to measure distances that the ball travels.

Bounce plate The bounce plate is designed so that it is mounted at a 45 degree angle. When a falling ball hits it, the balls motion becomes horizontal.

Drop plate This plate is used to define the height from which the ball will drop.

Carbon paper This paper is used to mark the locations where the ball hits the table.

Newsprint The large pieces of newsprint are placed on the desk to record the locations where the ball hits.

Range versus height

25

Lab Procedures

I. Measuring scatter in data When making a measurement, it is important to be both accurate and precise. For example, if you

measure the length of the room a number of times and get the following results, 10.64 m, 10.65 m, 10.64 m, 10.63 m, and 10.64 m, you would say this is a precise measurement because the variation between the different measurements is only ±0.01 m. However, if I told you the room was actually 11.00 m long you would know that your measurements were not very accurate. Similarly, if you measure the length of a meter stick 5 times and got the following results, 0.90 m, 1.10 m, 0.80 m, 1.20 m and 1.00 m you would say the measurement is not very precise because there is a large variation between the measurements. However, the average of all your measurements is exactly 1.00 m, which is the correct value. So, the average of the measurements is accurate. The goal of any experiment is to have both accurate and precise measurements. In this lab, we will be exploring ways to quantify the uncertainty in our measurements and how to improve them. In the first part of the lab, we will be dropping balls and measuring their final position. Each additional drop of the ball will reduce the uncertainty in our measurements. • Set up the bounce plate so that it is 20 cm above the table. • Set up the drop plate so that it is 20 cm above the bounce plate and directly above it. When it is

properly aligned, a ball will fall through the drop plate, hit the bounce plate and then land on your table.

• Place a large sheet of newsprint on your table where the balls will land. • Put a piece of carbon paper above the newsprint so that when the ball hits it, it leaves a mark

indicating the location of the ball. When everything is setup properly, it should look like the picture below.

Newsprint

Carbon paper

Drop plate

Bounce plate

Range versus height

26

You are going to do series of drops of the ball and observe how the average position changes as a function of the number of drops.

• Have the ball drop through the drop plate, bounce off the bounce plate and hit the carbon paper.

§ Measure the horizontal distance that the ball lands from directly below the bounce plate. Enter this value as the range in the first row of the table in the worksheet.

§ Calculate the sum of all the ranges for the drops up to this point. § Calculate the average of all the ranges up to this point. § Repeat this process until you have dropped the ball 18 times. For each drop, you will be filling

in one row of the table in the worksheet. You need to perform the measurements and calculations after each drop and not only after dropping the ball 18 times. After completing all 18 drops you will notice that there is a spread in the position where the ball has landed. The data should look similar to the figure on the right. The distribution of dots shows you how precise your procedure was for dropping the ball. The closer the spots are together the less uncertainty there is in the landing position. This uncertainty is usually expressed as a standard deviation, 𝜎 in the measurement. For a normal distribution, you expect about 2/3 of the points to lie within ±1𝜎 of the mean. This means 1/6 of the points should be farther away than the mean plus one standard deviation and 1/6 of the points should be closer than the mean minus one standard deviation. So, by excluding the 3 farthest and 3 shortest distances, you obtain approximately all of the data that is within ±1𝜎 of the mean. You can then measure the standard deviation directly with your ruler.

§ By looking at the spread of the landing spots, graphically determine the standard deviation.

II. Minimizing errors To get as accurate of a measurement as possible, we want to reduce the standard deviation in the

data. In this section, we will explore different methods of dropping the ball to get the minimum standard deviation. • Try a different way of releasing the ball from the drop plate. Some suggestions are using a ruler

underneath the ball to hold it in place and then slide it way. Use the magnet to hold the ball and then move the magnet away to let the ball fall. Or come up with your own method

• Drop the ball 18 times for two different methods (This will give three methods in total with the first part of the lab) § By looking at the spread of the landing spots, graphically determine the standard deviation for

each method.

For the rest of the lab, you should use whichever method gives the smallest standard deviation.

�m

Mean

Range versus height

27

III. Determining range versus height In the rest of the lab, we will determine the relationship between the range of the ball and the

height from which it is dropped. The range of the ball is given by the horizontal distance that it travels. It depends on both how far the ball falls before hitting the bounce plate and how high the bounce plate is from the table. We will specifically explore the relationship between the range and the drop plate height above the bounce plate. • Make sure that the bounce plate is still 20 cm above the table. • Adjust the height of the drop plate so that it is 30 cm above the bounce plate. You need to make

sure that the ball hits the bounce plate when it falls through the drop plate. • Drop the ball 12 times from the drop plate. You can drop the ball all 12 times before making the

measurements. Just make sure that you get 12 spots on the newsprint. § Measure the average range of the ball and the standard deviation. You should make a table

showing the range for all twelve drops of the ball. On the worksheet, you only need to include the average range and the standard deviation.

• Repeat the same measurement as above for the following series of heights for the drop plate, 25 cm, 20 cm, 15 cm, 10 cm, 5 cm. You should have a total of 6 heights when you are finished.

Name:_________________________

Date:__________________________

28

Range versus height worksheet I. Measuring scatter in data

Put your measurements in the table below. Do not forget to include units will all the values. The

table should include the range of the current drop, the sum of the ranges for all of the drops up until

this point and the current average range.

Assuming that you have dropped the ball n times, the sum of the ranges is given by

𝑅!"# = 𝑅!

!

!!!

(1)

where Ri is the range for drop i. The current average for drop n is then given by

𝑅!"# = 𝑅!"# 𝑛 (2)

Drop # Range Sum of Ranges, 𝑅!"# Current Average, 𝑅!"#

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

What was the standard deviation of your measurement?____________________________

Worksheet: Range versus height

29

II. Minimizing errors Describe the two different methods that you used to drop the ball and the resulting standard deviation for each one. Method 1. Method 2.

III. Determining range versus height In your lab notebook, you should keep all the data for the different drops. On the worksheet you just

need to fill out the final value for the average range and standard deviation. Do not forget to include

units with all the values.

Drop plate

height

(cm)

Average Range Standard Deviation

30

25

20

15

10

5

Make a plot of the average range versus the drop plate height. This should be labeled as GRAPH #1. You should include error bars from your standard deviation measurements. These plots can be computer graphed. To make it easier to analyze the data, we would like the points to lie along a straight line so that we can fit a line to them. In order to make the data lie along a straight line, you will need to change the x-axis of the graph. You should replot the data versus height squared or square root of the height, whichever one gives better results. This should be labeled as GRAPH #2. Fit a straight line to this graph and determine its slope.___________________

Worksheet: Range versus height

30

Questions to address in your lab report 1. Describe what happens to the average landing position of the ball as you drop the ball more times.

2. Describe the different methods that you used to drop the ball. Which one gives the best results?

3. Derive the expression for the range of the ball as a function of its height. To do this you should

use the following steps. In the first part of the motion, the ball undergoes vertical free fall until it hits the bounce plate. At this point, its final velocity in the y-direction becomes the initial velocity in the x-direction after hitting the plate. Now calculate the time it takes the ball to hit the ground after hitting the bounce plate. Lastly, find the distance that the ball will travel in the horizontal direction in this time. This is the range of the ball.

4. How does the data you graphed for the range versus height compare to the expression that you derived in question #3? Does your data have the correct functional dependence on the height of the drop plate? Does the data have the correct slope?

5. What was the largest source of error in your measurements? How could the accuracy of your data be improved?

31

Vectors and Forces

Newton’s first law states that an object at rest remains at rest unless acted upon by an outside force. Another way of saying this is that the forces on an object must add up to zero if it is at rest. This is a condition known as static equilibrium and can be expressed with the equation

𝐹!!

= 0 (1)

where 𝐹! are the forces acting on the object.

Suppose two people push on opposite sides of a box with equal magnitude forces. The box will not move because the sum of the two forces is zero. However, if the two people push with the same magnitude of force but one in the x direction and one in the y direction, the box will move. This is because forces are vectors and therefore have both a magnitude and direction associated with them. This means that when adding forces we need to use the rules for adding vectors. There are two methods that you can use to add vectors, (1) graphically add them tip to tail or (2) resolve them into x and y components and then add the components.

A vector can be resolved into its x and y components using

trigonometry. The drawing to the right shows a vector, 𝐹 and its resulting components. The x component of the vector is given by

𝐹! = 𝐹 cos𝜃 = 𝐹 sin𝜙 (2)

where 𝐹 is the magnitude of the vector and 𝜃 is the angle with respect to the x axis and 𝜙 is the angle with respect to the y axis. Similarly the y component of the vector is given by

𝐹! = 𝐹 sin𝜃 = 𝐹 cos𝜙. (3)

In this lab, we will be using a force table to graphically resolve vectors into their x and y

components. We will also use the table to add vectors and explore the conditions necessary for an object to be in static equilibrium.

Goals of the this lab: • Learn how to resolve vectors into components • Graphically add vectors • Understand the concept of static equilibrium

Fx

FyF

e

q

Vectors and Forces

32

Lab equipment: Force table This circular table has a ring in the center and the ability to hang three sets of masses off the sides. These hanging masses act as forces on the central ring.

Hangers These hangers are attached by strings to the central ring on the force table. You can place masses up to 200 g on each one.

Mass Set This box contains a set of masses to put on the hangers ranging from 1-200 g.

String String to attach the hangers to the central ring of the force table.

Vectors and Forces

33

Lab Procedures

I. One-dimensional equilibrium If there are two forces acting on an object, the only way for their sum to be zero is if they have

equal magnitudes but opposite directions. In the first part of the lab, we are going to test this assertion. Throughout the lab, you will be using a force table to measure the magnitude and direction of a series of forces. The force table is shown in the picture below. It consists of a large metal plate that has angular markings on it. There is a set of three pulleys that can be placed at any angle around the rim of the plate. In the center of the plate is a ring with three strings tied to it that run over each of the pulleys. At the end of these strings is a hanger on which you can put various masses. These masses apply a force to the central ring that is equal to their weight.

The magnitude of the force on the central ring from a particular string is the magnitude of the

weight of the hanging object. The direction of the force on the central ring from a given string is outward in the direction of

the string.

• Set up the force table so that the 0 degree mark points to the right. We will call this the positive x direction.

• Adjust one of the strings to go along the x direction. • Place a 200 gram mass on the hanger in the x direction.

At this point the central ring should be pulled in the positive x direction (towards the right) by a 2

N force. This is because the mass of the hanging object is 5 grams for the hanger and 200 grams for

Centering Ring

Hanger

Pulley

Metal plate

Vectors and Forces

34

the added mass. The weight is given by 𝑊 = 𝑚𝑔 where 𝑚 is the mass and 𝑔 = 9.8 m/s2 is the acceleration due to gravity. § At what angle should you hang additional weights to bring the ring back to the center of the

circle? Put your answer on the worksheet. § What mass do you need to put on the second hanger to bring the ring back to the center of the

circle? Put your answer on the worksheet and then add mass to the hanger until the ring is centered. Do you get the expected value for the added mass?

• To determine if the ring is centered you should look from straight above and use the black circle as a guide.

II. Resolving a force into components Any two-dimensional vector can be resolved into its two components along perpendicular

directions. Using the force table, you will find the components for a series of vectors. • There should already be one pulley at 0 degrees from part I. Move one of the other pulleys so

that it is at 90 degrees. This will be the positive y direction. You will use the third pulley to apply a known force to the central ring. Then by adding masses

to the hangers in the x and y direction, you can find the components of the force in each of these directions.

§ Place the third pulley at the angle given in the worksheet. § Add the given mass to the hanger on the third pulley. Remember that the hanger has an

additional mass of 5 grams. It is necessary to remember this mass when calculating the weight of the hanger.

§ Add mass to the hangers in the x and y direction until the ring is centered on the circle. At this point all three forces sum to zero. Then the x and y components of the force are given by the weight on the 0 and 90 degree pulleys.

§ Repeat this procedure for all three forces given in the second part of the worksheet.

III. Adding perpendicular forces To add two vectors together you must resolve them into components and then add the individual x

and y components. The result will give the x and y components of the resulting vector. This process is simple when the two forces are perpendicular because one vector has only an x component and the other one has only a y component. On the force table, you can find the resulting force by determining the mass needed on the third pulley and its angle to center the ring on the table. • Make sure that one pulley is at 0 degrees and the other is at 90 degrees. § Place masses on the two pulleys according to the values given in the third section of the

worksheet. § Adjust the angle of the third pulley and the mass on it until the ring is centered on the force

table. § On the worksheet, record the magnitude of the force given by the weight on the third pulley

along with its direction. § Repeat this procedure for all three sets of forces given in the third section of the worksheet.

Vectors and Forces

35

IV. Adding non-perpendicular forces Adding vectors that are not perpendicular is more difficult mathematically because both vectors

may have x and y components. However, on the force table the procedure is the same as for perpendicular forces. § Adjust the angle of two pulleys and the mass on their hangers according to the values given in

the fourth section of the worksheet. § Adjust the angle of the third pulley and the mass on it until the ring is centered on the force

table. § On the worksheet, record the magnitude of the force given by the weight on the third pulley

along with its direction. § Repeat this procedure for all fives sets of forces given in the fourth section of the worksheet.

Name:_________________________

Date:__________________________

36

Vectors and Forces Worksheet I. One-dimensional equilibrium

What should be the angle of the second force?

What should be the magnitude of the second force?

II. Resolving a force into components For each of the tables below, you should adjust the pulley to be at the given angle. Then put the

mass on the hanger. You are expected to find the force necessary to center the ring. These values should be entered in the “Force Table” box. You should calculate the components of the force using trigonometry and enter it in the given box. As a last step, you should make a drawing of the force vector and graphically resolve it into x and y components. Your drawings must be to scale. Please use 2 cm to correspond to the weight from a 100 gram mass.

Angle of pulley: 225 degrees Mass added to hanger: 100 grams

Graph

Force Table Fx:

Fy:

Calculation Fx:

Fy:


Graph

Force Table Fx:

Fy:

Calculation Fx:

Fy:

Worksheet: Vectors and Forces

37


Graph

Force Table Fx:

Fy:

Caclulation Fx:

Fy:

III. Adding perpendicular forces In this part of the lab, you are adding two forces, 𝐹! and 𝐹!, which are perpendicular. You may

assume that 𝐹! is in the x direction and 𝐹! is in the y direction.

What is the expression for the magnitude of the resulting force 𝐹! + 𝐹!?

What is the angle of the resulting force, 𝐹! + 𝐹!?

Similarly to the previous part, you should fill out the table for each set of forces. You should write

the experimentally determined values from the force table, the values determined from trigonometry

and make a drawing showing the graphical addition of the two forces. The mass given for each

hanger is the additional mass that you should add to the hanger. Your drawings must be to scale.

Please use 2 cm to correspond to the weight from a 50 gram mass.

Hanger 1: 30 grams at 0 degrees Hanger 2: 40 grams at 90 degrees

Graph

Force Table Magnitude: 0.51 N Direction: 52.1 degrees

Calculation Magnitude: 0.49 N

Direction: 53.1 degrees


38


Graph

Force Table Magnitude:

Direction:

Calculation Magnitude:

Direction:


Graph


Direction:


Direction:


Graph


Direction:


Direction:

If you keep the pulleys in their current position, is it possible to have the resulting force be outside the third quadrant (angles between 180 and 270 degrees)?

How would you adjust the two pulleys so that the sum of the forces is in the fourth quadrant (angles between 270 and 360 degrees)?


39

IV. Adding non-perpendicular forces

Similarly to the previous part, you should fill out the table for each set of forces. You should write

the experimentally determined values from the force table, the values determined from trigonometry

and make a drawing showing the graphical addition of the two forces. The mass given for each

hanger is the additional mass that you should add to the hanger. Your drawings must be to scale.

Please use 2 cm to correspond to the weight of a 100 gram mass.


Graph


Direction:


Direction:


Graph


Direction:


Direction:


40


Graph


Direction:


Direction:


Graph


Direction:


Direction:


Graph


Direction:


Direction:

Questions

1. Describe the sources of any errors in your measurement. In other words, why do your calculated and measured values differ?

41

Newton’s Second Law

Newton’s second law states that a net force on an object will cause it to accelerate. The acceleration of the object is inversely proportional to its mass and in the direction of the net force. We can write this as the following equation,

𝐹! = 𝑚𝑎 (1)

where 𝐹! are the forces acting on the object, 𝑚 is its mass and 𝑎 is its acceleration.

In this lab, you will confirm Newton’s second law by measuring the acceleration of a cart. We will assume that there is only one force in the system in which case we can rewrite Newton’s second law as

𝑎 =𝐹𝑚.

(2)

Written in this way we can make three statements about the acceleration of an object. (1) The acceleration is in the same direction as the force. This should be obvious to anyone who

has pushed an object. It moves in the direction that you push.

(2) The acceleration of an object is proportional to the magnitude of the applied force. The harder you push, the greater the acceleration of the object.

(3) The acceleration of an object is inversely proportional to its mass. The more massive an object the less it will accelerate for a given force.

In this lab, we will be measuring the acceleration of a cart as a function of its mass and the

applied force. Using these results we will confirm the second and third statements above.

Goals of the this lab: • Measure the acceleration of a cart as a function of the applied force. • Measure the acceleration as a function of mass.


42

Lab equipment: Track The cart will travel along this 1.2 meter long track.

Cart This cart will move along the track. It has a mass of approximately 500 grams.

Mass Set This box contains a set of masses to put on the hangers and cart ranging from 1-200 g.

Photogates The photogates have an infrared sensor in them that detects the motion of objects through them. They will be used to measure time and velocity with the Xplorer GLX.

Xplorer GLX The Xplorer GLX has the ability to measure times and velocities from up to 8 sensors at once.

String String to attach the hanging mass to the cart.


43

Lab Procedures

If you want to move an object of mass 𝑚, the harder you push it the greater its acceleration will be. Newton’s second law gives the relationship between the acceleration 𝑎 of the object and the force 𝐹 that you apply.

𝑎 =𝐹𝑚.

(3)

In this part of the lab, you will be varying the applied force while keeping the total mass of the system constant. The system that you will consider is a cart on a track and a hanging mass at one end. These two objects are connected by a string to ensure that they move together as one unit. The setup should look like the picture below.

In this configuration, we need to consider the total mass of the system consisting of the cart and the hanging mass. This total mass is given by

𝑀!"!#$ = 𝑀!"#$ +𝑚!!"#$"# (4)

where 𝑀!"#$ is the mass of the cart and 𝑚!!"#$"# is the mass on the hanger. The net external force acting on the system comes from gravity acting on the hanging mass. Therefore the magnitude of the force is given by the weight of the hanging mass,

𝐹 = 𝑚!!"#$"#𝑔 (5)

Combining these two equations with our expression for the acceleration gives

𝑎 =ForceMass =

𝑚!!"#$"#𝑔𝑀!"#$ +𝑚!!"#$"#

. (6)

Photogates are unable to measure the acceleration directly. They are able to either measure the

time it takes to travel from one gate to the second one or the time that the cart spends blocking one of the gates which gives the instantaneous velocity. Therefore, we will have to calculate the acceleration of the cart from these measurements. We will use two different methods to calculate the acceleration.

