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Table of Contents COURSE SYLLABUS & POLICIES ...................................................................................................1 KINEMATICS.......................................................................................................................................5 Kinematics Assignment Sheet..................................................................................................... 6-7 Graphing Fundamentals with a TI Calculator ..............................................................................8 Constant Speed vs. Constant Acceleration ...................................................................................10 Acceleration Due to Gravity .........................................................................................................15 Vector Field Trip ...........................................................................................................................20 Projectile Motion ...........................................................................................................................22 Projectile Motion at an Angle.......................................................................................................24 Kinematics Review Sheet ........................................................................................................ 25-27 DYNAMICS.........................................................................................................................................35 Dynamics Assignment Sheet..................................................................................................... 36-7 Gravitation Practice Problems......................................................................................................38 Newton’s Second Law ...................................................................................................................39 Newton’s Second Law - Revisited.................................................................................................43 Free Body Diagrams .....................................................................................................................46 Hooke’s Law..................................................................................................................................48 Friction And The Graphing Calculator .......................................................................................51 Torque and Static Equilibrium .....................................................................................................56 Centripetal Force ..........................................................................................................................58 Dynamics Review Sheet .......................................................................................................... 60-61 ENERGY, MOMENTUM, AND RELATIVITY ................................................................................68 Energy And Momentum Assignment Sheet ........................................................................... 69-70 Personal Power .............................................................................................................................71 Work and Energy ..........................................................................................................................73 Conservation of Energy ................................................................................................................77 Conservation of Momentum .........................................................................................................80 Model Rockets ...............................................................................................................................84 Energy, Momentum, and Relativity Review Sheet .......................................................................87 Mousetrap Racecar Project ..........................................................................................................89 APPENDIX A: UNIT CONVERSIONS ............................................................................................97 APPENDIX B: USEFUL INFORMATION ......................................................................................99 APPENDIX C: SECOND SEMESTER EQUATIONS ...................................................................100 Course Syllabus & Hello! Welcome to physics! This year you will be introduced to the greatest of all the sciences, the foundation upon which all other sciences are built. You will learn about some of the most influential and significant ideas developed by humankind, empowering you to have a much better understanding of the physical universe. In fact, a famous physicist named Ernest Rutherford once said “All science is either physics or stamp collecting.” Regardless of natural talents in the sciences, or your penchant for art, or literature, or something else, you will be expected to acquire fundamental knowledge and genuine fondness of physics. SYLLABUS Fall Semester 1. Optics A. Reflection & Mirrors B. Refraction & Lenses C. Optical Phenomena D. Human Vision 2. Waves A. Wave Properties B. Sound C. Diffraction & Interference D. Wave Phenomena 3. Electricity A. Static Electricity B. Current Electricity C. Electrical Circuits Spring Semester 4. Kinematics A. Motion in One Dimension B. Motion in Two Dimensions C. Relative Motion (Honors) 5. Dynamics A. Newton’s Law of Motion B. Torque and Rotation C. Circular Motion D. Gravitation 6. Energy, Momentum, and Relativity A. Work and Energy B. Impulse and Momentum C. Conservation Laws D. Special and General Relativity PHYSICS LAB BOOK & OTHER MATERIALS Every day you will need to bring: • Your lab manual…this is a must…please make it a priority…it doesn’t weigh that much! • A calculator with trigonometric functions (preferably a TI-84, or a low-cost TI-36) • On quiz days it is recommended that you bring your textbook for homework review GRADING Grading is based on total accumulated points within weighted categories. The categories and percentages are as follows: • Laboratory (25%) • Homework (15%) • Quizzes (20%) • Midterm Exams (20%) • Final Exam (20%) The grading scale is shown below A+ 97% C+ 77% A 93% C 73% A90% C70% B+ 87% D+ 67% B 83% D 63% B80% D60% F (0 credits) under 60% Note: grades are NOT rounded up. For example, 89.5% is NOT rounded to an A- 1 HOMEWORK, QUIZZES, AND UNIT EXAMS HOMEWORK is due weekly, usually on Tuesdays, from every student for the first unit. After that, homework will be collected based on the grade leading into the next unit of study (about 6 weeks). Students with a B (or lower) at the start of a new unit are required to turn in homework each week for the entire unit. Students with a B+ (or higher) may opt to turn in homework each week, but are not required to. After each unit this policy is repeated. In Honors Physics, this cutoff grade is an A–. Homework problems will not be reviewed each day, following each assignment. Some of the problems will be addressed each week, and additional review will occur the day of a quiz. However, there will be a homework solution binder available in class to check your work at any time. 10% penalty per day for late homework. Do not make late homework a habit! QUIZZES are a way to check to see if you have done your homework, including textbook reading and problem solving, as well as lab analysis and questions. Typically there are about 4-5 multiplechoice problems and 2-3 computational problems to solve, showing all work for credit. Quizzes will be given every other week and will cover all work from the previous two weeks, and may include questions related to older topics. MIDTERM EXAMS will occur at the end of each unit. The last unit of study in the semester will be covered in a cumulative final exam, not a separate midterm exam. A review sheet will be provided before each midterm exam. The review sheets are optional, but highly recommended. All quizzes, midterm exams, and the final exam will be closed notebook, except for equation sheets. TARDIES AND UNEXCUSED ABSENCES Tardies and unexcused absences are very disrupting to the class and to a student’s learning. A tardy is unexcused unless you bring a signed note from the Redwood staff member who caused the tardy. Please make an effort to be on time and eliminate absences of any kind. Ask your parents to NOT plan vacation time on school days. Also, college visits should be scheduled around a staff development day so that students avoid absences. TARDIES Poor attendance will result in grade reduction for the semester grade according to the attendance chart below. 0 1 2 3 4 5 6 7 8 9 10 11 12 0 0 0 0 0 0 0 -1/3 -1/3 -1/3 -2/3 -2/3 -2/3 -1 UNEXCUSED ABSENCES 1 2 3 0 -1/3 -2/3 0 -1/3 -2/3 0 -1/3 -2/3 -1/3 -2/3 -2/3 -1/3 -2/3 -2/3 -1/3 -2/3 -2/3 -2/3 -2/3 -2/3 -2/3 -2/3 -2/3 -2/3 -2/3 -2/3 -2/3 -2/3 -2/3 -2/3 -2/3 -2/3 -2/3 -2/3 -2/3 -1 -1 -1 4 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 5 -1* -1* -1* -1* -1* -1* -1* -1* -1* -1* -1* -1* -1* * District policy: 5 unexcused absences will also result in loss of 1 credit 2 MAKE UP FOR ABSENCES LABS: If you miss a lab, you must make it up at lunch, after school, or during office hours, but no later than SMART period the following week. You cannot just get data from a friend, and expect to get credit for the lab. In fact, you need to bring a friend in with you so that you have a lab partner to help explain the procedure and use the equipment to collect data. When the lab is completed, it is your responsibility to show me the results in order to receive a grade. QUIZZES: If you miss a quiz your only opportunity to take it will be before the quizzes are returned to other students, which means you have no more than two days. A quiz given on a Tuesday is usually returned by Friday. Smart period is often the best time for a quiz makeup. If you do not make up a quiz, it will be marked as “missing” in the grade book, which will result in zero points for that grade until the end of the unit, when the midterm exam score for the unit will replace your missing quiz. This is NOT a good idea, so avoid if possible. MIDTERM EXAMS: If you miss a midterm exam please email immediately ([email protected]) to explain your absence and to arrange a make-up exam. There are only two midterms (and a final) per semester, so missing any one of these days is highly unusual and requires a confirmed and legitimate excused absence involving a discussion with me and your parents. Missed work caused by extended absences (more than three school days) for extreme sickness or trauma will be handled on an individual basis. Please realize that poor attendance will have a negative effect on your understanding of physics, and on your semester grade. BEHAVIOR AND CLASS RULES In the physics classroom, you should feel comfortable, positive, and enthusiastic. You are expected to work diligently, conscientiously, and consistently, at all times. A few comments about student/teacher respect and rules about classroom behavior will soon be discussed during class. Food and drinks in the classroom are tolerated with three simple rules that apply: 1) it must not be a distraction to other students, 2) you must clean up and recycle when finished, and 3) if it looks tasty I get a bite - often called the “Nash tax”! Bathroom breaks are tolerated with three simple rules that apply: 1) you must ask me first - try to be subtle and not interrupt the class, 2) you must leave your cell phone behind - bathroom breaks are not text-messaging breaks! and 3) use of the bathroom must not be excessive, daily occurrence. Cell phones must be off and away during class. Please turn your cell phone off before you come to class. Per school policy, the teacher has the right to confiscate your cell phone if it rings in class. Expect only one warning, and then it will be taken away. LABORATORY WORK AND LAB ASSESSMENTS Each week the rubric on the following page will be used for lab assessment. This rubric has four areas of evaluation. Each area has a value up to five points for a total of 20 points. Generally, each lab group earns one group grade so choose your lab partners wisely. If a lab partner does considerably less than the others, that student will receive fewer points. This lab rubric is designed to minimize the busy work that is sometimes associated with doing labs in a science class. Physics labs require very little copying of information. Most time is spent analyzing data - often with a calculator, interpreting results, and answering relevant questions. It may take some time for you to get used to the way labs are graded, but soon you will be accustomed to this “real time grading”. You’ll quickly discover that labs are the best way to learn physics, and also the easiest way to improve your overall class grade. And remember, best of all, you will not have to turn in any lab reports in this class! 