PER Gr. 12 Physics Student Workbook (June 2011)

York Mills Collegiate Institute

SPH4U Course Notes 2011-12

Student Name:

SPH4U: Syllabus Instructor: Mr. Meyer

Updated: June 2011

Website: http://moodle.abel.yorku.ca/course/view.php?id=2456 This syllabus contains a list of all classes, topics and homework in the Gr. 12 physics course. You should always take notes from the listed readings. The simulations and videos listed below are optional but quite interesting! All links are also found on the course website – bookmark this for regular use. An excellent set of videos offering help with lessons and homework questions can be found at: www.physicseh.com and http://www.khanacademy.org/#physics. The Physics Classroom has an excellent set of tutorials: http://www.physicsclassroom.com/Class/. Introduction Day Topics 0 Course Introduction How to Be Smart 1 Intro to Group Work 2

Taking Notes Measurement

3

Numbers, Estimations and Fermi Questions

Kinematics and Projectiles 1 Short quiz on Introduction The BIG 5 Equations Intro to Problem Solving 2

CGPS: Modeling Problem Solving

3 4

CGPS Representations of Motion

5 6 7

CGPS or review Intro to 2-D motion Vector components

8

Intro to projectiles

9

Projectiles Problem Solving

10

CGPS

Follow-Up Video: Top 10 Amazing Physics Videos Read: course handouts. Actually read the handouts. Notes: pg. 748-9 “Scientific Notation” Investigation: Numbers and Physics Video: Scientific Notation Investigation: Fermi and Numbers Lesson: Fermi Problems

Gr. 11 Review Questions: 1-D Kinematics Gr. 11 Review Lessons: Kinematics Notes: pg. 24-7 Problems: pg. 27, #20-21, pg. 37, #9 * Optional - Handbook: Review: Graphing Kinematics Problems: pg. 65 #18, 19 a, b Simulation: The Moving Man Notes: pg. 35-8, pg. 65 #21, 26 Notes: pg. 11-3, 20-4, pg. 14 #15, 16 Handbook: Converting Graphs of Motion Video: Converting Between Graphs - Slopes Video: Converting Between Graphs - Areas Problems: pg. 24 #11, pg. 65 #27, pg. 67 #46 Simulation: Maze Game Handbook: Vectors and Components,two pages Notes: pg. 756-7: Vectors – Components of Vectors Video: Components Video 1 Video: Components Video 2 Video: Components Video 3 Notes: pg.41-45 Problems: pg. 46 #3, 5 Video: Velocity Vector Components Notes: pg. 46-49 Handbook: Problems - Projectile Motion, finish Video: Projectile Problem Solving Problems: pg.50 #10 Simulation: Projectile Motion

1

11

Test on kinematics and projectiles

Dynamics 1 Introduction to Forces Free Body Diagrams 2

Forces in 1-D

3 4

2-D Forces 2-D Forces continued Problem Solving Strategies

5 6 7 8

CGPS Newton’s 3rd Law: Part I Newton’s 3rd Law: Part II Motion of Combined Objects Weight

9

Frames of Reference

10 Tension and Pulleys 11 CGPS 12 Friction 13 Friction

14 Circular Motion Intro

15 Circular Motion and Forces

16 Circular Motion 17 Universal Gravitation 18 Orbits 2

Review: Projectiles (try #1-14, 35-39, 42-45, 51-53, 66, 71) NOTE: 2-D motion and relative motion are not part of this unit.

Gr. 11 Review Lessons: Newton’s Laws Gr. 11 Review Questions: Newton’s Laws Notes: pg. 72-73 Problems: pg. 73 #4 Notes: pg. 77, 81-82 Problems: pg. 80 #1, 6a,b Problems: pg. 83 #10, 13 Simulation: Forces in 1 Dimension Video: Simple Force Example Problems: pg. 73 #7 Problems: pg. 92 #6, 7 Simulation: The Ramp: Forces and Motion Video: Forces with Angles 1 Video: Forces with Angles 2 Problems: pg. 96 #8, 9 Notes: pg. 93-94 Problems: Pg. 94 #10, pg. 95-96 #3, 7 Lesson: Multiple Bodies Notes: pg. 84, 148 Problems: pg. 84 #16-17, pg. 150 #7, 8, pg. 169 #27 Notes: ahead for next class. Video: Forces and Elevators Notes: pg. 108-110 Problems: pg. 110 # 1-3 Video: Frames of Reference Notes: pg. 91 Problems: pg. 87 #9, pg. 92 #5 Video: Atwood Machine Problems: pg. 95 #5, 10 Notes: pg. 97-101 Problems: Pg. 101 #3, 5, 7 Problems: Pg. 118 # 16, 25(a) Simulation: Forces and Motion Video: Inclines with Friction Video: Pulley, Incline, Friction Notes: 122-126 Problems: Pg. 126 #8, 10 Simulation: Ladybug Motion 2D Video: Circular Motion Idea Video: Circular Motion Idea 2 Notes: pg. 128-133 Problems: Pg. 133 #6, Pg. 138 #7 Simulation: Ladybug Revolution Video: Tension in Vertical Circle Notes: pg. 139-143 Video: Conical Pendulum Video: Frames of Reference Problems: Pg. 141 #3, pg. 143 #10, 12 Simulation: Lunar Lander Video: Universal Gravitation Notes: pg. 145-147 Problems: Pg. 147 #2, 6, table pg. 776

19 Orbits, continued 20 CGPS

Lesson: Satellites Simulation: My Solar System Video: Gravity in Orbit Test Review pg. 168 #16, 19, 20, 23, 24, 26 Lesson: Orbital Calculations Video: Dark Matter Problems: Pg. 117 #7, 10, 24 (text answer wrong) Review: 2-D Forces Review: Gravity and Circular Motion (skip Kepler’s Laws)

21 Test on Dynamics Energy and Momentum 1 Impulse and Momentum

2

Conservation of Momentum in 1-D CGPS

3

Systems and the Process of a Collision

4

Conservation of Momentum in 2-D

5

2-D Momentum Investigation

6

Energy and Work

7

Conservation of Mechanical Energy

8

Gravitational Potential Energy Conservation of Energy CGPS Gravitational Fields: Part I

9

10 Gravitational Fields: Part II 11 Gravitational Fields: Part III 12 Spring Force and Energy Investigation

13 Elastic Energy Review

Notes: pg. 232-7, Problems: pg. 237 #6, 7, 10a Video: Impulse Lesson: Impulse Notes: pg. 239-44, Problems: pg. 243 #5, 7, 8 Video: Intro to Momentum Lesson: Conservation of Momentum Notes: pg. 246-8, Problems: pg. 248 #5, pg. 251 #11-13 Lesson: Isolated Systems Video: Slow Motion Collision Video: Smart Car Collision Handbook: 2-D Momentum Problem Solving Notes: pg. 254-8, pg. 257 #3, 4abd Video: 2-D Collision Problems: Pg. 257 #5 Simulation: Collision Lab Notes: pg. 178-83, Problems: pg. 181 #5, 7, pg. 183 #7 Notes: pg. 184-7 Problems: Pg. 186 #4, 6, 10 Lesson: Work Notes: pg. 189-91 and 195-201 Problems: Pg. 191 #1, 4, pg. 197 #6, pg. 200 #13, pg. 201 #6 Simulation: The Ramp Video: Roller Coasters Simulation: Energy Skate Park

Notes: pg 285 – 294, Problems: pg. 287 #3,5 Video: Ek and GPE Problems: Pg. 293 #8 Video: Binding Energy Problems: Pg. 293 #9, 12 Video: Energy in Orbit Notes: pg. 203-206, Problems: pg. 206 #5 Notes: pg. 207-210, Problems: pg. 211 #10, 12 Video: Hooke’s Law Video: Energy in Springs Video: Energy in Springs 2 Review: pg. 307 #4, 5, 10, 18, 20, 26, 30, 37, 48 Simulation: Masses and Springs

3

14 CGPS

Review: Momentum Review: Work and Energy

15 Test Special Relativity 1 Intro to Special Relativity 2

Postulates of Special Relativity Time Dilation

3

Length Contraction Visualizing Relativity

4

Energy and Relativity

5

Energy and Relativity

6

* Journey to Flatland

Notes: pg. 562 - 566 Video: In Search of the Edge Notes: pg. 569-72 Problems: Pg. 573 #2, 3 Video: Special Relativity Notes: pg. 573-575 Problems: Pg. 576 #5, 8 Video: Visualizing Relativity Notes: pg. 580 – 584 Problems: Pg. 583 #3, pg. 584 #5, 9 Video: Einstein Talks Notes: pg. 690-691 Video: Large Hadron Collider (LHC) Video: Proton Antiproton Collision Notes: pg. 730-731 Video: Dr. Quantum Visits Flatland Video: Hypercube

* = optional Electric and Magnetic Fields 1 Charge 2

Coulomb’s Law

3 4

Coulomb’s Law, continued Collections of Charges

5

Picturing Electric Forces

6

The Electric Field Concept

7

Fields, CGPS

8

* Energy in Electric Fields

9

Magnetic Fields

10 CGPS 11 Magnetic Forces on Charges

12 Electromagnetic Waves

13 Light and Polarization 4

Notes: pg. 318 – 322, Problems: pg. 324 #2, pg. 325 #5, 7 Notes: pg. 327-30 Problems: pg. 330 #3, 5, 7 Video: Charge and Coulomb’s Law Notes: pg. 331-334 Problems: Pg. 334 #8 Notes: pg. 337-343 Simulation: Charges and Fields Lesson: Electric Force and Fields Problems: pg. 343 #2, 4, 6, 7 Simulation: 3-D Electric Fields Lesson: Electric Field Lesson: Calculating Electric Fields Lesson: Calculating Electric Fields 2 Simulation: Electric Field Hockey Notes: pg. 349-353 Problems: pg. 358 #6 (practice), 1 Notes: pg. 384-7 Problems: pg. 391 #4, 7 Video: Magnetism Review Problems: pg. 358 #6 (questions), 7 Notes: pg. 392–5 Problems: Pg. 396 #2-4, pg. 403 #4, 7 Simulation: 3-D Magnetic Fields Video: Force on Moving Charges 1 Video: Force on Moving Charges 2 Notes: pg. 530-4, Problems: pg. 534 #1, 2 Simulation: EM Waves Simulation: EM Waves 2 Notes: pg. 494, Figure 3 Problems: pg. 498 # 4

14 Examples of Polarization 15 Test

Lesson: Polarization Video: Polarizing Filters Video: Polarization and Internal Stress Review: Electrostatic Forces and Fields Prepare next topic: 1-D waves Problems: Pg. 440 #7, 8, 11, 12

* = optional Light 1 Properties of 2-D Waves – diffraction, interference 2

Spatial interference: Part I

3

Spatial Interference: Part II

4

Young’s Double-Slit

5

CGPS

Gr. 11 Review: Waves Notes: pg. 453-4, 464, 467 Simulation: Wave Interference Video: Waves Review Notes: pg. 455-60, pg. 459 #3 Simulation: Sound Spatial Interference Simulation: 2-D Waves Lesson: Two Point Interference Video: Path Length Difference Problems: pg. 460 #4, 9 Simulation: Interference Lesson: Path Length Difference Notes: pg. 469-73, Problems: pg. 473 #1, 2 Lesson: Young’s Experiment Video: Double-Slit

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SPH4U – Physics, Grade 12 University Preparation York Mills Collegiate Institute Teacher: Mr. Meyer Course Website: http://moodle.abel.yorku.ca/course/view.php?id=2456 An Inquiry-Based Course Welcome to the wonderful world of physics! SPH3U is an introduction to the world of physics and a prerequisite for the grade 12 course, SPH4U. This course is designed according to the principles of Physics Education Research (see http://www.compadre.org/PER/) which clearly demonstrate the power of learning through inquiry in a collaborative group format. Major Canadian and American universities (U of T, McGill, McMaster, MIT, Harvard, Stanford and more) are transforming their introductory physics courses by reducing or eliminating traditional lectures and replacing them with engaging activities that have a deep conceptual and practical focus. MIT: http://www.nytimes.com/2009/01/13/us/13physics.html U of T: http://www.upscale.utoronto.ca/Practicals/Overview/Overview.html Harvard: http://youtu.be/WwslBPj8GgI Text reading, Note-Taking and Homework The majority of the class time will be spent doing activities and discussing physics with your colleagues. To accommodate this, you are required to do about 30 minutes of textbook readings almost every night and to take reliable notes from those readings. In class I will clarify and amplify the text in the minimum time possible and allow you to then explore the concepts through activity and discussion. In addition to the readings, you will have about 30 minutes of questions to solve. Daily textbook notes and homework problems will be randomly checked for marks. After-school study sessions will take place every Tuesday and Thursday. Plan enough time for one hour of physics homework each day. Optional online lessons and resources are listed Potassium Channel - It's All Physics for each lesson. Assessment and Evaluation Due to the central role of group work in this course, the work you do in groups will account for an important portion of your mark. Daily group work will be randomly handed-in and marked. To help

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ensure that individual students are pulling their weight in groups, there will be regular quizzes based directly on group work. A bonus will be given to groups whose members all score at least 75%. The content from your group work and home study will, of course, also appear on tests. There will be regular tests that survey each unit of our physics course. There is a final exam that covers the course's entire material and a few small projects that will be announced throughout the course. Mark Breakdown The categories of Knowledge and Understanding, Thinking, Communication, and Application are a component of most of the evaluation tools used in this course – however some focus on certain categories more than others. The basic mark breakdown for the course is 70% term work and 30% final examination. The term mark is composed approximately as follows: Knowledge and Understanding Thinking

28%

Tests

14%

Communication

14%

Application

14%

Daily group work (7%) and Group work quizzes (7%) Tests (8%) and Note Checks (6%) Projects

Large Hadron Collider

Attendance and Punctuality Most of your work takes place in groups and by being either absent or late you handicap yourself and your group. Students are responsible for determining what was missed and making sure that they are caught up before the following class – exchange phone numbers and consult your group members as your first step. Any evaluations of group work a student is absent for will be awarded a zero unless a valid reason with appropriate documentation is presented when the student returns to school. Missed Tests If you miss a test you must: • Let me know in advance if it is due to a pre-arranged reason (i.e. appointment for surgery) • Call in to the school so your name goes on the daily “Absent List” in the main office. • Contact me immediately after setting foot in the school upon your return. • Provide the “I Missed a Test” form complete with a valid reason. • Provide a doctor's note if the reason is illness. • Do not discuss the test by any means with your colleagues. • Be prepared to write the test immediately, at my discretion. Failure to do any of these will result in a zero for that evaluation. Did Anybody Read This? Please sign below signifying that you have read this course introduction. __________________________________ Signature of parent, or student if 18 and over

__________________________________ Parent e-mail for regular updates

__________________________________ Print name 7

I Missed a Test Student's Name:

Test Topic: Date of Test: Time of Test:

Reason for Absence (attach a doctor's note if ill):

Time Period of Absence from School:

Contact Information (a parent if the student is under 18, an office or someone to confirm these details with if the student is 18 or over) Name: Phone Number: Convenient Time to Contact:

Date of Form's Completion: Signature (parent, or student if 18 and over):

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SPH4U: How Groups Work Each group needs a whiteboard, marker and cloth. Assign each group member one role: Manager, Recorder, or Speaker. If there are four people in a group, two will act as the speaker. Working well in a group is a bit like acting in a play, we all have roles to perform!

Recorder: __________________ Manager: __________________ Speaker: __________________ Com / Know / Th / App: 0 1 2 3 4 5

Manager: Ask the group members to read the following instructions for this activity. The majority of our work in Gr. 11 physics will take place in groups. Take a few moments to think about our experiences of working in groups. Think about your experiences in other courses and your experience so far in Gr. 11 physics. We will discuss these experiences, but please don’t mention anyone’s name! Manager: Ask the group to complete the next two questions individually, without any discussion. When you see that everyone has finished, have the group move on. Complete the following two questions individually. 1. In your experience, what are some of the enjoyable characteristics of working in groups?

2. In your experience, what are some of the less-enjoyable characteristics of working in groups?

Work together now. On your whiteboard compile a list of the group’s responses to each question. Manager: Organize the discussion and ask for ideas from each group member. Recorder: Neatly summarize the ideas on the whiteboard, write large enough so other groups could read it if you were to hold it up. Speaker: Be prepared to speak to the class about your points when your group is called upon – if any points are unclear, ask your group questions. Continue the following questions as a group. Manager: Read out the next question and ask the group for their ideas. Kindly ask everyone for their input. Recorder: Make sure what you write down on your own sheet accurately represents the group’s ideas – your teacher will be checking your copy. Ask the other members for clarification if you’re not sure you have it right. Speaker: Be prepared to speak on behalf of the group. If any ideas are not clear, ask the others for an explanation or ask specific questions. Make sure the group explanations would receive a mark of “5” – are they thorough and complete? 3. Despite your experiences, what are some possible benefits for students learning physics by working in groups?

Manager: When the group agrees that it has finished with question #3, ask them to read the next instructions. We have all experienced difficulties working in groups. Sometimes, the challenge comes from within – for whatever reason you, as an individual, are unable to contribute effectively to the group. Other times, another group member may make the proper functioning of the group difficult.

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4. Think about the reasons why a group might not function at its best. Make a list of the reasons in the chart below – be specific. However, do not mention the names of any individuals. This is not a critique of your current group or any others you have been in. Reason 1.

2.

3.

4.

5. Title the second column in the cart above “Actions”. Describe what actions could be taken to help the group work better in each case you listed above. Indicate which group member (R, M, S) would be best to carry out the action, or if it is an action for everyone (E). Check your results with your teacher. Manager: When the group decides it had finished question 5, call the teacher over. Keep an eye on the clock since we want to complete the whole activity in this period Recorder: The teacher will ask you to write up one example on the whiteboard for a class discussion. Have the others check this. Speaker: Be prepared to speak on behalf of your group when called upon. Make sure the action is clear and precise. Manager: Ask your group to read the next instructions. Work individually on the next question. Complete the part corresponding to your role in the group. When complete, share and discuss your response with the others. Complete a chart with three columns on the whiteboard which summarizes the duties. Check this with your teacher. Complete your individual chart. Manager: Lead this process. When your group agrees it is complete, ask your teacher to check the whiteboard. Recorder: Make sure everyone agrees with what is on the whiteboard. Speaker: Be prepared to speak. If any points are unclear, start a discussion. Are the responses “5”s? 6. Summarize the duties of each group role. Manager

10

Recorder

Speaker

SPH4U: Group Work The Idea Group work is the main teaching format of the Gr. 12 physics course. Think of your group as your learning team - the people who will help you learn physics. Group membership will change every 10 to 20 classes or major unit, depending on the flow of the units in the course and will always have a heterogeneous composition of students (all ability levels) of my design. Every student is expected to take-up a specific role within the group and to carry out the responsibilities listed below. Members of every group will evaluate one another on their performance in their respective roles. Roles within a group must change for each new task or activity. Group Roles Actions Manager • Make sure everyone has read the initial instructions before starting. • Direct the sequence of steps. • Keep your group "on-track." • Make sure everyone in your group participates. • Watch the time spent on each step. Recorder • Act as a scribe for your group. • Check for understanding of all members. • Make sure all members of your group agree on plans and actions. • Make sure names are on group products. Speaker • Speak on behalf of your group when called upon in class discussions • Help your group avoid coming to agreement too quickly. • Make sure all possibilities are explored. • Suggest alternative ideas. • Energize your group when motivation is low by suggesting a new idea Seating When working in groups, please sit at the tables as illustrated to the right. This helps ensure that all members are able to interact easily with one another. When sitting three in one row, usually one person at the side is left out. I will constantly harass you to do this.

What it sounds like “Has everyone had a chance to read this before we continue?” "Let's come back to this later if we have time." "We need to move on to the next step." "Ralph, what do you think about this idea?" "Do we all understand this diagram?" "Explain why you think that." "Are we in agreement on this?"

"What other possibilities are there?" "Let's try to look at this another way." "I'm not sure we're on the right track."

human

human Table

human

Whiteboards One of the best ways to share work and ideas is using a whiteboard and your group's common workspace. This is much easier than all huddling around one sheet of paper. Please use these regularly!

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SPH4U: How to Answer a Question? Here is a question given to a grade 11physics class after a kinematics activity where a car travels either directly or along a zig-zag path from its start to a bowling pin.