Hanging Mass

Track

Cart

Photogates

DistancePulley


44

Method #1: Measure Travel Time

In this method, you will measure the time it takes the cart to travel between the two gates. The acceleration is constant and therefore we can use the equations from 1D kinematics to relate the distance traveled to the time using 𝑥! = 𝑥! + 𝑣!𝑡 +

!!𝑎𝑡! (7)

where 𝑥! is the initial position, 𝑣! is the initial speed, 𝑎 is the acceleration and 𝑡 is the time. If you choose your starting position to be 𝑥! = 0 and start the cart from rest, 𝑣! = 0 then equation (7) simplifies to 𝑥! =

!!𝑎𝑡! (8)

Solving equation (8) for 𝑡! gives,

𝑡! =2𝑥𝑓𝑎 . (9)

Therefore, a plot of 𝑡! versus 𝑥! has a slope of 2 𝑎.

Method #2: Measure Final Velocity

In this method, you will measure the final velocity of the cart when it passes through the second photogate. Once again, the acceleration is constant and therefore we can use the equations from 1D kinematics to relate the final velocity to the time using 𝑣! = 𝑣! + 𝑎𝑡 (10)

where 𝑣! is the initial velocity, 𝑎 is the acceleration and 𝑡 is the time. If you ensure that the cart starts from rest, 𝑣! = 0 at the first photogate, then equation (10) simplifies to 𝑣! = 𝑎𝑡 (11)

Squaring this equation gives, 𝑣!! = 𝑎!𝑡! (12)

Plugging in the expression for 𝑡! from method #1 gives, 𝑣!! = 2𝑎𝑥! . (13)

I. Acceleration for fixed total mass

• Set up the track as shown in the picture above. The distance between the two photogates should be 15 cm. You should move the photogate on the right as close to the beginning of the track as possible (away from the pulley).

• Attach a hanger to the cart using string and run it over the pulley. You should adjust the length of the string so that the hanger is near the pulley when the cart is at the beginning of the track.

• Setup the Xplorer GLX to measure the time between the photogates and the velocity in the second gate. For instructions on setting the Xplorer GLX, refer to the appendix of the lab manual.

• Make sure that the cart has a 1 cm wide flag and that its width is set in the Xplorer GLX.


45

• Make sure that the hanging mass does not hit the floor before the cart passes through the photogates.

• For this part of the lab, we want the total mass of the system to be fixed at 600 grams. Put 20 grams on the hanger and whatever you need on the cart to make a total of 600 grams. Make sure that you measure the mass of both the hanger and cart. § Release the cart from rest at the first photogate and measure the time it takes to reach the

second photogate and the final velocity. Record these values into the table on the worksheet. • Repeat this measurement two more times with the photogates separated by 15 cm. § Calculate the average of the three measurements. Then use these averages to calculate the time

squared and the final velocity squared for the 15 cm separation and 20 gram mass. Record these values into the table on the worksheet.

• Change the mass on the hanger to be 40 grams, 60 grams and 80 grams. For each mass, you should repeat the measurement 3 times. § Similarly to the 20 gram mass measurements, you should create a table with your data.

• Change the separation between the two photogates to be 30 cm, 45 cm and 60 cm. For each separation of the gates, you should perform 3 measurements for each of the 4 hanging masses. § Enter the data for each separation into the data tables that you created in the earlier steps. In the end, you should have completed a total of 48 measurements for this part of the lab. This

data should be organized in 4 tables, one table for each value of the hanging mass.

II. Acceleration for a fixed force In this part of the lab, we will be examining what happens to the acceleration if we keep the

magnitude of the force constant and change the mass of the system. This means that we want to keep a constant mass on the hanger and vary the mass on the cart. • For this part of the lab, we want the mass on the hanger to be fixed at 100 grams. Put whatever

mass you need on the cart to make a total mass of 600 grams for the system (hanger plus cart). § Release the cart from rest at the first photogate and measure the time it takes to reach the

second photogate, 15 cm away and the final velocity. Record these values into a table like was done in the first part of the lab.

• Repeat this measurement two more times with the photogates separated by 15 cm. § Calculate the average of the three measurements. Then use these averages to calculate the time

squared and the final velocity squared for the 15 cm separation and 600 gram total mass. Record these values into the table on the worksheet.

• Change the total mass of the system to be 850 grams, 1100 grams and 1350 grams. For each mass, you should repeat the measurement 3 times. § Similarly to the 600 gram mass measurements, you should create a table with your data.

• Change the separation between the two photogates to be 30 cm, 45 cm and 60 cm. For each separation of the gates, you should perform 3 measurements for each value of the total mass. § Enter the data for each separation into the data tables that you created in the earlier steps. In the end, you should have completed a total of 48 measurements for this part of the lab. This

data should be organized in 4 tables, one table for each value of the total mass.

Name:_________________________

Date:__________________________

46

Newton’s Second Law Worksheet I. Acceleration for fixed total mass

You should make a table like the one below for each value of the hanging mass. There should be a total of four tables for your data in this part of the lab. These should be labeled TABLE #1-4 and included in your lab report.

Total mass: Hanging mass: Cart mass:

Distance Time

between

photogates

Final

Velocity

Average of

time

Average of

time squared

Average of

velocity

Average of

velocity

squared

15 cm

15 cm

15 cm

30 cm

30 cm

30 cm

45 cm

45 cm

45 cm

60 cm

60 cm

60 cm

You should make a plot of the time squared as a function of distance travelled for each value of the hanging mass. This graph should be labeled GRAPH#1 and include four sets of data with different symbols (one for each value of the hanging mass).

Fit straight lines to the data for each value of the hanging mass in GRAPH#1. The inverse slope

will be proportional to the acceleration. Using the accelerations that you find, make a new graph of the acceleration as a function of hanging mass. This graph should be labeled GRAPH#2.

You should also make a plot of the final velocity squared as a function of the distance between

the gates for each value of the hanging mass. This graph should be labeled GRAPH#3 and include four sets of data with different symbols (one for each value of the hanging mass).

Fit straight lines to the data for each value of the hanging mass in GRAPH#3. The slope will be

proportional to the acceleration. Using the accelerations that you find, make a new graph of the acceleration as a function of hanging mass. This graph should be labeled GRAPH#4.

Worksheet: Newton’s Second Law

47

II. Acceleration for a fixed force You should make tables like the ones in part I for each value of the total mass. There should be a

total of four tables for your data in this part of the lab. These should be labeled TABLE #5-8 and included in your lab report.

You should make a plot of the time squared as a function of distance travelled for each value of

the total mass. This graph should be labeled GRAPH#5 and include four sets of data with different symbols (one for each value of the total mass).

Fit straight lines to the data for each value of the total mass in GRAPH#5. The inverse slope will

be proportional to the acceleration. Using the accelerations that you find, make a new graph of the acceleration as a function of 1/total mass. This graph should be labeled GRAPH#6.

You should also make a plot of the final velocity squared as a function of the distance between

the gates for each value of the total mass. This graph should be labeled GRAPH#7 and include four sets of data with different symbols (one for each value of the total mass).

Fit straight lines to the data for each value of the total mass in GRAPH#7. The slope will be

proportional to the acceleration. Using the accelerations that you find, make a new graph of the acceleration as a function of 1/total mass. This graph should be labeled GRAPH#8.

Questions to address in your lab report 1. Draw free body diagrams for both the hanging mass and the cart. You may assume that the track

is frictionless for your drawing.

2. Apply Newton’s second law to each object separately. Then combine the equations to derive the expression for the acceleration of the cart as a function of the hanging mass and total mass.

3. Fit straight lines to your data for GRAPHS#2 and 4. What is the slope of these lines? Does the

slope agree with the equation that you derived for the acceleration of the cart for a fixed total mass?

4. Fit straight lines to your data for GRAPHS#6 and 8. What is the slope of these lines? Does the

slope agree with the equation that you derived for the acceleration of the cart for a fixed force? 5. Discuss any sources of errors in your measurements.

48

Friction

We saw in the previous lab that a net force on an object causes it to accelerate. However, you know that it is easier to push an object across a slippery surface than a rough floor. This is because there is friction between the object and the floor that opposes the motion of the object. If the object is stationary this is known as static friction and if the object is sliding it is known as kinetic friction.

Whenever an object does not move, the forces on it must sum to zero. So, if you apply a force on

an object and it does not move, there must be another force equal in magnitude but opposite in direction acting on the object. In this lab, that force will be static friction.

As you apply a larger and larger force to the object, the static friction force must increase if the

object is to remain stationary. Its magnitude must match the magnitude of your applied force for the object to remain at rest. If you are able to apply a large enough force, the object begins to move. At this point the static friction is not able to keep the object at rest, i.e. there is a maximum value of static friction. The force of static friction, 𝑓! is given by

𝑓! ≤ 𝜇!𝑁 (1)

where 𝜇! is the coefficient of static friction and 𝑁 is the magnitude of the normal force.

Once the object is moving, static friction no longer applies. Instead kinetic friction acts to oppose the motion of the object. The force of kinetic friction is a constant given by

𝑓! = 𝜇!𝑁 (2)

where 𝜇! is the coefficient of kinetic friction. When you are trying to move an object, it is harder to get it to start moving then it is to continue to move it. This is because the coefficient of kinetic friction is always less than the coefficient of static friction. Both are typically less than 1.

In this lab, you will be measuring the coefficient of static and kinetic friction for three different surfaces. You will measure the coefficient of static friction by measuring the force necessary to get a sled to move on a track. The smaller the coefficient of static friction, the smaller the force required to move the sled. To measure the coefficient of kinetic friction, you will measure the acceleration of the sled for a given applied force. The larger the coefficient of kinetic friction the smaller the acceleration will be.

Goals of the this lab: • Measure the coefficient of static friction for different surfaces. • Measure the coefficient of kinetic friction for different surfaces. • Understand the relationship between the coefficients of static and kinetic friction for a surface.

Friction

49


Sleds These sleds have different surfaces on the bottom which have different coefficients of static and kinetic friction.


Photogates The photogates have an infrared sensor in them that detects the motion of objects through them. They will be used to measure travel time with the Xplorer GLX.

Xplorer GLX The Xplorer GLX has the ability to measure times from up to 8 sensors at once.

String String to attach the hanging mass to the cart.

Friction

50

Lab Procedures

I. Measuring static friction on a flat track When you apply a force to an object and it does not move, static friction is acting in the opposite

direction to keep the object at rest. To measure the coefficient of static friction, we can continually increase the force on an object until it begins to move. At this point the applied force will be equal in magnitude to the maximum force from static friction. In this part of the lab, you will be varying the force applied to a sled on a flat track. This is similar to the setup used in the previous lab. However, in this lab, you will find the force necessary for the sled to begin to move. The picture below shows the setup for this part of the lab. The hanging mass creates tension on the string which tries to pull the sled to the left. Static friction

opposes the motion of the sled and therefore acts toward the right. As long as the sled does not move, these two forces are equal in magnitude. Therefore,

𝑚𝑔 = 𝑓! (3)

where 𝑚 is the hanging mass and 𝑓! is the force of static friction. The maximum value for the force of static friction is given by

𝑓! ≤ 𝜇!𝑁 = 𝜇!𝑀𝑔 (4)

where 𝜇! is the coefficient of static friction, 𝑁 is the normal force and 𝑀 is the mass of the sled. As we increase the hanging mass, the friction force must increase to keep the sled at rest. At the point that the sled begins to move, the friction force is a maximum. At this point, we can combine the two equations above to get the coefficient of static friction,

𝜇! =𝑚𝑀 (5)

• Set up the track as shown in the picture above. • Attach a hanger to the sled using string and run it over the pulley. You should adjust the length of

the string so that the hanger is near the pulley. Make sure that the string is short enough that the hanging mass will not hit the floor.

• For this part of the lab, we will be varying the mass of the sled and determining the hanging mass needed to begin to move it. Initially, the sled should have a mass of 250 grams.

Hanging Mass

Pulley Sled

Track

Friction

51

§ Add mass to the hanger until the cart just begins to move. It is important to keep the hanging mass as still as possible to minimize the error in your measurement. Record the value of the hanging mass in the table on the worksheet.

• Repeat this measurement one more time with the cart at a slightly different location on the track. § Calculate the average of these two measurements and find the value for the coefficient of static

friction. Record these values in the table on the worksheet. • Change the mass on the sled to be 500 grams and 750 grams. For each mass, you should repeat

the measurement 2 times at different locations on the track. § Similarly to the 250 gram mass measurements, you should record the value of the hanging

mass and coefficient of static friction in the table on the worksheet. • Change the sled to use the ones with different surfaces on the bottom. Make sure that you use all

three different sleds. For each sled, you should use three different masses and make two measurements for each mass. § Enter the data for each sled into the data table on the worksheet. In the end, you should have completed a total of 18 measurements for this part of the lab. This

data should be organized in 3 tables, one table for each sled surface.

II. Measuring static friction on an inclined track In this part of the lab, you will be varying the force applied to a sled on an inclined track. As you increase the angle of the incline, the component of gravity acting along the track increases. In order for the sled to remain stationary, the static friction force must increase. Eventually, the friction cannot increase anymore and the sled begins to move. You will determine the angle when the sled begins to move by measuring the height of the end of the track. The setup is shown in the picture below. One end of the track rests on the table while the other end is slowly lifted up until the sled moves. In this setup, the component of the gravitational force along the track is given by 𝑀𝑔 sin𝜃

where 𝑀 is the mass of the sled. As long as the sled does not move, the static friction force is equal in magnitude to the component of the gravitational force along the track. So,

𝑓! = 𝑀𝑔 sin𝜃 (6)

where 𝑓! is the force of friction. Once again, the maximum value for the force of static friction is given by

𝑓! ≤ 𝜇!𝑁 = 𝜇!𝑀𝑔 cos𝜃 (7)

LH

K

Sled e

Friction

52

where 𝜇! is the coefficient of static friction and 𝑁 is the normal force. Now the normal force is the component of the weight that is perpendicular to the track, 𝑁 = 𝑀𝑔 cos𝜃. As we increase the angle of the track, the friction force must increase to keep the sled at rest. At the point that the sled begins to move, the friction force is a maximum. We can solve the two previous equations for the coefficient of static friction,

𝜇! = tan𝜃 (8)

So, to measure the coefficient of static friction, we need to know the angle of the track when the sled begins to move.

You will be measuring the height of one end of the track and from this measurement determining the angle of the track. Let 𝐾 be the height of the track above the table when it is flat, and 𝐿 be the distance from the pivot point to the end of the track. Then you can find the angle, 𝜃 using trigonometry,

sin𝜃 =𝐻 − 𝐾𝐿 (9)

• Set up the track so that it is flat on the table. • For this part of the lab, we will be varying the mass of the sled and determining the angle needed

to begin to move it. Initially, the sled should have a mass of 250 grams. § Slowly raise one end of the track until the sled begins to move. Record the height of the track,

𝐻, when the sled begins to move in the table on the worksheet. • Repeat this measurement one more time with the cart at a slightly different location on the track. § Calculate the average of these two measurements and find the value for the coefficient of static

friction. Record these values into the table on the worksheet. • Change the mass on the sled to be 500 grams and 750 grams. For each mass, you should repeat

the measurement 2 times at different locations on the track. § Similarly to the 250 gram mass measurements, you should record the height of the track and

coefficient of static friction in the table on the worksheet. • Change the sled to use the ones with different surfaces on the bottom. Make sure that you use all

three different sleds. For each sled, you should use three different masses and make two measurements for each mass. § Enter the data for each sled into the data table on the worksheet. In the end, you should have completed a total of 18 measurements for this part of the lab. This


III. Measuring kinetic friction on a flat track When you apply a large enough force to move an object, the force that resists the motion is

kinetic friction. The kinetic friction is acting in the opposite direction of the motion and is trying to slow down the object. Therefore, it is decreasing the acceleration of the object. To measure the coefficient of kinetic friction, we will apply a large enough force to move the sled and measure its acceleration. The larger the kinetic friction is, the smaller the acceleration will be.

Friction

53

In this part of the lab, you will be measuring the acceleration of the sled on a flat track like was done in the previous lab. The picture below shows the setup for this part of the lab. The hanging mass creates tension on the string which pulls the sled to the left causing it to accelerate. The kinetic friction opposes the motion of the sled and therefore acts toward the right. To find the acceleration of

the cart, we need to consider separate free-body diagrams for the hanging mass and the sled. The result is that the acceleration is given by,

𝑎 =𝐹net

Total Mass =𝑚𝑔

𝑚 +𝑀 −𝜇!𝑀𝑔𝑚 +𝑀 (10)

where 𝑚 is the hanging mass, 𝑀 is the sled mass and 𝜇! is the coefficient of kinetic friction. You will notice that if the coefficient of kinetic friction is zero, you obtain the same equation for the acceleration as in the previous lab. Also, when the coefficient of kinetic friction is non-zero it acts to reduce the acceleration of the sled. In this lab, you will be measuring the acceleration and using that to find the coefficient of kinetic friction. To do this, we can solve the above equation for 𝜇!,

𝜇! =𝑚𝑀 −

𝑚 +𝑀𝑀

𝑎𝑔 (11)

• Set up the track as shown in the picture above. The distance between the two photogates should

be 15 cm. You should move the photogate on the right as close to the beginning of the track as possible (away from the pulley)

• Attach a hanger to the sled using string and run it over the pulley. You should adjust the length of the string so that the hanger is near the pulley when the sled is at the beginning of the track.

• Setup the Xplorer GLX to measure the time between the photogates. For instructions on setting the Xplorer GLX, refer to the appendix of the lab manual.

• For this part of the lab, we will be varying the distance between the photogates to determine the acceleration of the sled. You should have found in the previous parts of the lab that the mass of the sled does not change its coefficient of friction. So, we will only use a constant mass on the sled of 250 grams.

• You need to attach a large enough mass to the hanger so that the sled will move. § Release the cart from rest at the first photogate and measure the time it takes to reach the

second photogate. Record this value into the table on the worksheet. • Repeat this measurement two more times with the photogates separated by 15 cm.

Hanging Mass

Pulley

SledTrack

X

Photogates

Friction

54

§ Calculate the average of these three measurements. Then use these averages to calculate the time squared for the 15 cm separation. Record this value in the table on the worksheet.