3 LAB REPORT LAB TECHNIQUES USE OF TIME AND CLEANUP LISTENING TO INSTRUCTIONS LAB GRADING RUBRIC 5 4 • Group needs no supervision after the initial lab instructions are given. • Each partner can articulate the goal of the lab and can explain the rationale for the proposed lab procedure. • Lab partners ask coherent and relevant questions that often lead to improvements in the lab. • Group needs minor clarification after having moved to the lab area. • The group, as a whole, needs minor help articulating the goal of the lab and/or explaining the rationale for the lab procedure. • Lab partners sometimes ask coherent and relevant questions. 3 • Group needs major clarification after having moved to the lab area. • One or more partners can articulate the goal of the lab and explain the rationale for the lab procedure, but one or more partners is slightly confused. • Lab partners sometimes ask illogical or redundant questions. • Each partner works • Some partners work • One partner does most equally on the data harder than others on the of the work on the lab. collection and analysis. lab. • There is major • There is no unnecessary • There is minor socializing within the socializing within or unnecessary socializing group or between groups, between groups. within the group, which which detracts from the • When finished, each detracts from the lab lab experience. partner is working on experience. • When finished, one or physics until the bell rings. • When finished, each more partners stop • All lab areas are clean partner is working on working on physics and all equipment is physics until the bell rings. before the bell rings. replaced as directed so that • Not all lab areas are • Most lab areas are not it is ready for the next clean or all equipment not clean and not all class lab. replaced. equipment is replaced. • All lab stools are put • Some lab stools are put • Lab stools are not put back under the lab desks. back under the lab desks. back under the lab desks. • Data collection • Data collection • Data collection techniques by the group techniques lead to slightly techniques lead to lead to exceptionally inaccurate or imprecise considerably inaccurate precise and accurate data. data. or imprecise data. • Follows all directions • Follows most directions • Follows most directions for data collection. for data collection. for data collection. • Always implements • Makes an effort to avoid • Makes some effort to techniques to avoid most experimental error. avoid experimental error. experimental error. • A graphing calculator is • A graphing calculator • A graphing calculator is used often to minimize is used occasionally to used whenever possible to rounding errors. minimize rounding errors. minimize rounding errors. • Presentation is neat, • Presentation is mostly • Presentation is sequential, and clear, using neat, sequential, and clear somewhat neat, only pencil using only pencil sequential, and clear. • Calculations, and lab • Calculations, and lab • Calculations, and lab analysis, and questions are analysis, and questions are analysis, and questions all complete and detailed. mostly complete with are somewhat complete • All measurements and most details. with some details. results are presented with • Most measurements and • Some measurements appropriate variables and results are presented with and results are presented units. appropriate variables and with appropriate variables • All data tables, graphs, units. and units. drawings, etc. are • Data tables, graphs, • Some data tables, completed. drawings, etc. are nearly graphs, drawings, etc. are completed. incomplete. 4 2 • Group needs lab reexplained after having moved to the lab area. • One or more partners can articulate the goal of the lab and explain the rationale for the lab procedure, but one or more partners has major confusion. • Lab partners ask only illogical or redundant questions. • One partner does all of the work on the lab. • There is major socializing within the group or between groups, which detracts from the lab experience. • When finished, all partners stop working on physics before the bell rings. • Not all lab areas are clean and all equipment is not replaced. • Lab stools are not put back under the lab desks. • Data collection techniques lead to drastically inaccurate and imprecise data. • Ignores most directions for data collection. • Makes little effort to avoid experimental error. • A graphing calculator is used rarely to minimize rounding errors. • Presentation is not at all neat, sequential, and clear. • Calculations, and lab analysis, and questions are mostly incomplete with few details. • Few measurements and results are presented with appropriate variables and units. • Most data tables, graphs, drawings, etc. are incomplete. Kinematics “All science is either physics or stamp collecting.” – ERNEST RUTHERFORD “Common sense is nothing more than a deposit of prejudices laid down by the mind before you reach eighteen.” – ALBERT EINSTEIN 5 Kinematics Assignment Sheet Online: www.physicsclassroom.com/Physics-Tutorial/1-D-Kinematics (Assignments 1-8) www.physicsclassroom.com/class/vectors (Assignments 9-15) No. Topics Reading Homework 1 displacement, speed, average velocity 40-44 Ch 2: 6,9,11,13 2 instantaneous velocity, position vs. time graphs 45-47 Ch 2: 2,3,4,10,14 3 average acceleration, instantaneous acceleration 48-51 Ch 2: 17,19,30,54 4 kinematic equations with constant acceleration 51-58 Ch 2: 23,24,25,29 5 additional constant acceleration problem solving Ch 2: Sec Rev pg 59 1,2,3,4 6 additional constant acceleration problem solving Ch 2: 26,28,31,32 7 falling objects, freefall acceleration 8 more freefall acceleration problem solving 9 scalars and vectors, properties of vectors, graphical addition 84-87 Ch 3: 3,7,8,12 10 vector operations, vector components, analytical addition 88-97 Ch 3: 16,22,23,26,28 11 projectile motion (horizontally launched projectiles) 98-101 Ch 3: 31,34,36,37 12 projectile motion (projectiles launched at an angle) 102-104 Ch 3: 35,38,39,41 13 more projectile problem solving Ch 3: 33,56,58,69 14 projectile review problems Pg 104: P 1,3 Pg 118-9: P 61,67 15 relative velocity, frames of reference 60-64 Ch 2: 34,38,39,49 Ch 2: 36,46,48,55 106-109 6 no assignment Kinematics Assignment Sheet - Honors Online: www.physicsclassroom.com/Physics-Tutorial/1-D-Kinematics (Assignments 1-8) www.physicsclassroom.com/class/vectors (Assignments 9-15) No. Topics Reading Homework 1 displacement, speed, average velocity 2-1 2-2 Q1 P 5,9,10,11 2 instantaneous velocity, position vs. time graphs 2-3 2-8 Q 3,4 P 50,51,56abde 3 velocity vs. time graphs, average and instantaneous acceleration 2-4 Q 5,8 P 17,18,49,55 4 kinematic equations with constant acceleration 2-5 Q 10 P 19,21,22,26,27 5 additional constant acceleration problem solving 2-6 Q 11 P 28,63,74,77 6 additional constant acceleration problem solving 7 falling objects, freefall acceleration 8 more freefall acceleration problem solving 9 scalars and vectors, properties of vectors, graphical addition 3-1 3-2 Q 1,6 P 3,5,13(graphically) 10 vector operations, vector components, analytical addition 3-3 3-4 Q7 P 1,7,10,12 11 projectile motion (horizontally launched projectiles) (to pg 57) 12 projectile motion (projectiles launched at an angle) (to pg 61) 3-6 Q 20 P 19,20,24,26 13 more projectile problem solving 3-7 P 62,67,69,70 14 relative velocity, frames of reference 3-8 Q 13 P 37,39,41,42,49 15 review problems Q9 P 31,32,68,73 2-7 Q 13 P 34,35,37,39 Q 14 P 42,44,45,78,79 3-5&3-6 Q 18 P 18,21,27,31,64 Ch 2: 75,83,85 Ch 3: 53,58,65 7 Graphing Fundamentals with a TI Calculator PURPOSE To learn about graphing by hand and also using the TI Graphing Calculator. To understand the nature of constant speed, average speed, and instantaneous speed. PROCEDURE In physics the best way to see patterns and relationships is through graphing the data collected in an experiment. This is true in the following lab which is designed to help you learn to graph variables, and help you interpret the meanings of graphs. Follow the directions below in for all graphs. 1. Determine axis variables. 2. Label axes with variable and unit. 3. Title the graph. 4. Establish axis scales (stretch data out.) 5. Plot points in pencil. 6. Draw best fit. Don't "connect the dots"! DATA & ANALYSIS 1. The table of data below represents the motion of a toy car for five seconds of time. Follow the above steps 1-6 to create a completed graph. Toy Car (5 sec) t (s) d (cm) 0 0 1.0 190 2.0 395 3.0 605 4.0 810 5.0 1040 2. Enter the data into List columns in the graphing calculator by pressing STAT, then ENTER; put time data into L1 and distance into L2. 3. Press STAT PLOT (2nd button, then Y= button) and turn Plot 1 on, set to scatter plot, with Xlist to L1 and Ylist to L2. Press GRAPH, then ZOOM, then 9:ZoomStat to fit data into the display. 4. Press STAT, then move cursor to CALC, then 4: LinReg, then check Xlist:L1, and Ylist:L2, and Store RegEQ: Y1 (to get Y1 press VARS, then move cursor to Y-VARS, then press 1:Function, then 1:Y1 and finally ENTER. Press GRAPH to see the curve fit on this equation. (on the TI-83 is has to look like this on the screen – LinReg L1,L2,Y1). 5. Write the equation for the best-fit line below. Use variables for distance (d) and time (t). 6. The toy car moves with nearly constant speed. What is the value, with units, of the speed? 8 7. The table of data below represents the motion of the same toy car as before, but now for just the first half-second of time as it speeds up from rest. Follow the steps, as before, to create a graph, including drawing a best-fit curve. Toy Car (0.5 sec) t (s) d (mm) 0 0 0.10 25 0.20 75 0.30 175 0.40 325 0.50 500 8. Calculate the average speed of the toy car between 0.1 sec and 0.5 sec. Average speed is distance traveled divided by time elapsed. 9. Draw with a ruler a line between the data points at 0.1 sec and 0.5 sec. This line is now a secant line to the best-fit curve. What does the slope of this line tell you? 10. Calculate instantaneous speed of the toy car at 0.25 sec. First draw with a ruler a tangent line to the best-fit curve at 0.25 sec; then find the slope of the line using two points on the tangent line. Show work. 11. Now find the instantaneous speed using the graphing calculator. Follow steps 2-4 from the last page, but this time fit the data with QuadReg since the data nearly fits a quadratic function. Write this quadratic equation below, with proper variables 12. Next press GRAPH, then DRAW (2nd PRGM) then 5: Tangent. Enter .25 for the time t = 0.25 s. Press ENTER. The calculator draws the tangent and displays the equation in y = mx + b form, where m is the slope. Write this linear tangent equation below, with proper variables. 