Recorder: __________________ Manager: __________________ Speaker: __________________ Com / Know / Th / App: 0 1 2 3 4 5

Did today’s activity help to differentiate between the terms distance and displacement? Mark the student responses shown below using the following criteria: a) Does it clearly answer the question asked? b) Is it complete? c) Is the physics correct? d) Is it grammatically correct? “Yes, since the distance the car travelled was not in a straight line it increased distance. When the car was travelling straight it represented displacement.” 0 1 2 3 4 5 because _________________________________________________________________ “It did but since we weren’t measuring every way the car went, (not straight) we didn’t see the distance so that activity primarily focused on displacement. Displacement is the length from two points that an object travels and the change in the location of an object when it is moving between two points. Distance is how far the car went. I think today’s activity was great. ” 0 1 2 3 4 5 because _________________________________________________________________ “Today’s activity helped me to differentiate between the terms distance and displacement. Distance (Δd) is the

r

path length that the car traveled. Displacement ( Δd ) is the length of a straight line segment directed from the spot where we dropped the car to the bowling pin.” (Labelled diagram included) 0 1 2 3 4 5 because _________________________________________________________________ “Yes it does.” 0 1 2 3 4 5 because _________________________________________________________________ “Yes. The displacement stayed the same while the distance changed. This distance is the amount of space the wheels travelled during each run. ” 0 1 2 3 4 5 because _________________________________________________________________ “Yes. Distance is when the car went, hitting the target after going zig-zag across the table. And displacement was when the car went straight to hit its target.” (Labelled diagram included) 0 1 2 3 4 5 because _________________________________________________________________ “Distance – the length travelled in a motion (Δd) r Displacement – a straight line from initial position to a final position ( Δd )” 0 1 2 3 4 5 because _________________________________________________________________ “Yes because the car kept turning so it would be displacement but finally we got it to go straight.” 0 1 2 3 4 5 because _________________________________________________________________

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SPH4U: Taking Notes and Homework Problems Taking Notes: The Idea This course has been designed to maximize the value of our classroom time together. Since I will do very little lecturing in that time, the majority of the content, or information that you must learn will come from the textbook. In most cases, the text does an excellent job introducing and explaining the new topics, so you will create your class notes directly from that. There are some cases where the text is lacking, and so I will provide very short lessons or alternate readings to fill those gaps. Why You Will Take Good Notes Reading is great, but for most of us, it is a very passive activity. We tend not to read critically, meaning we do little to digest the ideas and confront them head-on - basically always asking the question, “Oh really, why?” Taking notes provides one step in the process of assimilating the new ideas. (One which is lost, by those who simply highlight “relevant” passages in a book.) Good notes will provide you with a valuable summary of the course contents when the time comes around for a test or an exam, precisely when time is in shortest supply. The final reason why you will take good notes is that I require you to! I will occasionally check on and mark your production of notes, with and without warning. How to Take Good Notes Read the entire assigned passage without taking notes. Always be asking yourself the question “Oh really, why?” as you go. Take your time. Organize your note page for maximum utility. They should all go like this: Sample Notes Pg. 12 Unit: Newton's Laws Topic: Newton's 2nd Law Text Reference: pg. 77, 81-2 Idea: Definition: Key Examples: Idea: Key Examples: Problem: pg. 80 #1 etc.

•

• •

Number your note pages to help keep them organized. Include the text reference so you can easily go back and double check things when necessary. Never just copy information out of the text. That is as useless as mindlessly copying down notes from a blackboard – we are trying to avoid this. Begin by deciding what the important idea or concept is, and use the text to help you describe it. This is how you start to assimilate ideas rather than just memorize them. In a given reading usually there is only one or maybe just a few key ideas. Be brief but use correct physics terminology. Write in point or sentence form. Emphasize the definitions of new, important physics terms. Provide key examples. This may be an explanation, description or a calculation. Try to decide on the most relevant or useful ones. The text is always there to provide the rest.

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• • •

Repeat for each important new idea you identify. Quickly scan over the final note and ensure you have covered the important ideas, not missed anything critical and that it makes sense. I will check whether these things are present in your notes. The exact layout is up to you (colour, underlining, etc.)

Homework Problems The purpose of these problems is to give you an opportunity to try out a new skill or wrestle with a new concept. Compared with the work I assigned in the past, I now give less than half the number of questions. This I do for an important reason. Quantity is not the key to understanding or building skills – care, thoughtfulness, and diligence are. Fewer questions leaves time for your note taking and good thinking. The chosen questions focus on the key ideas and skills of the topic, and are absolutely critical that they are completed right away. Do this: Solve your problems in your regular notes, immediately following your summary of the readings, as shown above. Why: This keeps you organized and encourages you to do it right away rather than wait until the week-end. Do this: It is not necessary to copy out the question – you can always look back at the text. What you should do is provide a complete GRASP process for each one and a simple diagram. This does not mean something lengthy, but rather something complete. Why: When it comes time to study for a test or the final exam, time is always in short supply. Don't skimp now on your work and leave yourself with something that is of only marginal use as a study aid. Four months from now at exam time, you may not remember how to solve such a problem. When you go to university and find out you do need to take Physics 101 and you look back at you high school notes, you may not remember how to solve such a problem. Do this: When solving your problem, pretend a good friend is sitting beside you - one who is reasonably intelligent, pretty good at math, but really knows nothing about physics (a shame, really). They have a particular habit of constantly asking one question over and over: “why?” or occasionally, “huh, how come?” And try to explain to this friend how you chose to do each step, why it works, or why it's important. Why: This is another step in the assimilation of new ideas. Students often memorize a routine or algorithm for solving a type of problem. However, if they are not aware of why we do what we do (and this I always check on tests / exams), one small alteration to a question from what they are used to spells disaster (often happens on tests/ exams). In this course memorization will not do - only understanding will (especially on tests / exams). Stuck? We often are, all of us. Even simple questions can seem tricky. But in class, there is little or no time for taking up homework – that is your responsibility. Talk and check with you group members regularly about the problems – especially if you finish your group tasks early. Visit the after-school study sessions. Or talk to me right away if you need help. Don't wait until the test. Remember the golden rule!

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SPH4U: The Art of Measurement Measurements are the backbone of all science. Any theory, no matter how slick, is only as good as the measurements that have confirmed it. Without careful measurements, science is mostly guess work and hunches.

Recorder: __________________ Manager: __________________ Speaker: _________________ Com / Know / Th / App: 0 1 2 3 4 5

A. The Meter Stick The most basic scientific tool is the meter stick. But, do you know how to use it? For this investigation you will need one meter stick 1.

Examine the markings on the meter stick. What is the smallest interval marked?

2.

Three students use the meter stick to measure the height of a desk and each reports their results: 95 cm, 94.8 cm, 95.03 cm respectively. Considering the intervals marked on the meter stick, which result illustrates the best use of this measuring device? Explain.

3.

How accurate a measurement can you make with a meter stick (i.e. to the nearest…)? This represents the reading error of the metre stick.

The term significant figures describes which digits in a number or measurement are physically meaningful or reliable. 4.

How many significant figures can you get from a typical meter stick measurement?

5.

Measure the height of your desk, write it with an appropriate number of significant figures and include the error. (i.e. 45.7 cm ± 0.1 cm)

B. The Stopwatch Now we will examine another common measuring device. You will need one stop watch A student drops a pencil from a 1.00 m height. Another student times the fall. The stopwatch readout looks like this after the timing: 0:00.45 1.

Write this reading as a number in standard notation with units of seconds (s).

2.

What is the accuracy of the stopwatch according to its display (i.e. to the nearest …)? This is the stop watch reading error.

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3.

Perform the measurement three times, record the times below and calculate an average time. Avg:

4.

Do all the time measurements fall within the reading error of the stopwatch? Suggest reasons why.

There are often many sources of error in an experiment which will have an additional effect on the accuracy of a measurement. 5.

How many significant figures and what error are reasonable for the calculated average? Explain.

General Guideline for Significant Figures: When performing calculations, write the intermediate results with one extra significant figure and the final answer with no more significant figures than the least accurate piece of data. This is a handy rule of thumb. In university you will learn a mathematical system for determining the error in your calculated results which will replace this handy rule. General Guideline for Errors: Whenever you record a measurement, include the error, or note the error for a set of similar measurements. When measured quantities are used in calculations, the best we can do in grade 12 is to estimate a reasonable final error based on significant figures. In university you will learn more reliable techniques. We often compare calculations or predictions with measurements and try to decide if the two numbers agree, or are “close”. This can only be done in a reasonable way when errors are taken into account. Roughly speaking, if a measurement is within its error range of the other quantity, they may be said to agree or are “close”. 6.

A student uses a stopwatch to measure the time for a pencil to drop and reports a reading of 0.41 s. Another student calculates that the time for the pencil to fall is 0.30 s. Does the measurement agree with the prediction? Explain.

When writing the results of certain measurements in standard notation, there can be some ambiguity about the number of significant figures. 7.

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A student measures a distance and reports the result: 2400 m. Two students read the result and one claims this measurement has two significant figures while the other claims that it has four. Explain how both may be correct based on how the measurement was made.

SPH4U: Numbers and Physics Physicists think about numbers different ways than most people and even mathematicians. For example, if you think about the idea of numbers (quantity), there are really three kinds: BIG numbers (greater than one), unity (equal to one), and small numbers (less than one). It is important to know how these numbers behave under mathematical operations. All your work on this page should be done without a calculator! 1. Estimate whether the result of each expression is BIG, small, or close to one. a) 1 / BIG b) 1 × BIG c) 1 / small d) 1 × small e) BIG + small f) BIG – small g) BIG × small h) BIG / small i) small / BIG j) BIG / (small + BIG) k) BIG × BIG Physicists are often interested in the general patterns illustrated by numbers rather than their specific values. Students and even some teachers rely too much on calculators to do their thinking about numbers. As a physicist you should feel comfortable thinking about and using numbers in scientific notation without a calculator in sight! 2. Describe an easy way to compute: 6×106 + 5×106 without a calculator.

3. Compute these expressions. No calculators! a) 6.5×105 + 3×105 = b) 6.4×1012 + 2.9×1012 = 4. Describe an easy way to compute: (3×102)(6×106) without a calculator.

5. Compute these expressions. × 2×104 a) 3×104 2 × 8×102 b) 6×10 a) 4.9×10340

÷

= =

7×1090 =

6. Describe how to use estimations and scientific notation to easily compute: 2 168 222 × 4 937 without a calculator.

7. Estimate the results of these expressions. a) 1 168 222 × 6 900 000 = b) 0.0529 × 8.0036 = 17

SPH4U: Fermi Questions

Recorder: __________________ Manager: __________________ Speaker: _________________

Use point-form explanations and simple calculations to justify your Com / Know / Th / App: 0 1 2 3 4 5 estimated answer for each problem. Important: whenever you simply state a number, unless it is a well-known fact, you must justify it with a simple statement. State the final answer in scientific notation with only one significant figure. 1. How many litres of water are used for drinking purposes each year in Canada?

2. How many cups of water are needed to fill a standard bathtub?

3. How many days would it take a person to walk a distance equal to that from Earth to the Moon?

4. What is the volume in cm3 of 1 mole of aluminum? (The density is 2.7 g/cm3. The mass number is 27)

Fermi about to press a button 18

5. How many students are enrolled in Canadian schools, colleges and universities?

6. What is the volume of an average adult Canadian female in cm3?

7. How many breaths would Julius Caesar have taken if he were still living today? Fermi calculating the size of his bald spot.

8. How many postage stamps would be required to cover a football field?

Write all you answers, here complete with units, and check your score with your teacher! 1.

2.

3.

4.

5.

6.

7.

8.

9.

19

SPH4U: Fermi and Numbers! Write a complete solution (don’t skimp on the steps) for the Fermi problem: How many litres of gas are consumed by Canadian cars each year?

How are your number skills? Getting better? Try some more! 1. Adding numbers using scientific notation a) 6.0 x 107 + 5.0 x 107 = b) 1.9 x 103 + 3.4 x 103 = c) 7.6 x 10-2 + 1.8 x 10-2 = d) 4.6 x 104 + 6 x103 = 2. Subtracting numbers in scientific notation a) 4 x 103 – 8 x 103 = b) 1 x 105 – 4 x 105 = c) 2.0 x 1011 – 1.4x1011 = d) 9.0 x 106 – 8.0 x 106 = 3. Multiplying numbers in scientific notation a) 4 x 107 * 2.0 x 103 = b) 1.0 x 103 * 6.4 x 10-1 = c) 6 x 10-2 * 7 x 10-4 = d) 1.2 x 104 * 4 x103 = 4. Test yourself! Do this in less than 2 minutes! a) 7.6 x 108 – 1.4x108 = b) 7.0 x 10-9 * 4.0 x 107 = c) 6.6 x 1013 ÷ 6.0 x 105 = d) 1.2 x 10362 + 8.4 x 10362 = i) 7.6 x 108 – 1.4x108 = j) 7.0 x 10-9 * 4.0 x 107 = k) 6.6 x 1013 ÷ 6.0 x 105 = l) 1.2 x 10362 + 8.4 x 10362 = m) 8.0 x 106 * 1.1 x 10-4 = n) 9 x 10-4 - 2 x 10-4 = o) 3.9 x 10-3 ÷ 1.3 x 10-8 = -5 -5 p) 2.9 x 10 + 3.1 x 10 =

20

e) f) g) h) e) f) g) h)

8.7 x 102 + 3.53 x 103 = 1.0 x 10-4 + 9.1 x 10-4 = 2.9 x 1012 + 5.1 x 1012 = 6.1 x 10-3 + 7 x 10-4 =

9 x 10-4 - 6 x 10-4 = 5.8 x 109 – 1.2 x 109 = 3.5 x 106 – 2 x 105 = 1.7 x 103 – 1.6 x 103 = e) f) g) h)

7 x 10-5 * 3 x 10-7 = 3.0 x 10-12 * 3.0 x 1012 = 4 x 1012 * 8 x 1018 = 1.5 x 10-3 * 3 x 10-2 =

e) 8.0 x 106 * 1.1 x 10-4 = f) 9 x 10-4 - 2 x 10-4 = g) 3.9 x 10-3 ÷ 1.3 x 10-8 = -5 -5 h) 2.9 x 10 + 3.1 x 10 =

SPH4U: Uniform Acceleration In grade 12 physics we will analyze in detail situations where an object’s acceleration is uniform.


A. Uniform Acceleration 1.

State a simple definition of uniform acceleration.

2.

Three situations are described below. Describe each interval of time in which the object is accelerating uniformly. If the acceleration changes, describe the moment in time when the change begins or ends. (a) a blob of Jello drops downwards, (b) a cyclist glides down a steady hill (c) a ball starts from rest in a strikes the ground and comes to (friction and air resistance are baseball player’s hand, is released, rest (air resistance is negligible) negligible) travels straight up and reaches a maximum height of 15 m

3.

Explain to your friend, who isn’t taking physics, a simple way to decide whether an object experiences uniform acceleration.

B. The BIG 5 The equations in the table to the right are affectionately known as the “BIG 5” equations of uniform acceleration. When an object is accelerating uniformly, the five kinematic quantities, Δd , v1 , v 2 , a , and Δt , for that time interval are related by the BIG 5 equations. Note that v1 and v 2 are the instantaneous velocities found at the start and end of the interval.

v v v v2 = v1 + aΔt v v v Δd = v1 Δt + 12 a ( Δt ) 2 v v v Δd = 12 (v1 + v 2 )Δt v v v Δd = v 2 Δt − 12 a ( Δt ) 2 v v v v v22 = v12 + 2aΔd

Δd

v1

v2

a

Δt

1.

Place an “X” in the column of any quantity that is not found in each equation.

2.

Describe how you can use the chart to help choose an equation to solve a problem with. For example, v1 = 5 m/s, a = 2.6 m/s2, Δd = 12.7 m, Δt = ?

3.

Using one of these equations, how many quantities (pieces of data) do you need to know in order to be able to solve for an unknown quantity?

4.

(together as a class) What is Mr. Meyer's the magic saying for solving problems using the “BIG 5”

21

Philip drops a rock from the top of a bridge and it lands on the ground below. Jennifer times the rock’s fall. They both want to find the distance the rock travelled. Philip says that they don’t need the time, since they know the final velocity (zero, it came to rest). Jennifer disagrees and says that they can’t use the value of zero for the final velocity. 5. Who do you agree with? Explain using the focus of today’s activity.

C. How to Write a Solution Solution writing is like writing an essay: not only must you have the right ideas, but they must be convincingly presented using proper grammar and form. The BIG 5 are vector equations. This means they take into account the direction of the kinematic quantities. A simple way to handle direction information is to use a sign convention and write down the BIG 5 as scalar equations. Full vector notation

Δd = 53 m [W] v1 = 12.4 m/s[E] a = 0.59 m/s2 [W] v v Δt = 1? v Δd = v1 Δt + 2 a ( Δt ) 2

Scalar notation with a sign-convention W+

↔-

E

Δd = 53 m v1 = - 12.4 m/s a = 0.59 m/s2 Δt = ? Δd = v1Δt + ½a(Δt)2

Each set up above is complete and correct; the scalar version is often simpler and quicker to write down. This will be our preferred method. A common method for writing solutions is the GRASP technique. This should be used for all our formal work and your homework questions. The five steps are: GIVENS Draw a diagram and attach all the given information along with symbols. Be sure to show the start and end moments of the problem. Indicate the coordinate system and sign convention. Perform any conversions. REQUIRED TO FIND Indicate on the diagram the quantity that you are looking for. APPROACH Write down the key equations you will use. SOLUTION Perform the mechanics of finding the answer – all the algebra. Then substitute numbers with units. Compute a final answer. PARAPHRASE State your result in a sentence which answers the question. Here is a sample problem to complete with the GRASP solution method. Work out your solution below and also write it on a whiteboard which will be used to share with the class. A world’s land speed record was set by Colonel John P. Stapp when in March 1954 he rode a rocket propelled sled that moved along a track at 1020 km/h. He and the sled were brought to a stop in 1.4 s. What acceleration, in m/s2 did he experience while stopping?

22

SPH4U

Review: Graphing Kinematics Definitions: Complete the definitions for the following quantities. Distance (Δd): r Position ( d ): r Displacement ( Δd ):

Determine the slope of the graph. d2 d

d1

t2

t1

t

The slope of a d-t graph represents ____________________. Define velocity mathematically: Summary: For any complicated graph, divide it up into sections at every point where its behaviour changes. Describe each section mathematically.

d

t O

A

B

C

D

Summary Graph Questions During which interval(s) is the object a) at rest b) moving forward d) moving quickest e) moving slowest g) furthest from the start

E

F

c) moving backward f) at zero displacement

23

Instantaneous and Average Velocity Determine the instantaneous velocity at t = 5 s.

d

5

5

t

The slope of a ______________to a d-t graph represents instantaneous velocity. s Define mathematically: v inst = Determine the average velocity from 4 to 6 s.

d

5

5

t

The slope of a ______________to a d-t graph represents average velocity. s Define mathematically: v avg =

24

Velocity-Time Graphs

Determine the slope of the graph. v2 v

v1 t

t2

t1

The slope of a v-t graph represents ____________________. Define acceleration mathematically: Describe the motion of this graph. v t

Acceleration In physics we avoid using the term deceleration. Instead we prefer the terms speeding up and slowing down. When is an object undergoing these types of motion? Complete the chart below. Speeding Up 9

Slowing Down

a

v

+

+

v-t Graph

d-t Graph

v

d t

+

-

-

+

-

-

t

25

Area Under Velocity-Time Graphs What is the position of the object at t = 0? If the object is traveling at speed v and for time t, determine the area under the graph carefully using units.

v t

Describe in words the motion of the object.

+5 2s

v (m/s)

4s

t

Determine the displacement of the object.

-5

The area under a v-t graph represents __________________. Summary: Describe each section of the following graph mathematically and with words.

+10

v (cm/s)

t (s) 2

5

8

12

16

18

-10

Summary Graph Questions 1. During which interval(s) is the object a) at rest b) moving forward c) moving backward d) moving quickest e) moving slowest f) at zero displacement g) furthest from the start h) moving with the largest acceleration 2. Determine r a) the displacement from 0 to 18 s, Δd (0 – 18 s) b) the distance traveled from 0 to 18 s, Δd (0 – 18 s) r c) the velocity from 0 to 18 s, v (0 – 18 s) d) the speed from 0 to 18 s, v (0 – 18 s) r e) Δd (2 – 12 s) r k) the displacement from 0 to 18 s, Δd (0 – 18 s) 26

f) g) h) i) j)

Δd (2 – 12 s) r v (2 – 12 s) v (2 – 12 s) r a (10 s) r a (14 s)

Acceleration-Time Graphs What is the velocity of the object at t = 0? If the object is traveling with acceleration a for time t, determine the area under the graph.

a t

Describe in words the motion of the object.

+7

a (m/s) -2

10 s

20 s

t

Determine the change in velocity of the object.

The area under an a-t graph represents _____________________________.