• Change the separation between the two photogates to be 30 cm, 45 cm and 60 cm. For each separation, you should repeat the measurement 3 times. § Enter the data for each separation in the table that you used in the earlier steps. § Graph the time squared as a function of the separation of the photogates. Fit a straight line to

this graph. The acceleration is given by

𝑎 = 2

slope (12)

• Change the sled to use the ones with different surfaces on the bottom. Make sure that you use all three different sleds. For each sled, you should use four different separations and make three measurements for each separation. § Enter the data for each sled in the table on the worksheet. In the end, you should have completed a total of 36 measurements for this part of the lab. This


Name:_________________________

Date:__________________________

55

Friction Worksheet I. Measuring static friction on a flat track

You should fill out the tables for each sled. There should be a total of three tables for your data in this part of the lab.

Surface:

Sled Mass Hanging Mass Average Hanging Mass Coefficient of static friction

250 g

250 g

500 g

500 g

750 g

750 g

Surface:


250 g

250 g

500 g

500 g

750 g

750 g

Surface:


250 g

250 g

500 g

500 g

750 g

750 g

Worksheet: Friction

56

II. Measuring static friction on an inclined track You should fill out the tables below for each of the sleds. There should be a total of three tables

for your data in this part of the lab.

Surface: L: K:

Sled Mass Height Average Height Average Angle Coefficient of static friction

250 g

250 g

500 g

500 g

750 g

750 g

Surface: L: K:


250 g

250 g

500 g

500 g

750 g

750 g

Surface: L: K:


250 g

250 g

500 g

500 g

750 g

750 g

Worksheet: Friction

57

III. Measuring kinetic friction on a flat track You should fill out the following tables for each of the sleds. There should be a total of three

tables for your data in this part of the lab.

Surface: Sled mass: Hanging mass:

Separation Time Average Time Time Squared 15 cm 15 cm 15 cm 30 cm 30 cm 30 cm 45 cm 45 cm 45 cm 60 cm 60 cm 60 cm

You should make a plot of the time squared as a function of the distance travelled. This graph should be labeled GRAPH#1 and attached to the worksheet. Fit a straight line to the data. The inverse slope will be proportional to the acceleration. The acceleration for this sled is ________________________. Using the acceleration, find the coefficient of kinetic friction for the sled,___________________.

Surface: Sled mass: Hanging mass:

Separation Time Average Time Time Squared 15 cm 15 cm 15 cm 30 cm 30 cm 30 cm 45 cm 45 cm 45 cm 60 cm 60 cm 60 cm

Worksheet: Friction

58

You should make a plot of the time squared as a function of the distance travelled. This graph should be labeled GRAPH#2 and attached to the worksheet. Fit a straight line to the data. The inverse slope will be proportional to the acceleration. The acceleration for this sled is ________________________. Using the acceleration, find the coefficient of kinetic friction for the sled,___________________. Surface: Sled mass: Hanging mass:

Separation Time Average Time Time Squared

15 cm 15 cm 15 cm 30 cm 30 cm 30 cm 45 cm 45 cm 45 cm 60 cm 60 cm 60 cm

You should make a plot of the time squared as a function of the distance travelled. This graph should be labeled GRAPH#3 and attached to the worksheet. Fit a straight line to the data. The inverse slope will be proportional to the acceleration. The acceleration for this sled is ________________________. Using the acceleration, find the coefficient of kinetic friction for the sled,___________________.

Worksheet: Friction

59

Questions 1. Looking at your results for part I, do you obtain the same value for the coefficient of static

friction for all of the different masses? Explain.

2. Looking at your results for part II, does the mass of the sled affect the maximum angle that you can raise the track before it moves? Explain.

3. How do the results you obtained in parts I and II compare with each other? Which surface has the

smallest coefficient of static friction? Which has the largest? 4. Compare the coefficients of static and kinetic friction that you found. Do the results make sense? 5. Discuss any sources of errors in your measurements.

6. Summarize your results in the following table. Your table should include estimates of the uncertainty in your measurements.

Sled Material Coefficient of static friction Coefficient of kinetic friction

60

Energy Conservation

Newton’s second law relates the force on an object to its acceleration. If we want to find the speed of the object or its position we must know the forces acting on it and solve for the motion. We were able to do this for simple systems like a block moving down an incline or a ball thrown off a cliff. However, if the forces on the system are more complicated this becomes very difficult. In this lab, we are going to consider the case of a roller coaster that has a complicated track that includes a loop. Calculating the speed of the roller coaster with Newton’s laws would be very tedious but by using energy methods it is rather straightforward.

There are many different forms of energy including kinetic, gravitational potential, spring potential and thermal. For an isolated system, the total energy of the system remains constant and it is described as conserved. However, the form of the energy can change. For example a ball at rest at the top of a building has no kinetic energy but it may have gravitational potential energy. As the ball falls, it gains speed and therefore its kinetic energy increases however its gravitational potential energy decreases. In this lab, we are only going to be concerned with two forms of energy, kinetic and gravitational potential energy. We can expression the conservation of energy as

𝐸 = 𝐾 + 𝑈 (1)

where 𝐸 is the total energy, 𝐾 is the kinetic energy and 𝑈 is the gravitational potential energy. Since 𝐸 must remain constant, if 𝑈 increases then 𝐾 decreases.

The expression for energy conservation only holds true when the system is isolated and there are no non-conservative forces such as friction acting on it. If there are non-conservative forces, then the energy of the system will change as a function of time.

In the first part of the lab, you will use energy conservation to find the speed of the roller coaster as a function of its height along the track. Then in the second part of the lab, you will observe that energy is conserved as the roller coaster goes around the track. In the third part of the lab, you will measure the amount of work done by non-conservative forces. In the last part of the lab, you will try to find the fastest route from the beginning of the track to the end.

Goals of the this lab: • Measure the velocity of a roller coaster car as a function of height. • Study the conversion of potential energy to kinetic energy • Measure the energy loss due to friction. • Determine the fastest path between two points.

Energy Conservation

61

Lab equipment: Roller Coaster The cart will travel along this 1.2 meter long track.

Car This car can travel on the roller coaster path. The flag is 4.7 mm wide and 10 mm between the two arms.


Photogates The photogates have an infrared sensor in them that detects the motion of objects through them. They will be used to measure time and velocity with the Xplorer GLX.

Xplorer GLX The Xplorer GLX has the ability to measure times and velocities from up to 8 sensors at once.

Energy Conservation

62

Lab Procedures

I. Velocity as a function of height As a roller coaster goes down a hill, gravitational potential energy is converted into kinetic

energy. This causes its speed to increase. In this part of the lab, you will be measuring the speed of a roller coaster car as a function of height. The picture below shows the setup for this part of the lab. There are a series of locations where you can place the photogates. The car will start from rest at the

top of the hill and therefore its kinetic energy is zero. However, its potential energy will be a maximum. From energy conservation, we know that

𝐾! + 𝑈! = 𝐾! + 𝑈! (2)

where 𝐾 is the kinetic energy and 𝑈 is the potential energy. The kinetic energy is given by

𝐾 =12𝑚𝑣

! (3)

where 𝑚 is the mass of the object and 𝑣 is its speed. In the case of the roller coaster car, the only potential energy is due to gravity. The gravitational potential energy is given by

𝑈 = 𝑚𝑔𝑦 (4)

where 𝑦 is the height of the object above the reference point. You are free to choose the reference point as any location and at this point the gravitational potential energy will be zero. Therefore, only changes in potential energy are relevant. Putting the equations for the kinetic energy and gravitational potential energy into the energy conservation equation allows us to solve for the speed of the car as a function of height. We will take the top of the hill as the point where the gravitational potential energy is zero. Then the speed is given by

𝑣 = 2𝑔ℎ (5)

where ℎ is the vertical distance of the car below the top of the hill which is always a positive number.

12

3

45

6 7 8

Car

Photogates

Track

Energy Conservation

63

• Set up the roller coaster track as shown in the picture above. • There should be four photogates attached to the top four pegs labeled 1-4. § Measure the vertical distance of the car below the initial starting height when it passes through

each of the photogates. • Setup the Xplorer GLX to measure the velocity in the photogates. Remember to set the width of

the flag to 4.7 mm. For instructions on setting the Xplorer GLX, refer to the appendix of the lab manual.

• You will need to use the table setting to record the data. As the roller coaster car goes through each photogate, it will record two velocities because the flag has two arms. You only need to record the first of these velocities.

• Use the roller coaster car without any additional mass. § Release the car from rest at the top of the hill on the left. Measure the velocity at each of the

photogates. These will be the odd numbered entries in the Xplorer GLX table. Record these values into the table on the worksheet.

• Repeat this measurement two more times. § Calculate the average of these three measurements. Record these values into the table on the

worksheet. • Add additional mass to the roller coaster car. You should use 50 g and 100 g extra masses. For

each mass, you should repeat the measurement 3 times. § Enter the data for each mass into a data table similar to the one that you used in the earlier

steps. • Change the locations of the photogates to measure the velocity at the bottom four locations

(labeled 5-8 in the picture on the previous page). Be careful when changing the photogates! The pegs with the photogates have a small bump on them that needs to line up with the slot in the hole. Repeat the same series of measurements of velocity and height 3 times for each mass. § Enter the data for each mass and height into the data tables that you created in the earlier steps.

In the end, you should have completed a total of 72 measurements for this part of the lab. This

data should be organized in 3 tables, one table for each mass.

II. Determine the kinetic and potential energy

In this part of the lab, you will be measuring the kinetic and gravitational potential energy for the roller coaster car. The track should be setup as in the picture below. The track goes down steeply at first and then through a loop. The last part of the track will rise slowly to the right. You should attach four photogates at the marked locations.

Energy Conservation

64

We can calculate the gravitational potential energy at each photogate by measuring its height relative to either the top or bottom of the board. To find the kinetic energy, we need to measure the speed of the car at each of the photogates.

• Set up the roller coaster track as shown in the picture above. • There should be four photogates attached at the four locations marked; the start, top of loop, flat

section after loop and end. § Measure the height of the car at all four photogates relative to the top of the board. Record

these values into the table on the worksheet. This will allow you to calculate the change in gravitational potential energy.

• Setup the Xplorer GLX to measure the velocity in the photogates. Remember to set the width of the flag to 4.7 mm. For instructions on setting the Xplorer GLX, refer to the appendix of the lab manual.


• Use the roller coaster car without any additional mass. § Release the cart from rest at the top of the hill on the left. Measure the velocity at each of the

photogates. These will be the odd numbered entries in the Xplorer GLX table. Record these values into the table on the worksheet.

• Repeat this measurement two more times. § Calculate the average of these three measurements. Record these values into the table on the

worksheet. • Add additional mass to the roller coaster car. You should use 50 g and 100 g extra masses. For

each mass, you should repeat the measurement 3 times. § Enter the data for each mass into the data table that you used in the earlier steps. In the end, you should have completed a total of 36 measurements for this part of the lab. This


Car

Photogates

Track

Catcher

Energy Conservation

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III. Energy loss due to friction Energy is conserved as long as there are no non-conservative forces acting on the system.

Friction is one example of a non-conservative force. The roller coaster is designed to minimize friction but it is still present as you may have seen in the previous part of the lab. As the roller coaster travels along the track some of its energy is lost due to the work done by friction, which we have ignored in parts I and II.

In this part of the lab, you will be determining the amount of work done by non-conservative forces. The picture below shows the setup for this part of the lab. The roller coaster car will start at the left hand side of the track and go down the hill and back up the right hand side. If there is no energy lost, it would return to exactly the same height as it started. However, some energy is lost so it is unable to return to the initial height. The general expression for energy conservation is given by

𝐾! + 𝑈! +𝑊!" = 𝐾! + 𝑈! (6)

where 𝐾 is the kinetic energy, 𝑈 is the potential energy and 𝑊!" is the work done by non-conservative forces. You saw in the earlier part of the lab, how to determine the kinetic and potential energy. In this part of the lab, we will be determining the work done by non-conservative forces at two different points. This is done by finding

𝑊!" = 𝐾! + 𝑈! − 𝐾! + 𝑈! = ∆𝐾 + ∆𝑈 (7)

• Set up the track as shown in the picture above. A photogate should be at the start, the bottom of the hill and at the highest point on the right hand side. § Measure the height of the car at all three photogates relative to the top of the board. Record

these values into the table on the worksheet. This will allow you to calculate the change in gravitational potential energy.

• Setup the Xplorer GLX to measure the velocity in the photogates. Remember to set the width of the flag to 4.7 mm. For instructions on setting the Xplorer GLX, refer to the appendix of the lab manual.


• Use the roller coaster car without any additional mass. § Release the cart from rest at the top of the hill on the left. Measure the velocity at the second

and third photogates. These will be the first and third velocities recorded on the Xplorer GLX. Record these values into the table on the worksheet.

Car

Photogates

Track

Catcher

Energy Conservation

66

• Repeat this measurement two more times. § Calculate the average of these three measurements. Then use these averages to calculate the

kinetic and potential energy at the three locations, start, bottom of hill and top of right hand hill. Record these values into the table on the worksheet.

§ Determine how much energy was lost going from the start to the bottom of the hill. Determine how much energy was lost going from the bottom of the hill to the top of the right hand hill. Record these values into the table on the worksheet.

• Add an additional 100 gram mass to the roller coaster car. Repeat the previous measurements 3 times. § Enter the data for each mass into the data table that you used in the earlier steps.

IV. Fastest path between two points

You may have heard the expression that the fastest way between two points is a straight line. In this part of the lab, we will explore whether this saying is really true. You will try to find the fastest path from one side of the roller coaster to the other. The picture below shows the setup for this part of the lab. You should set up one of the tracks to go in a straight line from the beginning to the end of the roller coaster. These points are marked by the letters A and B on the picture.

• Set up the track as shown in the picture above. You should move the photogate on the left as close to the beginning of the track as possible and the photogate on the right to the end of the track.

• Setup the Xplorer GLX to measure the time between the photogates. For instructions on setting the Xplorer GLX, refer to the appendix of the lab manual.

• For this part of the lab, we will use the cart without any additional mass. We have already seen that changing the mass does not change the time to go from point A to point B. § Release the cart from rest at the first photogate and measure the time it takes to reach the

second photogate. Record this value into the table on the worksheet. • Repeat this measurement five more times. § Calculate the average of these six measurements. Record this value into the table on the

worksheet. • Now, you should experiment with other track configurations to see if you can get the cart to go

from point A to point B faster. Try at least two other configurations. You can setup two tracks at the same time. This way you can easily see which configuration is faster. § For each configuration, you should measure the time for six trials and record the average time.

Be sure to sketch the different configurations that you used on the worksheet.

Car

Photogates

Track Catcher

AB

Name:_________________________

Date:__________________________

67

Energy Conservation Worksheet I. Velocity as a function of height

You should make a table like the one below for each mass. There should be a total of three tables for your data in this part of the lab.

Car total mass:

Height Velocity Average Velocity

You should make a plot of the velocity as a function of the square root of the height. This graph should be labeled GRAPH#1 and include three sets of data with different symbols (one for each value of the total mass).

Fit straight lines to the data for each value of the total mass in GRAPH#1. The slope will be proportional to the 𝑔. Report the values that you find for 𝑔 for each total mass.

Worksheet: Energy Conservation

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II. Determine the kinetic and potential energy You should make a table like the one below for each mass. There should be a total of 3 tables for

this part of the lab.

Car total mass:

Height Velocity Average Velocity Kinetic Energy Potential Energy

III. Energy loss due to friction You should make a table like the one below for each mass. There should be a total of 2 tables for

this part of the lab. Car total mass:

Height Velocity Average Velocity Kinetic Energy Potential Energy

Wnc start to bottom Wnc bottom to end


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IV. Fastest path between two points You should fill out the following tables for each configuration of the track. Also include a sketch

of the layout of the track for each configuration. There should be a total of three tables for your data in this part of the lab.

Configuration: Straight

Time Average Time Drawing

Configuration: 1


Configuration: 2



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Questions to address in your lab report 1. Looking at your results for part I, do you obtain the same value for 𝑔 for the different masses?

Explain why the car goes the same speed regardless of its mass.

2. Explain why it would be hard to use Newton’s law to get the velocity in part I.

3. Looking at your results for part II, does the mass of the car affect kinetic energy or gravitational potential energy? Explain.

4. In part III, does the amount of energy lost depend on the mass of the car? In the introduction, we

said that energy is conserved for an isolated system. If that is the case, what is happening to the energy in this situation?

5. In part IV, you should have been able to find a quicker path from A to B then just a straight line.

Explain why your path is faster even though energy conservation tells you that the initial and final speeds must be the same.

6. Discuss any sources of errors in your measurements.

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Collisions

Previously, we have explored the motion of single objects but often multiple objects interact with each other. When two objects collide, it is difficult to calculate the force between them as a function of time. However, we can use the concept of linear momentum to understand their motion. The linear momentum of an object of mass 𝑚 moving with velocity 𝑣 is given by

𝑝 = 𝑚𝑣. (1)

Linear momentum is a vector quantity that points in the same direction as the velocity. If we consider the change in linear momentum of an object (assuming its mass is constant) with respect to time, we see that

∆𝑝∆𝑡 = 𝑚

∆𝑣∆𝑡 = 𝑚𝑎 = 𝐹net (2)

So, the change in linear momentum per unit time is given by the net force on the object. Therefore, if the net force is zero the linear momentum remains constant.

Now we consider the case of two objects colliding with each other. We will assume that during the collision the only force acting on the objects is a force between them. However, Newton’s third law states that the force on object 1 from object 2 must be equal in magnitude but opposite in direction to the force on object 2 from object 1. This means that the net force on the system must be zero. Therefore, in any collision with no outside forces the total momentum of the system must remain constant. This does not mean that the momentum of each object must remain constant but rather the total momentum of the system is constant. For example, the first object can slow down while the second object speeds up.

In contrast to the linear momentum of the system, the kinetic energy of the system does not need to remain constant. As an example, consider two cars having a head-on collision and stopping. Before the collision, they were moving and therefore both had kinetic energy but after the collision they are stopped. So, kinetic energy must have been lost in the collision. A collision in which the kinetic energy decreases is called an inelastic collision. A collision in which the kinetic energy remains constant is called an elastic collision. While the kinetic energy can change during a collision, the total energy is conserved. The collision just changes the kinetic energy into some other form of energy.

In this lab, we will be colliding carts on a track to observe the conservation of linear momentum. The carts can either undergo elastic or inelastic collisions. We will be measuring the linear momentum and energy before and after the collisions and therefore can determine whether a collision is elastic or inelastic.

Goals of the this lab: • Demonstrate linear momentum conservation for collisions. • Classify collisions as elastic or inelastic. • Measure energy loss in inelastic collisions.