12. Calculate the percent error between the experimental (step 10) and the known (step 11). 9 Constant Speed vs. Constant Acceleration PURPOSE To use graphical methods with distance and time data to analyze two types of motion: constant speed and constant acceleration. EQUIPMENT - dynamics track - dynamics cart - ticker tape timer - scotch tape - meter stick, rulers - support rod & stand PROCEDURE PART A - CONSTANT SPEED 1. Check that the dynamics track is level. If it is not, adjust the leveling screw underneath one end of the dynamics track and check for level again. See diagram below. 2. Feed one end of a 1-meter long piece of ticker tape through a ticker tape timer. The ticker tape goes through the metal brackets, over the carbon, and under the metal strike plate. 3. Use scotch tape to attach the end of ticker tape to the back end of a dynamics cart. 4. Set the plunger on the dynamics cart in 2 clicks. It works best if you press it in and slightly upwards at the same time so the spring is set correctly. Do a “dry run” to see if the cart will make it across the track with constant speed. Tap the plunger release with the metal block. If the cart slows down noticeably, set the plunger to 3 clicks and do another trial to check for constant speed. 5. Now it’s time to collect data! Turn on the ticker tape timer to the 40 Hz setting and then tap the plunger release lightly with the metal block. This takes a little practice and coordination. Don’t tap the release until the timer starts ticking. ticker tape timer plunger plunger release dynamics cart dynamics track leveling screw ticker tape end stop metal block DATA & ANALYSIS 1. Remove the ticker tape from the dynamics track and scotch tape it to the lab table. Circle the dot that represents the starting time (t = 0) where the dots begin uniform spacing. Ignore any dots before the circled dot. See diagram below. 2. The timer records 40 dots every second. Mark off every four spaces on your tape. These marks will represent time increments of tenths of seconds. Again, see diagram below. 3. With the meter stick, measure the distances from the circled dot (t = 0) to each of the marks on your tape. These will represent the total distance covered at the end of each tenth of a second. Measure each distance to the nearest 0.1 centimeters. Sample Ticker Tape d1 d2 10 4. Record the data in the table below. The data may not fit the table exactly. 5. Save the ticker tape. You may need it to check for mistakes. 6. Make a graph of distance vs. time below. Draw with a ruler a best-fit line for the data points. t (s) 0 d (cm) 0 QUESTIONS & CALCULATIONS 1. Calculate the constant speed of the cart as it moves across the dynamics track. Use the graphing calculator to do this, and refer to the “Graphing Fundamentals With a TI Calculator” activity. Write the equation for the best-fit line below. Use variables for distance (d) and time (t). 11 PART B - CONSTANT ACCELERATION 1. In this part you will make the dynamics cart roll down an inclined track. Attach the dynamics track to the support rod using the metal clamp. Raise the left end of the track so that the bottom of the track is about 15 cm above the table. See diagram below. ticker tape timer dynamics cart dynamics track end stop ticker tape 2. Feed the ticker tape through tape timer and attach it to dynamics track, as it PART I. 3. Turn on the ticker tape timer to the 40 Hz setting and then immediately release the dynamics cart. The spring plunger is NOT used here. Don’t start the timer late! DATA & ANALYSIS 1. Remove the ticker tape from the dynamics track and scotch tape it to the lab table. Circle the dot that represents the starting time (t = 0) where the dots clearly begin to get further apart. Ignore any dots before the circled dot. See diagram below. 2. The timer records 40 dots every second. Mark off every four spaces on your tape. These marks will represent time increments of tenths of seconds. 3. With the meter stick, measure the distances from the circled dot (t = 0) to each of the marks on your tape. These will represent the total distance covered at the end of each tenth of a second. d1 d2 4. Record the data in the table below. The data may not fit the table exactly. 5. Save both pieces of ticker tape. You may need them to check for mistakes. 6. Make a graph of distance vs. time. Draw a best-fit curve for the data points. 7. Use the graphing calculator to enter and analyze this new data, referring again to the “Graphing Fundamentals With a TI Calculator” activity. 12 t (s) 0 d (cm) 0 Write the best fit quadratic equation below, with appropriate variables for distance and time. 13 QUESTIONS & CALCULATIONS (these refer to the curved graph on previous page) Show all work and evidence for answers 1. Calculate the average speed of the cart for the entire time it moves down the ramp. On the graph, lightly draw with a ruler the secant line showing how you determined this average speed. 2. Calculate the average speed of the cart for the final 0.3 seconds of its motion. You do not need to draw the secant line showing how you determined this average speed. 3. Calculate the experimental instantaneous speed of the cart when it is halfway down the ramp in terms of time elapsed. That is, if your last recorded time is 1.4 s, then find the instantaneous speed at 0.7 seconds. On the graph, lightly draw with a ruler the tangent line that shows how you determined this instantaneous speed. 4. Use the graphing calculator to draw the tangent line for determining the known instantaneous speed, again referring to the “Graphing Fundamentals With a TI Calculator” activity. Find the percent error between the known and experimental values. 5. Honors Only: Calculate the instantaneous speed of the cart when it is half the recorded distance down the ramp (not necessarily the midpoint of the dynamics track). Lightly sketch the tangent line that shows how you determined this instantaneous speed. (Think about why this question has a different answer from the previous question.) 14 Acceleration Due to Gravity PURPOSE The purpose of this experiment is to investigate the nature acceleration due to gravity and determine the position, velocity, and acceleration of an object in freefall. EQUIPMENT - tape timer - ticker tape Direction of motion Paper tape (held by hand) Tape timer (held by hand) - meter stick - scotch tape - 200 gram mass PROCEDURE 1. Cut a piece of paper ticker tape about 2 meters long. Make a loop with one end of the ticker tape by folding it over and securing it with a piece of masking tape. Hook the 200 g mass on the loop end of the paper tape. PASCO TAPE TIMER 40 10 Hooked mass 2. Thread the other end of the paper ticker tape through the Tape Timer until the mass is at the level of the Timer. 3. Hold the Tape Timer about 2 meters off the ground by standing on the lab tables. Align the Tape Timer on its side so the tape passes through the Tape Timer vertically. 4. Hold the paper ticker tape so it can easily thread itself through the Tape Timer. This is best accomplished by having one partner hold the Tape Timer while the other partner holds the tape above the timer. 5. Turn on the Tape Timer to 40 Hz and drop the tape, allowing the mass to fall. The 40 Hz setting means that a dot will be marked on the ticket tape every 40th of a second. 6. Tape the paper ticker tape to a lab table. Identify and circle the first dot where freefall begins, that is, where zero time has elapsed. If you had very steady hands when performing the freefall this dot will be the very first one on the ticker tape. If you’re hands were shaky, the first true freefall dot may not be obvious. Ask for help if needed. 7. Now circle every fourth dot from the first dot. The time elapsed between these circled dots is 0.1 (one-tenth) second. Check for accuracy by noting that the second circled dot is at least 3.5 cm but no more than 6.5 cm from the first circled dot. (A typical position at time t = 0.1 seconds is 5.0 cm.) 8. Measure the positions of these circled dots to the nearest millimeter and record the data in Table 1. Remember that each dot’s position is measured from the first circled dot. All positions are negative! (Why?) 15 DATA & ANALYSIS Position versus Time 1. Make a pencil and paper graph below of position vs. time for the freefalling mass. This is a graph in the fourth quadrant of an x-y plane. Why? Draw a best-fit curve for these data points. Don’t connect the dots. Sketch a curve that represents the data well. 2. Enter position data into a table on the TI Graphing Calculator First clear existing data in calculator’s List columns by pressing STAT, then 5:SetUpEditor, and then ENTER. Next enter your new lab data. Press STAT, then ENTER. The independent variable goes into the first column L1, the dependent variable into the second column, L2. The point (0,0) is entered as the first data point. Table 1 Time (sec) Position (cm) 0 0 3. Graph the data from the table on the TI Graphing Calculator Press STAT PLOT (2nd button, then Y= button) and turn Plot 1 on, set to scatter plot, with Xlist to L1 and Ylist to L2. Press ZOOM, then 9:ZoomStat. This should automatically fit the data onto the WINDOW and display the graph. What does the data look like? What function might fit that data? 4. Fit a curve to your graphed data on the TI Graphing Calculator Press STAT, then move the cursor to CALC, then 5: QuadReg, then check Xlist:L1, and Ylist:L2, and Store RegEQ: Y1 (to get Y1 press VARS, then move cursor to Y-VARS, then press 1:Function, then 1:Y1 and finally ENTER. Press GRAPH to see the curve fit on this equation. (Note: on the TI-83, after 5: QuadReg it has to look like this: LinReg L1,L2,Y1) 5. Write the equation for the best-fit curve on the graph in your lab manual. Use appropriate variables for vertical position (y) and time (t). best fit equation: 16 Velocity versus Time 1. Now we must calculate the instantaneous velocity of the freefalling mass at various points in time. Recall that the slope of a tangent line to the curve on a position graph is the instantaneous velocity at that point in time. We’ll use the TI graphing calculator to find the slope of a tangent line at each 0.1-second interval. To avoid rounding errors, record the instantaneous velocities to four significant figures and use all four significant figures in further steps. 2. Find slopes of tangent lines to your best-fit position curve While viewing the GRAPH window, press DRAW (2nd PGRM), then 5: Tangent. Type in 0.1 for the first time t = 0.1 Press ENTER. The calculator draws the tangent and displays the equation for the line in y = mx + b form, where m is the slope. Repeat for each 0.1 s interval and record all instantaneous velocities in Table 2. 3. Make a graph below of velocity vs. time for the freefalling mass. Use a clear ruler to draw a best-fit line for these data points. 4. Enter velocity data into a table on the TI Graphing Calculator Press STAT, then ENTER, and put the values for velocities into L3. Press STAT PLOT and turn on Plot 2, set Xlist to L1 and Ylist to L3. Repeat the steps you used for position graphing, but now fit a line to your velocity data using STAT → CALC, 4: LinReg, then L1,L3,Y1. 5. Write the equation for the best-fit line on the graph in your lab manual. Use appropriate variables for velocity and time. best fit equation: 17 Table 2 Time (sec) Inst. Vel (cm/s) 0 0 QUESTIONS & CALCULATIONS 1. From your velocity vs. time best fit equation, determine the acceleration of the freefalling 200 gram mass (recall that acceleration is the slope of a velocity vs. time graph.) This acceleration due to gravity is so special in physics that it is given its own letter, “g”. Write down below your experimentally determined value for g with units of cm/s2. 