27

Summary

Complete each set of graphs from the one given.

d

d

d

d

d

v

v

v

v

v

a

a

a

a

a

In the chart, very briefly describe how to obtain the listed quantity from the given graph. For example, “the slope”, is a sufficient answer for one or more of these! Technique Quantity position displacement inst. velocity avg. velocity change in velocity acceleration

28

Position-Time Graph

Velocity-Time Graph

Acceleration-Time Graph

SPH4U: Sample Kinematics Test

Name:

Graphing calculators are not permitted. Use the GRASP problem solving method to show your work for full marks. The multiple choice questions will be collected five minutes before the end of the test.

Multiple Choice Mark:

1. (10 marks) The average NBA player can make a vertical leap of 71 cm above

/38

/12 Total Test mark:

the ground. To do this, the player’s crouches by 48 cm before jumping. Determine the acceleration of the player while in contact with the ground.

2. (6 marks) You throw a ball straight up in the air. At time t=0, the ball is at rest, at your side. It starts moving and is released from your hand near headheight, travels upwards and falls to the ground. Sketch a d-t and v-t graph for the ball. Note the moments in time marked on the graph. (A) The ball begins to move upwards. (B) The ball is released from your hand. (C) The ball reaches its maximum height. (D) The ball reaches the ground.

A

B

C

D

d

t

v

t

29

3. (5 marks) Fermi Question. After a successful semester of Gr. 12 physics you decide to try a career as a stunt man/woman/thing. For practice, you stand on top of a speeding train, step off, and land in a swimming pool that passes by in the ground right beside the train. Calculate the distance in advance of the pool when you step off the train.

4. (5 marks)

Please circle as a group those items that tie for the same ranking

30

Sample Multiple Choice: Kinematics 1. Each of the situations below illustrates one or more of the physics terms in the chart. Choose the terms that are clearly exemplified by each situation. You may choose a term more than once, some may not be chosen at all, and more than one may apply to one situation. Situation Physics Terms i. You see the flashing lights of a police car A. Displacement in your mirror. A quick glance at your dashboard reveals you are travelling at 80 km/h. ii. A sky diver lands on the ground and come B. Instantaneous to rest. velocity iii. A rock is falling through the air. C. Constant velocity iv. You walk 2 km, to school and back, in 25 D. Average velocity minutes because you forgot your lunch. v. The traffic is very light and you make good E. Uniform time travelling at about 110 km/h the acceleration whole way home. vi. You see a person walk out in front of your F. Speeding up car so you slam on the brakes. vii. A bus is accelerating west at a steady rate. G. Slowing down viii. A car travels from the shopping mall parking lot to a house. 2. The position of a car during a test crash is recorded using a high speed camera to track a yellow dot painted on its side. The position of the dots from each successive frame is shown in the image below, where the car was moving to the right. The rightward direction is positive and the origin of the position measurements is the first dot. Choose the correct graph below in response to each question. Choose (G) if none are suitable. • (A)

• (B)

(D)

(E)

i. ii. iii.

•

• • • • • • • • (C)

(F)

Which graph represents the position of the car? Which graph represents the velocity of the car? According to the pattern of dots, is the car, at any time, (A) speeding up, (B) slowing down, (C) moving at a constant velocity, or (D) at rest. Choose more than one if appropriate. 31

Writing About Testing Worries Boosts Exam Performance in the Classroom Gerardo Ramirez and Sian L. Beilock Department of Psychology, University of Chicago “We demonstrate that a short expressive writing intervention reduces performance deficits commonly associated with highpressure testing situations. The benefits of expressive writing are especially apparent for students who are habitually anxious about taking tests. Expressive writing eliminates the relation commonly seen between test anxiety and poor test performance. Moreover, it is not any writing that benefits performance, but expressing worries about an upcoming high-pressure situation that accounts for enhanced exam scores under pressure.” From: Science 14 January 2011: Vol. 331 no. 6014 pp. 211-213

The following is the exercise used by students in the study. Before each major test and the final exam (immediately before if possible), find a quiet space and complete the exercise. We would like YOU to take the next 10 minutes to write as openly as possible about your thoughts and feelings regarding the test you are about to take. In your writing, I want you to really let yourself go and explore your emotions and thoughts as you are getting ready to start the exam. You might relate your current thoughts to the way you have felt during other similar situations at school or in other situations in your life. Please try to be as open as possible as you write about your thoughts at this time.

Cognitive Test Anxiety Scale Contemporary Educational Psychology 27, 270–295 (2002) Answer each question using the scale: 1- Not at all typical of me, 2 - Only somewhat typical of me, 3 - Quite typical of me, and 4 - Very typical of me. Read the scoring instructions at the bottom when you have answered all the questions. 1. I lose sleep over worrying about examinations.

*

*

* * *

*

2. While taking an important examination, I find myself wondering whether the other students are doing better than I am. 3. I have less difficulty than the average college student in getting test instructions straight. 4. I tend to freeze up on things like intelligence tests and final exams. 5. I am less nervous about tests than the average college student. 6. During tests, I find myself thinking of the consequences of failing. 7. At the beginning of a test, I am so nervous that I often can’t think straight. 8. The prospect of taking a test in one of my courses would not cause me to worry. 9. I am more calm in test situations than the average college student. 10. I have less difficulty than the average college student in learning assigned chapters in textbooks. 11. My mind goes blank when I am pressured for an answer on a test. 12. During tests, the thought frequently occurs to me that I may not be too bright. 13. I do well in speed tests in which there are time limits. 14. During a course examination, I get so nervous that I forget facts I really know.

15. After taking a test, I feel I could have done better than I actually did. 16. I worry more about doing well on tests than I should. *

17. Before taking a test, I feel confident and relaxed.

*

18. While taking a test, I feel confident and relaxed. 19. During tests, I have the feeling that I am not doing well. 20. When I take a test that is difficult, I feel defeated before I even start. 21. Finding unexpected questions on a test causes me to feel challenged rather than panicky. 22. I am a poor test taker in the sense that my performance on a test does not show how much I really know about a topic. 23. I am not good at taking tests. 24. When I first get my copy of a test, it takes me a while to calm down to the point where I can begin to think straight. 25. I feel under a lot of pressure to get good grades on tests. 26. I do not perform well on tests. 27. When I take a test, my nervousness causes me to make careless errors. Total score = Average score = Total / 27 =

Scoring: Reverse the scoring for each box marked with a *. Add up your scores and divide by 27. 1 = little text anxiety, 4 = high test anxiety

32

SPH4U: Introduction to Cooperative Group Problem Solving The purpose of this problem solving strategy is to help you learn the more sophisticated techniques that experts use. The focus here is thinking about the problem and planning before starting the math work. It is very important that you complete the set-up (parts A, B and C) before you do the math work (parts D and E). The manager has the critical role of ensuring this. The solution sheet will help to keep your group on track – be sure to fill it out as you go. Best way to work: As a group, start each step by jotting down ideas on the whiteboard. When they are agreed upon and understood, the recorder writes out the good version on the solution sheet. The group should agree on and understand everything that is written on the solution sheet.

Set-up

A. The Picture • Draw a clear diagram showing what’s happening • Attach the important information to the diagram using simple phrases • Make any important measurements • Attach the unknowns to the diagram if possible • Indicate the coordinate system and sign convention If this step is complete, you should never have to refer to the problem statement again. B. The Question • Create a specific physics question that will give the answer to the problem. • Indicate which quantities will allow you to answer the question. C. The Plan • Summarize how it works, mentioning the key physics ideas involved • Outline the key steps in solving the problem • List any useful “textbook” equations (i.e. found in bold / in a box in a text) and any other relationships you will use

Execution

Have you carefully completed all the previous steps? If not, go back! Note that you should not have done any real math work yet. D. The Work • Create the specific equations you will use –write them down with a simple statement explaining what you are doing. You should only use symbols that appear in Part A. • Perform the algebraic manipulations first, whenever practical. • Verify the units of the final derived expressions (the ones you will substitute numbers into). • No number crunching yet! E. The Results • Substitute numbers into your manipulated equations and calculate a result. • State the final answer in response to the question you created. • Write brief statements explaining why the answer seems reasonable in size, direction and units.

33

SPH4U: Representations of Motion


A cart travels along a track under a variety of situations. Use the Com / Know / Th / App: 0 1 2 3 4 5 information provided to complete all the other representations of motion. A coordinate system with the positive direction to the right is used and origin at the left end of the track. For the “Arrows” box, draw a vector representing the direction of each vector. For the “Velocity Vectors” box, draw a velocity vector at four evenly spaced intervals of time. 1

Description of Motion

Top View Strobe Diagram

Arrows Direction of v:

The cart moves in the positive direction and speeds up. Direction of a:

track Velocity Vectors

v1 v2 v3 v4 2

d vs t

v vs t

d

a vs t

v

a

t


t

t


Arrows

The cart moves in the negative direction with a constant speed.

Direction of v:


v1 v2 v3 v4 3

d vs t

Direction of a:

v vs t

d

a vs t

v

a

t


t

t

Top View Strobe Diagram 1

3

2

Arrows 4

Direction of v:


v1 v2 v3 v4 34

d vs t

Direction of a:

v vs t

d

a vs t

v t

a t

t

4





v1 v2 v3 v4 5

d vs t

Direction of a:

v vs t

d

a vs t

v

a

t


t

t



track

Velocity Vectors

v1 v2 v3 v4 6

d vs t

Direction of a:

v vs t

d

a vs t

v

a

t


t

t




v1 v2 v3 v4

d vs t

Direction of a:

v vs t

d

a vs t

v t

a t

t

Describe a different, real-life situation that illustrates the motion in the following examples from above: 1:__________________________________________________________________________________ 3:__________________________________________________________________________________ 6:__________________________________________________________________________________ from McDermott, L. Tutorials in Introductory Physics. Homework. Prentice Hall, 2002

35

SPH4U: Converting Kinematics Graphs 1. Given that the initial position and velocity are zero, complete the following velocity-time and position-time graphs from the data contained in the acceleration-time graph: position-time graph

position (m [f])

1600 1400 1200 1000 800 600 400 200 0

time (s) 0

velocity (m/s [f])

20

40

60

80

100

120

100

120 time (s)

100

120 time (s)

velocity-time graph 60 50 40 30 20 10 0 -10 0

20

40

60

80

-20 -30 -40

acceleration-time graph

acceleration 3 (m/s2 [f]) 2.67 2 1 0 0 -1 -2 -3

36

20

40

60

80

3. Given that the initial position is zero, and the following velocity-time graph, plot the corresponding position-time and acceleration-time graphs: velocity-time graph

velocity (m/s [up])

30 25 20 15 10 5 0

time (s)

-5 0

1

2

3

4

5

-10 -15 -20

4. Given that the initial position is zero, and the following velocity-time graph, plot the corresponding position-time and acceleration-time graphs: velocity-time graph

velocity 30 (m/s [W]) 25 20 15 10 5 0 0

10

20

30

40

50 time (s)

37

SPH4U: 2-D Motion


A: One Tap

r

Com / Know / Th / App: 0 1 2 3 4 5

Consider a hover puck moving with an constant initial velocity, v1 , as shown below. It is given one short ‘tap’ in a direction that is perpendicular to its motion. Note: All ‘taps’ in these scenarios are always directed east! When asked to explain, try to use the word ‘force’ where appropriate. 1. How will the ‘tap’ affect the puck’s velocity in the ydirection? Explain.

r v1

tap

W (+y)

2. How will the tap affect the velocity of the puck in the x-direction? Explain.

S

N (+x) E

3. Describe how the overall speed of the object has changed.

r

4. Predict and draw the path of the puck after that ‘tap’. Draw a vector, v 2 , representing the instantaneous velocity after the tap. You do not need to use a scale, but show the relative sizes carefully.

B: Off to the Races! Consider a race between two pucks. Each starts with the same initial velocity to the north. The puck A travels on a level surface. The puck B moves on an incline tilted to the east. They race to the finish line. To help think about what happens, imagine that the puck on the incline receives a continuous series of taps to the east. 1. Imagine using a strobe light, flashing at regular time intervals, to take a picture of the pucks at three moments in time. Draw the path of each puck and its position for each flash.

38

A B

finish line

r v1 r v1 W (+y)

S

N (+x) E

2. Describe how the velocity of the two pucks in the x-direction will compare.

3. Describe how the velocity of the two pucks in the y-direction will compare.

4. Based on your answers to questions 2 and 3 above, which puck will cross the finish line first?

C: Rematch! Consider another race. Both pucks are on the incline tilted east. The puck on the left starts from rest, the puck on the right has an initial velocity north.

A

r

B v1

1. Describe how you think puck A will move when it is released.

W (+y) S

N (+x) E finish line

2. Imagine using a strobe light, flashing at regular time intervals, to take a picture of the pucks at three moments in time. Draw the path of each puck and its position for each flash. 3. Compare the motion of the two pucks at the time they reach the east edge of the grid. a. Which puck has traveled a greater distance? Explain.

b. Which puck has reached a greater speed? Explain. Don’t assume anything about time!

4. Which puck will cross the finish line first? Explain carefully how the factors of distance and speed affect the result.

39


SPH4U: Vector Components It is time to analyze the motion of the puck on the ramp. Careful measurement were made by a student who found that the puck moved through a displacement of 4.0 m [N 30o E].

Com / Know / Th / App: 0 1 2 3 4 5

1. Draw this vector on the grid starting from the image of the puck using the scale: 1.0 cm = 1.0 m and label it

r Δd

2. During the 2.0 seconds of its motion, the puck moved both in the x- and y-directions. Draw a right-

r

angle triangle on the grid using Δd that shows how far the puck travelled in those directions. We will call this the component triangle. 3. Use a ruler and make a measurement to determine how far the puck travelled in the a) x-direction b) y-direction

W (+y) S

N (+x) E

A vector is a quantity with two parts: a magnitude (size) and direction and is usually notated with a vector sign on

r r

r

top ( Δd , v ). If we want to refer to the magnitude (size) of the vector only, we write v in absolute value signs. However, out of convenience, if it is understood that the quantity involved is a vector, we usually write v with no vector sign to indicate the magnitude. A vector can always be determined from the values of its components. The two small sides of the triangle you drew in question 2 represent the components of that vector which we can call Δd x and Δd y . The components are written without a vector sign. Each component must also show direction (N or S, E or W). This is done using a sign convention and by making the components either positive or negative.

r

4. Write down the components you found for Δd .

Δd x =

Δd y = r

5. Draw the velocity vector, v = 2.0 m/s [N 30o E] on the grid using the scale 1 cm = 0.50 m/s. Label the vector and construct its component triangle. a) Use the Pythagorean Theorem to find an equation r that relates the magnitude of v to the value of its xand y-components vx and vy.

W (+y) S

N (+x) E

r

b) Use your diagram to directly measure the values of the components of v vx =

vy =

r

c) Use the measured values of vx and vy to calculate the magnitude, v, of vector v . How does this calculated value compare to the original magnitude?

40

d) Use the definition of the sine and cosine functions to show that if the angle, θ, inside the triangle at the r tail of the vector v , is known then the values of vx and vy can be calculated from v. Note: Do not memorize these expressions! Always deduce them from your component triangle.

r

r

e) Use the angle at the tail of v and the given magnitude of v to calculate the values of vx and vy. How do these values compare with those you measured directly for vx and vy?

f) Show that if the vector components vx and vy are known, then the angle, θ, at the tail of the vector can be determined using the inverse tangent function.

g) How does the calculated value of the angle, θ, compare with the original value?

6. If a vector was reversed – flipped to the opposite direction – how would the values of the components change?

r

7. The same vector, Δd , is shown below with respect to two different coordinate systems A and B. Complete the chart below using measurements from the diagrams. Scale: 1.0 cm = 10.0 cm W (+y)

A r Δd

S

B N (+x)

E

r Δd = _______ cm [

Coordinate System A

N (+x)

r Δd

E

W (+y)

S

Coordinate System B

]

Angle with respect to the x-axis x-component y-component

41

SPH4U: Vectors and Components 1. Find the components of each vector with respect to the given coordinate system. Be sure to use the given sign convention. N(+ b) a) r r o E(+x A = 30 m [N 40 E] Δd = 20 km [E] y y Ax = Δdx =

x

x Ay =

c)

y

r F = 45 N [S 20o E]

x

Δdy =

d)

y

r v = 7.8 m/s [S 50o W]

x

Fx =

vx = vy =

Fy =

Answers: (19 m, 23 m), (20 km, 0), (15 N, -42 N), (-6.0 m/s, -5.0 m/s) 2. Given the following components and sign convention, reconstruct the original vectors. Sketch the vector diagram showing the addition of the components into the resultant vector. r a) Δdx = 300 m, Δdy = 400 m, Δd =

r b) vx = -12.4 m/s, vy = 10.0 m/s, v =

r c) ax = 3.5 m/s2, ay = - 1.0 m/s2, a =

r d) Fx = -549 N, Fy = -1007 N, F =

Answers: 500 m [E 53o N], 15.9 m/s [W 39o N], 3.6 m/s2 [E 16o S], 1150 N [W 61o S]

42

3. For each example below, use the following coordinate system to resolve the vector into components. r r b) v1 = 50 m/s [E 45o S] a) Fn = 20 N [N 30o W] N(+ E(+x

r c) a = 1.5 m/s2 [S 80o W]

r d) Δd t = 3.0 x 104 m [N 10o E]

5. For each example below, use the following coordinate system to resolve the vector into components. r r b) v 2 = 250 m/s [U 40o B] a) Fn = 1.40 N [F 60o D] U(+ F(+x

r c) a = 9.8 m/s2 [D 20o F]

r d) Δd1 = 180 km [U 82o F]

6. Reconstruct each vector using the given components and coordinate system. a) ∆dx = 20 m, ∆dy = -15 m b) Fx = -160 N, Fy = 300 N

N(+ E(+x

c) v1x = 32 m/s, v1y = -45 m/s

d) Ftx= 2.25 N, Fty = 3.0 N

e) ax = -4.95 m/s2, ay = -3.05 m/s2

f) px = -40 kg m/s, py = 57 kg m/s

43

SPH4U: Studying Projectiles Part A: Tracking a Projectile


1. Choose a convenient reference point on the ball to help track its motion. Measure the x- and y-components of the position of the ball at each moment in time. The coordinate system for your measurements is drawn on the picture. The strobe light for the photo flashed at 10 Hz. Complete the chart below. Image No. 1

t(s)

dx (cm)

dy (cm)

2 3 4 5 6 7 8 9

+y

10

+x

Part B: Horizontal Motion Examine the horizontal motion of the projectile. 1. Plot a graph of dx vs. t dx

2. Refer to the pattern of data from the graph to explain the type of motion the projectile experiences in the x-direction.

(cm) 8.0 6.0 4.0

3. Use an appropriate graphical technique to determine the horizontal component of the velocity, vx.

2.0 0

0

0.2

0.4 t (s)

0.6

0.8

1.0

4. Use the graph to write an equation for the horizontal position of the ball. Be sure to include units!

5. Use the data of the graph to explain whether there is any significant evidence for a force acting in the horizontal direction?

44

6. If the ball continued moving for 10.0 s, what would its horizontal position be?

Part B: Vertical Motion Examine the vertical motion of the projectile. This one is a bit more complicated! A. Plot a graph of dy vs. t dy

B. Refer to the pattern of data in the graph to help explain what type of motion the ball experiences in the vertical direction.

(cm) 8.0 6.0 4.0 2.0

C. Use the data to describe the effects of the force the ball experiences in the vertical direction (pretend you don’t know it is gravity).

0

0

0.2

0.4 t (s)

0.6

0.8

1.0

D. Sketch a vy-t and an ay-t graph based on the dy-t graph.

E. On the sketch for question #4 above and on the original photo, highlight the region in time when the ball is speeding up, slowing down and has a vertical speed of zero. F. Will the projectile ever be found falling straight down? Explain.

G. After 10.0 s of projectile motion, vy = - 112 cm/s, determine the velocity vector at that time.

H. Describe how the two graphs would change if the ball traveled straight up and down in the air.

45

SPH4U: Projectile Problem Solving


The key idea which allows us to solve projectile problem is the independence Com / Know / Th / App: 0 1 2 3 4 5 of the horizontal and vertical motions. Since the vertical physics does not affect the horizontal physics, we can treat a single projectile problem as two related kinematics problems – one for each direction. When we set up our work, it is helpful to organize the givens into separate groups for the horizontal and vertical aspects of the problem. A convenient way to show the direction of the velocities used to describe projectile motion is to simply indicate the angle and use a sign convention with positive for above the horizontal and negative for below. For example: 12 m/s [32o] or 150 km/h [-12o].

A. The Ski Jump The ski jump is an exciting and death-defying event that turns humans into projectiles! Let’s study the physics of the craziest winter sport as featured at the Vancouver Winter Olympics in 2010. A typical ski jumper will leave the inrun hill with a speed of 26 m/s. What is hard to notice from pictures is that the launch angle is below grade! For the Vancouver hills, the angle was 11.25o below the horizontal. 1.