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Carts The carts have different ends for elastic and inelastic collisions.

Mass Set This box contains a set of masses to put on the cart ranging from 1-200 g.

Rotary Motion Sensor The rotary motion sensors can measure the position or velocity of the carts as a function of time.

Xplorer GLX The Xplorer GLX will be used to measure the velocities of the two carts as a function of time.

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Lab Procedures

I. Elastic collision with stationary cart When two objects undergo an elastic collision, their total momentum and kinetic energy is

conserved. In this part of the lab, you will be creating an elastic collision between a moving cart and a stationary one. You will be varying the masses on the carts. The picture below shows the setup for this part of the lab. Both carts are attached to rotary motion sensors which will measure their

velocities as a function of time. The cart on the left has a plunger that will push against the wall to give it an initial velocity while the cart in the center will be at rest. After the collision the second cart will be moving and the velocity of the first cart will change. We will call the initially moving cart, cart 1 and it will have a mass of 𝑚!. The second cart of mass 𝑚! is initially at rest and therefore the total momentum of the system before the collision is

𝑝! = 𝑚!𝑣!,! +𝑚!𝑣!,! = 𝑚!𝑣!,! . (3)

After the collision, the total momentum of the system is given by

𝑝! = 𝑚!𝑣!,! +𝑚!𝑣!,! . (4)

If momentum is conserved during the collision, the initial and final momenta should be equal. We can also calculate the kinetic energy of the system before and after the collision. Since only

the first cart is moving before the collision, the initial kinetic energy is given by

KE! =12𝑚!𝑣!,!! (5)

After the collision, both carts may be moving and the final kinetic energy is given by

KE! =12𝑚!𝑣!,!! +

12𝑚!𝑣!,!! (6)

As this is an elastic collision, we expect that the final kinetic energy should be equal to the initial kinetic energy.

• Set up the track as shown in the picture above. The carts are attached to the rotary motion sensors

using string. There should be one cart attached to each sensor. • You should make sure that the track is level. This is critical to get accurate results because

otherwise gravity will tend to accelerate the carts. • For this part of the lab, we will be varying the masses of the carts and determining the momentum

and energy before and after the collisions. Initially, the carts should have no added mass. • Setup the carts so that their magnetic sides will collide.

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• Make sure that the end of the cart with the plunger is against the wall at the left end of the track. • The second cart should be near the middle of the track and stationary. • Setup the Xplorer GLX to measure the linear velocity of the carts using the rotary motion sensors.

You should be in the graph mode of operation on the Xplorer GLX. See the appendix of the lab manual for instructions on setting up the Xplorer GLX.

• Push the plunger all the way inward to give the first cart the highest initial velocity. • Start recording data with the Xplorer GLX. • Release the plunger so that the first cart begins moving and collides with the second stationary

cart. § Record the speed of the carts just before and just after the collision. This can be done using the

cursors in the graph mode on the Xplorer GLX. • Repeat this measurement two more times and record the speeds for each trial. § Calculate the average of these three measurements. Use this information to calculate the initial

and final momenta and kinetic energies. Record these values in the table on the worksheet. • Add additional mass to the carts. You should make two additional combinations of masses, 500

grams extra on the moving cart and 500 grams extra on the stationary cart. § Similarly to the equal mass measurements, you should perform three trials and record the

initial and final velocities of both carts. Using the average of these values for each combination of mass, fill out the remaining data in the table on the worksheet.


data should be organized in one table in the format shown in the worksheet.

II. Inelastic collision with stationary cart When two objects undergo an inelastic collision, their total momentum is still conserved but their

total kinetic energy changes. In this part of the lab, you will be creating an inelastic collision between a moving cart and a stationary one. You will be varying the masses on the carts to explore the fraction of energy that is lost. The setup is basically the same as the previous part of the lab except the second cart should be turned around so that the Velcro is facing the first cart. After the collision the two carts will stick together and will be moving with the same velocity.

The equations from the first part of the lab for the initial momentum and kinetic energy will remain the same because the second cart is not moving. However, the equation for the final momentum of the system can be simplified because the carts have the same final speed. Therefore, 𝑣!,! = 𝑣!,! = 𝑣! and the final momentum is given by

𝑝! = 𝑚! +𝑚! 𝑣! . (7)

We can make a similar simplification for the equation for the final kinetic energy to give,

KE! =12 𝑚! +𝑚! 𝑣!! (8)

This is an inelastic collision and therefore the final kinetic energy should be less than the initial kinetic energy. However, there are no outside forces acting on the system and therefore the momentum should remain constant.

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75

• Setup the carts so that their Velcro sides will collide. • Make sure that the end of the cart with the plunger is against the wall at the end of the track. • The second cart should be stationary near the middle of the track. • Setup the Xplorer GLX to measure the linear velocity of the carts using the rotary motion sensors.


• Push the plunger all the way inward to give the first cart the highest initial velocity. • Start recording data with the Xplorer GLX. • Release the plunger so that the first cart begins moving and collides with the second stationary

cart. § Record the speed of the carts just before and just after the collision.

• Repeat this measurement two more times and record the speeds for each trial. § Calculate the average of these three measurements. Use this information to calculate the initial

and final momenta and kinetic energies. Record these values in the table on the worksheet. • Add additional mass to the carts. You should make two additional combinations of masses, 500

grams extra on the moving cart and 500 grams extra on the stationary cart. § Similarly to the equal mass measurements, you should perform three trials and record the

initial and final velocities of both carts. Using the average of these values for each combination of mass, fill out the remaining data in the table on the worksheet.



III. Explosion • Setup the carts so that the plunger of one cart is against the second cart. • The two carts should be near the middle of the track. • Setup the Xplorer GLX to measure the linear velocity of the carts using the rotary motion sensors.


• Push the plunger all the way inward to give the carts the highest final velocity. • Start recording data with the Xplorer GLX. • Release the plunger so that the carts fly apart from each other. § Record the speed of the carts just after they fly apart.

• Repeat this measurement two more times and record the speeds for each trial. § Calculate the average of these three measurements. Use this information to calculate the final

momenta for each cart. Record these values in the table on the worksheet. • Add additional mass to the carts. You should make two additional combinations of masses, 500

grams extra on the moving cart and 500 grams extra on the stationary cart. § Similarly to the equal mass measurements, you should perform three trials and record the final

velocities of both carts. Using the average of these values for each combination of mass, fill out the remaining data in the table on the worksheet.



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76

IV. Collision with two moving carts

In this part of the lab, we are going to explore what happens with both carts are initially moving. We will explore what happens when there is a head-on collision and also when one cart hits the other from behind. This will be done for both elastic and inelastic collisions. We can use the same equations from the earlier parts of the lab but now we must remember that both carts will have an initial velocity. This means that the initial momentum is given by

𝑝! = 𝑚!𝑣!,! +𝑚!𝑣!,! (9)

and the initial kinetic energy is given by

KE! =12𝑚!𝑣!,!! +

12𝑚!𝑣!,!! (10)

• Turn the carts so that their magnetic sides are facing each other to have an elastic collision. • Setup the Xplorer GLX to measure the linear velocity of the carts using the rotary motion sensors.


• Start recording data with the Xplorer GLX. • Gently push the carts towards each other. § Record the speed of the carts just before and just after the collision. Use this information to

calculate the initial and final momenta for each cart as will as the kinetic energy of the system. Record these values in the table on the worksheet.

• Add additional mass to the carts. You should make two additional combinations of masses, 500 grams extra on the moving cart and 500 grams extra on the stationary cart. § Similarly to the equal mass measurements, you should record the initial and final velocities of

both carts. Using the average of these values for each combination of mass, fill out the remaining data in the table on the worksheet.

• Repeat the measurements for all three mass combinations but now have the carts initially moving in the same direction. So, if they are moving to the right, the left hand cart must be moving faster so that it will hit the right hand cart from behind. § Record the speed of the carts just before and just after the collision. Use this information to

calculate the initial and final momenta for each cart as will as the kinetic energy of the system. Record these values in the table on the worksheet.

§ Turn the carts around and repeat all six measurements with the carts having an inelastic collision.



Name:_________________________

Date:__________________________

77

Collision Worksheet I. Elastic collision with stationary cart

You should create a table with the given columns for each combination of masses on the carts.

m1 m2 v1,initial v1,final v2,final p1,initial p1,final p2,final Δp KEinitial KEfinal ΔKE

II. Inelastic collision with stationary cart You should create a table with the given columns for each combination of masses on the carts.

m1 m2 v1,initial v1,final v2,final p1,initial p1,final p2,final Δp KEinitial KEfinal ΔKE

III. Explosion You should create a table with the given columns for each combination of masses on the carts. m1 m2 v1,final v2,final p1,final p2,final Δp

IV. Collision with two moving carts You should create a table with the given columns for each combination of masses on the carts.

m1 m2 v1,initial v2,initial v1,final v2,final p1,initial p2,initial p1,final p2,final Δp KEinitial KEfinal ΔKE

Worksheet: Collision

78

Questions to address in your lab report 1. Consider each part of the lab separately. To within the uncertainty of your measurements was the

momentum conserved? Explain.

2. Looking at your results for part I, was the energy conserved? What about for the elastic collisions in part IV? Explain.

3. Describe what happens when a cart has an elastic collision with an identical stationary cart. What

happens when a light cart has an elastic collision with a stationary heavier cart? 4. Where does the energy go in part II and part IV for the inelastic collisions?

5. Where does the energy come from in part III to make the carts move? 6. Discuss any sources of errors in your measurements.

79

Circular Motion

When an object is accelerating its velocity must be changing. However, this does not mean that its speed is changing. If we consider an object travelling around in a circle at uniform speed, then it must always be accelerating even though its speed is constant. This is because the direction of its velocity is constantly changing. Newton’s second law relates the force on an object to its acceleration. So, this means that there must be a net force that causes an object to go in a circle. If there is no net force, the object will travel in a straight path.

Consider the case of a ball at the end of a string that is being twirled in a horizontal circle on a table at uniform speed. In order for the ball to travel in a circle, there must be a constant acceleration pointing towards the center of the circle. This is known as the centripetal acceleration and it always points radially inward. Newton’s second law states that the net force must be equal to the mass of the object times its acceleration. So, in this case

𝐹net = 𝑚𝑎! =𝑚𝑣!

𝑟 (1)

where 𝑎! is the centripetal acceleration, 𝑣 is the speed of the ball and 𝑟 is the radius of the circle. We see that the faster the ball is moving the larger the centripetal acceleration. Similarly, the smaller the circle the larger the acceleration must be because the direction must be changing more quickly. In the case of the ball, the net force is provided by the tension in the rope. If the rope was suddenly cut, the ball would fly off in an initially straight path.

In the first part of the lab, you will be measuring the force required to keep different masses rotating at constant speeds. In the second part of the lab, you will be examining the motion of a roller coaster when it is upside down. In this case, the roller coaster must be moving fast enough that the centripetal acceleration is larger than the acceleration due to gravity. You will find the minimum height that the roller coaster must start from in order to complete a loop without falling off the track.

Goals of the this lab: • Understand the concept of centripetal acceleration. • Measure the force required for uniform circular motion. • Determine the minimum height for a roller coaster to complete a loop.

Megan

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Megan

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Rectangle

Megan

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Lab equipment: Rotating arm The rotating arm is spun by hand. There is a rotary motion sensor on the bottom and two masses on it.

Force sensor This force sensor can measure forces up to 50 N. It will be used to measure the force on the rotating mass.

Rotary motion sensor This rotary motion sensor measures the angular velocity of the rotating arm.

Roller Coaster The car will travel along the roller coaster track.

Car This car can travel on the roller coaster path. The flag is 4.7 mm wide and 10 mm between the two arms.

Photogates The photogates have an infrared sensor in them that will be used to measure the velocity of the car with the Xplorer GLX.

Xplorer GLX The Xplorer GLX has the ability to measure times, velocities and forces from up to 8 sensors at once.

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Lab Procedures

I. Force versus rotation speed As an object is spun faster in a circle, its acceleration must increase. In order to increase the

acceleration, the force acting on the object must increase. The picture below shows the setup for this part of the lab. There is a force sensor attached by a string to a mass sitting on a rotating arm. As the arm rotates, the string will ensure that the mass moves in a circle of constant radius. You are going to

be measuring the tension in the string as a function of the rotation speed. There are a total of three forces acting on the mass; gravity, the normal force and the tension from the string. The first two are both in the vertical direction and will cancel each other. This leaves only the tension which always points along the string, towards the center of the circle. Newton’s second law gives,

𝐹net = 𝑇 = 𝑚𝑎! =𝑚𝑣!

𝑟 (2)

where 𝑇 is the tension in the string, 𝑚 is the mass, 𝑣 is the magnitude of the linear velocity and 𝑟 is the radius. You can measure the mass of the object and its radius. The force sensor measures the tension. This leaves only the magnitude of the linear velocity to measure. The rotating arm is connected to a rotary motion sensor which can measure the angular velocity of the arm. The angular velocity, 𝜔 of an object is related to its linear velocity by

𝜔 =𝑣𝑟 (3)

Putting equation (3) into the expression for the tension (equation (2)) gives

𝑇 = 𝑚𝑟𝜔! (4)

Force Sensor

Rotating Arm Mass

Counterweight

Rotary MotionSensor

String

Megan

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Rectangle

Megan

Rectangle

Megan

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• Set up the rotating arm as shown in the picture above. • Setup the Xplorer GLX to measure the force sensor and the rotary motion sensor. For

instructions on setting the Xplorer GLX, refer to the appendix of the lab manual. • You will need to use the table setting to record the data. One column of the table should measure

the angular velocity and the second column should measure the force. • Change the data properties to display 4 digits for the force and angular velocity. • Pick a mass of around 200 grams and choose a position near the middle of the arm. You can

adjust the position of the mass by moving the stand with the force sensor up or down. § Measure the mass of the object and its distance from the center. Don’t forget to measure the

mass of the black slider and the additional mass. Record these values into the table on the worksheet.

• Make sure that the rotating arm is well balanced by moving the counter weight on the other side. When it is properly balanced, the arm should not turn by itself when tilted.

• Press the zero button on the force sensor. • Start turning the arm until it is rotating at its maximum speed which should be around 𝜔 =

10 rad/s. § Start the Xplorer GLX and record data while the arm slows to a stop. § After the arm has stopped rotating, look back at the data in the table. You should record the

force and angular velocity at equal intervals of angular velocity in the table on the worksheet. You should have at least 10 different values of the angular velocity.

II. Force versus mass

In this part of the lab, you will be varying the mass and recording the tension for a fixed radius. You will need to take all of the measurements at the same angular velocity because as you saw in the first section, the angular velocity also changes the force. The setup is exactly the same as the previous part.

• Pick a convenient angular velocity to use for the measurement and choose a position near the

middle of the arm for the mass. § Measure the distance from the center. Record this value and the angular velocity that you will

use into the table on the worksheet. • You are going to use five different masses in this part of the lab. When you put each mass on,

you need to make sure that the rotating arm is well balanced by moving the counter weight on the other side. When it is properly balanced, the arm should not turn by itself when tilted.

• Press the zero button on the force sensor. • Start turning the arm until it is rotating faster than the angular velocity that you are using for your

measurements. § Start the Xplorer GLX and record data while the arm slows to a stop. § After the arm has stopped rotating, look back at the data in the table. You should find the

entry with the angular velocity that you are using. Then record the force at this angular velocity in the table on the worksheet.

§ Repeat this measurement with the other masses until you have measured 5 different masses.

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III. Force versus radius In this part of the lab, you will be varying the distance of the mass and recording the tension.

You will need to take all of the measurements with the same mass and at the same angular velocity because as you saw in the previous two parts, the angular velocity and mass also changes the force. The setup is exactly the same as the previous parts.

• Pick a convenient angular velocity to use for the measurement and choose a mass around 200

grams. § Measure the mass. Record this value and the angular velocity that you will use into the table

on the worksheet. • You are going to use five different distances in this part of the lab. You adjust the distance by

moving the stand with the force sensor up or down. After doing this, you need to make sure that the rotating arm is well balanced by moving the counter weight on the other side. When it is properly balanced, the arm should not turn by itself when tilted.

• Press the zero button on the force sensor. • Start turning the arm until it is rotating faster than the angular velocity that you are using for your

measurements. § Start the Xplorer GLX and record data while the arm slows to a stop. § After the arm has stopped rotating, look back at the data in the table. You should find the

entry with the angular velocity that you are using. Then record the force at this angular velocity in the table on the worksheet.

§ Repeat this measurement with the other distances until you have measured 5 different distances. These distances should span the range from the central axis to the far end of the rotating arm.

IV. Minimum height for roller coaster loop In an earlier lab, we saw that the roller coaster car can make it around the loop without falling off

the track. Now, we are going to figure out how high the car must start from in order to stay on the track all the way around the loop. The setup for this portion of the lab is shown in the picture below.

Car

TrackPhotogate

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84

If the roller coaster car stopped at the top of the loop, gravity would be acting downward and it would fall. However, if the car is moving as it goes around the loop it must have a centripetal acceleration. When the car is moving quickly enough this centripetal acceleration is greater than the acceleration due to gravity. In this case, the normal force from the track must also push the car and it stays on the track. The minimum speed the roller coaster can have is when the normal force from the track goes to 0. In this case,

𝐹net = 𝑚𝑔 = 𝑚𝑎! =𝑚𝑣!

𝑟 (5)

where 𝑣 is the minimum speed of the roller coaster at the top of the loop, 𝑚 is its mass and 𝑟 is the radius of the loop. Solving this equation for the minimum speed at the top of the loop gives,

𝑣 = 𝑔𝑟 (6)

Now that we know the minimum speed at the top of the loop, we can use energy conservation to find the height of the roller coaster when its speed is 0. This will be a greater height than the top of the loop. The height of the roller coaster when its speed is 0 is the minimum height that it must start from in order to complete the loop.

• Set up the track as shown in the picture above. A photogate should be at the top of the loop to

measure the speed of the car at that point. • Setup the Xplorer GLX to measure the velocity in the photogate. Remember to set the width of

the flag to 4.7 mm. For instructions on setting the Xplorer GLX, refer to the appendix of the lab manual.

• You can use the meter setting to watch the velocity of the car. • Use the roller coaster car without any additional mass. § Experiment with releasing the car from different locations. Determine the minimum height

that the car must start from to complete the loop. Measure the height of this location relative to the bottom of the track. Record this height in the worksheet.

§ Measure the radius of the circle that the car travels in around the loop. This is not the radius of the loop because the car sits on top of the track. You should measure to the center of the car.

§ Release the car and measure the speed at the top of the loop. This is the minimum speed needed to complete the loop.