2. Convert your value for g to units of m/s2. Show the correct unit analysis. 3. How does your experimental value of g compare with the theoretical (known) value of g? To compare, find the percent error between the your experimental value with the known value of -9.80 m/s2. Use the percent error formula shown below to calculate the error. Known-Experimental Percent Error = × 100% Known 4. Why do students performing this lab typically find an experimental value for the acceleration due to gravity that is lower (in magnitude) than the accepted (known) value of - 9.8 m/s2? There are several good reasons that mostly center around one key idea. 5. Honors Only: Velocity graphs can be used to find acceleration (see above), but they can also be used to find information about position. Recall that the area under a velocity vs. time graph determines an object’s change in position during that time interval. From your velocity versus time graph find the area above the line up to the last time measured. How does this compare with the final position measured from the ticker tape? Show all work. 18 6. Galileo was the first to quantitatively analyze the acceleration due to gravity. Unfortunately, the only timing devices he had available were things like an hourglass or a simple pendulum. This made it difficult to measure the position or the velocity of a freefalling object. Do some quick textbook or internet research and find out how Galileo was able to ingeniously “dilute” or “slow down” gravity in order to collect lab data. Briefly describe what you learned below. 7. A device that measures acceleration due to gravity (or any other kind of acceleration) is called an accelerometer. Do some quick textbook or internet research to find out how accelerometers are used in a practical ways by scientists and engineers. Briefly describe what you learned below. 19 Vector Field Trip PURPOSE To add displacement vectors and determine a resultant vector using graphical methods PROCEDURE 1. In groups of two or three, learn to use a compass to find a bearing. 2. Establish one partner as a walker, and count steps for walking the length of the lunch plaza from the first square tree planter to the last square tree planter (a total of seven, not the eight near the building.) Walk this length twice for accuracy. This distance has been carefully measured with a long tape measure, and it is 140.5 feet. It will be used later to convert steps to meters. 3. From a starting position at the lunch plaza - where the Redwood Tree is painted on the bricks outside the exit doors to the South Lawn, walk in straight displacement vectors, being careful to take an accurate bearing and count your exact number of steps. You must stay on the asphalt path around the South Lawn, moving in a counterclockwise direction, ending up at the CEA doors. 4. Take the first vector bearing and walk the first vector length. Record the data in the table. 5. Continue walking straight-line displacement vectors until you reach the double doors of the CEA, being sure to record the data for each displacement vector. DATA & ANALYSIS 1. Using a TI Graphing Calculator, press STAT then ENTER. Clear out the first three columns: L1, L2, and L3. (If columns are missing press STAT 5:SetUpEditor, ENTER.) Type in column L1 the length, in steps, of your vectors. 2. Using data in Appendix B, calculate the ratio of meters per step in the space below, and then use this conversion in L2 to convert steps into meters. Record the distances in the table below. 3. Now scale the distance column, so that the vectors can be drawn on a computer to fit on a sheet of 8.5” X 11” paper. A good scale is 5 meter = 1 centimeters. Convert your vector distance (m) to a scaled distance (cm) on the calculator, using column L3. Record the scaled distances in the table below. _____________ steps to walk the lunch plaza, which is 140.5 feet Bearing (degrees) Length (# of Steps) Distance (m) 20 Scaled Distance (cm) 4. Create a scale drawing of your vector walk. Each student must complete their own drawing and turn it in for grading. You can use Microsoft Word, PowerPoint, or any other software you want, or you can even do the drawing with pencil and paper. All drawings must have the following: a. Show the proper axes for bearing angle. This means showing a north-south vertical axis and an east-west horizontal axis somewhere on the page. (You can import a graphic easily if you type “compass” or “compass with degrees” into an image search on the computer.) b. Draw each vector to scale, and show each vector in order of the vector walk, using “head to tail” addition. If you are using a computer each vector can be drawn as a line with an arrowhead, usually found as a Shape under the Insert menu. To “nudge” the vectors until they are head to tail use the cursor arrows, holding Option (Mac) or Ctrl (PC) for precision. c. Label each vector with magnitude in meters and direction in degrees. For example, your first vector should be something like this: 18.2 m, 167˚. d. Show a sample conversion of one of your displacement vectors - from steps to meters, and then meters to scaled centimeters. e. Draw the resultant vector, from the tail of the first vector (where you started on the bricks) to the head of the last vector (where you ended at the CEA doors.) Make this resultant vector in bold, or a different color, to show it is not just another displacement vector from the data. f. Determine the magnitude and direction of your resultant vector. This can be measured with a ruler and protractor, or determined analytically if you used a computer by finding out the height and width of the resultant vector. g. Calculate the percent error between your vector’s magnitude and the known magnitude, which will be given in class. Next, determine the percent error between your vector’s direction and the known direction, which will be given in class. 5. Print two copies of your vector drawing: one for your lab manual and one to turn in for a grade. It should look like the drawing below, but with all vectors labeled, conversion and errors shown. 21 Projectile Motion PURPOSE To understand the nature of projectile motion and to predict the landing spot for a horizontally launched object. EQUIPMENT - photogate timer with accessory photogate - steel ball, metal ramp and clamp - plumb bob - meter stick photogates PROCEDURE 1. Measure the distance between the photogates (centerline to centerline), in meters. It's best to measure this distance at the bottom of the two photogates, where the ball rolls through. Also, measure once from each side of the photogates to be sure they are parallel to each other. Record the distance, in meters, in the table. 2. Measure the height, in meters of the lab table. Record the height in the table. 3. Set the photogate timer to PULSE mode. This will measure the time elapsed as an object passes between the photogates. This allows the photogate timer to show 4 significant figures instead of three. The time displayed is in seconds. 4. Roll the steel ball from the top of the ramp, across the lab table, and through the photogates. Do not let the ball land on the floor. Your peers will catch you if you do! 5. Record the time measured by the photogates in the table below. Repeat four more times, or more if you have to discard any outliers. Don’t change the photogate locations! Distance between photogates: ___________ Height of lab table: ___________ time through photogates average time DATA & ANALYSIS 1. Calculate the average speed of the ball through the photogates. This will determine the horizontal velocity (vx) of the marble when it leaves the table. 2. Calculate the time of flight (t) for the steel ball. Hint: use an equation for vertical motion. 22 3. Calculate the horizontal displacement (Δx) of the steel ball when it lands on the ground. 4. Use the plumb bob to find the point on the floor that is directly below the launch point. Use a pencil to mark this point on the floor. Then locate and mark the predicted landing point on the floor. Draw a large x-y axis at this landing point to align the target sheet. 5. After you have shown me the correct calculations it will be your turn to compete! Score up to 20 points on the total of two tries. 6. Clean up! Erase the pencil marks, turn off and unplug the photogates, and return the steel ball to the lab table. QUESTIONS & CALCULATIONS 1. Describe and explain the horizontal and vertical motion of the steel ball while it is in the air. Be specific and use mathematical expressions. 2. If you are standing near the edge of a cliff how can you determine the height of the cliff using a rock and a stopwatch? 3. Is the method you described in Question 2 always accurate in a real application? If yes, explain why. If no, explain the inaccuracy. Be specific. 23 Projectile Motion at an Angle PURPOSE To use the equations of motion and the ideas of projectile motion to predict the range of a steel marble projected at an angle above the horizontal. EQUIPMENT - projectile launcher, marble - white paper, carbon paper - plumb bob - meter sticks PROCEDURE 1. Set the projectile launcher to zero degrees so that it is horizontal to the ground. 2. Use the plumb bob to find the point on the floor that is directly below the launch point. Use a pencil to mark this point on the floor. Circle the point so it is clearly visible. 3. Measure the height, in meters, of the ball at launch point. Measure from the floor to the bottom of the marble, ignoring the launch position marked on the side of the launcher. Height of ball at launch (m) ___________ 4. Use the ramrod to load the steel marble to the middle setting (2 out of 3 “clicks”). Be sure one lab partner is in position to “field” the marble after it lands so the marble is not lost. 5. Launch the marble and note carefully where it landed. Look at the tiles on the floor and check to see that the marble is going straight, and not off to one side. If needed, pivot the projectile launcher to adjust the aim so it will launch straight, but don’t loosen the clamp. 6. Tape a piece of paper to the floor centered where the marble landed, and then place a half sheet of carbon paper (dark side down) on top of the paper. (Don’t tape the carbon paper to the floor.) 7. Launch the marble five times onto the carbon paper. 8. Measure the horizontal range, in meters, for each trial and record the results in the table. Horizontal Range of Marble, Δx (m) average: DATA & ANALYSIS (Show general equations, substitutions, calculations, and units.) 1. Using the height of the launcher, the initial vertical velocity, gravity, and the appropriate equation of projectile motion, calculate the time of flight for the projectile. 2. Using the time of flight, and the average horizontal range, calculate the launch velocity of the marble. 24 Now that you know the launch velocity of the marble you can use this information to launch the marble at any angle and predict the time of flight and range of the projectile. Use a 70˚ angle. 3. Calculate the horizontal and vertical components of the marble’s velocity when it is fired at 70˚ above horizontal. Remember that the launch velocity that you found in the previous calculation is now the hypotenuse of a right triangle trigonometry problem. 4. Using the height of the launcher, the initial vertical velocity (not zero now), gravity, and the appropriate equation of projectile motion, calculate the time of flight for the projectile at a 70˚ angle. This will require using the quadratic formula to solve a quadratic equation. 5. Using the time of flight, and the horizontal velocity component, calculate the range (horizontal displacement) of the marble. 