Set up horizontal and vertical lists of kinematic information for the ski jumper at the moment of launch. Include a sign convention. Horizontal

Vertical

2.

Explain what steps are involved in finding the jumper’s velocity vector after a time interval Δt.

3.

Determine the jumper’s velocity vector after 4.8 s of flight.

4.

The jumper lands after descending a certain vertical distance. Explain how to find the horizontal distance the jumper travelled before landing.

5.

The jumper descends 35.8 m. How far horizontally did she travel?

46

6.

Challenge (if you get stuck after awhile, move on to part B): Now let’s model the actual landing hill. Do not use information from questions 3 and 5.Model the hill using a triangle that makes an angle of 35o with the horizontal. Explain how to find the distance the jumper travels along the incline before landing.

7.

Calculate the distance the jumper travels along the incline before landing.

B. The Great Jumper Sondre Norheim (1825 – 1897) was a ski jumping champion and the designer of the modern ski used for ski jumping. The modern ski acts like a wing, providing the jumper with lift. In our work here, for simplicity, we ignore all effects of the air. The story goes that Sondre wowed a group of spectators by jumping over a very tall rock. Let’s explore the physics of this daredevil event. We will suppose that he launched from a ramp with a speed of 18.0 m/s at an angle of 28o above the horizontal. The edge of the ramp was 1.5 m above the ground level. The tallest point of the rock was located 13.8 m horizontally from the edge of the ramp and was 8.7 m above the ground. The ground in this area is quite level. If Sondre knew a bit of physics he might have tried a calculation before jumping! 1.

Sketch this scene and construct a list of horizontal and vertical givens. Include a sign convention.

2.

Explain the steps you would use to perform a calculation that will determine whether or not he would make it over the rock. In particular, think about what quantity you would have to compare with the known information to decide whether he can make it.

3.

Perform a calculation to determine whether or not he will make it over the rock.

4.

The ground is level. How far horizontally does he travel while in the air.

47

SPH 4U: Representing Forces – Free Body Diagrams A free body diagram (FBD) is a tool that helps us understand the total effect Recorder: __________________ of all the forces acting on an object. There are a few steps that you should Manager: __________________ always go through when you draw a FBD. Speaker: __________________ • Model the object as a point-particle (its centre of mass) Com / Know / Th / App: 0 1 2 3 4 5 • Represent the external forces acting on the object using vectors that start at the object’s centre of mass. As a guide, always ask the question: “At this moment in time, what is pushing or pulling on the object?” Note: the force vectors do not need to be drawn to scale, but should be drawn roughly according to their relative magnitudes. • Include a separate wiggly acceleration vector whenever possible. • Draw a coordinate system with a sign convention that lines-up with the acceleration. A. The Falling Rock Consider the situation shown to the right of a falling rock. 1. At this moment in time, what is pushing or pulling on the rock?

A rock is falling.

FBD

The Air Resistance Rule: For our purposes, we will always assume there is no air resistance unless it is mentioned or the situation does not make sense without it. 2. Draw the FBD. The purpose of the FBD is to help us understand the total effect of all the forces acting on the object. This total is called the net force, Fnet. To calculate the net force, we must add up all the force vectors found on the FBD. 3. Write a vector equation for the net force the falling rock experiences.

r r Fnet = ΣF =

Working with a vector equation is often inconvenient. Instead, we construct two scalar equations that represent the x- and and y-components of the net force. To do this, follow these steps • Choose a sign convention for the coordinate system with the direction of acceleration as positive. • Write the scalar equation using the force sign convention. The Force Sign Convention: When we write a scalar equation involving forces, the force symbols, such as Fg, are all positive quantities. Show the directions of the forces by using a sign convention and adding or subtracting the appropriate magnitudes. For example: Fnet = Fn – Fw 4. Draw a coordinate system and sign convention for the falling rock. Include this in the box. 5. Write a scalar equation for the net force in the y-direction. Include this in the box. B. Terminal Velocity A rock is falling at a constant speed. This is called the object’s terminal velocity. 1.

Draw a FBD. Write an expression for net force in the ydirection. (Be sure to follow all the steps!)

2.

A student asks, “How can the rock be moving if the acceleration is zero?” Offer an explanation to the student.

48

A rock is falling at terminal velocity.

FBD

C. The Rock Toss A student throws a rock which travels in an arc through the air. The rock was released and is now travelling upwards. FBD

1.

Draw a FBD for the rock after it is released and write an expression for net force in the y-direction.

2.

A student asks, “Why isn’t there a force pointing in the direction the rock is moving - why else would it be going that way?” Offer an explanation to the student.

3.

Compare the FBDs from parts A and C. Based on your comparison, what information can we not find from a FBD.

D. The Sliding Rock A smooth rock slides along a smooth level surface without speeding up or slowing down. Sliding at a steady speed.

FBD

1.

Draw a FBD for the rock and write an expression for net force in the x- and y-directions.

2.

Student A asks, “I know the surfaces are smooth – but, is there another way we can tell that there is no friction? It doesn’t actually say no friction.” Respond.

3.

Student B asks, “How can the rock be moving to the right without slowing down when there is no force pushing on it to the right?”

E. The Finale A rock is being pulled by a string along a rough surface. It is gradually slowing down. 1.

Draw a FBD for the rock and write an expression for net force in the xand y-directions.

2.

Compare the magnitudes of the forces in the horizontal and vertical directions.

Slowing down on a rough surface

FBD

49

SPH4U: Forces in 1-D For these investigations you will need: two identical spring scales (not the 50 N ones!), one dynamics cart, one triple beam balance with extra counter weight


Spring Scale Tips 1) Hold the scale in the vertical or horizontal position in which it will be used. Calibrate it to read 0 N with no forces applied. 2) Do not twist it when forces are applied. The internal pieces bind and give a false reading.

A. The Hanging Cart The cart is suspended from a spring scale and hangs at rest.

Free-body diagram

1. Draw the FBD for the cart.

y x

2. Which force on the FBD does the spring scale reading correspond to? 3. Write a scalar equation for the net force in the y-direction. What is the net force equal to? Fnety = 4. Use your expression from question 2 to explain how to find the size of the force of gravity acting on the cart.

5. What is the mass of the cart? B. Tug-of-War

r r The dynamics cart experiences two opposite horizontal forces ( F1 , F2 ) acting on it and remains at rest on a level surface. F1 = 5 N. 1. Explain what you think the magnitude of F2 should be. Free-body diagram y

x 2. Test your prediction: use two spring scales to create F1 and measure F2. How does your prediction hold up?

3. Write a scalar equation for the net force in the x- and y-directions. Fnetx =

50

Fnety =

4. What values are Fnetx and Fnety equal to in this situation? Explain.

5. Use one of your expressions from #4 to determine the size of the normal force acting on the cart.

C. Upended The cart experiences an upwards force, FA of 3 N, while standing on its end, at rest upon a level surface.

Free-body diagram

y x

1. Explain what you think the magnitude of the normal force should be.

2. Write a scalar equation for the net force in the y-direction. Indicate what the total is equal to. Fnety = 3. Solve the previous expression for the magnitude of Fn.

4. Now you will model this situation on a balance scale. Place the cart upon a triple-beam balance using a counterweight. Apply the 3 N force upwards using a spring scale. Readjust the balance and record the new reading. This may be tricky! Demonstrate this for your teacher. Balance reading = 5. Has the mass of the cart actually changed? Explain. 6. The balance gives us a reading in the units of grams. What is the equivalent reading in terms of force? Explain how you find this. (Hint: Scales always measure force and not mass. How do you think the scale company decided how to calibrate their device? What force is pushing down on the cart?)

7. What force does the balance scale measure? Compare your reading with your force calculations and explain.

51

SPH4U: Forces in 2-D For these investigations you will need: two spring scales (not the 50 N ones!), one dynamics cart, one protractor A. Tilted Forces

r


r

The cart experiences a force, F1 = 5 N [Left] and a force, F2 , which is directed [Right 30o Up]. It remains at rest on a level surface.

r

Free-body diagram

y

r

x

1. How do you think the magnitude of F2 compares with F1 ? Explain.

r

r

2. Model this situation using two spring scales for F1 and F2 , and a r protractor. Measure the size of F2 .

r

3. Draw a component triangle for F2 . Which side of the triangle does the spring scale measurement give? Calculate the components of r F2 using its magnitude and direction.

Component triangle of

r F2

r

4. Use your results from #3 to explain which component of F2

r

balances F1 .

5. List all the forces or components of forces that act in the x-direction.

6. Write a scalar equation for the net force in the x-direction. Show the trigonometry for any components. Fnetx = 7. Determine the magnitude of F2 directly from the equation in question 6. How does this compare with your measurement?

8. List all the forces or components of forces that act in the y-direction.

9. Explain without using equations how the size of the normal force, Fn, compares with Fg, the force of gravity.

52

10. Write a scalar equation for the net force in the y-direction. Use trigonometry for any components. Fnety = 11. Determine the size of the normal force from your previous expression.

B. Forces on a Tilt

r

The cart is at rest on a surface inclined at 30o. It is held in place by a force, Fa , that is parallel to the incline. You need one ramp, one retort stand and one clamp. 1. Fill in the chart showing which forces, or components of a force, act in each direction. Note the special coordinate system chosen!

Free-body diagram

y x

+x

-x

+y

-y

2. Which forces, or components of a force, balance each other? 3. Which force or component of a force pulls the cart down along the ramp?

y

r

4. Draw the component triangle for the force of gravity, Fg , relative to the tilted coordinate system. Locate the 30o angle in the triangle.

r

Write down the scalar components of Fg using sin or cos.

x r

5. Predict the size of Fa .

r

6. Measure the size of Fa . How does this compare with your prediction?

7. Write a scalar equation for the net force in the x- and y-directions. Fnetx = Fnety = 8. Explain how Fa would change if the angle of the incline is increased.

r

9. Determine the size of Fn from your equation. 10. Explain how Fn would change if the angle of the incline is increased.

53


SPH4U: Components and Free Body Diagrams A. Warm-Up For each situation, draw the FBD and write out a complete expression for the net force in the x- and y-directions.

A rock is sliding without friction.

FBD

Com / Know / Inq / App: 0 1 2 3 4 5

Fnetx =

Fnety =

θ Friction prevents the rock from sliding.

FBD

Fnetx =

Fnety =

θ

B. The Rough Incline A rock slides up and then down an incline. There is a constant force of friction, but not quite enough to prevent the rock from moving on the incline. Two students discuss the situation. Student A says, “The magnitude of the rock’s acceleration will be the same when it is going up the incline as when it is going down, since it’s the same forces.” Student B says, “Even though there are no new forces, the magnitudes will be different going up compared with going down.” Who do you agree with? Explain with the help of FBDs and equations. Going up.

FBD

θ

Going down

θ

54

FBD

Explanation

C. The Floating Rock A rock is tied to a string. Student A says, “I can pull the rock with the string at an angle, like this, so it moves horizontally through the air.” Student B replies, “That’s not possible, even for a short period of time. The forces simply don’t work out properly.” Do you agree with student A or B? Explain with the assistance of FBDs and equations. Can the rock travel horizontally?

θ

FBD

Explanation

r v

D. The Toss A rock is being thrown horizontally and is shown in the diagram as it is speeding up. Student A says, “Let’s choose a coordinate system along the direction of the velocity.” Student B says, “No, I want to choose one that lines up with the angle of the hand.” Which student has the better suggestion? Explain with the assistance of a FBD and equations. A rock speeds up as it is being thrown. It does not slide against the hand.

FBD

Explanation

r v θ

E. The Magic Rock Student A says, “Look! I can make the rock stay in place against my vertical hand without me holding on to it!” Student B says, “Nonsense! Gravity is guaranteed to pull it down.” Who do you agree with? Explain. Can the rock “stick” to a person’s hand?

FBD

Explanation

r v

55

SPH4U: Newton’s Third Law


56

57

Bonus Section:

58

from McDermott, L. Tutorials in Introductory Physics. Prentice Hall, 2002

SPH4U: Composite Objects


Five identical blocks, each of mass m, are pulled across a table by a Com / Know / Th / App: 0 1 2 3 4 5 steady force as shown. Use the approximations that the table is frictionless and the strings are massless and do not stretch. You may assume the blocks are stuck together. 1. Compare the motion of each of the systems A, B, and C.

System A (one block) String P

System B (two blocks)

String Q

System C (two blocks)

String R

2. Draw and label the separate free-body diagrams for the separate systems A, B, and C. Indicate the mass of each system. Draw an acceleration vector for each. System A

System B

System C

ma =

mb =

mc =

3. Rank in increasing size the magnitudes of the net forces on systems A, B, and C. Explain.

4. Draw a free-body diagram for a two new systems. System D represents system B and C together. System E represents all the five blocks. Remember, only draw external forces! Indicate the mass of each system.

System D

System E

md =

me =

5. How does the acceleration of system D and E compare with the A, B, and C? Explain.

6. Which system yields the easiest free-body diagram if you want to find FTP, FTQ or FTR? 7. Rank in increasing size your predictions for the magnitudes of the three tension forces. 8. Consider a situation where m = 1 kg and FTP = 15 N. Find the size of FTQ and FTR.

59

SPH4U: Weight and Acceleration


In the world of physics, weight has a simple definition: Weight = Fg = mg. Where the gravitational field strength, g = 9.8 N/kg. This Know / Com / Th / App: 0 1 2 3 4 5 sometimes differs from our everyday experience of weight which we associate with the reading of a bathroom scale. To distinguish this, we will call the scale reading the apparent weight. We have already learned that this reading is actually a measurement of the force required to support the object (often the normal force). You need one spring scale, one mass (~500 g) and a little bit of tape to attach the weight A. The Elevator You may have notices a curious sensation while travelling in an elevator. At certain times, it feels like your weight is changing. Since you are such a curious student, you decide to investigate this. You step into an elevator at the ground floor of a tall building. You place a 2 kg rock a bathroom scale that gives readings in newtons. You haven’t pushed any buttons yet and you look down at the scale.

FBD

1. Draw a FBD for the rock in the elevator according to the description above. 2. A student, Albert, suggests: “There should be another force on the FBD showing the upwards pull of the elevator.” Respond to Albert.

Scale

3. Use Newton’s 2nd law to determine the apparent weight of the rock.

B. Moving On Now we press the button in the elevator and go for a ride! The elevator starts speeding up as it begins your trip to the 20th floor. You notice the reading on the scale changes. 1. Explain how the FBD you drew earlier and the forces involved might change while the elevator speeds up.

2. Explain how the reading of the scale will change. Support your explanation with some simple observations using the spring scale.

3. The elevator is accelerating at a rate of 1.5 m/s2. Use Newton’s 2nd law to determine the apparent weight of the rock.

60

4. Based on this discussion, why is it important to distinguish between weight and apparent weight?

Somewhere around the 2rd floor, you notice the scale reading returns to normal. 5. Why does this happen?

6. A student, Marie, comments, “I think the upwards force must still be larger than the downwards, or else the elevator would not be moving upwards.” Respond to Marie and use simple observations from your spring scale.

The elevator travels past the 19th floor. You notice another change to the scale reading as the elevator is slowing down. 7. Explain how the reading of the scale will change. Support your explanation with some simple observations using the spring scale.

8. The elevator slows at a rate of 3.4 m/s2. Determine the apparent weight of the rock.

9. Explain in general how apparent weight is related to the acceleration of an object.

C. A Strange Elevator You wake up to find yourself in a very strange elevator with no buttons or lights. The rock is floating just above the scale. The scale itself reads zero. You notice you are floating too and have lost the sensation of weight. Offer two possible explanations for this very curious situation. 1)

2)

61

SPH4U: Frames of Reference


Your friend is standing on a bus that is travelling east and speeding Com / Know / Th / App: 0 1 2 3 4 5 up at a uniform rate along a level road. While this is happening she holds up a rope with a ball attached to the end of it. The ball is allowed to hang freely. Assume east is to the right. Answer the following questions while the bus is accelerating and the ball hangs in a steady way (not swinging around!)

3. Draw a FBD for the ball from each frame of reference. Do not include fictitious forces yet. Include an acceleration vector!

Your frame

Her frame

Your frame

Her frame

Diagram of ball and rope

Her frame

Description of ball’s motion

2. Describe the motion of the ball relative to frame of reference while the bus accelerates in a steady way.

Your frame

Free Body Diuagram

1. Draw a diagram showing the orientation of the ball and rope while the bus is accelerating in a steady way.

4. There is a problem with one of the free body diagrams. Explain.

5. Draw a modified FBD for your friend’s frame of reference. Add a fictitious force, Ffict, such that FBD agrees with her description of the ball’s motion. How does the direction of the fictitious force compare with the acceleration?

New FBD

6. Use Newton’s 2nd law to write an expression to determine the size of the fictitious force, given the ball’s mass, m, and the acceleration of the bus, a. 7. From your own frame of reference, write an expression that allows you to determine the acceleration given the angle the rope makes with the vertical.

62

SPH4U: Tension and Pulleys


Part A: Tension and Pulleys

Com / Know / Th / App: 0 1 2 3 4 5

Suppose you were to hang a mass of 0.5 kg in the various configurations shown below. You may assume the scales have no mass.

A B C

m

m

m

1. Draw a free body diagram for the spring scale itself in example A. Determine the size of the forces involved. Predict the reading on the scale.

E m

D

m

m

2.

3.

4.

The readings on the spring scales reflect the forces of tension in the strings. Predict and measure the tension in the string for configurations A through E. In each example, how much force is pulling on the end of the strings? How does this compare with the force of tension inside the string?

m

Predicted

Measured

FTA =

FTA =

FTB =

FTB =

FTC =

FTC =

FTD =

FTD =

FTE =

FTE =

What effect does a pulley have on the force of tension?

Part B: Atwood Machine An Atwood machine consists of two weights tied together and suspended over a pulley. 1.

Assume the two masses are equal. You give one gentle push to a mass and release it. What type of motion will occur after it is released? mb ma

from Laws P. Workshop Physics Activity Guide. Volume 1. John Wiley, 2004

63

2.

As a class, observe the motion of the masses as described in question #1. How do your observations compare with your prediction?

3.

Suppose that mb is greater than ma. Will mass A accelerate? If so, in what direction? Will mass B accelerate? If so, in what direction? How will those accelerations compare in magnitude? Explain.

4.

As a class, observe the motion of the masses as described in question #3. Describe the motion you observe.

5.

Draw a FBD for mass A and for mass B.(ma > mb). Indicate the acceleration of each mass and choose your sign conventions so they agree with the acceleration of their mass!

6.

For the FBDs, do you need to use a different symbol for the magnitude of the acceleration of mass A and mass B? What about for the forces of tension and gravity acting on each mass? Explain.

7.

Use Newton’s 2nd Law to write down a scalar equation that relates the forces acting on mass A to the acceleration of mass A. (Don’t solve for a.)

8.

Use Newton’s 2nd Law to write down a scalar equation that relates the forces acting on mass B to the acceleration of mass B.

9.

Algebraically eliminate FT from the above two equations and solve for the acceleration of the masses.

Mass A

Mass B

10. If ma = 0.4 kg and mb = 0.7 kg, find a.

12. Use an equation from question 8 or 9 to solve for the magnitude of the force of tension. How does this value compare with Fga and Fgb?

64


SPH4U: Friction Your goal is to get a sense for how friction works.

Com / Know / Th / App: 0 1 2 3 4 5

A: Looking for Friction in All the Right Places

For this activity you will need: • 1 wood block with hook • 1 spring scale (10 N range) • bunch of masses Attach your spring scale to a heavy object. Very gradually begin to pull on the object until it moves at a slow, steady speed. Do this a few times until clearly notice when the object becomes “unstuck”. 1. Starting from zero, describe what happens to the readings on the spring scale as you very gradually increase the force.

3. Sketch a graph showing the size of the force of friction according to your scale readings as you gradually increased the size of your force.

Force of Friction

2. What happens to the object when you notice the sudden change of the spring scale reading?

Time 4. In which of the following situations does the object experience a force of friction? Draw a free body diagram for the object in the three following situations: a) at rest, you exert no forces on it

b) at rest, you are starting to pull

c) moving at a constant velocity due to your pull

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5. Compare the size of the force of friction with the force you exerted in situations a, b, and c. 6. There are two kinds of friction: static friction for stationary objects and kinetic friction for moving objects. Label the free body diagrams above as static or kinetic. Label your graph from question 3 to show the static and kinetic friction regions. 7. When the object is at rest, what happens to the size of static friction as you very gradually increase your pull? Try this with a heavy object. B: How the Normal Force Affects Kinetic Friction

Kinetic friction arises when two objects are in contact and are sliding relative to one another. Being in contact, they exert normal forces on each other. How does the size of the normal force affect the force of kinetic friction? 1. On a horizontal surface how does the normal force compare with the object’s weight? (Remember, weight = Fg = mg) 2. Using the equipment from the first part, devise an experiment, take data and plot a graph to answer the question of Part 2. Explain your procedure for this experiment.