• Repeat this measurement two more times. § Calculate the average of these three measurements and record the result in the worksheet. § Add an additional 100 gram mass to the roller coaster car. Repeat the previous measurements

of the minimum starting height and speed at the top of the loop.

Name:_________________________

Date:__________________________

85

Circular Motion Worksheet I. Force versus rotation speed

You should fill out the table below (don’t forget the units).

Mass: Radius:

Angular Velocity Force

You should make a plot of the force as a function of the angular velocity squared. This graph should be labeled GRAPH#1.

Fit a straight line to the data in GRAPH#1. The slope should be equal to 𝑚𝑟. Measured value of the slope. ___________________. Calculated value of the slope from measurement of mass and radius. ____________________

II. Force versus mass


Angular Velocity: Radius:

Mass Force

Worksheet: Circular Motion

86

You should make a plot of the force as a function of the mass. This graph should be labeled GRAPH#2.

Fit a straight line to the data in GRAPH#2. The slope should be equal to 𝑟𝜔!. Measured value of the slope. ___________________ Calculated value of the slope from measurement of radius and angular velocity. ______________

III. Force versus radius


Mass: Angular Velocity:

Radius Force

You should make a plot of the force as a function of the radius. This graph should be labeled

GRAPH#3. Fit a straight line to the data in GRAPH#3. The slope should be equal to 𝑚𝜔!. Measured value of the slope. ___________________ Calculated value from the measurement of the mass and angular velocity. ___________________

IV. Minimum height for roller coaster loop

Measured minimum height for roller coaster car. __________________

Measured minimum speed at top of loop. _________________

What changes when the mass of the car is increased?

Worksheet: Circular Motion

87

Questions 1. Draw the free body diagram for the mass on the rotating arm.

2. Compare your measured slopes for GRAPHS#1-3 with the expected values. Are they in good agreement? Explain any discrepancies that you observe.

3. Compare your measured value for the minimum speed at the top of the roller coaster loop with

the calculated value? Explain any discrepancies that you observe. 4. Using energy conservation, find the minimum height of the roller coaster in order to complete the

loop in terms of the radius of the loop, 𝑟. How does this calculated value compare to your measured value?


88

Pendula and Springs

An object undergoes simple harmonic motion if the restoring force on it is proportional to its displacement. Two examples of simple harmonic motion are a mass on a spring and a pendulum if its angular displacement from vertical is small.

In the case of a mass on a spring, the restoring force is given by

𝐹 = −𝑘𝑥 (1)

where 𝑘 is the spring constant and 𝑥 is the displacement from equilibrium. This is known as Hooke’s law. Newton’s second law states that the net force must equal the mass times the acceleration. So, we can write

𝑚𝑎 = −𝑘𝑥 (2)

The acceleration is defined as the change in velocity with respect to time. Similarly, the velocity is determined by how quickly the position changes with respect to time. Therefore, we can solve equation (2) to get the position of the mass as a function of time. The solution is given by

𝑥 𝑡 = 𝐴 cos 𝜔𝑡 (3)

where 𝐴 is the amplitude of the motion and 𝜔 = !!

is the angular frequency. From this equation we

see that the mass will oscillate back and forth as a function of time. The amplitude of the motion gives the maximum displacement away from equilibrium. The period 𝑇 of oscillation is given by the time required to make one complete cycle of the motion. This is the inverse of the frequency 𝑓 giving

𝑇 =1𝑓 =

2𝜋𝜔 (4)

For a pendulum, the restoring force is due to the component of gravity along the tangent to the circle which wants to bring the pendulum back to the vertical position. As the angle of the pendulum increases away from vertical, the restoring force increases. Therefore, the equation that describes the angle that a pendulum makes with the vertical axis is given by

Θ 𝑡 = 𝐴 cos 𝜔𝑡 (5)

with the angular frequency given by 𝜔 = !! where 𝐿 is the length of the pendulum.

In the first part of the lab, you will be examining the motion of pendula. You will determine which physical parameters such as mass, radius, and length affect the period of the pendulum. In the second half of the lab, you will measure the spring constant of unknown springs using their extension as a function of the applied force and their frequency of oscillation for a known mass.

Goals of the this lab: • Understand the concept of simple harmonic motion. • Measure the period of a pendulum as a function of its length. • Determine the spring constant for a spring using two different methods.

Pendula and Springs

89

Lab equipment: Bobs for pendulum These bobs all have the same size but different masses.

Balls for pendulum These balls are of various diameters for using on the pendulum.

Stopwatch The stopwatch is used to measure the period of the pendulum.

Photogates The photogates have an infrared sensor in them that will be used to measure the period of the pendulum with the Xplorer GLX.

Springs Springs of various spring constants to measure.

Force sensor This force sensor can measure forces up to 50 N. It will be used to measure the force on the spring as a function of time.

Xplorer GLX The Xplorer GLX has the ability to measure times, velocities and forces from up to 8 sensors at once.

Pendula and Springs

90

Lab Procedures

I. Period versus mass In this first part of the lab, you are going to measure the period of pendula of different mass. There are four different pendulum bobs that have an identical shape but different mass. You are to measure the period of each bob. The picture below shows the setup for this part of the lab. The pendulum bob hangs from two strings. You will pull back the bob and release it to start the pendulum swinging. Then using a stopwatch you will measure the period of the pendulum.

• Set up the pendulum as shown in the picture above. • You will be measuring the period of the pendulum using a stopwatch. • Start the pendulum swinging. You want to release the pendulum from the same angular

displacement in each measurement. You should also use the same length of string for all of the measurements. § Find the period of the pendulum by measuring the time it takes to undergo 10 complete cycles

of its motion and dividing by 10. Record this value and the mass of the bob into the table on the worksheet.

§ Repeat the measurement two more times with the same bob. • Perform the same measurements with the other three bobs. At the end of this section, you should have measured the period of four different bobs, three times each for a total of 12 measurements.

Knobs

Pendulum Bob

String

Pendula and Springs

91

II. Period versus size In this part of the lab, you will be varying the size of the pendulum bob and observing its affect

on the period. The setup is exactly the same as the previous part but now you will use wooden balls of varying diameters.

• Set up the pendulum as in the first section of the lab. • You will be measuring the period of the pendulum using a stopwatch. • Start the pendulum swinging. You want to release the pendulum from the same angular

displacement in each measurement. You should use the same length of string for all of the measurements. § Find the period of the pendulum by measuring the time it takes to undergo 10 complete cycles

of its motion and dividing by 10. Record this value and the diameter of the ball into the table on the worksheet.

§ Repeat the measurement two more times with the same ball. • Perform the same measurements with the other three balls. At the end of this section, you should have measured the period of four different balls, three times each for a total of 12 measurements.

III. Period versus length In the first two parts of the lab, you explored the effect of mass and diameter on the period of a

pendulum. You should have found that the period was not constant for all of the measurements in the previous section when the size of the ball was changing. In this section of the lab, you will measure the relationship between the period of a pendulum and its length. In order to do this, you need a more accurate measurement of the period of the pendulum. So, we will be using photogates to measure the

Knobs

Pendulum BobString

Photogate

Xplorer GLX

Pendula and Springs

92

period. The setup for this part of the lab is shown in the picture below. The pendulum should swing through the photogate and block the infrared sensor.

• Set up the pendulum as shown in the picture above. • You will be measuring the period of the pendulum using the photogate. • Make sure that the Xplorer GLX is setup to measure the period of a pendulum using the

photogate. For instructions on configuring the Xplorer GLX see the appendix of the lab manual. • Put the Xplorer GLX in the digital display mode. • Start the pendulum swinging. Make sure that the pendulum bob will pass through the photogate

without hitting it. § Make 3 measurements of the period of the pendulum for each length. You should try to start

the pendulum from the same angular displacement in each trial. Also, let the pendulum swing several times before recording the period. Record these values into the table on the worksheet.

• Adjust the length of the pendulum by using the knobs to change the length of the string. § Repeat the measurement for a total of 7 different lengths of the pendulum.

IV. Measure spring constant using Hooke’s law

Springs have a restoring force that is proportional to their extension. In this part of the lab, you are going to measure this proportionality by applying a known force to different springs and measuring their extension. The setup for this portion of the lab is shown in the picture below.

Force sensor

Spring

Mass

Xplorer GLX

Pendula and Springs

93

• Set up the spring as shown in the picture above. • You will be measuring the spring constant for 3 different springs. • Hang the spring from the force sensor without any mass on the end. § Measure the extension of the spring and denote this as its equilibrium length.

• Add a mass to the end of the spring. § Measure the change in extension of the spring and the mass on the end. Record these values

into the table in the worksheet. • Repeat this measurement 4 more times with different masses on the end of the spring. § Similarly to the first measurement, record the mass on the spring and its extension into the

table in the worksheet. • Repeat the measurements of the extension versus mass for two other springs.

At the end of this section, you should have measured the extension of three different springs for five different masses on each for a total of 15 measurements.

V. Measure spring constant using the period

If you displace the mass on the end of the spring from its equilibrium position and release it, it will undergo simple harmonic motion. The position of the mass as a function of time is given by

𝑦 𝑡 = 𝐴 sin 𝜔𝑡 + 𝜙 (6)

where 𝐴 is the amplitude, 𝜔 is the angular frequency and 𝜙 is the initial phase. The amplitude of the motion depends on how far from equilibrium the spring is displaced. The initial phase is related to the initial displacement of the mass. The angular frequency is related to the mass and the spring constant by

𝜔 =𝑘𝑚 (7)

where 𝑘 is the spring constant and 𝑚 is the mass on the end of the spring. • Set up the spring and mass system in the same way as the previous section. • You will be measuring the spring constant for the same 3 springs as in the previous part. • Set up the Xplorer GLX to measure the force sensor. For instructions on configuring the Xplorer

GLX see the appendix of the lab manual. • Put the Xplorer GLX in the graph mode and plot the force as a function of time. • Press the zero button to reset the force sensor. • Hang a mass of about 50 grams from the end of the spring. • Gently pull the mass downward and release it. This should start the mass moving vertically in

simple harmonic motion. • Begin recording the force as a function of time § Measure at least 10 oscillations of the mass. From the graph on the Xplorer GLX, determine

the period of the oscillation. § Use the period of oscillation that you find to determine the spring constant of the spring.

• Repeat this measurement for the other 2 springs (press the zero button before each spring).

Name:_________________________

Date:__________________________

94

Pendula and Springs Worksheet I. Period versus mass


Mass Measured Period Average Period

Measured value for the length of the pendulum. _____________________

II. Period versus diameter


Diameter Measured Period Average Period

Measured value for the length of the pendulum. _____________________

III. Period versus length


Length Measured Period Average Period

You should make a plot of the period as a function of the square root of the length. This graph

should be labeled GRAPH#1.

Worksheet: Pendula and Springs

95

Fit a straight line to the data in GRAPH#1. The slope should be equal to !!!

.

Measured value of the slope. ___________________ Calculated value for the gravitational constant. ___________________

IV. Measure spring constant using Hooke’s law


Equilibrium Length

Spring 1 Spring 2 Spring 3

Extension

Mass Spring 1 Spring 2 Spring 3

You should make a plot of the extension as a function of the mass for each of the three springs.

This graph should be labeled GRAPH#2.

Fit straight lines to the data in GRAPH#2. The slopes should be equal to !!

. Measured values of the slope. ____________ ____________ ____________ Calculated values for the spring constants. ____________ ____________ ____________

V. Measure spring constant using the period You should fill out the table below (don’t forget the units).

Spring 1 Spring 2 Spring 3 Period

Spring Constant

Worksheet: Pendula and Springs

96

Questions to address in your lab report 1. Draw the free body diagram for the pendulum.

2. Theoretically, how does the mass of the pendulum affect its period? Do your measurements for

the period agree with theory? Explain any discrepancies that you observe. 3. Theoretically, how does the diameter of the pendulum bob affect its period? Do your

measurements for the period agree with theory? You should think about how the diameter of the ball affects the length of the pendulum. Explain any discrepancies that you observe.

4. Draw the free body diagram for the spring and mass system. 5. Compare the two methods that you used to measure the spring constant of the springs. Which

one gives more accurate results? Discuss the errors associated with each measurement technique. 6. Why does the period of the pendulum not depend on the mass but the period of a mass on a spring

does depend on the mass?

7. Explain why it is better to measure many periods of the oscillations than measuring just one period.


97

Standing Waves

A standing wave forms from the interference of many traveling waves. In the case of a string clamped at the ends, there are only certain wavelengths that are allowed for standing waves. This is because the ends are fixed and therefore these locations must have no movement. Locations where the string does not move are called nodes. Between each node, there is an anti-node where the string has its maximum displacement. In order to have nodes at the end of the string, its length 𝐿 must be an integer number of half-wavelengths,

𝐿 = 𝑛𝜆2 (1)

This means that the allowed wavelengths are given by

𝜆! =2𝐿𝑛 (2)

where 𝑛 denotes the mode number. The frequency 𝑓 at which a string vibrates is given by

𝑓 =𝑣𝜆 (3)

where 𝑣 is the speed of the wave and 𝜆 is the wavelength of the wave. The speed of a wave on a string is related to the forces on the string and its inertia. The speed is given by

𝑣 =𝐹𝜇 (4)

where 𝐹 is the tension in the string and 𝜇 is its mass density (mass per unit length). Putting equations (2)-(4) together gives an expression for the frequencies that standing waves on

a string can have,

𝑓 =𝑛2𝐿

𝐹𝜇 (5)

In this lab, you will be examining the frequency of standing waves on a string as a function of the mode number, tension and length. From all three of these measurements you can determine the mass density of the string.

Goals of the this lab: • Understand the concept of standing waves. • Measure the frequency of standing waves as a function of mode number, tension and length. • Determine the mass density of a string.

Standing Waves

98

Lab equipment: String The string is connected to the string vibrator to create standing waves.

String vibrator The string vibrator oscillates the string using the voltage from the sine wave generator.

Sine wave generator The sine wave generator creates a sine wave with an adjustable frequency and amplitude.

Masses The masses are hung on the end of the string to change its tension.

Standing Waves

99

Lab Procedures

I. Frequency versus number of nodes If you drive a string with the correct frequency, you can create a standing wave. This is

characterized by a series of nodes where the string does not move and anti-nodes where the string has a maximum displacement. In this part of the lab, you will be changing the driving frequency on the string and observing the number of nodes. The setup for all of the sections of the lab is shown in the picture below. In this picture there is one node and two anti-nodes. Therefore, this oscillation corresponds to the second harmonic which is called the 𝑛 = 2 mode.

• The output of the sine wave generator should be connected to the input of the string vibrator. § Cut about 2 meters of string and measure its mass. From this measurement, determine its

linear mass density. Record this value in the worksheet. • Connect the string to the string vibrator and then run it over the pulley. • Hang a mass from the end of the string to create tension in the string. You should use a mass of

about 500 g. • The distance between the string vibrator and the pulley should be about 1.5 m. § Measure the length of the string and the tension in the string. Record these values into the

table on the worksheet. • Turn on the sine wave generator and increase its amplitude to the maximum value. • Adjust the frequency until you obtain an oscillation with one anti-node and zero nodes (the

fundamental mode, 𝑛 = 1). § Record the frequency from the output of the sine wave generator in the table on the worksheet.

• Adjust the frequency on the sine wave generator to obtain an oscillation with two anti-nodes and one node (the second harmonic mode, 𝑛 = 2). The mode should look like the picture above. § Record the frequency from the output of the sine wave generator in the table on the worksheet.

• Continue to increase the frequency on the sine wave generator until you have measured the frequency of the different modes up to 𝑛 = 11.

II. Frequency versus tension In this part of the lab, you will be keeping the number of nodes on the string and its length fixed

while varying the tension. The goal is to measure what frequency is needed to produce a standing wave. The setup is exactly the same as the previous part but now you will be changing the mass on

Mass

String

Sine Wave Generator

Pulley

String Vibrator

Node

Anti-Nodes

Standing Waves

100

the end of the string.

• Set up the vibrating string as in the first section of the lab. • If you are using the same string, you do not need to repeat the mass density measurement. If you

have changed strings, repeat the mass density measurement. • Hang a mass of 100 g from the end of the string. • The distance between the string vibrator and pulley should be fixed and about 1.5 m. § Record the length of the string in the table on the worksheet.

• Turn on the sine wave generator and increase its amplitude to the maximum value. • Adjust the frequency until you obtain an oscillation with 3 nodes (the 𝑛 = 4 mode). § Record the frequency from the output of the sine wave generator in the table on the worksheet.

• Repeat the measurement with different masses on the end of the string. You should measure the frequency of the 𝑛 = 4 mode for a total of 10 masses ranging from 100 g to 1000 g in 100 g increments.

III. Frequency versus length In this part of the lab, you will be keeping the number of nodes on the string and its tension fixed

while varying its length. The goal is to measure what frequency is needed to produce a standing wave. The setup is exactly the same as the previous parts but now you will be changing the length of the string.

• Set up the vibrating string as in the previous sections of the lab. • If you are using the same string, you do not need to repeat the mass density measurement. If you

have changed strings, repeat the mass density measurement. • Hang a mass of 500 g from the end of the string. • The distance between the string vibrator and pulley will be changed but start with 1.5 m. • Turn on the sine wave generator and increase its amplitude to the maximum value. • Adjust the frequency until you obtain an oscillation with 3 nodes (the 𝑛 = 4 mode). § Record the frequency from the output of the sine wave generator and the distance from the

string vibrator to the pulley in the table on the worksheet. • Repeat the measurement with different lengths of string. You can change the length by bringing

the string vibrator closer to the pulley. You should measure the frequency of the 𝑛 = 4 mode for a total of 5 different lengths, 1.5 m, 1.25 m, 1.0 m, 0.75 m and 0.5 m.

Name:_________________________

Date:__________________________

101

Standing Waves Worksheet I. Frequency versus number of nodes


String Mass Density:

Tension: Length:

Mode Frequency

1 2 3 4 5 6 7 8 9 10 11

You should make a plot of the frequency as a function of the mode number. This graph should

be labeled GRAPH#1.


!!

.

Measured value of the slope. ___________________ Calculated value for the string’s mass density. ___________________

Worksheet: Standing Waves

102

II. Frequency versus tension You should fill out the table below (don’t forget the units).


Mode: Length:

Mass (g) Tension Frequency

100 200 300 400 500 600 700 800 900 1000

You should make a plot of the frequency as a function of the square root of the tension. This

graph should be labeled GRAPH#2.


!!

.



103

III. Frequency versus length You should fill out the table below (don’t forget the units).


Tension: Mode:

Length Frequency

You should make a plot of the frequency as a function of one over the length. This graph should

be labeled GRAPH#3.

Fit a straight line to the data in GRAPH#3. The slope should be equal to !!

!!

.