6. Because air resistance affects path of the projectile, in this lab let’s try to account for it with a simple approximation. Assume that the actual range is 96% of the theoretical range. Show the calculation below. Use answer to mark the target clearly on the floor. 7. OK, time to test your physics! After you have shown me the correct calculations it will be your turn to compete! Score up to 20 points on the total of two tries. 8. Clean up: set the launcher back to 0 degrees, place the ramrod and marble on the launcher, replace the paper, recycle used paper, set the plumb bob on the carbon/paper. 25 QUESTIONS & CALCULATIONS 1. Continuing with the data from the lab, calculate the time it takes for the projectile to achieve maximum altitude, when launched at a 70˚ angle. 2. Calculate the maximum altitude above the floor for the projectile, when launched at a 70˚ angle. 3. Calculate the horizontal distance to where the projectile’s altitude is at a maximum, when launched at a 70˚ angle. 4. What accounts for the errors in this lab? Why are the results of this lab less accurate and less precise than in the last lab? 26 Kinematics Review Sheet Multiple choice. Circle the best answer. 1. Consider the position vs. time graph on the right. At which lettered point or points is the object a) moving the fastest? b) moving to the left? c) speeding up? d) slowing down? e) turning around? 2. 3. 4. 5. 10. A penny is dropped near the surface of the earth and freefalls without any air resistance. After 1 second the displacement and the velocity of the penny are (A) -4.9 m, -4.9 m/s (C) -4.9 m, -9.8 m/s (B) -9.8 m, -4.9 m/s (D) -9.8 m, -9.8 m/s 11. A rock falls from rest 0.72 m down near the surface of a planet in 0.63 s. The planet’s acceleration due to gravity is (A) -1.1 m/ s2 (C) -3.6 m/s2 2 (B) -2.3 m/s (D) -9.8 m/s2 Consider the velocity vs. time graph on the right. During which segment(s) is the object a) moving with constant velocity? b) speeding up? c) slowing down? d) standing still? e) moving to the right 12. Two vectors having magnitudes of 5 and 8 cannot have a resultant with a magnitude of (A) 3 (C) 13 (B) 7 (D) 15 On a highway, a car is driven 80 km/h during the first 1.0 hour of travel, 50 km/h during the next 0.5 hour, and 40 km/h in the final 0.5 hour. What is the car’s average speed for the entire trip? (A) 45 km/h (C) 85 km/h (B) 62.5 km/h (D) 170 km/h As a car is driven south in a straight line with decreasing speed, the acceleration of the car must be (A) directed northward (C) zero (B) directed southward (D) constant, but not zero A race car starting from rest accelerates uniformly at a rate of 4.9 m/s2. What is the car’s speed after it has traveled 200 m? (A) 1960 m/s (C) 44.3 m/s (B) 62.6 m/s (D) 31.3 m/s . 6. Velocity is to speed as displacement is to (A) acceleration (C) momentum (B) time (D) distance For the next three questions consider the graph below, which shows the motion of three cars. Car A v(m/s) 15 Car B 10 Car C 5 10 20 30 40 t(s) 7. (Honors only) From 0 s to 20 s, which car has been displaced the most? (A) car A (B) car B (C) car C 8. At 30 s which car has the most velocity? (A) car A (B) car B (C) car C 9. At 40 s which car has the most acceleration? (A) car A (B) car B (C) car C 13. When a softball, thrown vertically upwards, gets to the top of its path, it has (A) velocity = 0 m/s; acceleration = 0 m/s2 (B) velocity = -9.8 m/s; acceleration = 0 m/s2 (C) velocity = 0 m/s; acceleration = -9.8 m/s (D) velocity = -9.8 m/s; acceleration = -9.8 m/s 2 2 14. Starting from rest a coconut falls from a tree. Its average speed during 8 s of freefall is (A) 19.6 m/s (C) 39.2 m/s (B) 9.8 m/s2 (D) 9.8 m/s 15. A golfer putts the ball 5.0 m due east, then 2.1 m due north, and then finally 0.5 m due west, into the hole. What is the magnitude and direction of the resultant vector for the three putts? (A) 5.89 m, at 69.0˚ from north (B) 5.89 m, at 21.0˚ from north (C) 4.97 m, at 65.0˚ from north (D) 4.97 m, at 25.0˚ from north 16. A ball thrown horizontally from a 20 m high building strikes the ground 5.0 m from the base of the building. With what velocity was the ball thrown? (A) 4.9 m/s (C) 3.0 m/s (B) 3.3 m/s (D) 2.5 m/s 17. A cannon, elevated at 40˚ is fired at a wall 300 m away on level ground, as shown. The initial speed of the cannonball is 89 m/s How long does it take for the ball to hit the wall? (A) 1.3 s (B) 3.3 s (C) 4.4 s (D) 6.8 s 18. Referring to the previous question, at what height h does the ball hit the wall? (A) 39 m (B) 47 m (C) 137 m (D) 157 m 27 Kinematics Review Sheet Problem solving. On separate paper, show all your work, (general equation, substitutions, calculations, and units). 19. A car accelerates from rest at 4.50 m/s2. a. What is its velocity after it’s been displaced 100 m? b. If the car now brakes and comes to rest after 6 s of braking, how far did it travel in total from its start? c. How much time elapsed from start to stop? 28. A racecar completes one lap around a track at 200 miles per hour, and the second lap at 220 miles per hour. What is the average speed of the racecar? (Hint: you don’t know the length of one lap, but show how that distance d cancels in the solution. By the way, the answer is called a harmonic average, and it’s not the arithmetic average!) 20. A hot air balloon descends vertically at a constant speed of 2 m/s. When it is 40 m above the ground, a rock is dropped from the balloon and freefalls to the ground. a. What is the rock’s velocity when it strikes ground? b. What is the balloon’s height above the ground when the rock hits the ground? 29. A motorist is driving at 20 m/s when she sees that a traffic light 200 m ahead has just turned red. She knows that this light stays red for 15 s, and she wants to reach the light just as it turns green again. It takes her 1.0 s to step on the brakes and begins slowing at a constant acceleration. What is her speed as she reaches the light at the instant it turns green? 21. During a fireworks display, a rocket is launched with an initial velocity of 35 m/s at an angle of 75° above ground. a. What is the rocket’s height when it explodes in 2.0 s? b. What is angle (from horizontal) of the rocket’s velocity just before the explosion occurred? 22. A baseball is thrown at an unknown angle, from 2.0 m above the ground, and it lands at a horizontal distance of 90, away. If the baseball takes 3.5 s to hit the ground: a. What is the horizontal component of the ball's velocity when it leaves the hand? b. What is the vertical component of the ball's velocity when it leaves the hand? c. What is the resultant velocity of the ball when it leaves your hand? 23. Draw a head-to-tail addition of the four vectors below, and show the resultant vector. Then draw them in reverse order. Tracing is the best method. A D B C 24. A football is kicked from the ground at an angle of 40˚ and a speed of 23 m/s. To score a field goal, the football has to clear a 3.05 m high crossbar. How far away can the kick be made to for the football to just clear the crossbar (on its way downward)? 25. A rocket is fired straight up and accelerates from rest at 30 m/s2 for 5 s, and then it runs out of fuel. Assuming no air friction, a) What is the rocket’s maximum altitude? b) How long in total does it take the rocket to return to the ground? (Hint: it free falls up, then down to the ground) Questions 26-33 are Honors only 26. A hiker walks vectors (θ in bearing) 20 m, 60˚; 30 m, 0˚; 40 m, 270˚; 20 m, 135˚. Find the resultant (θ in bearing.) 27. On the Apollo 14 mission to the moon, astronaut Alan Shepard hit a golf ball that went 330 m on level ground. The balls left the “club” at 25 m/s. The free fall acceleration on the moon is about 1/6 of its value on Earth. At what angle was the golf ball hit? (Do this problem without using the “range equation”.) 30. A boat that can travel at 9.0 km/hr in still water is used to cross a river flowing at a speed of 4.0 km/hr. a. At what angle must the boat head so that its motion is straight across the river? b. Find the resultant speed relative to shore. 31. A stunt car rolls down a 24˚ incline, starting from rest, accelerating at 4.0 m/s2, for 50 m. It then rolls off the incline and freefalls into water, which is 30 m below the incline. How far forward (range) does the stunt car land? 32. A stone is thrown downward at 8 m/s from a height of 22 m. At the same time, a stone is thrown upward from 2 m the ground with a speed of 17 m/s. a. At what point in time do their paths intersect? b. At what height do their paths intersect? 33. Extra Challenge: In a 100-m race, a sprinter accelerates from rest at 2.68 m/s2. After reaching a top speed, he runs the rest of the race at constant speed, finishing the race in 12 s total. a. For how much time does the runner accelerate? (time to reach max. speed) b. How far does he run during while maintaining constant speed? Answers 1. a) D b) C,D,E c) C d) A,E e) B 2. a) A,D b) C c) B,E d) F e) A,B 3. B 4. A 5. C 6. D 7. C 8. A 9. B 10. C 11. C 12. D 13. C 14. C 15. C 16. D 17. C 18. D 19. a. 30.0 m/s b. 190 m c. 12.67 s 20. a. -28.1 m/s b. 34.7 m 21. a. 48.0 m b. 57.5˚ 22. a. 25.7 m/s b. 16.6 m/s c. 30.6 m/s at 32.8˚ from horizontal 23. see solutions page for vector drawings 24. 49.2 m 25. a. 1523 m b. 37.9 s 26. 27.2 m, 341.3˚ 27. 29.8˚ 28. 209.5 km/h 29. 5.71 m/s 30. a. 63.6˚ from river b. 8.06 km/hr 31. 32.5 m 32. a. 0.8 s b. 12.5 m 33. a. 3.67 s b. 81.9 m 28 Notes 29 Notes 30 Notes 31 Notes 32 Notes 33 Notes 34 Dynamics “If I have seen farther than others, it has been by standing on the shoulders of giants.” – SIR ISAAC NEWTON “The most incomprehensible thing about the universe is that it is comprehensible.” – ALBERT EINSTEIN 35 Dynamics Assignment Sheet Online: www.physicsclassroom.com/Physics-Tutorial/Newton-s-Laws (Assignments 1-6) www.physicsclassroom.com/class/vectors (Assignments 9-11) www.physicsclassroom.com/class/circles (Assignments 12-15) No. Topics Reading Homework 1 Newton’s First Law of Motion “The Law of Inertia” 130-135 Ch 4: 1,2,3,4,6 2 Newton’s Second Law of Motion “The Law of Acceleration” 136-138 Ch 4: 14,19,20,21 3 Newton’s Third Law “The Law of Action-Reaction” 138-140 Ch 4: 13,17,18,33 4 definition and units of force, free-body diagrams 124-125 Ch 4: 7,8,9,10,11 5 common forces: weight, normal, friction, tension 141-142 Ch 4: 15,26,45,47,49 6 more on common forces: weight, normal, friction, tension 7 static and kinetic friction; coefficients of friction 8 more problems on Second Law of Motion with force of friction Ch 4: 36,53,63ab,64a 9 more problems on Second Law of Motion Ch 4: 35,50,51,67 10 torque, force, lever arm, and rotational motion 278-282 Ch 8: 1,8,9,39 11 rotational equilibrium 286-288 Ch 8: 18,20,45,48 12 circular motion and centripetal acceleration 257-259 Ch 7: 16,26,47a,50 13 centripetal forces that cause circular motion 260-262 Ch 7: 19,43,47b,48 14 Newton’s Law of Universal Gravitation 263-265 Ch 7: 29,35,39,40,49 15 supplemental problems on gravitation In the lab book Practice problems 2,3,4,7,8,9 Ch 4: 25,56,57,58 FBDs in lab manual 143-148 36 Ch 4: 34,37,44,52 Dynamics Assignment Sheet - Honors Online: www.physicsclassroom.com/Physics-Tutorial/Newton-s-Laws (Assignments 1-6) www.physicsclassroom.com/class/vectors (Assignments 9-11) www.physicsclassroom.com/class/circles (Assignments 12-15) No. Topics Reading Homework 1 Force; Newton’s First Law of Motion: “The Law of Inertia” 4-1 4-2 Q 1,2,3,9,18 2 Mass; Newton’s Second Law of Motion: “The Law of Acceleration” 4-3 4-4 Q 4,10 P 3,6,9,11 3 Newton’s Third Law: “The Law of Action-Reaction” 4-5 Q 6,7,15,16,17 4 Force of gravity (weight); Normal force 4-6 Q 14 P 4,5,13,15,20 5 Free body diagrams; solving problems with Newton’s laws 4-7 P 24,25,29,31 6 More problem solving with Newton’s laws 7 Static and kinetic friction; coefficients of friction 8 More problem solving with friction and inclined planes P 41,42,45,56,60 9 Newton’s law review problems Q 13 P 67,69,76,86,87 10 Torque and rotational motion 8-4 Q6 P 22,23,24,26 11 Rotational equilibrium 9-1 9-2 Q8 P 6,7,16,20,21 12 Uniform circular motion and centripetal acceleration 5-1 Q6 P 2,5,6,72 13 Centripetal forces; banked and unbanked curves 5-2 5-3 Q 1,8 P 7,9,11,14 14 Newton’s law of universal gravitation 5-6 5-7 Q 11 P 28,30,33,39,70 15 Satellite motion; weightlessness; Kepler’s laws 5-8 5-9 Q 20 P 43,48,58,83,90 FBDs in lab manual P 23,26,32,33,35 4-8 37 P 37,39,46,48,61 Gravitation Practice Problems 1. The mass of Pluto was not accurately known until a satellite of the planet was discovered. Explain why? What law helps answer this question? 