3. Write headings for your table, list your data (fill up the table!) and plot it on the graph.

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4. Draw a free body diagram showing your block while you are pulling it. Under what condition will your applied force be equal to the force of kinetic friction?

5. Describe in words the type of mathematical relationship between the Fn and Ffk.

6. Determine the slope of this graph. This value is called the coefficient of kinetic friction (μk). What characteristics of your experiment affect the size of the slope?

7. Does the mass of the object have any significant effect on the coefficient? What can you conclude from this?

8. Make a prediction. What will the force of friction be if your mass weighed 400 N? 2 000 N?

C: How the Type of Materials Affects Kinetic Friction

1. What are some possible causes of friction between two surfaces?

2. Predict what kinds of surfaces will produce little friction and what kind will produce great friction.

In the next activity you will investigate what combination of surfaces will produce the most friction. Make sure that you use a clean surface, otherwise you will be measuring the forces from grinding dirt. You will need: • 1 block with hook • heavy masses • 1 spring scale • an overhead acetate • 4 surfaces to draw the object over (lab table, glass, floor, one more of your choice) 67

3. What is the mass of your block? 4. List all possible combinations of upper and lower surfaces. Predict which will yield high, medium or low friction. Bottom Surface

Upper Surface

Prediction

Force of

Coefficient

Friction (Ff)

(μk)

5. Drag the loaded wood block over each surface and measure the force of kinetic friction. Attach the acetate to the bottom of the block and drag the block over each surface. Record your results in the chart. Remember to keep your surfaces clean! Use some paper towel to wipe things off don’t let the acetate get scratched. 6. Are there any discrepancies between your predictions and measured results? Can you offer any reasons why?

7. An important quantity for the discussion of friction is the coefficient of kinetic friction. This value depends only upon the nature of the two surfaces and is found by the expression:

μk =

force of friction normal force

Calculate the coefficient of kinetic friction for the combination of surfaces in your experiment and add this to the table. BONUS: What will happen to the size of the force of friction with the same combination of materials on an incline? Draw a free body diagram and use the expression for the net force to support your answer.

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SPH4U: Going in Circles A: Picturing Circular Motion


Com / Know / Th / App: 0 1 2 3 4 5 Attach a piece of string roughly the length of your forearm to a group of eight washers. Swing the washers at a consistent, steady rate in a circle, flat against the surface of a table or floor. Note that the string will twist up after a while – you will need to untwist it or its length changes! To describe the rate of circular motion we use the term frequency (f), which describes the number of cycles per second of the rotating of object. This is closely related to the quantity period (T) which is the time taken by one cycle of the circular motion. The two quantities are related by the expression T = 1/f. The units of frequency are hertz (Hz). 1.

Determine the frequency and period of the washers’ motion.

2.

Determine the radius of the washers’ circular path and the speed along its circular path.

3.

Draw a picture of the washers travelling along their circular path at the 12, 3, 6, and 9 o’clock positions. Practice releasing the washers exactly at those positions – longer string lengths makes this easier. (Warning! Don’t release the washers at people!) Observe carefully the direction in which they travel immediately after release. Use your observations to draw an instantaneous velocity vector for the washers. In general, describe direction of the instantaneous velocity with respect to the circular path of the object.

4.

Two students are drawing a FBD for the washers at the 12 o’clock position as they spin at a steady rate. Isaac says, “I think there must be a force tangent to the circular path – that’s the direction it’s moving in.” Marie says, “I’m not sure there is. I can’t think of what type of force it could be.” Who do you agree with? Explain.

5.

Draw a FBD for the washers, at 9 and 12 o’clock. Draw the FBDs looking down from above (bird’s eye view). Always ask yourself: At this moment in time, what is pushing or pulling on the object? Assume that friction is small enough to be ignored. Include the acceleration vector.

6.

How does the direction of the force that keeps the washers moving in a circle compare with the object’s velocity at each moment in time? How does this force affect the object’s speed and direction?

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An object moving along a circular path always experiences force towards the centre of the circle and must therefore always be accelerating towards the center of the circle. We give the label centripetal force to any force or component of a force which points towards the centre of on object’s circular path. Warning: Never use the symbol Fc in your work! The term centripetal force is simply a category for our familiar forces (Ft, Fg, Fn, etc.) that provide the centripetal acceleration. Special Case: In Uniform Circular Motion, the net force an object experiences points directly to the centre of its circular path. There are no components of forces tangent to the circle, so only the direction of the object’s velocity changes and not its speed. 7.

Albert has been trying to follow your discussions about circular motion, but is stuck on something. Albert says, “I just don’t understand how the object moves in a circle when the acceleration points towards the centre. Wouldn’t tension just pull it straight into the centre?” Help Albert understand by describing and explaining the following diagrams to him. (1)

(1) Explanation

(2)

(2) Explanation

r v1 r ac

r Δv

r ac

Small time interval Δt

Time t1 (3)

(3) Explanation

r v1

(4)

r ac r v2

(4) Explanation

r v 2

r Δv

Time t2

Part B: Feel the Force Find a small tube, string (at least 1 m), rubber stopper, and spring scale (5 N). Attach the string to the stopper and thread it through the tube. Have one person hold the tube and another person hold a spring scale connected to the bottom of the string. Find some space where you can swing the stopper in a horizontal circle in the air above your heads. Don’t hit people! Practice swinging it at a steady rate.

spring scale

1.

Draw a FBD for the stopper from the side view. Include the small angle, θ, between the string and the horizontal.

2.

Hideki says, “I’m not sure the FBD makes sense. There is nothing to balance gravity here.” Do you agree or disagree with Hideki? Explain.

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3.

Swing the rubber stopper in a horizontal circle. Change the frequency of the circular motion while keeping the radius of the motion constant. How does this force of tension change when the frequency of the circular motion changes.

4.

Change the radius of the circular motion while trying to keep the frequency constant. How does the force you supply change? State how this force is related to the radius of the circular motion.

5.

We can write a complete expression for the acceleration of an object moving in a circle at a constant speed: ac = 4π2Rf 2. Use the measurements you made in part A to a) Calculate the acceleration of the washers.

b) Use Newton’s 2nd Law to calculate the force of tension in the string.

Note that there are two other ways of writing the acceleration equation involving period or speed:

ac = 6.

4π 2 r v2 a and = c r T2

Determine which quantities change as the washers rotate at a steady rate and provide a brief explanation or example for each. a.

period / frequency

b.

speed

c.

velocity

d.

magnitude of acceleration

e.

acceleration vector

f.

magnitude of net force

g.

magnitude of force of tension

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SPH4U: Forces and Circular Motion Let’s use our knowledge of forces and circular motion to explore a few different situations. In this activity will be asked for an explanation for each situation below. Explain carefully and be sure to include a FBD and a force equation.


A. The Balancing Act A rock is swinging in a vertical circle. The rock reaches the lowest point in its trip. A student, Niels, says, “I think the vertical forces don’t balance.” Another student, Werner, says, “Since the object is moving to the left at this moment, the two vertical forces must balance.” Who do you agree with? Explain. FBD

Explanation

B. The Loose Rock A student swings a rock in a vertical circle at a slow but steady rate. Erwin says, “At the very top, I felt the string go slack. The tension must be zero at that point.” Max says, “I don’t understand how that’s possible. At that moment, there has to be tension to supply a centripetal force or else the rock would fly off.” Explain to Max how this is possible. FBD

Explanation

C. Low Circles A student swings a rock in a horizontal circle such that the string makes an acute angle with the vertical. Marie says, “Tension is the centripetal force, so the centripetal acceleration is pointing at an angle to the vertical, along the string.” Emelie says, “I think the centripetal acceleration pointing to the centre of the rock’s circular path.” FBD

θ

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Explanation

D. Rocks at 33½ A student places a rock on a rotating platform, just like an old fashioned record turntable. Friction prevents it from slipping. Student A says, “At this moment in time the instantaneous velocity of the rock is downwards, from this point of view, so friction will act in the opposite direction, upwards.” Student B says, “I’m not sure. Since the rock is moving in a circle, I think the force of friction must act towards the centre.” Top view

FBD

Explanation

E. The Ball and the Funnel Albert recently visited the Ontario Science Centre and saw an exhibit where a marble rolling in a funnel glides around in horizontal circles with very little vertical motion (friction is negligible). Albert says, “That was cool!” Alice says, “But I don’t understand. Wouldn’t gravity pull it down to the bottom pretty quick? I don’t see how it could travel in a circle.” Explain to Alice how this is possible. FBD

Explanation

θ

F. The Pendulum A pendulum moves back and forth in a circular arc. The motion of a pendulum is an example of circular motion, but more complicated than what we have studied so far. Let’s study it FBD carefully. 1. Draw a FBD for the pendulum at this moment in time. Leave out the acceleration vector. Write an expression for the components of the net force. Fnety =

Fnetx =

y θ

x

2. What evidence is there for a non-zero net force in the y(radial) direction? What type of acceleration will it produce?

3. There is an unbalanced force in the x- or tangential direction. Explain the effect this will have on the speed of the object.

4. Explain how the motion of the pendulum differs from uniform circular motion. Draw an acceleration vector on the FBD for the radial and tangential accelerations.

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SPH4U: Universal Gravitation Good old Sir Isaac Newton determined the relationship between the force of gravity, mass (m) and the separation between the centres of the two objects (r): Gm1 m 2 , where G = 6.67x10-11 Nm2/kg2. Your friend, who is now an F = g


r2

astronaut (mA = 100 kg), is currently standing on the Earth (mE = 5.98x1024 kg, rE = 6.38 x 106 m). 1.

Draw a diagram of your friend and Earth. Label the quantity, r. Draw a FBD for your friend and Earth.

2.

Calculate the size of the force of gravity the earth and the astronaut exert on one another using the new expression for universal gravitation.

3.

Diagram

Astronaut FBD

Earth FBD

Fg E − A =

Fg A − E =

Your friend blasts off and travels away from the earth. Complete the table showing the size of the force of gravity due to the earth on the astronaut at different distances from the centre of the earth. Sketch a graph of Fg vs. r Location

Distance

Earth’s Surface

6.38 x 106 m

Space Shuttle Orbit Geosynchronous Orbit Lunar Transfer Orbit Moon’s Orbit

6.39 x 106 m

Mars’ Orbit

7.9 x 1010 m

Fg

4.22 x 107 m 2.17 x 108 m 3.85 x 108 m

0

| 8 1 x 10

| | 8 8 3 x 10 2 x 10 distance (m)

| 8 4 x 10

4.

Describe in words how the size of the force of gravity varies with the separation of the objects.

5.

How far does Earth’s gravitational force extend into the universe? Explain.

6.

Albert says, “I understand that there still is gravity out in space, sometime lots of it, but why do we see astronauts floating in the Space Shuttle – they seem weightless.” Explain to Albert why.

BONUS: Imagine a hole is dug straight through the centre of the Earth. Describe your motion if you fall in.

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SPH4U: Orbits Consider an object (mo) travelling in a circular orbit around the earth.

mE = 5.98x1024 kg rE = 6.38 x 106 m G = 6.67x10-11 Nm2/kg2

Com / Know / Th / App: 0 1 2 3 4 5

1. Label the altitude (h), the radius of the circular orbit (r), the radius of Earth (rE), matching them with the letters A, B, and C. 2. Which quantity, A, B, or C, does r in the expression for universal gravitation represent? Explain.

B

Gm1 m 2 Fg = r2

A

C

3. Which quantity, A, B, or C, does r in the expression for the centripetal acceleration represent? Explain. 4π 2 r ac = 2 T 4. Draw a free body diagram for the object. 5. Use Newton’s 2nd Law, universal gravitation and an expression for centripetal acceleration to create an equation that relates the radius of the orbit and the period of the orbit. Solve this for T. Be careful with the labels for the masses!

FBD

6. Complete the chart below which compares orbital velocities, altitude, radii and periods. Location Space Shuttle Orbit Geostationary Orbit (satellite stationary above Earth’s surface) Moon’s Orbit

Altitude

r (m)

T (s)

600 km 24 h = 27 d =

Earth’s Orbit around Sun (msun = 1.98 x 1030 kg)

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7. Earth’s gravity is always pulling objects towards the centre of the earth. Why don’t objects in orbit fall straight down and crash into Earth?

8. Sketch and label a scale diagram of the radial distance of the first two orbits from question 6 in terms of earth radii. Draw the earth itself. The given point represents the centre of the earth. 4 boxes = 1 rE.

•

9. Using this scale, where would the Moon be located? Carefully find this position and show your teacher. Consider a binary star system (YM alpha and YM beta) consisting of two equally massive stars which orbit one another. 10. How do you decide which star will orbit around which? Explain.

11. Label the radius of the orbit (ro) and the separation between the centres of mass of each star (d), matching them with the letters A (radius) or B (diameter). C is the common centre around which both stars orbit.

α

C •

12. Use Newton’s 2nd law to write a complete expression relating the period and radius for the stars’ orbits. Be sure to use the given symbols! B β

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A

SPH4U: “Oomph” This tutorial introduces momentum conservation. Equally important, using momentum as an example, this tutorial explores the extent to which formulas relate to common sense.


I. What’s your view? A. (Work individually) Which of the following best expresses your view about the relationship between physics formulas and common sense? (You can choose more than one.) i.

Many physics concepts make a lot of sense and connect to everyday experience; but formulas are more of a problem-solving tool than a sense-making tool.

ii.

It really depends on the formula. Some of them make sense, but you shouldn’t expect them to make sense as a general rule.

iii.

In general, physics formulas express some kind of common-sense ideas.

B. Compare your answers with the rest of your group. If there was disagreement, have a debate—not to convince each other, but to understand each others’ views. If someone makes a good point that disagrees with what you initially thought, summarize that point here.

II. Figuring out the formula for oomph An important physical quantity, the name of which we’ll give later, corresponds to the intuitive idea of oomph. The more oomph something has, the harder it is to stop, and the more ability it has to knock other things over. Let’s figure out the formula for oomph. If you already know the formula from a previous class, please “play along” and don’t give it away. We’ve structured this tutorial so that you’ll learn something even if you already know the formula. We know some of you don’t like mucking around with intuitions, but trust us, in this tutorial it’ll lead somewhere quickly, and you’ll end up practicing some physics. A. (Work together) A small pebble and a larger rock are thrown at the same speed. 1. Which one has more oomph? Why?

2. The rock is twice as massive as the pebble. Intuitively, how does the rock’s oomph compare to the pebble’s oomph? Is it twice as big? Half as big? Three times as big?

B. (Work together) Picture two identical bowling balls, one of which is rolling faster than the other. 1. Which ball, the faster or slower one, has more oomph? Why?

2. The faster ball is exactly 7 times as fast as the slower one. Intuitively, how does the faster ball’s oomph compare to the slower ball’s oomph?

© University of Maryland Physics Education Research Group, Fall 2004

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C. (Work together) The physics concept corresponding to oomph is momentum. Building on your above answers, figure out a formula for momentum (oomph) in terms of mass and velocity. Explain how the formula expresses your intuitions from parts A and B above. (For nutty historical reasons, physicists use the letter p for momentum.).

** check with your teacher at this point ** III. Intuitions about collisions Above, your intuitions about oomph led to a formula for momentum. Now let’s see if your intuitions about collisions lead to similar progress. A. (Work together) A 1 kg cart, rolling with negligible friction at 6 meters per second, collides with and sticks to an identical cart. So, after colliding, the carts roll together as a single unit.

6 m/s

1 kg

BEFORE

1 kg

AFTER

?

1. Using your intuitions, guess the post-collision speed of the two carts. Briefly explain your reasoning.

2. According to the intuitive guess you just made, is the overall momentum of the two-cart system after the collision greater than, less than, or equal to the overall momentum before the collision? Work this out using the momentum formula you figured out above and plugging in the relevant numbers.

B. (Work together) In a similar experiment, the 1 kg cart collides with a 3 kg cart but doesn’t stick to it. Instead, the 3 kg cart gets knock forward by the 1 kg cart, which comes to rest after the collision.

6 m/s

BEFORE

1 kg

3 kg

1. Again using intuitions, guess the post-collision speed of the 3 kg cart. AFTER

?

stopped

2. According to the intuitive guess you just made, is the overall momentum of the two-cart system after the collision greater than, less than, or equal to the overall momentum before the collision?

C. (Work together) Based on your work above, state a general rule about how the total momentum of a system changes during a collision.

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D. Let’s look at one more collision. Two identical blocks, both of mass 0.5 kg and covered with Velcro™, slide toward each other at equal speeds, 6 m/s. The blocks stick together. 1. Intuitively, after the collision, how fast do the blocks move and in what direction?

6 m/s

BEFORE

0.5 kg

6 m/s

0.5 kg

AFTER ?

2. In the cart collisions from parts A and B above, momentum was conserved; it was the same before and after the collision. Because conserved quantities are useful in problem-solving, it would be cool if we could define momentum in such a way that it’s always conserved in collisions (between objects that are free to move). Is there some way to modify or clarify the momentum formula you figured out at the top of page 2 so that momentum is conserved in the head-on collision between the two blocks? (Hint: Maybe oomph “cares” about direction.)

** check with your teacher at this point ** IV. Conservation of momentum Conservation of momentum is a fundamental physical law. Among other things, it says that when two objects collide, the total momentum of the system immediately after the collision equals the total momentum of the system immediately before the collision: Conservation of momentum:

m1v1 before + m2v2 before = m1v1 after + m2v2 after ,

Since p = mv, and since velocity “cares” about direction, so does momentum. So, a negative oomph (momentum) can partially or fully cancel a positive oomph, as the Velcro™ blocks demonstrated.

A. (Work together) Let’s practice using momentum conservation. On a safety test course, a 1000 kg car heading north at 5 m/s collides head-on with an 800 kg car heading south at 4 m/s. At these low speeds, the new high-tech bumpers prevent the cars from crumpling; they bounce off each other. After the bounce, the 1000 kg car is heading northward at 1 m/s. We’re going to ask you for the post-collision speed and direction of motion of the other car. 1. What’s a good first step in this problem, one that will help you avoid mistakes? After coming to consensus, do that step.

2. Without doing calculations, “guess” the final direction of motion of the lighter car. Briefly explain your reasoning.

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3. Now calculate the lighter car’s speed and direction of motion after the collision. Make sure everyone in the group is comfortable with this.

4. If you didn’t already do so, check for consistency between your intuition in question 2 and your formal answer in question 3.

B. (Work together) Two students, after listening to one of their physics teacher’s sermons about sense-making, are arguing about the conservation of momentum equation from the previous page (in the gray box). FRANK: I don’t think the equation expresses common sense, because if it did, then we could have used common sense directly—instead of the equation—to solve the problem about the colliding cars. But we really needed the equation there to get the exact numbers. ERNEST: Sure we needed the equation. But the equation kind of expresses the common-sense idea that oomph stays the same in a collision. The whole point of expressing common-sense ideas in equations is to get something more generally useable, something we can use when common sense alone can’t deal with the situation. FRANK: You’re admitting that the equation goes beyond common sense, because we can use it when common sense alone wouldn’t get us anywhere. So, at least in those cases, the equation doesn’t express common sense! In what ways do you agree or disagree with Frank? With Ernest? What’s your stance on this issue?

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SPH4U: Types of Collisions Let’s try to predict the results of some collisions using our knowledge of momentum. Two carts of equal mass (m) collide. Cart A

Com / Know / Th / App: 0 1 2 3 4 5

Cart B

For each case below: 1. Sketch your prediction for the results of the collision. Use arrows to indicate the direction and magnitude of the carts’ velocities after the collision. Note that we use the prime symbol (i.e. v ′2 ) to show quantities after the collision. 2. Observe the actual collision using the dynamics carts set-up in the room. Ignore small changes in velocity due to friction and vibrations. Do your predictions agree? If not, draw a second set of vectors in your sketch. 3. In the middle box, compare the magnitude of the velocities ( , = ) before and after the collision.. 4. Determine the total momentum of the entire system before and after the collision. To keep the numbers simple, assume the mass of the cart is 0.5 kg and the initial velocities are 1.0 m/s. Case 1: The Bouncy Collision A The two carts collide using repelling magnets. va = -vb This is an example of an elastic collision. After v v′ a

a

vb

vb′

ptotal = ′ = ptotal

Case 2: The Bouncy Collision B The two carts collide using repelling magnets. vb = 0. This is an example of an elastic collision. After v v′ a

b


Case 3: The Sticky Collision! The two carts collide using velcro. This is an example of an inelastic collision. va = -vb After v v′ a

a

vb

vb′


Case 4: The Explosion! The two carts collide with loaded springs. This is an example of a superelastic collision. va = -vb After v v′ a

a

vb

vb′


Does momentum appear to be conserved in each case? Is the final total momentum the same as the initial total momentum of the two-cart system?