104

Questions 1. Draw a free body diagram for the system to relate the tension in the string to the applied mass.

2. From GRAPH#1, does the slope of your graph agree with your measurement of the mass density?

Explain any discrepancies that you observe. 3. From GRAPH#2, does the slope of your graph agree with your measurement of the mass density?

Explain any discrepancies that you observe. Explain why the frequency increases as the tension on the string increases.

4. From GRAPH#3, does the slope of your graph agree with your measurement of the mass density?

Explain any discrepancies that you observe. Explain why the frequency increases as the length of the string is shortened.


105

Archimedes’ Principle

If you have ever held a helium filled balloon or tried to submerge a ball in water, you will have experienced Archimedes’ principle. It states that a fluid exerts an upward buoyant force on an object equal to the weight of the displaced fluid. In air this force is usually small because the density of air is small but in water the force can be quite large. Suppose you submerge an object of volume 𝑉 then the buoyant force 𝐵 acting on the object is given by

𝐵 = 𝜌fluid𝑉𝑔 (1)

where 𝜌fluid is the density of the fluid. If the density of the object is greater than the density of the fluid, the object will sink. However, if the density of the fluid is greater than the object, the object will float. This is the case of a helium balloon in air or a ball in water.

If an object is floating without accelerating, then the other forces on the object must exactly balance the buoyant force. For an object floating in a liquid, the only other force is the weight of the object and therefore,

𝑚object𝑔 = 𝐵 = 𝜌fluid𝑉sub𝑔 (2)

where 𝑚object is the mass of the object and 𝑉sub is the volume of the object that is submerged. The density of the object can be written as

𝜌object =𝑚object

𝑉 (3)

Combining equations (2) and (3) shows that for an object to be completely submerged and float its density must be the same as the fluid’s. If the object’s density is less than the fluid it will float at the top of the fluid with only a portion of the object submerged. This is the case for an ice cube in water because the density of ice is less than the density of water.

In this lab, you will be investigating the buoyant force and using Archimedes’ principle to measure the density of water and a series of assorted objects.

Goals of the this lab: • Understand the buoyant force on an object. • Measure the density of water. • Determine the density of unknown objects using Archimedes’ principle.


106

Lab equipment: Assorted Masses A collection of assorted masses whose densities you will determine.

Beaker A beaker with heights marked on the side to use in determining the volume of the objects placed in the water.

Graduated Cylinder Graduated cylinder used for floating in the beaker of water.

Washers Assorted washers to put in the bottom of the graduated cylinder in order to make it float upright in the beaker.

Golf Ball on Spring The golf ball is attached to a spring. The ball is placed in water to observe a change in the buoyant force.

Spring Scale Scale used for measuring the force on the assorted masses. Digital Scale A digital scale to measure the mass of objects and the beaker of water.


107

Lab Procedures

I. Golf Ball in Water Before beginning the experiments to measure the density of various objects, we will get a feel for the buoyant force. This is done by slowing lowering a golf ball into water as shown in the picture below. The golf ball is connected to a spring whose extension is proportional to the net force on the ball.

• Fill the bucket about 2/3 of the way full with water. • Hold the spring by the end so that the golf ball hangs downward and the spring is fully extended. § Slowly lower the golf ball into the water. Describe what happens on the worksheet.

Water

Bucket

Golf Ball

Spring


108

II. Density of Water Before we can measure the density of unknown objects, we will determine the density of water.

This is done by varying the mass of an object and watching the change in the amount of water that it displaces. The picture below shows the setup for this part of the lab. The graduated cylinder is floating in the beaker of water. Placing washers inside the cylinder changes its mass.

• Empty the bucket of water from the previous part of the lab. • Place the beaker inside the bucket. The bucket is only used to catch any water that spills out of

the beaker. • Fill the beaker about 2/3 of the way full of water. § Determine the mass of the empty cylinder and the mass of a washer.

• Place about 10 washers at the bottom of the cylinder. Make sure you count the exact number. • Put the cylinder into the beaker, so that it is floating and upright. If the cylinder is not upright,

add more washers. § Record the number of washers inside the cylinder and its volume that is submerged. § Add a washer to the cylinder and repeat the previous two measurements. § Continue to add washers until the cylinder sinks.

From equation (2), we know that the mass of the object must be equal to the density of the fluid times the volume submerged,

𝑚object = 𝜌fluid𝑉sub (4)

In this case, the mass of the object is the sum of the mass of the washers plus the mass of the cylinder 𝑚cyl. This can be written as

𝑚object = 𝑚cyl + 𝑛𝑚washer (5)

Water

Bucket

Beaker

Cylinder


109

where 𝑛 is the number of washers and 𝑚washer is the mass of one washer. The markings on the side of the cylinder give the volume inside it. The total volume that is submerged is given by the volume of the base 𝑉base plus the reading on the side of the cylinder at the water level, 𝑉side.

𝑉sub = 𝑉base + 𝑉side (6)

Therefore, if you plot the volume reading from the side of the cylinder as a function of the number of washers, you should be able to determine the density of the water.

𝑉side =𝑚washer

𝜌fluid𝑛 +

𝑚cyl

𝜌fluid− 𝑉base (7)

III. Buoyant Force Versus Volume Submerged

In this section of the lab, we are going to measure the buoyant force as a function of the volume submerged using a wooden block. You will be slowly lowering a wooden block into a container of water and monitoring the force on it. The setup for this part of the lab is shown in the picture below.

• Hang the wooden block from the spring scale. § Record the tension on the spring scale with the wooden block completely out of the water. § Adjust the height of the block by lowering the support stand. You should record the reading

on the scale as a function of the volume of the wooden block that is submerged. § You should record at least 5 different submerged volumes. Use this data to show that the buoyant force linearly increases with the volume of the object that

is submerged.

Water

Beaker

Wooden Block

Spring Scale


110

IV. Density of Objects In this last section of the lab, you will determine the specific density of an assortment of different masses. The specific density of an object 𝑠object is defined as the ratio of its density to the density of water. It is given by

𝑠object =𝜌object𝜌water

= 𝑤object

𝑤system − 𝑤water (8)

where 𝑤object is the weight of the object, 𝑤water is the weight of the water and 𝑤system is the weight of the combined system when the object is suspended in the water. In order to improve the accuracy of your measurements, we will use a digital scale to measure the mass of a beaker of water as different objects are lowered into it. The picture below shows the setup for this section of the lab.

• Place a beaker filled with water on the digital scale. The scale can read up to 1000 g, so you want

to make sure that there is not too much water in the beaker. § Measure the mass of water without the object in it. § Measure the mass of the object.

• Hang the object whose density you will determine, from the stand. Make sure that the object is completely submerged in the water but does not touch the bottom of the beaker. § Record the reading on the scale for the combined water and object. § Determine the specific density of the object. § Repeat the measurement for all of the different objects listed on the worksheet.

Water

Beaker

Object

Scale

Name:_________________________

Date:__________________________

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Archimedes’ Principle Worksheet

I. Golf Ball in Water Describe what happens when you lower the golf ball into the water. Is the golf ball more or less dense than water? Explain your reasoning.

II. Density of Water Make a table with two columns, the number of washers in the cylinder and the volume of the

cylinder that is submerged. Don’t forget the units.

Mass Cylinder: Mass washer: Volume base:

Washers in cylinder Volume submerged

You should make a plot of the volume submerged as a function of the number of washers in the

cylinder. This graph should be labeled GRAPH#1.

Fit a straight line to the data in GRAPH#1. The slope should be equal to !washer!fluid

. Measured value of the slope. _______________________ Calculated value for the density of the water. _____________________

Worksheet: Archimedes’ Principle

112

III. Buoyant Force Versus Volume Submerged You should fill out the table below (don’t forget the units).

Tension on spring scale Volume submerged

Make a plot of the buoyant force versus the volume submerged. Note that the reading on the

spring scale does not directly give you the buoyant force. You will need to calculate it using the free-body diagram from question 3. This graph should be labeled GRAPH#2.

IV. Density of Objects

You should fill out the tables below (don’t forget the units).

Name Description Specific Density

Wood #1 Small Diameter

Wood #2 Large Diameter

Slag Rock

This table is for the seven one-inch cubes. You should measure the specific density and calculate the density of the cubes from their mass and volume.

Name Description Specific Density Density

PVC Gray

Nylon White

Acrylic Clear

Copper Red-Brown

Brass Gold

Steel Black (painted)

Aluminum Silver

Worksheet: Archimedes’ Principle

113

Questions to address in your lab report 1. How does your measured value for the density of water compare to the expected value, which is

1 g/cm3? 2. Does the slope of your data in GRAPH#2 agree with the expected value that you get from

equation (1)? Explain any discrepancies that you find. 3. Draw a free body diagram for the wooden block in section III when it is partially submerged in

the water. 4. Draw the free body diagram for one of the objects in section IV when it is completely submerged

in the water. Use this to derive the expression for the specific density of an object that is shown in equation (8).

5. How does your measurement of specific density for the 1-inch cubes compare to the expected

value from their calculated density? 6. Discuss any sources of errors in your measurements.

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Ideal Gas Law and Heat Engine

The ideal gas law describes the relationship between the pressure, volume, temperature and number of moles of a gas. The law is described as ideal because it is assumed that there are no interactions between the different gas molecules. They move around freely in a container of volume, 𝑉 and only collide with other molecules or the walls. The collisions create a force on the walls of the container that leads to a pressure, 𝑃. The temperature 𝑇 of the molecules determines their average speed. If there are 𝑛 moles of molecules, then the ideal gas law states that

𝑃𝑉 = 𝑛𝑅𝑇 (1)

where 𝑅 is the gas constant, 8.31 J/(mol K). From this relationship, we see that if the temperature and number of moles of gas are held constant, then as the pressure increases the volume must decrease. This is known as Boyle’s law,

𝑃 ∝1𝑉 (2)

Similarly, if the pressure and number of moles of gas are held constant, we have Charles’ Law. This states that

𝑉 ∝ 𝑇 (3)

The last combination occurs if the volume and number of moles of gas are held constant. Then the Gay-Lussac Law states that

𝑃 ∝ 𝑇 (4)

When the volume of a gas is changed, there must be work done to create this change of volume. For example, pushing down on a syringe to reduce its volume requires some amount of mechanical work. A heat engine consists of several steps where the pressure and volume of a gas are changed in a cyclic manner. The work 𝑊 done during each step in the process is given by the area under the curve in a graph of the pressure versus volume. As the engine completes a cycle, its pressure and volume change such that it traces a closed path in a pressure versus volume graph. The net work done in the cycle is the area enclosed inside this graph.

In this lab, you will be using two different sets of devices to explore the ideal gas law and a simple engine. Using a fixed volume cylinder, you will change its temperature and measure the pressure and therefore verify the Gay-Lussac Law. Then, you will verify the other two laws associated with the ideal gas law. After completing these tasks, you will measure the work done by a simple engine that operates using these basic processes.

Goals of the this lab: • Explore the relation between volume and temperature at constant pressure, Charles’ Law. • Explore the relation between pressure and volume at constant temperature, Boyle’s Law. • Explore the relation between pressure and temperature at constant volume, Gay-Lussac Law. • Determine the work done by a heat engine.


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Lab equipment: Bunsen Burner The Bunsen burner is used to heat water. Begin heating a beaker of water as soon as lab starts.

Metal Beaker The water should be heated in the metal beaker.

Gloves and Goggles Gloves and goggles should always be worn when handling the hot water.

Gas Law Apparatus – Base Unit The gas law apparatus consists of a glass cylinder with a piston inside it. There are two pressure connections at the bottom which can be closed using the clamps inside.

Gas Law Apparatus – Air Chamber The closed vessel is a fixed volume with a tube connected to it. The tube can be connected to a pressure sensor or the gas law apparatus.


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Gas Law Syringe The syringe is used to change the volume of gas. There are two connections to the syringe, one is a thermocouple for measuring the temperature and the second one is a tube for measuring the pressure.

Thermocouple The thermocouple can measure the temperature of the water by being placed on the outside of the closed vessel.

Pressure and Temperature Sensor This sensor attaches to the Xplorer GLX and can measure the temperature and pressure of the syringe or gas law apparatus.

Xplorer GLX The Xplorer GLX has the ability to measure temperature and pressure with the proper sensor attachment.


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Lab Procedures

I. Gas Law Syringe The gas law syringe allows us to test some of the basic elements of the ideal gas law. The picture

below shows the setup for this part of the lab. There is a temperature and pressure sensor attached to the Xplorer GLX. The gas law syringe is attached to this sensor. There are two outputs to the syringe, one is connected to a thermocouple to measure the temperature inside and the second one is

connected to a tube to measure the pressure. The syringe is a variable volume of up to 60 mL which you can control with the plunger.

In this first part of the lab, the idea is to explore the ideal gas law with the syringe and describe your findings.

• Set up the gas law syringe and sensor as shown in the picture above. It is also possible to connect

the temperature and pressure sensor to the Explorer GLX with an extension cable to give more freedom for moving the syringe.

• Setup the Xplorer GLX to measure the temperature and pressure sensor. For instructions on setting the Xplorer GLX, refer to the appendix of the lab manual.

• Initially set the Xplorer GLX in the digits mode and measure both the temperature and pressure. § Hold the syringe tightly in your hands to warm it up. In the worksheet describe what happens. § Release the syringe and let it cool back down to room temperature. In the worksheet describe

what happens. • Set the Xplorer GLX to graph mode. You should be graphing the temperature as a function of

time. For instructions on setting the Xplorer GLX, refer to the appendix of the lab manual.

TemperatureProbe

PressurePort

TemperaturePressure Sensor

Xplorer GLX

Gas Law Syringe


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§ Repeat the warming and cooling of the syringe. In the worksheet sketch the graph of the temperature as a function of time.

• Disconnect the pressure probe and set the volume of the syringe to 40 mL. Then reconnect the pressure probe. This will ensure that you have atmospheric pressure when the volume is 40 mL.

• For the next parts of the lab, you will want to use both the digits mode and the graph mode on the Xplorer GLX. You will need to repeat your measurements several times to use each mode. § Compress the syringe. In the worksheet describe what happens and sketch the graph of the

temperature as a function of time and pressure as a function of time. § Expand the syringe so that its volume increases. In the worksheet describe what happens and

sketch the graph of the temperature as a function of time and pressure as a function of time.

II. Gas Law Apparatus The gas law syringe nicely demonstrates the qualitative relations of the ideal gas law. However,

the friction in the syringe makes quantitative measurements difficult. In this part of the lab, you will use the gas law apparatus to overcome this limitation. It consists of two pieces, an air chamber and a base unit. The air chamber is an aluminum cylinder with a rubber stopper at one end with an air hose. The base unit is a precision glass cylinder with a piston with a platform for holding masses. At the bottom of the base unit are two hose connections that can be opened or closed as needed.

Gay-Lussac Law

We will be exploring the variation of temperature and pressure at a fixed volume. This relation is known as the Gay-Lussac Law. The image below shows the setup for this portion of the lab. Only the air chamber will be used.

• Set up the air chamber and sensor as shown in the picture above. You should tape the

thermocouple to the outside of the air chamber to measure its temperature.

Temperature

Probe

Temperature

Pressure Sensor

Xplorer GLX

Air Chamber

Hot Water


119

• You should make all the connections between the air chamber and Explorer GLX with the chamber sitting on the table.

• Connect the air chamber directly to the pressure sensor and connect the thermocouple to the temperature sensor.

• Setup the Xplorer GLX to measure the temperature and pressure sensor. For instructions on setting the Xplorer GLX, refer to the appendix of the lab manual.

• Set the Xplorer GLX in the digits mode and measure both the temperature and pressure. § Place the air chamber in a beaker of hot water. In the worksheet record the temperature and

pressure as the water cools. You can speed up the cooling process by adding ice and stirring the water.

Boyle’s Law

We will be exploring the variation of pressure and volume at a fixed temperature. This is known as Boyle’s Law. The image below shows the setup for this portion of the lab. Only the base unit will be used.

• Set up the base unit as shown in the picture above. You should connect one of its ports to the pressure sensor on the Xplorer GLX.

• You should raise the piston to its highest position, giving the base unit the largest volume. • With the piston at the highest point, the other pressure port should be closed. This can either be

done with the clamp in the back of the base unit or attaching another hose with a clamp on it to the second port.

• Setup the Xplorer GLX to measure the pressure sensor. For instructions on setting the Xplorer GLX, refer to the appendix of the lab manual.

• Set the Xplorer GLX in the digits mode and measure the pressure.

Piston

TemperaturePressure Sensor

Xplorer GLX

Base Unit


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§ Press down on the piston. When the pressure is constant (after the temperature inside the cylinder has reached equilibrium) record the pressure and volume of the cylinder in the worksheet.

§ Repeat this measurement for at least 5 different heights of the piston.

Charles’ Law We will be exploring the variation of volume and temperature at a fixed pressure. This is known

as Charles’ Law. The image below shows the setup for this portion of the lab. Both the air chamber and base unit will be used.

• Set up the base unit and air chamber as shown in the picture above. You should connect one of its ports to the pressure sensor on the Xplorer GLX. The other connection should go to the air chamber.

• The base unit should be carefully placed on its side. Please be careful as the glass is fragile. • Tape the thermocouple to the outside of the air chamber. • Setup the Xplorer GLX to measure the temperature and pressure sensor. For instructions on

setting the Xplorer GLX, refer to the appendix of the lab manual. • Set the Xplorer GLX in the digits mode and measure the temperature and pressure. § Place the air chamber in a beaker of hot water. In the worksheet record the temperature and

volume as the water cools. You can speed up the cooling process by adding ice and stirring the water.

Temperature

Pressure Sensor

Xplorer GLX

Base Unit

Air Chamber

Hot Water

Temperature

Probe


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III. Heat Engine A heat engine has the ability to use heat to perform work. In this case, we will use heat from the

hot water to perform work by lifting a mass on the piston. The setup for this portion of the lab is shown in the figure below. The setup is very similar to the Charles’ Law experiment except that the base unit is placed upright. There is also an additional cup of very cold water.

• Set up the base unit and air chamber as shown in the picture above. You should connect one of the base unit ports to the pressure sensor on the Xplorer GLX. The other connection should go to the air chamber.

• Tape the thermocouple to the outside of the air chamber. • The heat engine will work best if there is a large temperature difference between the hot and cold

water. For the cold water, you should melt ice to have it as cold as possible. • Setup the Xplorer GLX to measure the temperature and pressure sensor. For instructions on

setting the Xplorer GLX, refer to the appendix of the lab manual. • Set the Xplorer GLX in the digits mode and measure the temperature and pressure. • Place the air chamber in the cold water. • You will be taking the piston through a cycle by placing the air chamber in the hot and cold water

and also changing the mass on the piston. • The cycle will consist of the following 4 steps.