2. Respond carefully to the question, "What keeps a satellite up?" 3. Mimas, a moon of Saturn, has an orbital radius of 1.87 × 108 m and an orbital period of about 23 hours. Use Newton's variation of Kepler's third law to find the mass of Saturn. 4. A geosynchronous satellite appears to remain over one spot on Earth. Such a satellite has an orbital radius of 4.23 × 107 m. a. Calculate its period in hours (the answer explains why the satellite’s orbit is “geosynchronous”) b. Calculate its orbital speed. 5. Jupiter has about 300 times the mass of Earth and about 10 times earth's radius. Estimate the size of g on the surface of Jupiter. 6. On July 19, 1969, Apollo 11's orbit around the moon had an average altitude of 111 km above the moon. The moon’s radius is 1785 km and the moon’s mass is 7.3 × 1022 kg. a. How many minutes did it take to orbit once? b. At what velocity did it orbit the moon? 7. If the space shuttle goes into a higher orbit, what happens to the shuttle’s period? 8. If a mass located in Earth's gravitational field is doubled, what happens to the force exerted by the field upon this mass? Is the Earth’s field constant at such a location? 9 Yesterday, Sally had a mass of 50.0 kg. This morning she stepped on the scale and found that she gained weight. a. What happened if anything to Sally's mass? b. What happened, if anything, to the ratio of Sally's weight to her mass? 10. Earth's gravitational field is 7.83 N/kg at the altitude of the space shuttle. What is the size of the force of attraction between a student, mass of 45.0 kg, and Earth? Answers: 3. 5.65 × 1026 kg 4. a) 24 hours (86,400 s) b) 3070 m/s 5. about 3 times g on Earth 6. a) 124 minutes b) 1600 m/s 10. 352 N 38 Newton’s Second Law PURPOSE The purpose is to investigate the relationship between force and acceleration, on a fixed (constant) mass. EQUIPMENT - dynamics track and cart - photogate timer and accessory photogate - mass set and hanger - meter stick plastic flag end stop dynamics cart with block photogate timer m1 accessory photogate pulley mass hanger with mass m2 PROCEDURE 1. Place two 100 g, a 50 g, a 20 g, and a 10 g masses in the cart. Place a 5 g and a 10 g mass on the mass hanger. The hanger itself is 5.0 g. The cart has a mass of 500 g, and the black metal block in the cart also has a mass of 500 g. The clear plastic “flag” with the black tape is 17 g. Calculate the total known mass of the system, in kilograms, and record this value in your lab book above the data table. 2. Record the hanging mass, in kilograms, in the mass column of the data table. Remember that the hanger itself has a mass of 5.0 g so initially the total hanging mass is 20.0 g. There should only be a small acceleration as the 20.0 g is nearly balanced by the force of friction. (You will calculate force later.) 3. Pull the cart back until it touches the rubber bumper on the dynamics track. This will be the release point for the cart for the entire lab. Move the photogate timer back so that the flag on the cart will immediately trigger the timer when the cart is released. The initial velocity of the cart will now be zero. Place the other photogate 70.0 cm away from the photogate timer. Record the distance, in meters, in your lab book above the data table. 4. Turn the photogate timer to Pulse Mode, which measures the time between gates. With the cart against the rubber bumper, release the cart and then catch it before it hits the end stop. Repeat three more times. Record the times across the first row in your data table. Ignore obvious outliers and take additional time trials if necessary. Watch carefully for masses that may fall off the hanger onto the floor. Also be sure to check that the end stop doesn’t interfere with the string. 5. Now move one 10 g mass out of the cart and place it on the mass hanger. The mass of the system has not changed, but the hanging mass (and therefore the net force) has changed. Record this new 30 g mass, in kilograms, in the mass column of the data table. Measure and record four new time trials in the next row of the data table 6. Keep adding to the hanging mass in increments of 10 g until 100 g of mass are moved. Think about it! You have a 50 g, a 20 g, two 10 g, and two 5 g masses so increments of 10 g is certainly possible. Measure and record four new time trials for each new hanging mass value. 39 DATA & ANALYSIS 1. Press STAT, then 5:SetUpEditor, then ENTER to reset the Lists in your calculator. Then press STAT, then ENTER to view the Lists. If there is any data in Lists L1 through L6 delete the data by moving the cursor to the top of the list and pressing CLEAR, ENTER. 2. Enter the time trials into your graphing calculator in Lists L1 through L4. Then calculate the average times for the four time trials by moving the cursor to the top of L5 (in the column heading) and type (L1+L2+L3+L4)/4, then ENTER. Record these average times in the table in your lab book. 3. Enter the hanging mass data into List L6 in your calculator. Then create a new List column by going to the column heading of List L6 and move the cursor one space to the right. Name this new column “F” for force using the F key on the calculator. 4. Calculate the force acting on the system for each set of trials. To convert hanging mass into a force (its weight) in newtons use: force (N) = hanging mass (kg) × 9.8 ( ) N kg In the top of the “F” List write L6 × 9.8, then ENTER. Record the forces in the data table. 5. Create another new column in your calculator and name the List “A” for acceleration. Since the cart started from rest, and you measured the time it takes to travel a known distance, you can use an appropriate kinematics equation to determine the acceleration. Solve the equation for acceleration, and then write the equation in the top of List “A”. Record the accelerations in the table in your lab book. 6. Create a graph of acceleration versus force in your lab book. Make sure that the independent variable is on the x-axis. Ask yourself: “does the cart’s acceleration depend on the force applied or does the force applied depend on the cart’s acceleration?” 7. Graph the acceleration vs. force on the calculator by turning on a STAT PLOT and adjusting the Xlist to “F” and the Ylist to “A”. To access a List, use the LIST menu (2nd STAT) and then move the cursor down to the appropriate list. 8. Looking at your force and acceleration data, use the WINDOW menu to determine the Xmin, Xmax, Xscl, Ymin, Ymax, Yscl so the graph can be viewed clearly. Or, just press ZoomStat to automatically fit the data onto the graph. 9. To find the best-fit line, press STAT, then move the cursor over to CALC and use 4:LinReg(ax+b). Then adjust the menu to read Xlist:A, and Ylist:F, and Store RegEQ: Y1. (To get “A” and “F” press 2nd STAT for Lists and cursor down; to get Y1 press VARS, then move cursor to Y-VARS, then press 1:Function, then 1:Y1 and finally ENTER) Note: on the TI-83 this should appear on screen as: LinReg(ax+b) LF, LA, Y1 then press ENTER. 10. Draw the best-fit line on your graph. Write the equation of the best-fit line on the graph in your lab book. Use appropriate variables in the equation. DON’T use y and x! 40 Photogate distance (in meters): __________ t1 (s) (L1) t2 (s) (L2) t3 (s) (L3) Known system mass (in kg): __________ t4 (s) (L4) tav (s) (L5) best fit equation: 41 hanging mass (kg) (L6) force (N) (F) acc (m/s2) (A) QUESTIONS & CALCULATIONS 1. a) From your equation for the best-fit line, predict what force is needed to accelerate the cart at a rate of 2.0 m/s2? b) How many grams (not kilograms!) of hanging mass are necessary to do this? 2. The line of best fit should cross the positive x-axis. This x-intercept has a very real physical significance in the lab. What is the significance of the x-intercept on your graph? 3. The slope of your graph should be a straight line. What does this imply about the relationship between acceleration and the force that causes it? 4 If acceleration is proportional to force and inversely proportional to mass then we could propose that a = mF . That could also be expressed as a = m1 ⋅ F . That means that the experimental mass of the system is the reciprocal of the slope of the graph of a vs. F. Calculate the experimental system mass. 5. Find the percent error between the experimental system mass and the known system mass. 42 Newton's Second Law - Revisited (Honors Physics only) PURPOSE The purpose is to investigate the relationship between mass and acceleration using a constant net force. EQUIPMENT - dynamics track and cart - photogate timer and accessory photogate - mass set and hanger - meter stick plastic flag end stop dynamics cart with block photogate timer m1 accessory photogate pulley mass hanger with mass m2 PROCEDURE 1. Place three 100 g and two 50 g masses in the cart. The cart has a mass of 500 g and the black metal block in the cart also has a mass of 500 g. The clear plastic “flag” with the black tape is 17 g. Calculate the total cart mass in kilograms, not grams. 2. Place two 20 g and a 5 g mass on the mass hanger. Remember that the plastic mass hanger is 5 g. Record the total hanging mass (in kilograms) in the data table. Then record the system mass (in kilograms) in the first row of the data table. 3. Pull the cart back until it touches the rubber bumper on the dynamics track. This will be the release point for the cart for the entire lab. Move the photogate timer back to the cart so that the flag on the cart will immediately trigger the timer when the cart is released. The initial velocity of the cart will now be zero. Place the other photogate about 70.0 cm away from the photogate timer, Record the distance, in meters, in your lab book above the data table. 4. Turn the photogate timer to Pulse Mode, which measures the time between gates. Release the cart, but catch it before it hits the end. Record the time in your lab book. Repeat three more times. Ignore obvious outliers. 5. Now remove 100 g from the cart, and record the new system mass in the table. The hanging mass remains at 50 g. Remember, you are not changing the hanging mass, only the cart mass, so put the 100 g mass that came out of the cart back in the mass set box. 6. Repeat steps 3 & 4. Record these four new times in the second row of the data table. 7. Keep removing 100 g at a time until the cart is completely unloaded, including the block. Repeat steps 3 & 4 again and record new times for each new cart mass value. WATCH FOR MASSES THAT MAY JUMP OFF! 43 DATA AND ANALYSIS 1. Press STAT, then 5:SetUpEditor, then ENTER to reset the Lists in your calculator. Then press STAT, then ENTER to view the Lists. If there is any data in Lists L1 through L6 delete the data by moving the cursor to the top of the list and pressing CLEAR, ENTER. 