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SPH4U: Momentum and Isolated Systems There are some situations where momentum seems to appear and disappear. Let’s study one of these situations carefully.


The Slowing Block A 1.0 kg block initially sliding at 1.5 m/s along a rough surface comes to a stop. pb =

p′b =

Δpb =

1. Calculate the block’s change in momentum. Describe how the momentum of the block has changed. 2. In considering the system of the block, does momentum appear to be lost? Explain.

3. What external forces acted on the block while slowing? Which of these forces is responsible for the block’s change in momentum? Is the block an isolated system?

4. Newton’s 3rd law is closely related to the conservation of momentum. Consider the force responsible for slowing the block. What is the other force in a 3rd law pair with that force? Use that other force to help you guess where the block’s momentum went. Make a guess and move on!

Now the block slides along a rough, level track that is supported on wheels and is free to move (no friction). The block is initially sliding just as before, starting at 1.5 m/s. The track has a mass of 2.3 kg and is initially at rest.

5. Describe what will happen after the block is released.

6. Is the combination of block and track an isolated system? Explain. 7. Calculate the final velocity of the track. What type of collision is this? 8. Imagine the mass of the track was increased enormously to equal that of the earth. Describe what would be different.

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SPH4U: The Process of a Collision


Collisions often occur very quickly so we don’t usually notice what is actually happening during a collision. In this example, cart A (3 kg) collided with a Com / Know / Th / App: 0 1 2 3 4 5 smaller cart B (2 kg) using an uncompressed spring. The velocity of each cart was recorded at 8 moments in time and used to calculate the momentum and kinetic energy. A third line on each graph represents the total momentum and total kinetic energy of the system of two carts. Before Cart A

2.

Cart B

Draw a vertical line on each graph labelled “A” to indicate the moment in time when the collision begins and one labelled “C” to indicate when the collision ends. What would we observe about the spring at moments “A” and “C”?

3.

What is the duration of the collision?

4.

Determine how much momentum is lost from the system during the collision. Where did this momentum go?

Cart B

Cart A

Momentum during a Collision

Momentum (kg m/s)

1.

After

18 16 14 12 10 8 6 4 2 0

PA PB Ptotal

0

0.02

0.04

0.06

0.08

Time (s)

Kinetic Energy during a Collision

6.

How much kinetic energy is lost from the system from 0 to 0.08 s? Where did the energy go?

Why does the total kinetic energy dip down in the middle of (during) the collision? Where did it go?

Kinetic Energy (J)

5.

50 45 40 35 30 25 20 15 10

Ek A Ek B Ek total

5 0 0

0.02

0.04

0.06

0.08

Tim e (s)

7.

Draw a vertical line on each graph labelled “B” to indicate the moment in time when the spring was at its maximum compression. How much energy was stored in the spring at this moment?

8.

Is kinetic energy conserved during a collision? Use the graphs to explain.

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SPH4U: Car Crash!

Recorder: __________________ Manager: __________________ Speaker: _________________ Com / Know / Th / App: 0 1 2 3 4 5 For each diagram shown below to the left, draw a dotted line in the direction you believe the two vehicles will move after the collision. Provide an explanation for your reasoning. (a)

(b)

(c)

(d)

Draw solid lines on your diagrams showing the observed results of the collisions. What rules can you devise to predict more or less what is going to happen as a result of a 2-D collision?

From Laws, P. Workshop Physics Activity Guide. Volume II. John Wiley, 2004

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SPH4U: Momentum in 2-D Momentum is a vector quantity and the Law of Conservation of Momentum is a vector equation. An object’s momentum can be broken up into components and so can the law, meaning that momentum is conserved in each component direction.


Consider a simple example where a small, fast moving mass (ma = 2.0 kg) collides with a large stationary mass (mb = 4.0 kg)

y va' = 2.0 m/s

1. Quickly use you intuition to predict the direction mb will travel after the collision. Draw a vector arrow in the diagram to show this. Briefly explain your reasoning.

x

60o

ma

va = 6.0 m/s

mb vb = 0

2. For simplicity we chose a coordinate system that lines up with va. Determine the initial x- and y-components of the momentum of each mass. Show your work and be sure to use the given sign convention. pax =

pbx =

pay =

pby =

p′ax =

p′ay =

3. Complete the tables below. Calculate the total momentum in each direction before the collision. Use a scale of 1kgm/s = 1 square to draw the vector components of the momentum on the grid. The positive direction is to the right. pax pay pbx

pby

ptx

pty

4. Complete the table below for the momentum components after the collision. p′ax p′ay

r r r ′ will look like. Briefly explain your reasoning. predict what the vector pbx

5. Compare the vector ptx with p ′ax . Based on your drawings for these vectors, ptx p′ax p′bx The conservation of momentum in the x-direction tells us that: pax + pbx = p′ax + p′bx 6. Use the conservation of momentum in the x-direction to solve for p′bx.

85

7. A student remarks that the magnitude of p′bx is quite large compared with the other components. “It must be going really fast.” Do you agree or disagree? Explain.

r

pty

p′ay

p′by

r

8. Compare the vector pty with p ′ay . Based on your drawings for these vectors, r ′ will look like. Briefly explain your reasoning. predict what the vector pby

The conservation of momentum in the y-direction tells us that: pay + pby = p′ay + p′by 8. Use the conservation of momentum in the y-direction to solve for p′by.

9. Looking back at your work so far, why was it helpful to choose a coordinate system that lined up with va.

r

r

r

′ to draw pb′ . Compare this with your prediction. ′ and pby 10. Use the components pbx r

r

11. Calculate pb′ and θ. Use these to determine pb′ .

r

12. Determine vb′ .

13. Complete the tables below. Calculate the total momentum in each direction after the collision. Compare the totals after with the totals before. pty p′ay p′by p′ty p′ax p′bx p′tx ptx

14. What does it mean to say, “Momentum is conserved in each direction”? Use the results from the previous question in your answer.

86

SPH4U: 2-D Momentum Problem Solving A typical problem involving the conservation of momentum in 2-D is often challenging for students, usually due to lack of organization and careless mistakes. Problem Two hover pucks glide towards each other, collide and then glide away. Puck A (5.0 kg) was initially travelling at 2.0 m/s [E 25o N]. Puck B (3.0 kg) was initially travelling at 4.0 m/s [E 30o S]. After the collision, puck A travelled at 1.6 m/s [E 30o S]. Determine sketch the velocity of Puck B after the collision. 1.

Draw a sketch of the collision, including a prediction for the motion of puck A after the collision. Use symbols to label the important quantities.

2.

Draw the coordinate system clearly showing the positive x- and y-directions.

3.

Create a chart showing all the known velocities and their components. Use your sign convention at this step.

4.

Write down an algebraic expression for the law of conservation of momentum in the x-direction. Solve this for the unknown. m1v1x +

5.

Write down an algebraic expression for the law of conservation of momentum in the y-direction. Show explicitly the masses and velocity components. Solve this for the unknown. m1v1y +

6.

If your final result is a vector, draw a vector triangle and construct the unknown quantity.

7.

State your final answer in terms of the original problem.

87

88

A

pay‘ =

pax‘ =

Eka‘ =

vay‘ =

vax‘ =

Puck A

Ekb =

Pbx =

pay =

pax =

Eka =

Vbx =

vay =

vax =

Puck A

Ekb‘ =

Pbx‘ =

Vbx‘ =

Totals

Ekt =

pty =

ptx =

Pby‘ =

Vby‘ =

Puck B

After the Collision

Pby =

Vby =

Puck B

Before the Collision

Ekt‘ =

pty‘ =

ptx‘ =

Totals

B

On this page is pictured part of a collision between two dry-ice pucks A and B (ma = mb = 300 g). The collision takes place using magnets and is captured using a strobe light that flashes 10 times a second. Ignore the numbers in the diagram! The centre of each puck is indicated using a small circle and the path of puck B has been erased. Puck B was initially at rest and was positioned at the small circle near the letter B. The image is shown to scale. Show your results on the this page and your work on the following page.

SPH4U: 2-D Collisions Recorder: __________________ Manager: __________________ Speaker: _________________ Com / Know / Th / App: 0 1 2 3 4 5

1. Label the points where the collision appears to begin and end. Explain how you can tell.

2. Student A says, “Let’s choose a coordinate system that lines up with the page – that will be the most helpful.” Student B says, “I think we should choose one that lines up with the initial velocity of puck A. That will be easiest.” Who do you agree with? Explain.

3. Student A says, “Let’s use units of cm/s with our calculations – that will be convenient.” Student B says, “Hmm … maybe we should use m/s, even though the numbers will be very small – they are S.I. units.” Who do you agree with? Explain.

4. Write a scalar equation for the conservation of momentum in the x-direction where vBx’ is the unknown. Solve this for vBx’. Note: the equation will simplify. Explain why.

5.

Write a scalar equation for the conservation of momentum in the y-direction where vBy’ is the unknown. Solve this for vBy’.

r 6. Determine the vector vB ’. Draw the vector beginning at the circle under the “B”. ** Check this with your teacher! **

7. Use kinetic energy calculations to help explain what type of collision this is?

89

SPH4U: Working the Angles How should work be calculated when the external force and the displacement of an object are not parallel?


You will need: a spring scale (5 N or 10 N), a dynamics cart, a smooth surface, lots of mass (2 or 3 kg), a protractor

Add lots of mass to your dynamics cart and practice pulling it approximately 0.5 m with a small, constant rate of acceleration. Repeat until you are good at it! 1. Hold the spring scale horizontal to the table and use it to pull the cart a distance of 0.5 meters along the horizontal surface in such that it moves with at a small, constant rate of acceleration. Record the magnitude of force, (F1) and the distance in the space below and calculate the work done on the block (W = FΔd).

2. Repeat the measurement, except pull on the cart at 30o above the horizontal. Is the magnitude of the force, F2, larger or smaller that you measured in #1? Without calculating, how does the work done in this trial compare with the work done in #1.

3. For a given force, at what angle θ between the force and the direction of motion would the work be a maximum? A minimum? (in absolute values!) At what angle is the cosine function a maximum? A minimum? (in absolute values!)

4. Modify the simplistic expression, W = FΔd, to incorporate what you have learned about angles. Draw a diagram showing the two important vectors, displacement and force, and the anglesθ .

5. What would happen to the work if the angle was 180o? What would be happening to the object?

90

from Laws, P. Workshop Physics Actigity Guide. Module II. John Wiley, 2004

SPH4U: Work and Kinetic Energy


A: Two Types of Work A block is moving to the left on a frictionless, horizontal table. A hand exerts a constant, horizontal force on the block. 1.

Suppose that the work done on the hand is positive. In the space to the right, draw arrows to show the direction of the block and the direction of the force by the hand.

Com / Know / Th / App: 0 1 2 3 4 5

Displacement of block

Force on block by hand

a) Explain how you chose the direction of the force on the block by the hand.

b) Is the block speeding up, slowing down, or moving with a constant speed? What is happening to the kinetic energy of the block?

2.

Suppose the block again moves to the left but now the work done by the hand is negative. In the space to the right, draw arrows to show the direction of the block and the direction of the force by the hand.

Displacement of block

Force on block by hand

a) Explain how you chose the direction of the force on the block by the hand. What is the angle between the force and displacement vectors?

b) Is the block speeding up, slowing down, or moving with a constant speed? What is happening to the kinetic energy of the block?

c) Does the sign of the work in these two examples (#1 and #2) depend on a choice of sign convention (like with vector quantities)? Explain.

** Demonstrate an example of the two situations above for your teacher. **

B: Comparing the Amount of Work In a new experiment, two hands push horizontally on the same block. Hand 1 pushes to the left on the block and Hand 2 pushes to the right. Initially the block is moving to the right and is speeding up. 1.

In the box to the right, draw a complete FBD for the block that shows the horizontal and vertical forces exerted on the block.

2.

Determine whether the work done by each force is positive, negative, or zero. Indicate this writing a +, –, or 0 above (or beside) the force vector.

3.

Which forces add kinetic energy to the block and which forces remove kinetic energy? Which have no effect on the kinetic energy? Explain.

(a) The block is moving to the right and is speeding up

Wnet =

91

To find the total effect of the forces on the kinetic energy of the block, we determine the total or net work. 4.

Determine whether the net work, Wnet = Wh1 + Wh2 + Wg + Wn, is positive, negative, or zero.

5.

Consider three more situations that occurred as part of the experiment above with the two hands pushing on the block. For each situation: • draw a complete FBD for the block that shows the horizontal and vertical forces exerted on the block; • determine whether the work done by each force is positive, negative, or zero; • determine whether the net work, Wnet, is positive, negative, or zero.

(b) The block is moving to the left and is speeding up

(c) The block is moving to the right and is slowing down

(d) The block is moving to the left with a constant speed.

Wnet =

Wnet =

Wnet =

6.

There are two ways to calculate the net work. You can add up all the individual works, like in #5, or you can use the net force. Use algebraic expressions to show that either will give the same results for the example of this experiment. Choose situation (a) and you may assume the block moves through a displacement of Δd. You may notice a minor difference in the two expressions. Explain the reason for the difference. (a) Wnet = Wh1 + Wh2 + Wg + Wn =

(b) Wnet = FnetΔd =

7.

Generalize your answers from the four situations (a – d) above and describe how the speed and kinetic energy of an object change if the net work done on the object is: (a) positive

(b) negative

(c) zero

from McDermott, L. Tutorials in Introductory Physics. Prentice Hall, 2002 92

SPH4U: Conservation of Energy on a Ramp

93

Ek and Eg

from Sokoloff and Thornton. .Interactive Lecture Demonstrations. John Wiley, 2004

94

SPH4U: Energy and Frames of Reference


How do people in different reference frames near the surface of the earth view the same event with regard to mechanical energy associated Com / Know / Th / App: 0 1 2 3 4 5 with a mass and its conservation? Suppose the president of your college drops a 2.0-kg water balloon from the second floor of the administration building (10.0 meters above the ground). The president takes the origin of his or her vertical axis to be even with the level of the second floor. A student standing on the ground below considers the origin of his coordinate system to be at ground level. Have a discussion with your group and try your hand at answering the questions below.

Part A: Potential Energy What is the value of the potential energy of the balloon before and after it is dropped according to the president? According to the student? Show your calculations and don't forget to include units! The president's perspective: y = 0.0 m at t=0.0 s and y= -10.0 m when the balloon hits the student: Eg1 =

Eg2 =

The student's perspective is that y = 10.0 m at t = 0.0 s and that y = 0.0 m when the balloon hits the student: Eg1 =

Eg2 =

Part B: Kinetic Energy What is the value of the kinetic energy of the balloon before and after it is dropped according to the president? According to the student? Show your calculations. Hint: Use a kinematic equation to find the velocity of the balloon at ground level. President’s perspective: Ek1 =

Ek2 =

Student’s perspective: Ek1 =

Ek2 =

Part C: Total Mechanical Energy What is the value of the total mechanical energy of the balloon before and after it is dropped according to the president? According to the student? Show your calculations. Note: If you get the same values for the student and the president for the total energies, you are on the wrong track. President’s perspective: Et1 =

Et2 =

Student’s perspective: Et1 =

Et2 =

1. Why don’t the two observers calculate the same values for the total mechanical energy of the water balloon? 2. Why do the two observers agree that mechanical energy is conserved?

from Laws, P. Workshop Physics Activity Guide. Volume II. John Wiley. 2004

95


SPH4U: Rocket Science! Or, the science of movement in a gravitational field.

Com / Know / Th / App: 0 1 2 3 4 5

Part A: Ascent, Descent Consider a rock traveling straight up and in down in the earth’s gravitational field under the influence of gravity alone near the surface of the earth. Indicate in the chart whether the quantity is increasing or decreasing in magnitude. For velocity and work indicate the sign.

+

Trip Up

v

Trip Down

1. Describe the energy transformations during: (a) the trip up

Ek

Eg ET (Total Energy)

(b) the trip down

Work by Force of Gravity 2. When we calculate Eg, we use the value g = 9.8 N/kg. What happens to this value as we get quite far from the earth? What is the reason for this change? 3. Write down the equation for Newton’s Law of Universal Gravitation. Sketch a graph of the force of gravity over great distances. Mark two separate positions, d1 and d2. Use the text to help.

Fg =

4. Explain how you graphically represent the work done on an object moved from position d1 to d2. Show this on the graph.

96

Part B: Gravitational Potential Energy Mathematically we describe the gravitational potential energy (GPE) at a given position when using the GMm equation: Eg = − which is the result of the calculus work describing the area under the graph in r Part A Q#3.

1. Fill in the chart showing the gravitational potential energies for a 1 kg rock at different positions from the centre of the earth. (re = 6.4 x 106 m. Me = 6.0 x 1024 kg, G = 6.67x10-11 Nm2/kg2) Sketch this as a graph.

Distance from centre of earth re

Eg

2re 3re 4re 6re 8re Really, really far Note! Really, really far means an infinitely large distance

We will be comparing energies that are may be either positive or negative values. Always imagine a number line with the negatives on the left. We will compare two energy values using their position along the number line. If an energy value is found further to the right on the number line, we will say that it is greater than the other. If it is found further to the left on the number line it is less than the other. If a change causes a quantity to move to the right we will say it is increasing. If the change causes a quantity to move to the left we will say it is decreasing. 2. Use greater than or less than symbols to compare following quantities: Eg(re) Eg(2re) Eg(really, really far) Eg(re) 3. According to your graph, at what position would the greatest amount of energy be stored in the gravitational field? What is the greatest value possible for GPE?

4. Calculate the change in the GPE, ΔEg, in each case. State whether the GPE is increasing or decreasing, and whether it is being stored in or released from Earth’s gravitational field. (a) 2re to 6re (b) 8re to 4re (c) re to infinity (d) infinity to re

97

5. Imagine a small rocket engine is attached to the rock. How much work is done by the rocket engines to move the rock, without changing its speed, from: a) 2re to 6re, b) re to infinity Part C: Velocity and Gravitational Fields 1. Complete the chart for the 1 kg rock and its trip through the earth’s gravitational field (use some of your results from before!). The rock was powerfully launched straight up from the surface of the earth at 1.0 x 104 m/s. Note: An object’s kinetic energy must be a positive value – a negative value is a physical impossibility. Distance from Eg velocity Ek ET centre of earth re

2re 3re 4re 6re 8re Really, really far 2. Is it possible to find the rock at a distance of 6re? What happens to the rock between 4 and 6re? Explain.

3. Plot on one graph Eg, Ek and ET as a function of distance. Use smooth curves and straight lines to highlight the patterns. Indicate on the graph the maximum distance the rock will travel. 8 6

Energy (107 J)

4 2 0 -2 -4 -6 -8

98

Position (re)

4. Explain how you could use the values of ET and Eg on the graph to determine the rock’s furthest position from the earth. Use the conservation of energy to confirm this.

5. How much kinetic energy should the rock be launched with to reach a maximum distance of: i. 3re ii. 6re 6. With what velocity should the rock be launched to reach a maximum distance of 4re?

7. Complete the chart again, but now the rock has an initial velocity of 1.2 x 104 m/s. Distance from Eg Ek velocity ET centre of earth re 2re 3re 4re 6re 8re Really, really far 8. Plot a graph that shows Eg, Ek and ET as a function of distance. 8 6

Energy (107 J)

4 2 0

Position (re)

-2 -4 -6 -8

9. At what position will the rock in this example finally come to rest? Explain. 99

10. The rock is now launched from Earth such that it will come to rest when it has travelled an infinite distance from Earth. Explain how to find the kinetic energy required to accomplish this.

11. Derive an algebraic expression that gives the launch velocity for any object to reach a very great distance (infinity) from the earth with essentially no (zero) kinetic energy? This is called the escape velocity.

12. Complete the chart for escape velocities. Object Radius (m) Mass (kg) vescape (m/s) 6

5.98x10

Moon

6

1.74x10

7.35x1022

Jupiter

7.15x107

1.90x1027

Earth

6.38x10

vescape (km/h)

24

13. Imagine the earth and all its matter is compressed to a much smaller size a) What size of radius for the earth would give the maximum possible escape velocity (c = 3.0 x 108 m/s, the speed of light)?

b) What would happen to all objects trying to escape if the earth was compressed even further? What has been created?

14. What is the total energy of an object moving (consult your charts Q#1 and 7) a) at its escape velocity? b) slower than its escape velocity? c) faster than its escape velocity?

100

Part D: Orbits 1. An object is moving in a circular orbit around the earth with some radius r. Use the Law of Universal Gravitation and Newton’s 2nd Law to determine the object’s orbital velocity.