1. Add 200 gram mass with air chamber in cold water. 2. Move air chamber to hot water. 3. Remove 200 gram mass with air chamber in hot water. 4. Move air chamber to cold water.

• The cycle can be repeated many times. • At each step, you will need to measure the pressure, volume and piston height. • You should practice a couple of times to make sure that you can do the steps quickly. § Perform the cycle and enter the pressure, volume and piston height on the worksheet.

Xplorer GLX

Base Unit

Air Chamber

Hot Water

Temperature

Probe

Cold Water

Name:_________________________

Date:__________________________

122

Ideal Gas Law and Heat Engine Worksheet I. Gas Law Syringe

Describe what happens when you hold the syringe in your hand. Make a plot of the temperature as a function of time when it is warming.

Describe what happens when the syringe is cooling. Make a plot of the temperature as a function

of time when it is cooling.

Describe what happens when the syringe is compressed. Make a sketch of the temperature and

pressure as a function of time.

Describe what happens when the syringe is expanded. Make a sketch of the temperature and

pressure as a function of time.

Worksheet: Ideal Gas Law and Heat Engine

123

II. Gas Law Apparatus Gay-Lussac Law

Make a plot of the pressure versus temperature and attach it to the worksheet. Boyle’s Law

Record the pressure and volume for at least 5 different heights of the piston.

Pressure Volume

Make a plot of the pressure versus volume and attach it to the worksheet.

Charles’ Law Make a plot of the temperature versus volume and attach it to the worksheet.

III. Heat Engine


Step Volume Pressure Height

1

2

3

4

Describe what happens at each step of the heat engine cycle. Make a plot of the pressure versus

volume.

Worksheet: Ideal Gas Law and Heat Engine

124

Calculate the work that the engine has done on the 200 gram mass in lifting it up and down.

Calculate the area enclosed by the path in your pressure versus volume graph. How does this area compare to the work done by lifting up and down the 200 gram mass?

125

Heat Capacity

One way to change the temperature of an object is to either add or remove heat from it. The heat capacity, C of the object is a measure of how difficult it is to change its temperature. It is defined as the amount of heat that must be added to an object to increase its temperature. It is given by

𝐶 =𝑄∆𝑇 (1)

where 𝑄 is the heat flowing into the object and ∆𝑇 is the change in temperature. This quantity depends on how much mass the object has. So, the usual way to express the heat capacity is per unit mass. This is known as the specific heat, 𝑐. This is given by

𝑐 =𝐶𝑚 =

𝑄𝑚∆𝑇 (2)

Each material has a different value for the specific heat. Materials that require a lot of heat to change their temperature have a large specific heat while ones that are easier to change have lower specific heats. The calorie is defined such that 1 calorie of heat will raise the temperature of 1 gram of water by 1 degree Celsius. Therefore, the specific heat of water is given by

𝑐 = 1000 calkg C (3)

These expressions for the specific heat apply when a material is not changing its phase. When an object changes phase the heat needed is given by the latent heat, L. The latent heat of an object is given by

𝐿 =𝑄𝑚 (4)

where 𝑄 is the heat needed to change the phase and 𝑚 is the mass of the object. This means that you must add an amount of heat, 𝑄 = 𝑚𝐿 to change the phase of an object, to say go from ice to water without changing the temperature. Boiling water requires more heat than if you are just raising the temperature. This is why a pot of water seems to take forever to boil after the temperature is raised to nearly the boiling point. You must still add a large amount of heat to convert the water to steam.

In this lab, you will be measuring the latent heat of vaporization of liquid Nitrogen. Before doing this, you will need to calibrate your measurements using water which has a known specific heat.

Goals of the this lab: • Understand the concept of specific heat. • Measure the specific heat of water. • Measure the latent heat of vaporization for liquid Nitrogen.

Heat Capacity

126

Lab equipment: Heating element The electrical heater is used to heat the water and liquid nitrogen.

Liquid Nitrogen The liquid nitrogen for the experiment is stored in the large dewar.

Gloves and Goggles Gloves and goggles should always be worn when handing liquid nitrogen as it can cause severe burns.

Power Supply The power supply is used to apply voltage and current to the heating element.

Thermocouple The thermocouple is used to measure the temperature of the water as it is heated.

Heat Capacity

127

Thermos A thermos is used when heating the water or liquid nitrogen to keep it isolated from the surrounding environment.

Stopwatch The stopwatch is used to keep track of the time while heating the liquids.

Scale The scale is used to measure the mass of water or liquid nitrogen as it undergoes a phase change.

Xplorer GLX The Xplorer GLX has the ability to measure the temperature from the thermocouple as a function of time.

Heat Capacity

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Lab Procedures

I. Specific Heat of Water In this lab, you will be using an electrical heating element to add heat to water and liquid nitrogen

to measure their heat capacity and latent heat. The first step is to determine how much heat the heater adds to the system as a function of time. The picture below shows the setup for this portion of the lab. The power supply applies a current and voltage to the heating element to warm it up. The power

that is applied to the heating element is given by

𝑃 = 𝐼𝑉 (5)

where 𝑃 is the power, 𝐼 is the current and 𝑉 is the voltage. If the current is read in Amps and the voltage is in Volts, then the power will be in Watts. Since Watts is energy per unit time, we need to multiply the power by a time to get the amount of heat or energy. Therefore, the heat added to the water is given by

𝑄 = 𝑃∆𝑡 = 𝐼𝑉∆𝑡 (6)

where ∆𝑡 is the time that the heater is on in seconds. Now that the amount of heat added by the heating element is known, we can determine the

specific heat of the water. The heat added to the system is used to raise the temperature of the water so

𝑄 = 𝑚𝑐water∆𝑇 (7)

where m is the mass of water, 𝑐water is the specific heat and ∆𝑇 is the change in temperature. Therefore, if you measure the change in temperature as a function of time, you can find the specific heat because

∆𝑇 =𝐼𝑉

𝑚𝑐water∆𝑡 (8)

Xplorer GLX

Water

ScalePower Supply

Heating Element

Heat Capacity

129

• Set up the system as shown in the picture above. • Place the thermocouple in the water and connect it to the Xplorer GLX. • Setup the Xplorer GLX to measure the temperature. For instructions on setting the Xplorer GLX,

refer to the appendix of the lab manual. • Set the Xplorer GLX in the digits mode and read the temperature. • There should be two wires going from the power supply to the heating element, one from the

positive terminal and one from the negative terminal. • Pick a mass of around 200 grams of water. Submerge the heating element in the water. You can

adjust the height of the heater by moving the stand. § Turn on the power supply and set the current and voltage. The product of these two values

needs to be around 20 Watts to heat the water. Record these values in the worksheet. § Start the stopwatch. Record the temperature of the water and time at regular intervals until the

water temperature has changed by at least 20 degrees. You should keep stirring the water. • Repeat the measurement with two different amounts of water. Start with fresh water each time so

that the initial temperature is relatively constant.

II. Latent Heat of Vaporization of Liquid Nitrogen In this part of the lab, you will use a similar setup to the previous part. However, you will replace

the water with liquid nitrogen. The goal is to measure the latent heat of vaporization of liquid nitrogen. This will be done by measuring the heat added to the system as well as the change in mass of the liquid nitrogen. Then the latent heat is given by

𝐿 =𝑄𝑚 (9)

where 𝑄 is the heat added and 𝑚 is the mass of liquid nitrogen that boils. This mass will be determined by measuring the mass before and after boiling.

• Set up the system as shown in the picture above. When pouring the liquid nitrogen into the

thermos be very careful not to splash it. Always wears goggles and gloves because the liquid nitrogen can cause severe burns if it comes in contact with your skin.

• There should be two wires going from the power supply to the heating element, one from the positive terminal and one from the negative terminal.

• Fill the thermos with liquid nitrogen. Submerge the heating element in the liquid nitrogen. You can adjust the height of the heater by moving the stand. Make sure that it does not touch the thermos or else the scale reading will be incorrect.

• If you just leave the liquid nitrogen in the thermos it will boil away because of the heat added from the room. The amount of heat added in this way is difficult to measure but we must take it into account. § After the heater is in thermal equilibrium with the liquid nitrogen, measure the change in mass

of the liquid nitrogen every 10 seconds. Record these values until the change of mass with time is constant. This is called the static loss rate.

§ Turn on the power supply and set the current and voltage. § Start the stopwatch. Record the change in mass of the liquid nitrogen every 10 seconds.

Record these values until the change of mass with time is constant.

Heat Capacity

130

• Repeat the measurement two more times with different amounts of power from the heating element.

• To determine the change of the mass of liquid nitrogen per time caused the heater, you must subtract the static loss rate from the rate that you found with the heater on.

Once you have determined the rate of mass change caused just by the heater, you can determine the latent heat of vaporization for the liquid nitrogen using

𝑚heater = 𝑄heater𝐿 =

𝐼𝑉𝐿 ∆𝑡

(10)

Name:_________________________

Date:__________________________

131

Heat Capacity Worksheet I. Specific Heat of Water

You should record your data in tables similar to below (don’t forget the units).

Mass: Current: Voltage:

Time Temperature

You should make a plot of the change in temperature as a function of the change in time for the

three different masses of water. This graph should be labeled GRAPH#1.

Fit straight lines to the data in GRAPH#1. The slope should be equal to !"

!!water.

Measured value of the slopes. __________ __________ __________ Calculated value of the specific heat from the measured slopes. ____________________

II. Latent Heat of Vaporization of Liquid Nitrogen

You should record your data in tables similar to below (don’t forget the units).

Static Loss Rate: Current: Voltage:

Time Mass

You should make a plot of the change in mass caused by the heater as a function of the change in

time for the three different amounts of power. This graph should be labeled GRAPH#2.

Fit straight lines to the data in GRAPH#2. The slope should be equal to !"!

. Measured values of the slope. __________ __________ __________ Calculated value of the latent heat of vaporization from the measured slopes. ________________

Worksheet: Heat Capacity

132

Questions to address in your lab report or worksheet 1. Compare your measured value of the specific heat of water with the expected value. Note that the

conversion factor from Calories to Joules is 4.184 J = 1 cal.

2. What are some possible origins for the discrepancy between your measured value of the specific heat and the known value?

3. Compare your measured value of the latent heat of vaporization of liquid nitrogen with the

expected value, 200 J/g. What are the sources of error in your measurement? 4. How does your error for the specific heat of water compare to your error for the latent heat of

vaporization? Which measurement was more accurate? Explain how the different measurement techniques led to different errors.

133

Xplorer GLX Instructions

We will be using Xplorer GLX devices throughout the semester. They are capable of sensing, recording, and graphing information using a wide range of add-on sensors including photogates, force sensors, rotary motion sensors, electronic thermometers, and pressure gauges. Plots and data can be directly displayed on the Xplorer GLX, or data can be saved to a USB memory stick for later analysis. A complete manual from the manufacturer, Pasco can be downloaded from the course site on D2L. However, this manual contains the basic information that you will need for operating the Xplorer GLX in the Physics 181 lab. The image below shows the Xplorer GLX with labels on the most important buttons.

Run

HomeEsc

Power

Power input

Sensor inputs

Select


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I. General Settings A. Power

Before turning on the Xplorer GLX, make sure that the sensor(s) that you will be using for the experiment are connected. Usually the Xplorer will correctly configure the sensors for your measurements. The power button is located on the lower right hand side of the Xplorer.

• If the Xplorer does not turn on, check that the power supply is connected to the power input jack on the right hand side.

• If the unit is frozen or won’t turn on try pressing the escape key, “Esc”, and the “Home” button.

• If this does not fix the Xplorer, ask the TA to reset the Xplorer. Pressing a paperclip on the reset button on the back will restart the Xplorer.

Whenever possible leave the Xplorer plugged into the power supply. This keeps the batteries fully charged. The current charge level of the batteries is indicated on the top right of the screen.

B. Display The contrast of the display can be adjusted by holding down the “Home” button and pressing the up or down arrow keys. The backlight on the display can be toggled on and off by holding down the “Home” button and pressing the “Select” button.

C. Data collection You use the “Run” button to start and stop the data collection. The small symbol at the top of the display indicates whether data is currently being collected or not. If data is being collected, it is a circular dial with a spinning indicator like a clock. If data collection is stopped, it is a picture of the “Run” button in a circle. See the figures to the right for the two different icons.

D. Displaying data The Xplorer GLX can display the data as a digital readout, analog meter, a graph or in a table. To choose which method to use to display the data, you should start at the main menu (press the “Home” button). Then use the arrow keys to highlight the display method (Digits, Meter, Graph or Table) that you want to use. Press the “Select” key to enter that mode. The four different display modes are highlighted in the figure on the right. After selecting the display mode, you can customize its appearance using the following procedure. • Press the “Select” button to highlight the labels. • Use the arrow key to highlight the label that you would like to change and then push the

“Select” key.

Running Stopped


135

• A menu listing the possible parameters and data properties will appear. Select the parameter that you would like to display or select Data Properties to change things like significant figures and units.

Specific details for the different sensors are given in the later sections of the manual. The displays for the four different modes are shown below. 1) Digital Display From the home screen, use the arrow keys to select the

digits display. This will bring up a screen with a digital read out from the photogate sensors. The basic screen is shown on the right. If you want to change one of the parameters that are displayed, you should follow the general procedure above.

2) Analog Meter From the home screen, use the arrow keys to select the

meter display. You should see a display like the image on the right. If you want to change which parameter is being measured or the number of significant figures displayed, you should press the select key. The method for changing the parameters is the same as the general procedure described above.

3) Table From the home screen, use the arrow keys to select the

table display. You should see a display like the image on the right. The method for changing the parameters that are displayed in the table is the same as the general procedure discussed above.

4) Graph

From the home screen, use the arrow keys to select the graph display. You should see a display like the image on the right. The method for changing the parameters is the same as the general procedure outlined above.


136

E. Saving data Data can be saved to a USB memory stick using the USB port on the right hand side of the Xplorer GLX. This feature can be used to save the data displayed in a table for later analysis. The procedure is as follows. 1. Plug the USB memory stick into the Xplorer GLX. 2. Press the “Home” button and then press F2 (Table) to go into the Table mode. 3. Press F4 (Tables) to enter the table options menu. 4. Select option number 8, “Export All Data…”. 5. Press F1 (Ok) to start the transfer. 6. Press F1 (Ok) after the transfer is complete.

F. Avoiding crashes

When the memory of the Xplorer GLX gets full, it has a tendency to freeze or crash. There are a few steps to follow to minimize the chance of this occurring. 1. Delete unknown data files when starting. This is done by selecting “Data Files” from the main

menu. Then select the data file that you want to delete. 2. Use a reasonable sample rate. 3. Only acquire data when needed. Turn off the data collection when you are not using it. 4. Turn off the Xplorer GLX when not using it for a long period of time. Otherwise, it may

overheat.

If you are still having trouble with crashes, report the “bad” Xplorer GLX to your TA and they can give you a replacement.

II. Photogates Photogates have the ability to measure the time that an object is blocking the infrared sensor, the time between two successive blocking events or the period of a pendulum. You will use photogates with the Xplorer GLX in the following labs: Newton’s Law, Friction, Energy Conservation, Circular Motion and Pendulums and Springs. The setting “Time between gates” is used to measure the time between two successive blocking events of the infrared beam. This mode is usually used when you want to find the time it takes a cart to pass from one photogate to a second one. The setting “Velocity in gate” is used to measure the velocity of an object when it is passing through one of the gate. The photogates determine the velocity of an object by measuring the time that the infrared beam is blocked. Then the velocity is given by

𝑣 = ∆𝑥∆𝑡

where ∆𝑥 is the width of the object (flag) that is blocking the beam. You must set this width on the Xplorer GLX in order to get accurate measurements.


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For either mode of operation of the photogates you must set them up on the Xplorer GLX. The basic setup is as follows:

1. Plug in the digital adapters in the sensor inputs. There can be two photogates attached to each adapter.

2. Plug the photogates into the digital adapters using the supplied cables.

3. Turn on the Xplorer GLX. 4. Select “Photogate Timing.” 5. The timing menu should appear. It looks like the figure on the

right. If it does not appear, you can access the menu by using the arrow keys to select “Timing” from the main menu.

A. Measure time between gates

If you are going to measure the time between gates, you need to make sure that the “Time Between Gates” setting is set to visible.

B. Measure velocity in gate

If you are going to measure the velocity using the photogates, you need to set the correct “Flag Length” and make sure that “Velocity in Gate” is set to visible. The Flag Length depends on the size of the flag you are using on the cart. For the roller coaster carts, the flag is 0.0047 m. When you are making your own flags, you should try to make flags that are about 0.01 m in width.

To use the one photogate to measure the period of a pendulum, you follow the following steps:

1. Plug in a digital adapter in one of the sensor inputs. 2. Plug a photogate into the digital adapters using the supplied

cable. 3. Turn on the Xplorer GLX. 4. Select “Photogate and Pendulum.” 5. The timing menu should appear. It looks like the figure on the

right. If it does not appear, you can access the menu by using the arrow keys to select “Timing” from the main menu.

6. Set the width of the pendulum that you will be using. Once you have setup the timing using the photogates, you need to set how you want the data to display. There are three different modes depending on the lab, an analog meter, a digital display or in a table. To set the number of significant figures to display, perform the following steps.

1. Press the “Select” button. 2. Use the arrow keys to highlight the parameter that you are measuring and then press “Select”

again. 3. Use the arrow keys to highlight “Data Properties…” and press “Select.” 4. Use the arrow keys to select “Number of Digits” and set it to 3. 5. Press F1, “Ok” to confirm the selection.


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III. Rotary Motion Sensors The rotary motion sensors have the ability to measure the linear position, linear velocity, angular position or angular velocity. You will use the rotary motion sensors for the Collisions and Circular Motion labs. The basic steps for setting up the rotary motion sensors are as follows:

1. Plug in the rotary motion sensors to the Xplorer GLX sensor inputs. 2. Turn on the Xplorer GLX. 3. The timing menu should appear. The display should look like

the figure on the right. If it does not appear, you can access the menu by using the arrow keys to select “Timing” from the main menu.

4. Set the Sample Rate Unit to “samples/s.” 5. Set the Sample Rate to 50. 6. Set the Linear Position Scale to “Lg. Pulley (Groove).” 7. Set Zero Automatically On Start to “On.”

A. Graph Position Versus Time

For the collisions lab, you will want to make a graph of the linear position as a function of time for the two carts. This is done as follows: 1. From the main menu, select the graph mode (press F1). 2. When the graph mode first loads it looks like the image

on the right that shows only one graph. You want to display the position for each cart. So, you need to make two graphs.