2. Enter the time trials into your graphing calculator in Lists L1 through L4. Then calculate the average times for the four time trials by moving the cursor to the top of L5, type (L1+L2+L3+L4)/4, then ENTER. Record these average times in the table in your lab book. 3. Enter the system mass data (sum of cart mass and hanging mass) in kilograms into List L6 in your calculator. 4. Then create a new List column by going to the top of List L6 and move the cursor one space to the right. Name this new column “INM” for inverse of mass. 5. Calculate the inverse mass of the system for each set of trials. To convert mass into an inverse mass, go to the top of the “INM” List and write L6-1, then ENTER. Record the inverse mass in the data table. 6. Create another new column in your calculator and name the List “A” for acceleration. Since the cart started from rest, and you measured the time it takes to travel a known distance, you can use an appropriate kinematics equation to determine the acceleration. Solve the equation for acceleration, and then write the equation in the top of List “A”. Record the accelerations in the table in your lab book. 7. Create a graph of acceleration versus inverse mass in your lab book. Make sure that the independent variable is on the x-axis. Ask yourself: “does the cart’s acceleration depend on the mass or does the mass depend on the cart’s acceleration?” 8. Graph the acceleration vs. inverse mass on the calculator by turning on a STAT PLOT and adjusting the Xlist to “INM” and the Ylist to “A”. To access a List, use the LIST menu (2nd STAT) and then move the cursor down to the appropriate list. 9. Looking at your force and acceleration data, use the WINDOW menu to determine the Xmin, Xmax, Xscl, Ymin, Ymax, Yscl so the graph can be viewed clearly. Or, just press ZoomStat to automatically fit the data onto the graph. 10. To find the best-fit line, press STAT, then move the cursor over to CALC and use 4:LinReg(ax+b). Then adjust the menu to read Xlist:INM, and Ylist:A, and Store RegEQ: Y2. (To get “INM” and “A” press 2nd STAT for Lists and cursor down; to get Y2 press VARS, then move cursor to Y-VARS, then press 2:Function, then 2:Y2 and finally ENTER) Note: on the TI-83 this appear on screen as: LinReg(ax+b) LINM, LA, Y2 then press ENTER. 11. Draw the best-fit line on your graph. Write the equation of the best-fit line on the graph in your lab book. Use appropriate variables in the equation. DON’T use y and x! 12. Calculate the known value of the net force acting on the system by converting the hanging mass into a force (its weight) in newtons. Use: force (N) = hanging mass (kg) × 9.8 ( ) N kg 13. Convince yourself that the slope for the graph a vs. 1/m yields the experimental value for the net force on the system. Compare this with the known value for the net force on the system from the last step. Calculate percent error below. 44 Photogate Distance (in meters): ____________ t1 (s) (L1) t2 (s) (L2) t3 (s) (L3) Hanging Mass (in kg): ____________ t4 (s) (L4) tav (s) (L5) best fit equation: 45 system mass-1 (kg-1) acc (m/s2) mass (kg) (INM) (A) (L6) Free Body Diagrams This lesson is designed to help students apply Newton’s 2nd Law to solve problems, which is the central theme within the study of dynamics. Thus, the need to identify, draw, and label a variety of forces is essential. Read through the list of forces on the right. Details about these forces, like the difference between static and kinetic friction, will be described later. For now, you are only drawing and labeling forces. Follow these rules: 1) Draw forces on the dot, to the right of each picture. 2) Draw forces with a ruler, and put an arrow on each, to show direction. 3) Draw forces that are proportionally correct. That is, if you intend to show equilibrium, then forces are balanced. If not, then the net force must be in the direction of acceleration. 4) Label each force according to the list shown to the right. 5) Draw an acceleration arrow nearby (unless it’s zero) 1 2 3 4 5 6 46 Common Forces Fg = force of gravity (weight) Fn = normal force (support) Fr = air resistance (drag) Fs = static friction Fk = kinetic friction FT = tension Fsp = spring force 7 8 9 10 11 12 13 The next four are Honors Physics only 14 15 16 47 Hooke's Law . PURPOSE To investigate Hooke's Law, the relationship between spring force and displacement for a stretched spring; to investigate how multiple springs behave in different arrangements. SET UP Δx series parallel single PROCEDURE Do NOT exceed the elastic limit of any spring by overloading it past a maximum stretch of 50 cm! PART A - SINGLE SPRING 1. Hang a single spring from the support. Use an upright (vertical) meter stick, with the 100 cm mark on the lab table, to measure the initial position of the lowest loop on the spring. Record this initial position. 2. Attach the 5.0 g mass hanger to the bottom of the spring. Add 35 more grams of mass for a total of 40 grams on the spring. Measure the final position of the bottom of the lowest coil on the spring (the one that’s turned vertically). Subtract initial position from final position to find the displacement of the masses. Record this displacement in the lab table. 3. Place another 20 grams of mass on the mass hanger for a total of 60 grams. Again, measure final position, calculate displacement, and record the data in the table. Add more mass in 20-gram increments up to a maximum of 140 grams and record all collected data. PART B – SERIES SPRINGS 4. Now hang a second spring from the bottom of the first one, an arrangement called "series." Again start with 40 grams of initial total mass, and add masses in 20-gram increments, up to a maximum of 140 grams. You may need to raise the support rod to allow for a greater maximum displacement. Check it with the 140-gram load to be sure. Measure final position, calculate displacement, and record the data in the table. PART C – PARALLEL SPRINGS 5. Now hang the second spring on the mass hanger next to the first one, an arrangement called "parallel." Use a small paper clip, bent into a triangle, to connect both springs. Again start with 40 grams of initial total mass, and add masses in 20-gram increments, up to a maximum of 140 grams. Measure final position, calculate displacement, and record the data in the table. DATA & ANALYSIS 1. Complete the data table of mass (in kilograms), displacement (in meters), and force (in newtons) for each part of the lab. Remember, to calculate force (weight), multiply the mass by g = 9.8 N/kg. 2. Graph the data for force vs. displacement for each part of the lab on the same graph, making sure to determine dependent and independent variables. Be sure to consider the maximum displacement and maximum force to set the graph scale so the graph is filled. 3. Enter the data into the TI Graphing Calculator using the Lists (L1 through L5) according to the columns in the data table. 4. Use the TI Graphing Calculator to find the equation for each best-fit line, and include the equations below the graph. Use appropriate variables - DON’T use y and x! 5. From the best-fit equation, determine the spring constant (in units of N/m) for each part of the lab. Label each best-fit line on the graph with this spring constant. Use k for the single spring, kseries for the series springs, and kparallel for the parallel springs. 48 Single Spring m (kg) (L1) F (N) (L2) Series Springs Δx (m) (L3) initial position (m) ________ Equations: single spring m (kg) (L1) F (N) (L2) Parallel Springs Δx (m) (L4) initial position (m) ________ springs in series 49 m (kg) (L1) F (N) (L2) Δx (m) (L5) initial position (m) ________ springs in parallel QUESTIONS & CALCULATIONS 1. The results from Part B confirm that for two identical springs in series, the spring constant is kseries = 12 ksingle . Now rearrange the equation and solve for the single spring constant, ksingle . Then find the percent difference (not percent error) between this value and the value for ksingle that you found from data on the single spring. Look up percent difference in the back of the lab manual. 2. The results from Part C confirm that for two identical springs in parallel, the spring constant is kparallel = 2ksingle . Now rearrange the equation and solve for the single spring constant, ksingle . Then find the percent difference (not percent error) between this value and the value for ksingle that you found from data on the single spring. 3. What is the physical significance of the x-intercept of the graphs you created? 4. Why does the spring constant get smaller when two springs are added together in series? Why does the spring constant get larger when two springs are added together in parallel? 5. A spring is cut exactly in half. What happens to the spring constant of the two equal-sized pieces compared with the spring constant of the original full-sized spring? (Hint: think about your answer to the previous question about series springs.) 50 Friction & The Graphing Calculator PURPOSE To measure the coefficient of static friction (µs) between a TI Graphing Calculator and a Dynamics Track, using a level plane and an inclined plane. EQUIPMENT - dynamics track - mass set and hanger - meter stick - calculator - cardboard PROCEDURE PART A - LEVEL PLANE m1 TI graphing calculator mass hanger m2 1. Measure the mass of the calculator on the electronic balance, and record the mass, in kilograms, in the data table. 2. Put cardboard on the floor below the hanging mass to ease the impact, especially when a lot of hanging mass is used. Attach the string to the calculator using the method shown in class. 3. In this part of the lab, you will test the coefficient of friction between the TI Graphing Calculator and a level Dynamics Track. Start with the calculator resting on the four friction pads of the calculator cover. Make sure the friction pads are not on the groves in the track. 4. While one lab partner pushes down lightly on the calculator (to give it more normal force) another lab partner loads mass on the mass hanger. Then the person pushing down on the calculator releases it to see if the calculator moves. Your job is to put enough mass on the hanger so the calculator just starts to move along the track. Trial and error is necessary. Record the hanging mass, in kilograms, for this first trial in the table below. 5. Repeat step 4 three more times with the calculator placed on a different section of the track. Record the data in the table. DO NOT let the hanging mass slam onto the floor! 6. Now repeat steps 4 and 5 with the calculator placed on its keyboard side. Record all data. mass of calculator, m1 (in kg) = ____________ m2 - trial #1 (kg) m2 - trial #2 (kg) calculator on 4 friction pads calculator on keyboard 51 m2 - trial #3 (kg) m2 - trial #4 (kg) m2 - average (kg) DATA & ANALYSIS 1. Carefully draw free body diagrams below for the calculator (m1) and the hanging mass (m2). Check that the forces are drawn qualitatively to scale, and are labeled properly. 2 Calculate the weight of the calculator, and the weight of the average hanging mass for both trials. 3. Look at the free body diagrams above. Determine the normal force on the calculator. To do this, recognize that the forces must keep the calculator’s mass in equilibrium. 4. Look at the free body diagrams above. Determine the static friction force on the calculator on the friction pad side and keyboard side. Again, recognize the forces keep the mass in equilibrium. friction pad: keyboard: 5. Calculate the coefficient of static friction for the friction pad side and the keyboard side. friction pad: keyboard: 52 PROCEDURE PART B - INCLINED PLANE meterstick m1 h L TI graphing calculator θ 1. Take the pulley off the end of the Dynamics Track and leave it on the table. Put the calculator on the track, with the calculator resting on the four friction pads of the case. Make sure the friction pads are not on the groves in the track. 