2. Write an expression for the kinetic energy of an object in a circular orbit. Write this using the symbol Eg.

3. Write an expression for the total energy of an object in a circular orbit. Write this using the symbol Eg.

4. To calculate the energy necessary to move from one situation to another we compare the total energy in each. How much work must be done by a rocket engine to transfer a 1000 kg satellite from a circular orbit of radius 2re to 3re?

5. How much work must be done by a rocket engine to lift a 1000 kg payload from rest on the earth’s surface to a circular orbit of radius 2re?

6. The binding energy is the work that must be done to allow an object to escape to infinity. What is the binding energy for an object: a) on the surface of the earth

b) in a circular orbit around the earth

101

SPH4U: Spring Force and Energy We would like to quantify the force and work needed to extend a spring as a function of its displacement from its unstretched position. Let’s start by carefully measuring the force and work needed to stretch a brass spring. You will need: •


1 metal spring, 1 spring scale (20 N), 1 C-clamp, 1 metre stick

We want to explore the relationship between the amount of force applied to the spring and the amount of stretch produced. You will use the C-clamp to hold one end of the spring in place and stretch the other end using the spring scale.

Part A: Force and Springs

1.

Before making any measurements, describe in an intuitive way (without numbers) how the stretch should depend on the amount of force applied. Provide a simple explanation to your younger brother for why you believe this.

2.

Make five measurements and complete the second column of the data chart below. Average Force Partial Work (J) Displacement Force Applied Partial (N) from Equilibrium (N) Displacement: (m) xn – xn-1 xo = 0.00 x1 = 0.10 x2 = 0.20 x3 = 0.30 x4 = 0.40 x5 = 0.50 Total Work (J)

102

3.

Plot a graph that shows how the force depends on the stretch. (Which axis should the force be on?)

4.

Draw a line of best-fit and find the slope of the line. Use the symbol k to represent the slope of the line. What is the value of k? What are its units? Note: k is known as the spring constant.

5.

What does a large or small spring constant tell us about the physical characteristics of a spring?

6.

Write an equation describing the relationship between the force you applied to the spring, Fapp, and the displacement, x, of the spring from its equilibrium position using the symbols Fapp, x, and k. Explain what this equation tells us about the relationship between the force applied and the stretch.

7.

According to Newton’s 3rd law, the spring exerts a restoring force on the spring scale or on your hand that is equal and opposite to the applied force, so Fspring =

If a restoring force on an object is proportional to its displacement, it is known as a Hooke’s Law force. This law is named after an erratic, contentious genius named Robert Hooke who was born in 1635. Part B: Energy and Springs

1. We used our spring scale to exert a force and stretch the spring, doing work. Describe the energy transformations that take place as you stretch the spring.

103

2. We want to find the amount of work done as we stretch the spring. Two students discuss how to do the calculation. Alice says, “I think we should use the value of the force at the start of the displacement and multiply that by the stretch since the force is always smaller at the beginning.” Bob says, “I think we should use the value of the force at the end of the displacement since it can become quite large, otherwise your result would be too small.” Who do you agree with? What force value would you use? Explain.

3. The chart for question A.2 contains three columns that will help us calculate the work. Explain the meaning of the quantity in each column. After that, complete the chart. a) Partial displacement: b) Average force: c) Partial work:

5. Now we are ready to calculate the total work done to stretch a spring from its equilibrium position to each displacement. To do this, consider that the work to displace the spring to 0.20 m is equals the work to go from 0 to 0.10 m plus the work to go from 0.10 to 0.20 m. Complete the chart to the right and plot a graph of total work vs. displacement. 6. Calculate the work need to stretch the spring to a displacement of 0.40 m by computing the area under the Fappx vs. x graph you created earlier. Compare this with your result from #1.

Displacement from Total Work (J) equilibrium (m) 0 0.10 0.20 0.30 0.40 0.50

7. Speculate on an equation that relates W and x. Hint: Consider how you computed the area to get the work. You may be guessing that this is indeed related to an integral!

from Laws, P. Workshop Physics Activity Guide. Volume II. John Wiley. 2004 104

Exploring Elastic Energy


from McDermott and Shaffer, Tutorials in Introductory Physics. Prentice Hall, 2002

105

SPH4U: Velocity and Frames of Reference All physics quantities that we measure depend on the frame of reference of the observer.


Alice is standing on the Earth and watches a train go by with a velocity of 150 km/h [E]. Inside the train stands Bob. Both Alice and Bob are physicists and make observations about each other’s motion. 1. Complete the chart showing the measured velocity of each object from each reference frame. Object Alice Bob Earth Train

Frame A (Alice)

Frame B (Bob)

2. Bob has a ball and throws it. He measures the velocity of the ball to be 40 km/h [E]. The train keeps going at its usual speed. Complete the chart showing the measured velocity of the ball from each reference frame. Explain how you found the velocity of the ball relative to frame A. Object Ball

Frame A (Alice)

Frame B (Bob)

3. Bob throws a second ball and measures the velocity to be 30 km/h [W]. Complete the chart showing the measured velocity of the ball from each reference frame. Explain how you found the velocity of the ball relative to frame A. Object Ball

Frame A (Alice)

Frame B (Bob)

4. Bob pulls out a flashlight, points it east and turns it on. Using a fancy apparatus he measures the velocity of a particle of light from his flashlight to be 300 000 000 m/s [E]. Using the previous logic, what is the velocity of the light relative to Frame A in m/s?

5. Imagine Bob was on an “express” train that travelled at 2 x 108 m/s [E] and turned on his flashlight just as in question 4. What is the velocity of the light relative to Frame A?

6. Alice now has her flashlight turned on and points it east. Bob’s same express train passes by. What is the velocity of the light from Alice’s flashlight relative to Bob?

106


SPH4U: The Light Clock

Bob is travelling in a spacecraft at a velocity, v, relative to the earth. He is Com / Know / Th / App: 0 1 2 3 4 5 carrying with him a light clock – a special kind of clock invented by Einstein. The clock consists of two perfect, smooth mirrors that face each other and are separated by a distance D. A particle of light (a photon) reflects back and forth between the mirrors which are lined up carefully so that the photon always reflects off the same points X and Y. The time it takes for the photon to travel between the mirrors represents the “tick’ of the clock. 1. Complete the chart of measurements from Bob’s frame of reference. Time for the light to travel from X to Y (one tick) Velocity of the light clock itself Distance the photon travels From X to Y Speed of the photon

to

X

•

2. Construct an equation that relates the speed of the photon to the distance, time it travels as measured in Frame B. Alice is standing on Earth watching Bob travel by in the rocket ship. She is able to make careful measurements of the light clock and its photon.

D

• Y

3. From Alice’s frame, we see the light clock at three moments in time corresponding to the photon at Y, the photon at X and then the photon back at Y (two ticks of the clock). You may assume the rocket is travelling quite fast! Draw the path of the photon through space. Label the interval XY as d.

4. Complete the chart of measurements from Alice’s frame of reference. No calculations are required! Time for the light to travel from X to Y (one tick) Velocity of the light clock Distance the photon travels in one tick Speed of the photon

t

5. Construct an equation that relates the speed of the photon to the distance, time it travels as measured in Frame A. 6. Compare the size of the results from each frame. Measurement Speed of photon Distance travelled from X to Y (d vs. D) Time for photon to travel from X to Y (t vs. to)

Comparison

Note that both observers must agree on the speed of light according to the first postulate of special relativity. 7. Speculate on the implications of your comparisons for the flow of time on the spacecraft!

107


SPH4U: Why Don’t We Notice? 1. Complete the chart below. Rewrite the first five speeds in terms of c. Calculate γ for each speed. Sketch a graph of γ vs v. Speed

Speed (in terms of c)

Com / Know / Th / App: 0 1 2 3 4 5

γ

Fast Runners, 10 m/s Fast Cars, 40 m/s Fast Jets, 600 m/s The Space Shuttle, 7 860 m/s Voyager Space Probe, 17 000 m/s 0.1 c 0.3 c 0.5 c 0.7 c 0.9 c 0.99 c X-Ray Machine Electrons 0.999 c LHC protons, 0.999 999 999 95 c TV screen electrons

20.0 —

15.0 —

10.0 —

5.0 —

0

| 0.2 c

| 0.4 c

| 0.6 c

| 0.8 c

| 1.0 c

2. Should the first five γ values you calculate be the same? Explain.

3. Based on the chart, offer a simple explanation for why relativistic effects are not noticed in daily life.

4. What happens to the size of γ as v approaches the value c?

5. What does this tell us about the flow of time for a highly relativistic object (speeds close to c)?

6. Relativistic effects are important for GPS satellites which orbit at a similar speed to the space shuttle relative to the ground. Precision timing is absolutely essential for determining an object’s location on the earth. For a GPS satellite observed from the earth, γ = 1.000 000 000 3. a) Over the course of one day, how much time in seconds does the GPS clock gain or lose compared to a ground clock? Watch your math!

b) How far does light travel during that time discrepancy? What would be the implication for the GPS system?

108

SPH4U: Distance and Velocity Consider the subatomic particle called the muon, μ, that is moving rapidly at a speed, v, relative to a metre stick in the direction of the stick’s length.


Part A – The Stick’s Frame 1. The diagram below shows the metre stick at rest. Draw the muon at two moments in time when: A – the muon is at the left end of the stick; and B, the muon is at the right end of the stick. Indicate the direction of motion of the muon. Label the distance the muon travels along the metre stick, Lo.

2. An observer in the stick’s frame of reference measures the time interval between the two events. Explain what type of time interval this is. What symbol should you use to represent it?

3. Write an equation that relates speed, distance and time of the muon as measured by an observer in the stick’s frame.

Part B – The Muon’s Frame 1. The diagram below shows the muon at rest. Draw the metre stick at two moments in time when: A – the tip of the stick first reaches the muon; and B, the opposite end of the meter stick reaches the muon. Label the distance the metre stick travels, L. Indicate the direction of motion of the metre stick.

μ 2. An observer in the muon’s frame of reference measures the time interval between the two events A and B. Explain what type of time interval this is. What symbol should you use to represent it? 3. How far does the metre stick travel in this frame of reference? 4. Write an equation that relates speed, distance and time for the metre stick as measured by an observer in the stick’s frame. Compare Compare the size of the results from each frame. Speed of muon / metre stick Time between events A and B Distance between ends of metre stick What can we conclude about the distance measurement of each observer?

109

SPH4U: Relativity and Energy The consequences of Einstein’s bold suggestion, that the speed of light is constant for all inertial reference frames, go far beyond just space and time – they also extend to our notions of energy. Using a clever argument, Einstein created the world’s most famous equation:


E = γmc2 where γ = (1-v2/c2)-½ This is usually written, for the general public, as Eo = mc2, where the “o” is usually left out! Sometimes physics ideas stretch beyond our common sense and we begin to rely on equations to help us understand how our universe works. Let’s explore this equation and try to figure out what it tells us about energy. Part A: The Mass-Energy Relationship 1. Describe carefully how this energy depends on the velocity of an object.

2. What other type of energy depends on an object’s velocity? What does this tell us about the type of energy Einstein’s equation describes?

3. Describe carefully what happens to the energy when the object is at rest. Is Einstein’s equation still describing kinetic energy?

4. In other situations, what are some examples of types of energy that may be present when an object is at rest? Excluding thermal energy, what broad category of energy do these types fall into?

5. Explain how the equation for E becomes the equation for Eo when the object is at rest. What quantity may change the amount of “rest energy”? How does it change?

An object at rest possesses a form of potential energy called its rest energy, Eo, Einstein’s complete expression (E = γmc2) gives the total energy of the object, which always includes the rest energy and possibly some kinetic energy depending on the object’s velocity. 6. Write an expression that shows the relationship between E, Eo and Ek

110

Part B: Relativistic Energy 1. Consider a 1.0 kg block moving at 0.6 c. Imagine we had learned nothing about relativity and determine the energies for the “Before Einstein” column in the chart below. Use Einstein’s equation to fill out the third column in the below. Before Einstein (B. E.)

After Einstein (A. E.)

Total Energy Rest Energy

zero

Kinetic Energy

2. Explain how you use Einstein’s equation to find the relativistic kinetic energy.

3. Based on the results in the chart, what should we conclude about the limitations of the traditional kinetic energy equation and Einstein’s equation?

Accelerating an object to speeds near that of light is extremely challenging and with our current technology, we can only accomplish this for atoms and sub-atomic particles. According to Newton, all we need to do is exert a steady force on something for long enough and the uniform acceleration will eventually cause the object to reach 3.0 x 108 m/s and our science fiction dreams will come true. According to Einstein, things are different.

4. How much work would a force have to do to accelerate a 1.0 kg rock to the speed of light? Explain the mathematical difficulty with performing this calculation? Explain to your kid sister how much energy would you “need”?

5. What does the difficulty of the previous calculation imply about the possibility of ever reaching or exceeding the speed of light?

This is the main reason why the latest and greatest particle accelerator, the Large Hadron Collider ($ 9 000 000 000), is such a colossal engineering feat. A tremendous amount of energy is required to accelerate the collider’s protons to 0.999 999 991 c.

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Part C: Particle Physics

1. A proton is a very small particle with a mass of 1.673 x 10-27 kg. How much energy is stored in the mass of the particle?

Subatomic particles usually possess very small quantities of energy. A new unit is needed to conveniently notate these small values. One electron volt (eV) is a unit of energy equivalent to 1.602 x 10-19 J. 2. Find the proton’s rest mass energy in terms of MeV (106 eV).

3. Physicists often write the mass of the proton as 938.3 MeV/c2. Use the rest-energy equation to help explain why this is a valid unit for mass. Explain why these units make it easy to calculate the rest energy.

Is it possible to release the energy stored in a particle’s mass? You may have already studied the process of nuclear fusion or fission in another course and have learned that, yes, this is possible. In a typical fusion reaction, a deuterium particle (1876 MeV/c2) fuse with a tritium particle (2809 MeV/c2) producing a helium nuclei (3729 MeV/c2) and a neutron (937 MeV/c2). D + T → He + n 4. How much energy is released in this process? Give your answer in joules and electron volts.

5. What does this imply about the conservation of mass?

The conversion of matter to energy can be total if a matter particle collides with a corresponding anti-matter particle. This is the purpose of the Large Hadron Collider: to collide protons and anti-protons, which releases a tremendous amount of energy. This is also the physics behind the medical imaging technique positron imaging tomography (PET scans), where an electron collides with a positron (the antielectron). In the case of the PET scan, radioactive materials are injected into the blood stream of a patient. The decay process releases a positron (via beta decay) which collides with an electron of a nearby atom. In the process, the two particles annihilate and produce two gamma-ray photons (γ). 112

e- + e+ → energy (two photons) 6. How much energy is released when an electron (0.511 MeV/c2) collides with a positron (same mass) and the two annihilate (leave no mass behind)?

7. In Star Trek, the main power source for the starship enterprise is a matter-antimatter engine. How much energy would be produced by annihilating 1.0 L of gasoline (0.720 kg) with 1.0 L of anti-gasoline (0.720 kg)? What form of energy is the annihilation energy transformed into? What speed would that accelerate a typical car (1200 kg) to (use ½mv2)?

8. The previous result is quite fast! We should confirm this with a more reliable calculation using Einstein’s equation to solve for γ and then v. Use the result for γ explain why the result from #7 was reasonably accurate.

BONUS: The reverse process can also take place! Energy can be converted into a particle – antiparticle pair. e- + e + → γ + γ → p + p In this case, the extra kinetic energy of the electron-positron pair is converted into the mass of the proton and anti-proton. This is exactly what used to happen at the LEP (Large Electron Positron) collider at CERN in Switzerland. What should the speed be of an electron and positron in the LEP to allow this to happen?

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SPH4U: Journey to Flatland


A: Lineland

Com / Know / Th / App: 0 1 2 3 4 5

Our first stop is a visit to a very unusual universe that has only one dimension – it is shaped like a line. This universe is inhabited by curious creatures called Liners. They are point-particle organisms that live on “The Line” (as they call their world.) This is the story of the Liners Alice, Bob, Chun-Ying, and Jennifer who are all neighbours on The Line, a portion of which is shown in the diagram below. Alice •

Bob •

Chun-Ying •

Jennifer •

The Line

1. Let’s think about how Liners can move about. In which directions can Chun-Ying move? What will happen when she reaches Bob? How many Liners has she ever physically met?

Bob likes to talk and one day he mentioned to both Alice and Chun-Ying that he likes Jennifer and wishes he could meet her one day. “But promise me you won’t tell Jennifer”, Bob says to Chun-Ying. “Sure, sure”, she replies. The next day Chun-Ying exclaims to Bob, “Jennifer knows you like her!” Bob, shocked, shouts back, “I told you not to tell her!” “I didn’t say anything – Jennifer said she heard it from her neighbour Jose”, replies Chun-Ying. Bob turns to Alice, “What’s going on? How could she find out?” Alice responds, “I don’t know, but I did tell my neighbour Elizabeth.” 2. How did Jennifer find out? What do the Liners not realize about the shape of Lineland itself?

Alice is a Liner physicist. She has studied the theoretical “second dimension” in school, but didn’t believe it existed until now. She explains to the others: “Now we know that the directions up and down do exist. But we can’t move into the second dimension or see it since it is perpendicular to The Line.” The others have never heard of this term: perpendicular. Alice continues, “but we can imagine what might happen if Bob could simply walk into the second dimension.” 3. Explain what Alice and Chun-Ying would observe if Bob magically moved in a direction perpendicular to the line. Why would this be a truly shocking experience for them?

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4. Bob returns to The Line, but surprisingly finds himself in between Chun-Ying and Jennifer (what luck!). Explain how Bob did this. How would Bob’s complete trip appear to Chun-Ying?

B: Flatland

It’s time to leave Lineland behind and travel to a new and more complex universe called Flatland. This universe has two dimensions of space and is populated by colourful and pancake-like creatures called Flatlanders - a portion of Flatland is shown to the right. Its inhabitants move freely in “The Plane” as they call it, moving North and South, East and West. Two Flatlanders, curiously called Alice and Bob, are chatting.

A B

1. Alice says to Bob, trying to make conversation, “You look nice today”, she says. How does Bob appear to Alice?

After some chit-chat, Alice holds three rulers and starts talking to Bob about a theoretical third dimension of space. Bob understands the idea of “perpendicular”, but then Alice continues by saying that “Up” and “Down” are directions perpendicular to both North-South and East-West. Bob looks very puzzled. 2. Explain why Bob doesn’t understand. Pretend you are Alice and demonstrate these ideas with rulers for your teacher.

A mysterious voice, seeming to come from every direction, is heard throughout The Plane. “Bob … I will help you understand.” A hand reaches into Flatland, grabs Bob and pulls him upwards, perpendicular to The Plane. 3. How would a set of fingers that intersects The Plane appear to Alice? What would she see when Bob is being kidnapped?

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Bob looks back at The Plane and shrieks, “Alice, is that you? You look hideous! You have spots! What am I seeing?” 4. Explain what is special about what Bob is seeing and why it is so surprising to him.

“Please put me back! I’m afraid”, yelps Bob. The mysterious hand obliges and places him back in The Plane. After a moment Bob gets back his senses and yells, “What’s wrong with me? Ahh!”

A

5. What happened to Bob during his trip through the third dimension?

C: Journey to the Fourth Dimension

Alice, your physicist friend, is now talking to you about a mysterious fourth dimension of space. She says we can’t see it, or move through it, but it is perpendicular to our three dimensions of space. She knows she can’t point to the fourth dimension using perpendicular rulers so she tries to show you another trick. “Let’s build a four dimensional cube, a hypercube”, she announces. “Okay, take a straight line and fold it in the second dimension four times, join the ends and you have a square!” 1. Annotate the diagram below, and use arrows to show where the segments move and which ends meet up with which other ends.

•

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•

•

•

•

Alice continues, “Now try taking a set of squares, fold them into the third dimension, join up the ends and you get a cube!” 2. Annotate the diagram to the right, and use arrows to show which edges meet up with which other edges.

“Now you’ve got it!” Alice exclaims. “And now, for the grand finale, take a collection of cubes, fold them in the fourth dimension, join up the surfaces and, voila, you’ve got a hypercube!” This collection of cubes is called a tesseract (see http://en.wikipedia.org/wiki/Tesseract). 3. Annotate the diagram to the right, and use arrows to show which faces meet up with which other faces.

Alice finishes by saying, “we can see 4-D objects in the same way that we can trace the 2-D shadows of a 3-D object on a piece of paper. Position the object or light differently, and you get a differently shaped shadow. Here are some of those ‘shadows’ for a hypercube.” For more, search YouTube for: Dr. Quantum Flatland. 4. What would it look like if a 4-D creature reached in to our 3-D universe, grabbed your neighbour and pulled him out? What would it look like as that slowly hand reaches in?