3. Press F4 and select “Two Graphs.” 4. Adjust the axes to display the linear position. First press

the “Select” button. 5. Move the highlighted label to the axis that you would like

to change and press the “Select” button. 6. Use the arrow keys to select, “Linear Velocity.” It is

probably under the “More” selection. The image to the right shows what it should look like.

7. Repeat the process for the second graph. Make sure that you select the linear velocity for the other sensor. It should be labeled “Linear Velocity 2”.

8. When the process is complete, you should have two graphs; one showing Linear Velocity versus Time and the other showing Linear Velocity 2 versus Time.

9. You can start/stop the data collection with the “Run” button. 10. After the carts collide, you will have two graphs

showing the velocity of each cart as a function of time. You can use the “Tools” (F3) to select cursors. You can use the arrow keys to move the cursors around the graphs. In this way, you can measure the velocity of each cart before and after the collision. The resulting data should look like the image on the right.

11. To change the graph that the cursor is on, press “Tools”, F3 and select “Toggle Active Data”.


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IV. Force Sensor The force sensor will measure the force applied to it. You will use the sensor in the Circular Motion and Pendulum and Springs labs. The basic steps for setting up the force sensor are as follows:

1. Plug in the force sensor to the Xplorer GLX sensor inputs. 2. Turn on the Xplorer GLX. 3. The timing menu should appear. The display should look

like the figure on the right. If it does not appear, you can access the menu by using the arrow keys to select “Timing” from the main menu.

4. Set the Sample Rate Unit to “samples/s.” 5. Set the Sample Rate to 10.

For the Circular Motion lab, you will need to record both the angular velocity and the force. It is easiest to do this using the table format. To setup the table perform the following steps.

1. From the home screen, use the arrow keys to select the table display or press the F3 button.

2. Press the “Select” button to highlight the first column. 3. You should be reading, “Force, push positive” 4. Select “Data Properties…” 5. Change the data properties to match the image on the right.

The only one that you should have to change is the “Number Of Digits” to be 4.

6. From the table screen, press the “Select” button and use the arrow keys to highlight the second column.

7. Press the “Select” button. 8. Change the parameter being measured to “Angular Velocity”.

It may be under the “More…” section. When the table is properly configured, your data should look like the table on the right. Before acquiring data with the force sensor, make sure that you press the Zero button. This will make sure that you are only measuring the change in force caused by the rotating or moving object.


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V. Temperature and Pressure Sensor The temperature and pressure sensor has the ability to measure the temperature of an object using a thermocouple and the pressure using the small plastic hoses. The sensor uses one of the sensor inputs on the Xplorer GLX. There are also two additional thermometer ports on the left hand side of the Xplorer GLX. You will use the temperature and pressure sensor in the Heat Engine and Heat Capacity labs. The basic steps for setting up the temperature and pressure sensor are as follows.

1. Plug in the temperature and pressure sensor to the Xplorer GLX sensor input.

2. Turn on the Xplorer GLX. 3. The timing menu should appear. The display should look like

the figure on the right. If it does not appear, you can access the menu by using the arrow keys to select “Timing” from the main menu.

4. Set the Sample Rate Unit to “samples/s.” 5. Set the Sample Rate to 10.

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Position, Velocity and Acceleration Lab Report

Brian LeRoy

Lab Partner: Whoever sat next to me

Course: PHYS181-001

TA: Unknown Graduate Student

Due Date: 8:00 AM on 06/04/2013

Position, velocity and acceleration report

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Abstract We measured the size, velocity and acceleration of various objects. We analyzed the motion

of carts on level and inclined tracks using carbon tape timers. We found that the cart moved with a nearly constant speed of 0.48 m/s on a level track. On the inclined track, the cart accelerated with an acceleration of 0.34 m/s2.

Introduction

In this lab, we are measuring the position, velocity and acceleration of a cart moving along a track. In the first portion of the lab, this is done with a level track where we expect that the velocity of the cart to be constant. In the second part of the lab, we used an inclined track so that the cart accelerates. We can measure the acceleration of this cart by observing its change in position as a function of time. The position, velocity and acceleration of an object can all be related using the kinematic equations of motion as long as the acceleration is constant. In this lab, we will be using these two equations to calculate the velocity and acceleration of the cart after measuring its position as a function of time.

Procedure

We wanted to measure the velocity and acceleration of carts using tape timers. The tape timers work by creating a carbon dot on a piece of paper at well-defined intervals of time. In this lab, we used the 10 Hz setting so that the timers made 10 marks per second on the paper. In the first part of the lab, we used a level track for the cart. The schematic of the setup is shown in Figure 1 below. The cart was started on the left hand side of the track. The tape timer was turned on and the cart was given a gentle push so that it moved to the right hand side. Once the cart reached the right hand side, the tape timer was turned off. We examined the tape to make sure that the timer created a series of dots. To find the position and velocity of the cart we measured the positions and separations of the dots on the tape.

For the second part of the lab, we used a slightly inclined track. The schematic of the setup for this portion of the lab is shown in Figure 2 below. The left hand side of the track was inclined using a 4 cm high block of wood. The tape timer was turned on and the cart was released from rest. When the cart reached the bottom of the track, the tape timer was turned off. Once again, we examined the tape to make sure that the timer created a series of dots on the tape. To find the position, velocity and acceleration of the cart, we measured the dots on the tape as discussed below.

Figure 1. Schematic for the level track experiment showing cart, tape timer, track and paper tape.


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Theory In this lab, we measured the separation between dots created by the tape timers. We needed

to convert these separations into the velocity and acceleration of the cart as a function of time. The tape timers create dots at constant intervals of time. Therefore, we can determine the average velocity 𝑣 of the cart between two dots as

𝑣 =𝛥𝑥!𝛥𝑡!

=𝑥! − 𝑥!!!𝑡! − 𝑡!!!

(1)

where Δ𝑥! is the separation between the nth and (n-1)th dot and 𝛥𝑡! is the change in time between those dots. Similarly, we can determine the average acceleration 𝑎 of the cart

𝑎 =𝛥𝑣!𝛥𝑡!

=𝑣! − 𝑣!!!𝑡! − 𝑡!!!

(2)

where Δ𝑣! is the change in velocity between the nth and n-1 dot and 𝛥𝑡! is the change in time between those dots.

We can also determine the velocity and acceleration of the cart using the kinematic equations. The position of the cart as a function of time is given by

𝛥𝑥 = 𝑥! − 𝑥! = 𝑣!𝑡 +!!𝑎𝑡! (3)

where 𝑥! is the initial position, 𝑥! is the final position and 𝑣! is the initial velocity. If we start the cart from the origin (𝑥! = 0) and on a level track with no acceleration (𝑎 = 0), this equation simplifies to

𝑥! = 𝑣!𝑡 (4)

So, we can obtain the velocity of the cart from the slope of the graph of position as a function of time.

When the cart is on an incline, it will accelerate. However, we start the cart from rest (𝑣! = 0), so equation (3) becomes

𝑥! =!!𝑎𝑡! (5)

Then from the slope of the graph of position versus time squared, we obtain the acceleration of the cart. We can also obtain the acceleration of the cart using the equation for the velocity of the cart as a function of time,

Figure 2. Schematic for the inclined track experiment showing cart, tape timer, track, paper tape and block.


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𝛥𝑣 = 𝑣! − 𝑣! = 𝑎𝑡 (6)

So, if the cart starts from rest 𝑣! = 0, we can simplify equation (6) to read,

𝑣! = 𝑎𝑡 (7)

Therefore, the slope of a plot of the velocity as a function of time gives the acceleration of the cart.

Sample Calculation and Results We have measured the position of the cart as a function of time for both the level track and

the inclined track. The raw data is shown in table #1 for the level track and table #2 for the inclined track. These tables are attached at the end of the report. As a dot was created every 0.1 seconds, we have calculated the time by multiplying the dot number by 0.1 s. We have measured the position of the dots from the first point to calculate the displacement of the cart.

For the level track data, we have plotted the position of the cart as a function of time in graph #1. The slope of the best fit line to this plot gave us the average velocity of the cart and it was 0.479± 0.004 m/s. We have calculated the average velocity of the cart between each pair of dots using equation (1). We have determined the velocity between the first pair of points as follows

𝑣! =𝛥𝑥!𝛥𝑡!

=𝑥! − 𝑥!𝑡! − 𝑡!

=0.052 m− 0 m0.1 s− 0 s =

0.0520.1

m s = 0.52m s (8)

The velocities at the other locations were calculated in a similar manner. In graph #2, we plotted this calculated velocity of the cart as a function of time. We see that the cart slows done slightly as a function of time. From our data the average velocity was 0.48± 0.03 m/s.

For the inclined track, we once again plotted the position of the cart as a function of time. This is shown in graph #3. This graph is parabolic showing that the cart is accelerating as it moves down the track. We have re-plotted this position data as a function of time squared in graph #4. This graph is now linear and we can fit a straight line to the data to give the average acceleration of the cart. We find that the best fit line has a slope of 0.188± 0.001 m/s2. The acceleration of the cart is twice this slope, so we find an acceleration of 0.376± 0.002 m/s2. We have calculated the velocity of the cart as a function of time using equation (1) in the same manner as the level track. The results are plotted in graph #5. We find that the velocity of the cart linearly increases with time. The slope of the best fit line gives the acceleration of the cart. We find the slope is 0.346± 0.002 m/s2. Lastly, we calculated the acceleration of the cart from the change in velocity using equation (2). We have determined the acceleration at the second point as follows

𝑎! =𝛥𝑣!𝛥𝑡!

=𝑣! − 𝑣!𝑡! − 𝑡!

=0.012 m s− 0.013 m s

0.2 s− 0.1 s =−0.0010.1

ms2 = −0.01m s2 (9)

The accelerations at the other locations were calculated in a similar manner. The results are plotted in graph #6. We see that the acceleration is nearly constant as a function of time but it is slightly smaller near the end of the data. We found an average acceleration of 0.34± 0.07 m/s2.


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Discussion and Conclusions We found that the velocity of the cart was relatively constant on the level track. We could

tell this because the separation between adjacent dots was constant throughout the length of the track as seen in graph #1. This means that the velocity determined from the slope of the position as a function of time was nearly the same as the values found by using equation (1) to calculate the velocity from the change in position. There was a slight difference because the cart was slowing down near the end of the track as can be seen in graph #2. This negative acceleration was probably due to some friction either in the wheels of the cart or because of the paper going through the tape timer.

When the cart was released from an inclined track, the velocity of the cart continually increased. The dots on the paper became more and more spread apart as the cart moved down the track. This means that its velocity was constantly increasing. In other words, the cart was accelerating. If the acceleration of the cart was higher, then the dots would become spaced out more quickly. We found that the acceleration of the cart as determined using graphs #4-6 were all consistent with each other. This means that any of the methods will work for determining the acceleration of the cart. However, the errors were largest when using equation (2) to calculate the acceleration. This is because our 1 mm uncertainty in determining the position of the cart gets amplified when it is divided by the time interval twice. Our value for the acceleration was in reasonable agreement with the expected value of 0.33 m/s2 from the angle of the incline. Our values were slightly higher which may be caused by the block being slightly higher than 4 cm or the track being shorter than 1.2 m. Either of these errors would cause the angle of the track to be slightly larger and therefore we would expect a larger acceleration.

From our graphs of the velocity as a function of time, graph #5, we can determine the initial velocity of the cart. This is given by the y-intercept of the graph which was approximately 0. Therefore, our cart started from rest. We could also find this initial velocity by looking at graph #3. The slope of this graph at 𝑡 = 0 gives the initial velocity of the cart.

There was an uncertainty of 1 mm for each of our measurements of the position of the cart. This uncertainty arose from the markings on the ruler that we used. The error bars on the graphs of position versus time (graph#1, #3 and #4) are all 1 mm in height. This 1 mm uncertainty in position becomes an error of 10 2 mm/s for the velocity that is plotted in graphs #2 and #5. The square root of 2 is because both position measurements that are used to calculate the velocity have a 1 mm error and these errors add in quadrature. The factor of 10 is because the time interval is 1/10 of a second. Lastly, we have an error of 200 mm/s2 for the acceleration. Once again this is due to both measurements of the velocity having the same error and adding them in quadrature.


146

0.6

0.4

0.2

0.0

Position (m)

1.61.4

1.21.0

0.80.6

0.40.2

0.0Tim

e (s)

GRAPH #1

Graph #1: This graph plots the position of the cart as a function of time for the level track. The points are equally spaced inposition indicating that the cart moved with a constant velocity. The red line indicates a best-fit line to the data.


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0.54

0.52

0.50

0.48

0.46

0.44

0.42

0.40

Velocity (m/s)

1.61.4

1.21.0

0.80.6

0.40.2

0.0

Time (s)

GRAPH #2

Graph #2: This graph plots the average velocity of the cart as a function of time for the level track. The velocity slows down the longer the cart travels.


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1.0

0.8

0.6

0.4

0.2

0.0

Position (m)

2.01.5

1.00.5

0.0

Time (s)

GRAPH #3

Graph #3: This graph plots the position of the cart as a function of time for the inclined track. The points become farther spaced inposition as a function of time indicating that the cart is moving with an increasing velocity.


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1.0

0.8

0.6

0.4

0.2

0.0

Position (m)

54

32

10

t 2 (s 2)

GRAPH #4

Graph #4: This graph plots the position of the cart as a function of time squared for the inclined track. The points lie on a straight lineindicating that the cart moves with a constant acceleration. The red line indicates a best-fit line to the data.


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0.8

0.6

0.4

0.2

0.0

Velocity (m/s)

2.01.5

1.00.5

0.0

Time (s)

GRAPH #5

Graph #5: This graph plots the average velocity of the cart as a function of time for the inclined track. The points are equally spaced invelocity indicating that the cart moved with a constant acceleration. The red line indicates a best-fit line to the data.


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0.5

0.4

0.3

0.2

0.1

0.0

Acceleration (m/s2)

2.01.5

1.00.5

0.0

Time (s)

GRAPH #6

Graph #6: This graph plots the average acceleration of the cart as a function of time for the inclined track.The average acceleration values are nearly constant.


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Position, velocity and acceleration worksheet

I. Measuring lengths Put your measurements in the table below. Do not forget to include units will all the values.

Length of room Width of room Your height “Cubit”

10.81 m 6.59 m 1.80 m 48 mm

Identify the objects measured and their size in the table below.

Object that is ~0.1 m Object that is ~0.01 m Object that is ~0.001 m

Index finger Fingernail 10 sheets of paper

II. Determining velocity using photogates Write the readings on the timer for the following velocities of your finger.

1 m/s 2 m/s ½ m/s

1.000 s 0.500 s 2.000 s

Put your measurements for the velocity of the meter stick in the following table. Do not forget to include units with all the values.

Width of meter stick 25.4 mm

Shortest time for width 0.0075 s

Greatest velocity for width 3.39 m/s

Thickness of meter stick 6.0 mm

Shortest time for thickness 0.0024 s

Greatest velocity for thickness 2.50 m/s


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IV. Level track data Fill out TABLE #1 below with the data from the level track. The units of each quantity can be given just once in the label of the column as shown. All of your graphs must be hand drawn.

Dot

number

tn

(s)

xn

(m)

Δxn

(m)

vn

(m/s)

0 0 0 ----- -----

1 0.10 0.052 0.052 0.52

2 0.20 0.104 0.052 0.52

3 0.30 0.155 0.051 0.51

4 0.40 0.206 0.051 0.51

5 0.50 0.256 0.050 0.50

6 0.60 0.305 0.049 0.49

7 0.70 0.354 0.049 0.49

8 0.80 0.402 0.048 0.48

9 0.90 0.449 0.047 0.47

10 1.00 0.496 0.047 0.47

11 1.10 0.542 0.046 0.46

12 1.20 0.587 0.045 0.45

13 1.30 0.631 0.044 0.44

14 1.40 0.675 0.044 0.44

15 1.50 0.717 0.042 0.42

Attach a plot of the position of the cart as a function of time. This should be labeled as GRAPH #1. Extract the slope of this plot._______ 𝟎.𝟒𝟕𝟗± 𝟎.𝟎𝟎𝟒 m/s _____________ Attach a plot of the velocity of the cart as a function of time. This should be labeled as GRAPH #2.


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V. Inclined track data Fill out TABLE #2 below with the data from the inclined track. Give the units of each quantity in the label of the column as done in TABLE #1. All of your graphs must be hand drawn.

Dot

number

tn

(s) tn2

(s2)

xn

(m)

Δxn

(m)

vn

(m/s)

Δvn

(m/s)

an

(m/s2)

0 0 0 0 ----- ----- ----- ----- 1 0.10 0.010

0.04

0.013 0.013 0.13 ----- ----- 2 0.20 0.040 0.025 0.012 0.12 -0.01 -0.1 3 0.30 0.090 0.040 0.015 0.15 0.03 0.3 4 0.40 0.16 0.059 0.019 0.19 0.04 0.4 5 0.50 0.25 0.081 0.022 0.22 0.03 0.3 6 0.60 0.36 0.107 0.026 0.26 0.04 0.4 7 0.70 0.49 0.136 0.029 0.29 0.03 0.3 8 0.80 0.64 0.169 0.033 0.33 0.04 0.4 9 0.90 0.81 0.206 0.037 0.37 0.04 0.4 10 1.00 1.00 0.246 0.040 0.40 0.03 0.3 11 1.10 1.21 0.289 0.043 0.43 0.03 0.3 12 1.20 1.44 0.336 0.047 0.47 0.04 0.4 13 1.30 1.69 0.387 0.051 0.51 0.04 0.4 14 1.40 1.96 0.441 0.054 0.54 0.03 0.3 15 1.50 2.25 0.498 0.057 0.57 0.03 0.3 16 1.60 2.56 0.559 0.061 0.59 0.04 0.4 17 1.70 2.89 0.624 0.065 0.65 0.04 0.4 18 1.80 3.24 0.691 0.067 0.67 0.02 0.2 19 1.90 3.61 0.762 0.071 0.71 0.04 0.4 20 2.00 4.00 0.836 0.074 0.74 0.03 0.3 21 2.10 4.41 0.912 0.076 0.76 0.02 0.2

Attach a plot of the position of the cart as a function of time. This should be labeled as GRAPH #3. Attach a plot of the position of the cart as a function of time squared. This should be labeled as GRAPH #4. Extract the slope of this plot.________ 𝟎.𝟏𝟖𝟖± 𝟎.𝟎𝟎𝟏 m/s2___________ Attach a plot of the velocity of the cart as a function of time. This should be labeled as GRAPH #5. Extract the slope of this plot._________ 𝟎.𝟑𝟒𝟔± 𝟎.𝟎𝟎𝟐 m/s2_________ Attach a plot of the acceleration of the cart as a function of time. This should be labeled as GRAPH #6.

phys181 lab manual spring 2015

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