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SPH4U: Getting a Charge

Electrical Interactions Com / Know / Th / App: 0 1 2 3 4 5 A. Press a piece of sticky tape, about 10-15 cm in length, firmly onto a smooth unpainted surface, for example, a tabletop. (For ease in handling, make "handles" by folding each end of the tape to form portions that are not sticky.) Then peel the tape off the table and hang it from a support (e.g., the edge of a table or pencil). Describe the behaviour of the tape as you bring objects toward it (e.g., a hand, a pen).

B. Make another piece of tape as described above. Bring the second tape toward the first (sticky sides apart!). Describe your observations.

It is important, as you perform the experiment above, that you keep your hands and other objects away from the tapes. Explain why this precaution is necessary.

How does the distance between the tapes affect the interaction between them?

C. Have two members of your group each press a tape onto the table and write a "B" (for bottom) on it. Then press another tape on top (completely overlapping) of each B tape and label it "T" (for top). Pull each pair of tapes off the table as a unit. After they are off the table, separate the T and B tapes. Hang one of the T tapes and one of the B tapes from the support at your table. Describe the interaction between the following pairs of tape when they are brought near one another. • two T tapes

• two B tapes

• a T and a B tape

D. Obtain an ebonite rod and a piece of fur. Rub the rod with the fur, and then hold the rod near newly made T and B tapes. Describe how the rod interacts with the two tapes. How do these interactions compare with the T and B tapes together (part C)?

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E. Base your answers to the following questions on the observations you have made thus far. 1. Is it possible that there is only one type of charge? If not, what is the minimum number of different types of charge needed to account for your observations thus far? Explain carefully.

2. Use the electrostatic series to explain what type of charge the ebonite rod acquires.

Which tape, T or B, has a positive charge? Explain.

Please remove all tape from the tabletop before continuing. Insulators, Conductors and Induction A. What happens when you bring a negatively-charged rod near an uncharged insulator consisting of a single Styrofoam ball? 1. Before the two objects touch: 2. After they touch: B Repeat observation a. using an uncharged metal-coated Styrofoam ball. Note: If it “works” you should see it “jump”. 1. Before the two objects touch: 2. After they touch: C. Use induction and the idea that charges move readily on conductors, to explain why the metal-covered ball is attracted to the rod before touching and repelled after touching it.

D. Use the idea that if an object is an insulator, its electrons will stay in their vicinity of their atoms, to explain why the uncoated Styrofoam ball is still attracted to the rod after touching it.


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SPH4U: Electrostatic Forces Your task is to determine the effects of the magnitude of charge and the separation of the charged objects on the strength of the electrostatic forces. You will need: • Two metal-coated Styrofoam balls, one in the middle of a thread • Two retort stands • One metre stick • One ebonite rod and fur • Anything else useful


Part A: Introduction Set-up your apparatus with the metre stick flat on the table and the ball hanging between two stands forming a tall “V”. Please do not cut or tie the thread – use tape to attach it and use its full length. Choose a combination of ebonite rod and fur that produces very noticeable repulsive static electric effects with the ball. Demonstrate this for your teacher. You will use this set-up to measure the size of the electrostatic force between the charged rod and the charged ball when the ball is deflected by the rod. Consider the bird’s-eye view illustration below. The ball was initially hanging above the 90 cm position and has been deflected to the left.

ball

rod

1. Explain how to determine the deflection of the ball. Draw an arrow on the illustration above showing the deflection and determine its value.

2. Explain how to determine the separation between the charged objects. Draw an arrow on the illustration above showing the separation and determine its value.

3. Describe how you can modify your apparatus to make the measurement of these quantities more accurate. Make these modifications!

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4. Produce a fairly steady situation with a large deflection of the ball. Continue to hold the rod in place. Describe what gradually happens. Explain how this knowledge is useful in designing your experiment.

5. Draw a FBD for the ball as viewed from the side, showing the deflection of the ball.

FBD

6. Use the FBD to help derive an expression for the magnitude of the electrostatic force, Fe, as a function of L, x, m and g. Note: you may assume that since θ is small, tan θ ≅ x/L

θ L

y

x

7. Measure L for your apparatus. What simple error might students make in determining this due to the “V” shape of the thread?

8. The ball is extremely light and its mass is difficult to determine. Look for the plastic case labelled “For Mass Measurement”. Describe a procedure that will allow you to make a more accurate mass measurement. Determine the mass of the ball in kg.

Part B: Magnitude of Charge and Force You will now begin your investigation into the effect of the magnitude of charge and the force. 1. When Coulomb first performed his experiments into electrostatic forces, he devised a clever trick to reduce the charge of a charged conductor by one half. He simply touched the charged conductor with an identical, neutral one. Explain how this works.

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2. Describe the procedure for your experiment.

3. Based on your initial observations with the equipment, make a prediction for the relationship between the magnitude of charge and force. Show your preparations to your teacher.

Practice conducting your experiment before taking the official data – make sure you are good and quick at it. 4. If one object is accidentally touched during the data taking, explain why you cannot simply recharge it and continue from where you left off.

5. Perform three successful repetitions of the experiment. If you can halve the charge further, do so. Charge

Run #1 Deflection, x, (m)

Force, Fe (N)


Force, Fe (N)


Force, Fe (N)

q q/2 q/4 q/8

6. Explain why you cannot simply average the force results from the different runs together, even though the charge is always decreasing by half.

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7. Plot the three runs together with separate lines on one graph. Be careful how you mark the axis scale for charge and the data points. 8. Explain how the data supports or contradicts your prediction.

0

0.25

0.5 Charge (q)

0.75

1.0

9. We determined the size of the electrostatic force acting on the ball. Explain how, with no further measurements, to determine the size of the electrostatic force acting on the rod.

Part C: Separation and Force You will now begin your investigation into the effect of the separation of the charged objects and the force. 1. Describe the procedure for your experiment.

2. Based on your initial observations with the equipment, make a prediction for the relationship between the separation of the charged objects and force.

Practice conducting your experiment before taking the official data – make sure you are good and quick at it.

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Perform three successful repetitions of the experiment. Run #1 Separation Deflection Force (m) (m) (N)

Run #2 Separation Deflection Force (m) (m) (N)

Run #3 Separation Deflection Force (m) (m) (N)

3. Plot the three runs together using separate curves on one graph. Be careful how you mark the data points. 4. Explain how the data supports or contradicts your prediction.

0

0.05 0.1 Separation (m)

0.15

5. Does the data best support a 1/r or a 1/r2 relationship? This may be hard to tell, but based on your data, try to describe the relationship between separation and force.

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SPH4U: Collections of Charges


Careful experimentation has shown that the strength of the Com / Know / Th / App: 0 1 2 3 4 5 electrostatic force between two point-like charges can be represented by Coulomb’s law: kq q Fe = 1 2 2 r It is our job now to learn how to use that equation. Consider the combination of charges in the following questions.


125

C. Two charges are located in a plane as shown below. qa = +2.0 x 10-9 C, qb = -2.0 x 10-9 C, k = 8.99 x 109. The scale for forces is 1 cm = 1.0 x 10-10 N and the scale for distances is 1 cm = 1 cm. Determine and draw the net force vector for a test charge, qt = +2.0 x 10-14 C, located at points A, B, C and D. For ease, use graphical techniques (measurements and scale vector diagrams) along with Coulomb’s law – don’t use components! Remember: one will attract and the other repel! Use a different colour for the net force vector. Show your calculations below. |Fe| aA bA aB bB aC C•

A•

bC aD

+

-

qa

qb

D

•

bD

|Fnet| B•

A B C D

from Laws, P. Workshop Physics Activity Guide Module 4. John Wiley, 2004

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SPH4U: Gravity vs. Electricity


A fact emerges that is quite amazing. Both types of forces, electrical Com / Know / Th / App: 0 1 2 3 4 5 and gravitational, are very similar. Essentially the same mathematics can be used to describe orbital and linear motions due to either electrical or gravitational interactions of the tiniest fundamental particles or the largest galaxies. A. Write down and examine the mathematical expression for the two force laws.

1. What is the same about the two force laws? 2. What is different? For example, is the force between two like masses attractive or repulsive? How about two like charges? What part of each equation determines whether the like charges or masses are attractive or repulsive?

Gravitational forces hold the planets in our solar system in orbit and account for the motions of matter in galaxies. Electrical forces serve to hold atoms and molecules together. If we consider two of the most common fundamental particles, the electron and the proton, how do their electrical and gravitational forces compare with each other? Let's peek into the hydrogen atom and compare the gravitational force on the electron due to interaction of its mass with that of the proton to the electrical force between the two particles as a result of their charge. In order to do the calculation you'll need to use some well-known constants. Electron: me = 9.l x 10-31 kg Proton: mp = 1.7 x 10-27 kg

qe = -1.6 x 10-19 C qp = +1.6 x 10-19 C

Distance between the electron and proton: r = 1.0 x 10-10 m a. Calculate the magnitude of the electrical force on the electron. Is attractive or repulsive?

b. Calculate the magnitude of the gravitational force on the electron. Is it attractive or repulsive?

c. Which is larger? By what factor (that is, what is the ratio)?

d. Which force are you more aware of on a daily basis? If your answer does not agree with that in part c, explain why.

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SPH4U: Picturing Electric Forces Individual Question – Draw force vectors showing the electric force acting on a test charge (always positive!) at each point surrounding the single positive and negative charges. Carefully show the relative size of the vectors!

• • •

• •

• •

+ •

• •

•

• •

-

•

• •

Class Question – Draw the electric field lines representing the forces surrounding the single positive and negative charges. Ignore the dashed circles.

+

-

1. Explain how far the electric field lines extend into space. Make sure your diagrams reflect this. 2. In a field diagram, we no longer see the length of force vectors to help us determine the strength of the electric effects. Use the two dashed circles to help explain another way to determine the strength.

3. The two field diagrams above illustrate two parts of an important rule. Create a rule that describes the type of charges at which field lines start and stop.

4. Use these ideas to predict the appearance of the electric field around a line of charges (a charged rod).

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Group Question – Predict the net electric force a test charge will experience when placed next to a charge dipole, two opposite but equal charges. Draw three vectors for each point (no calculations, just estimate!) – one for each charge and one for the net force. In the next diagram sketch your prediction for the electric field around the dipole.

• •

+

•

-

•

+

-

•

1. What is the charge dipole a basic model of?

2. How do we explain how objects composed of equal quantities of opposite charges can appear to be “neutral”.

Group Question: What will the electric field between two oppositely charged parallel plates look like? Use the ideas from the line of charge and the dipole help with this! If in doubt, think about the forces acting on a test charge!

Side View

Perspective View

+ + + + + + + + +

+ + + + + + + + +

- - - - - - - - - - - - - - - - -

This device (oppositely charged parallel plates) is called a capacitor and is a very important component on many electronic devices.

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SPH4U: Abstract Concepts – Fields


130

4

131

132

© University of Maryland Physics Education Research Group, Fall 2004

SPH4U: Electric Potential Difference


133

What happens to the sign of Welec? Explain.

134

Note that ΔVZY = -ΔVYZ


135

SPH4U: Magnetic Fields Recorder: __________________ Manager: __________________ Speaker: _________________ Com / Know / Th / App: 0 1 2 3 4 5

Refer to the text for the following information 1. Draw the magnetic field around the bar magnet below.

2. Magnetic field lines point from:

136


3. Important: Define the conventional current. 4. Draw the magnetic field around a conductor.

I

Practice 1. Close your text and try these! Current is travelling through a wire. Complete the diagrams by drawing the current direction or magnetic field lines. Note that “x” means current travelling into the surface and “•” means current travelling out of the surface.

x

137

SPH4U: Magnetic Forces on Charges A. Magnetic Forces on Static Charges Let’s start by investigating the forces on static electric charges. We will use the “Magic Tape” from a previous experiment as the charged object.


You will need: • 2 tape pieces • 1 bar magnet • 1 non-magnetized rod 1. Test to see if there is any force on the electrically charged tape from either pole of your magnet. Do the same test with a non-magnetized bar or rod. Summarize your findings.

2. If the charged tape is attracted to both the magnetic and non-magnetic bar in the same way, can you conclude that there is any special interaction or force between either of the magnetic poles and the tape?

B. Magnetic Forces on Moving Charges Let’s try something unusual. Let’s see if a magnet can exert forces on electrical charges that are moving. As a class we will use: • one oscilloscope tube • one high-tension power supply • one bar magnet 1. Move the north pole of your magnet parallel and then perpendicular to the electron beam in the oscilloscope. What is the direction of the displacement (and hence the force on the beam) in each case? Sketch vectors showing the direction of the magnetic field, the direction of the motion of the original electron beam before it was deflected, and the direction of the resultant force on the beam.

2. Draw an example of the right-hand rule for the motor principle for the above situation.

138

The force that results from the movement of charge in a magnetic field is known as the Lorentz force. This leads r r to a rather backwards mathematical definition of the magnetic field, B . The magnetic field, B , is defined as that

r

r

r

vector that, when crossed into the product of charge and its velocity, leads to a force, Fm = qv ⊗ B = qvB sin θ Summary 1. A charge is at rest in an electric field. There is no magnetic field. Explain what will happen.

+

2. A charge is at rest in a magnetic field. Explain what will happen.

+

N

3. A charge is moving in a magnetic field. a. What is the angle between v and B?

v

N b. What is the direction of the force on the charge?

S

+

S

c. If the charge was a single proton, v = 4 x 103 m/s and B = 1.5 x 10-3 T determine Fm.

C.

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d. When the net force is perpendicular to the object’s instantaneous velocity vector what type of motion results?

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From Laws, P. Workshop Physics Activity Guide Module 4. John Wiley, 2004

SPH4U: Electromagnetic Disturbances We have studied the creation of both electric and magnetic fields separately, but not it is time to build up the complete picture of what happens when current flows through a wire.


Part A: The Physics of a Wire Consider a wire connected between the negative and positive terminals of a power supply. A steady current travels through the wire. The diagram to the right illustrates this situation and show two sets of field lines. 1. Choose a colour to represent the conventional current. Draw an arrow that represents the flow of conventional current. 2. Choose a colour to represent the electric field. Explain how you decide which lines are the electric field lines. Label and colour code these lines.

-

+

3. Choose a colour to represent the magnetic field. Explain how you decide (as if you haven’t done question #2) which lines are the magnetic field lines.

Now imagine we zoom in to a small region of space located above the middle of the wire. The diagram to the right illustrates this along with the electric and magnetic field lines. Note that one field points in or out of the page and is represented with the circles. A positive test charge • is placed at rest in this region of space. In this investigation, we will ignore the effects of the magnetic field since they are relatively small.

•

4. Label and colour code the electric and magnetic field lines. Fill-in the circles with • or × to indicate out or in the page. 5. How do the directions of the electric and magnetic field lines compare according to the second diagram?

6. What will happen to the positive charge which is initially is at rest? Explain and draw vectors representing any forces.

141

Below is a third diagram showing the same region of space as above. But for this example, we have switched the polarity of the power supply, reversing the positive and negative terminals shown in the first diagram. 7. Describe the how the fields have changed due to the reversed terminals.

•

8. What will happen to the positive charge which is initially at rest? Explain.

Now imagine that we continuously vary the polarity of each terminal of the wire from positive to negative. 9. Describe the motion of the charges in the wire.

10. Describe the motion of the positive test charge as the polarity varies.

11. As a result, a smoothly changing pattern of electric and magnetic fields spreads outwards from the wire through space, alternating between the two pictures we drew. James Clerk Maxwell studied these phenomena in detail and mathematically described the wave-like motion of the oscillating fields. From his calculations he determined the outwards speed of the rippling fields to be given by:

v2 =

4πk

μo

Where k is Coulomb’s constant (k = 8.99 x 10 Nm /C ) and μo is a similar constant for magnetism (μo = 1.257 x 10-6 Tm/A). Determine the velocity of this wave including units (1A = 1C/s) (1T = 1 kg/Cs) 9

2

2

What we have discovered is an electromagnetic wave which works according to these ideas: 1) An EM wave is created by accelerating charges that slow down or turn (our varying current). 2) Energy from this wave may be absorbed by causing other charges to speed up (our distant charge). 3) The frequency of the wave’s oscillation and the properties of the object containing the distant charge determine how much that charge will respond to the wave, and how much energy it might absorb.

Ez = Eo sin(kx + ωt)

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By = Bosin(kx + ωt)

Shown below are mathematical and pictorial representations of an electromagnetic plane wave propagating through empty space. It shows the strength and direction of the electric and magnetic fields at each point in space along the x-axis. We can imagine our wire with its oscillating current oriented along the z-axis. The electric field is parallel to the z-axis and the magnetic field is parallel to the yaxis.

A •

• B

C •

1. Is the wave transverse or longitudinal? Explain in terms of the quantities that are oscillating.

2. A positive test charge is placed at rest at points A, B, and C. Describe the initial force experienced by the charge at each point.

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from McDermott and Shaffer, Tutorials in Introductory Physics. Prentice Hall, 2002 144

SPH4U: Polarization 1. Look at the room lights through one of the polarizing filters. Describe how the filter affects what you see.


2. Hold a second polarizing filter in front of the first, and look at the room lights again. Describe how the filter affects the light you see. How does rotating one of the filters with respect to the other affect what you see?

If the electric field in all parts of a light beam oscillates along a single axis, the light beam is said to be linearly polarized. The direction along which the electric field oscillates is called the direction of polarization of a light beam. If the electric field oscillated in different, random directions within the same beam, that beam is said to be unpolarized. Light transmitted by a polarizing filter (or polarizer) depends upon the relative orientation of the polarizer and the electric field in the light wave. Every polarizer has a direction of polarization, which is often marked by a line on it. The electric field of the transmitted wave is equal to the component of the electric field of the incident wave that is parallel to the direction of polarization of the polarizer. 3. Do the room lights produce polarized light? Explain how you can tell from your observations.

4. Describe how you should orient the polarizers with respect to one another so that the light transmitted through the polarizers would have (1) maximum intensity or (2) minimum intensity.

145

When two polarizers are oriented with respect to one another such that the light is at a minimum intensity, the polarizers are said to be crossed. 5. Suppose that you had a polarizer with its direction of polarization marked. How could you use this polarizer to determine the direction of polarization of another unmarked polarizer? Explain your reasoning.

6. A beam of light is incident on a polarizer, as shown in the side view diagram below. The direction of polarization of the light makes an angle θ with respect to the polarizer’s direction of polarization. (see front view diagram.) The amplitude of the electric field of the incident light is Eo. The magnetic field (not shown) has an amplitude Bo. Write an expression for the amplitude of the transmitted component of the electric field Incident light

Polarizer

Direction of polarization of filter

E vector θ

Side View Front view

7. BONUS: Explain how you can figure out an expression for the amplitude of the transmitted component of the magnetic field. Complete the diagram below (similar to the diagram above) to illustrate this.

Front view

from McDermott and Shaffer, Tutorials in Introductory Physics. Prentice Hall, 2002 146

SPH4U: Modelling 2-D Waves A wave moving in 2-D is a challenge to draw. Consider the illustration from the simulator: www.falstad.com/ripple/. We can draw a sketch of this wave by drawing lines of constant phase such as crests or troughs. These represent the wave fronts.


We want to model how a 2-D wave works. Christian Huygens proposed that each point along a wave front can be considered to be a point source of circular waves or wavelets. When these circular waves travel outwards their wave fronts join together and form the future wave front of the present wave. Use a penny and draw a dot every penny radius along the present wavefront. Think of a clever way to easily do this. Be sure to centre the penny at the two ends as well! Next, trace the outline of the penny when centred on each dot. Form the future wavefront by drawing a new surface tangent to the foreword edges of the wavelet circles. You can smooth this out, understanding that there are many more wavelets that we did not draw in between the ones we did.

present wavefront

future wavefront

1. What does the radius of the penny represent about the wave? We will use this technique to model what happens when a wave passes through an opening. Find the next three future wavefronts. Be sure to use points at the two ends!

A

past wavefront present wavefront

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Now try a really narrow opening. Find the next three wave fronts.

B

present wavefront

2. Describe what happens to the shape of the original wavefront after it passes through an opening.

3. In which example (A or B) was the spreading, or diffraction of the wave greatest? Compare the width of the opening with the wavelength.

What happens when a wave meets an obstacle? In particular we want to study the region behind the obstacle. Find the next three wave fronts. past wavefront present wavefronts

4. Explain what happens to the wave as it passes by a barrier. Mention in particular what happens in the “sheltered region” behind the barrier.

5. Explain how the results of this activity might differ if a wave has a very small wavelength compared to the penny-sized examples above.

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SPH4U: Two Source Interference


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SPH4U: The Double-Slit Interference Pattern


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