2. Lift the bumper end of the track up slowly until the calculator just starts to slide down the track. Trial and error is necessary. (Be sure to use the same calculator from Part I.) 3. Use the meter stick to measure the height of the track. Be sure you are measuring the under side of the track not the top. Record the height, in meters, in the table below. 4. Repeat steps 2 and 3 three more times with the calculator placed on a different section of the track. Record the data in the table. 5. Now repeat steps 2 and 3 with the calculator placed on its keyboard side. Record all data. 6. Determine the angle θ from the height (opposite) and length (hypotenuse) of the track. Show one sample calculation for finding the angle, θ, using one height average. mass of calculator, m1 (in kg) = __________ h - trial #1 (m) h - trial #2 (m) length of track, L (in meters) = __________ h - trial #3 (m) calculator on 4 friction pads calculator on keyboard 53 h - trial #4 (m) h – average (m) angle, θ DATA & ANALYSIS 1 Honors: Draw a free body diagram for the calculator in Part II. Be sure to check that the forces are drawn qualitatively to scale, and are labeled properly. 2 Honors: Use Newton’s 2nd Law to show that forces in the direction parallel to the inclined plane on the calculator are balanced. 3 Honors: Use Newton’s 2nd Law to show that forces in the direction perpendicular to the inclined plane on the calculator are balanced. 4 Honors: Combine the 2nd Law equations from the last step to prove that µ s = tan θ . 5. Using the result, µ s = tan θ , for the calculator on the inclined plane, calculate the coefficient of static friction for both the friction pad side and the keyboard side. friction pad: keyboard: 54 QUESTIONS & CALCULATIONS 1. Calculate the percent difference (not percent error) for the coefficient of static friction for the friction pad side in Part A versus Part B. Look up percent difference in the lab manual. Then calculate the percent difference for the keyboard side. friction pad: keyboard: 2. Some people just fidget a lot. They can’t help it. They tap their pencils on the desk, they twirl their hair, and they peel things off that should stay put, like the rubber feet on the bottom of their graphing calculator. Use data from the lab to describe why this is a bad habit for the calculator. 3. The TI Graphing Calculator only uses four small friction pads on the cover, but they effectively work about as well as a calculator that has the entire case covered with a large friction pad. Explain why these small pads work so well; that is, why surface area of contact appears to have little effect on frictional force. 4. The backside of the TI-83 has only two friction pads, and the backside of the TI-84 has three friction pads, while the case has four. Why the difference? (There are several good reasons, and only Texas Instruments can tell us for sure!) 55 Torque & Static Equilibrium PURPOSE To investigate the torques responsible for creating a system with rotational equilibrium and to calculate an unknown torque if all other torques in the system are known. d2 d1 m2 m1 dcm center of mass support stand PROCEDURE 1. Insert a meter stick into a balance clamp and place onto the support stand Adjust the position of the clamp until the meter stick balances level on the metal support stand. Record the center of mass for the meter stick exactly to the nearest millimeter. 2. Now move the clamp 10 cm to one side. The meter stick will no longer balance if supported at the clamp, but if weight is added to the light side, a balance can once again be achieved. Add two sets of “hanging masses” to the light side until the meter stick is balanced again. Note that the mass hangers have a mass of 5 grams. Record both masses. 3. Measure and record the distances from the point of rotation to each of the hanging masses and to the center of mass of the meter stick. 4. Measure and record the meter stick’s mass (without the clamp) using an electronic balance. DATA & ANALYSIS meter stick center of mass _______ (m) distance to center of mass, dcm ________ (m) hanging mass #1 _______ (kg) distance to first mass, d1 ________ (m) hanging mass #2 _______ (kg) distance to second mass, d2 ________ (m) meter stick mass _______ (kg) 1. There are now three torques creating rotational equilibrium. On the one side of the point of rotation there are two torques due to the weights (gravitational forces) of the hanging masses. On the other side, the torque comes from the weight of the meter stick. This meter stick weight can be considered to be concentrated at the center of mass. Using the relationship Fg = mg, convert the hanging masses into weights measured in newtons. 56 QUESTIONS & CALCULATIONS 1. Now calculate the torques produced by the hanging masses. Express these in newton-meters (N⋅m). 2. Using the sum of torques law, write an equation to determine the torque on the other side produced by the meter stick’s center of mass. 3. Calculate the mass of the meter stick. 4. Calculate the percent error between the experimental meter stick mass and the known mass. 5. Using a torque equation, calculate the normal force that the support stand exerts on the meter stick (while it is loaded with masses m1 and m2). Check your answer using a force equation. 6. Calculate the unknown mass and distance in the diagram of the mobile to the right so that it will balance properly in static equilibrium. 0.2 m 0.6 m ?m 0.1 m ? kg 2 kg 57 5 kg PURPOSE Centripetal Force To use the ideas of uniform circular motion to calculate the centripetal force acting on a rubber 4π 2 mr stopper moving in a circle using the equation Fc = . T2 PROCEDURE (PART 1) 1. Set up your apparatus as shown to the right. 2. Choose and measure a convenient radius “R” at which to swing the rubber stopper. Measure the radius to the center of the stopper. - Use a piece of tape on the string a bit below the plastic tube to help keep the radius constant (but don’t let it touch the bottom of the tube). - Don’t let any part of your body touch the string or the tape. 3. Use a stopwatch to measure the time for the stopper to go through 10 revolutions. Do this five times with both partners taking turns swinging the stopper. DATA Hanging mass: ___________ kg Radius: ___________ m Time for 10 revolutions (s) Average time for 10 revolutions: ___________ Period: ___________ QUESTIONS/CALCULATIONS (Show all work) 1. Calculate the weight of the hanging mass. 2. The weight you calculated in the last question provides the centripetal force to move the rubber stopper in a circle. Use this idea to calculate the mass, in grams, of the rubber stopper. Predicted Mass: __________ Actual Mass: __________ 58 Percent Error: __________ PROCEDURE (PART 2) Obtain a mystery mass from me with which to replace the hanging mass. Use the space below to provide data and calculations that determines the mystery mass (in grams). Predicted Mass: __________ Actual Mass: __________ QUESTIONS (PART 2) Complete the following free body diagrams for objects in circular motion A D B E C F 59 Percent Error: __________ Dynamics Review Sheet Match the Free Body Diagram with the correct description. 1. Book at rest on a level table 2. Freefalling body (no air friction) FT Fr 3. Person on elevator accelerating up 4. Car skidding to rest (no airndfriction) 5. Mass #2 from Newton’s 2 Law Lab Fg Fg 6. Skydiver falling at terminal velocity 7. Mass at rest on incline, held by friction 8. Football at peak of path (with air friction) A B Fs Fn Fk Fg C Fn Fn Fg Fg E D Fr Fn Fg Fg Fg F G H Multiple choice. Circle the best answer. 9. A 60-kilogram skydiver is falling at constant speed near the surface of Earth. The magnitude of the force of air friction acting on the skydiver is A. 0 N C. 58.8 N B. 5.88 N D. 588 N 14. If the sum of all the forces acting on a moving object is zero, the object will A. slow down and stop B. change the direction of its motion C. accelerate uniformly D. continue moving with constant velocity 10. Two masses exert a gravitational force F on each other. If the mass of each is doubled and the distance between them is tripled, the force between them is: 15. As a ball falls, the action force is the pull of the earth’s mass on the ball. The reaction force is the A. air resistance acting against the ball. B. acceleration of the ball. C. pull of the ball’s mass on the earth. D. non-existent in this case. A. 12F B. 4 9 F 3 4 4 3 F D. F C. 11. A child is riding on a merry-go-round. As the speed of the merry-go-round is doubled, the magnitude of the centripetal force acting on the child A. remains the same C. is halved B. is doubled D. is quadrupled 16. A net force of 10 newtons accelerates an object at 5 meters per second squared. What net force would be required to accelerate the same object at 1.0 meters per second squared? A. 1.0 N C. 5.0 N B. 2.0 N D. 50.0 N 12. A 1,200-kilogram car traveling at 10 meters per second hits a tree that is brought to rest in 0.10 second. What is the magnitude of the net force acting on the car to bring it to rest? A. 120 N C. 12,000 N B. 1200 N D. 120,000 N 17. (HONORS ONLY) A force applied to a 100-kilogram rocket gives it an upward acceleration 15 meters per second squared. The magnitude of the applied force is equal to: A. 520 N C. 2480 N B. 1500 N D. 14700 N 13. A satellite is observed to move in a circle about the earth at a constant speed. This means that the force acting upon it is: A. zero B. opposite of the satellite’s velocity C. perpendicular to the satellite’s velocity D. parallel to the satellite’s velocity 18. (HONORS ONLY) If a net force is applied to a 1000kilogram car traveling at 38 meters per second, the car is brought to rest in 100 meters. If the braking force is doubled, what maximum speed can the car have and still come to rest in 100 meters? A. 53.7 m/s C. 26.9 m/s B. 76.0 m/s D. 19 m/s Problem solving. Show all your work including general equation, substitutions, calculations & units. 19. An advertisement claims that a certain 1060-kilogram car 21. The gravitational field strength on the surface of the can be accelerated from rest to 80 kilometers per hour in moon is 1.63 N/kg. An astronaut weighs 960 N on earth. 9.4 seconds. How large a net force must act on the car to (a) What is the astronaut’s mass? (b) What does the give it this acceleration? astronaut weigh on the moon? (c) What is the astronaut’s mass on the moon? 20. A rope pulls upward on a bucket weighing 54 newtons. The bucket is accelerating upward at 0.77 m/s2. What is the tension in the rope? 22. Two identical spherical balls are placed so their centers are 2.1 m apart. The force between them is 3.4 x 10-11 N. What is the mass of each ball? 60 23. A 6-kg block is pushed with an force F of 75 N, as shown in the drawings to the right. The coefficient of kinetic friction between the block and the surface is 0.22. What is the force of friction in each case, and what is the acceleration in each case? 53˚ F F F 27. Our solar system is in the Milky Way galaxy. The nearest galaxy is Andromeda, a distance of 2 x 1022 m away. The masses of the Milky Way and Andromeda galaxies are 7 x 1041 kg and 6 x 1041 kg, respectively. Treat the galaxies as particles and find the magnitude of the gravitational force exerted on the Milky Way by the Andromeda galaxy. 37˚ HONORS ONLY HONORS ONLY 28. Titan, a moon of Saturn, has on orbital period of 15.95 days and an orbital radius of 1.22 x 109 m. From this data, determine the mass of Saturn. 24. The tension in the rope used to pull the two blocks shown in the drawing to the right is 58 N. (a) Find the acceleration of the blocks if there is no surface friction. (b) Find the acceleration of the blocks if the coefficient of kinetic friction between the blocks and the surface is 0.33. (c) HONORS ONLY: Find the tension in the rope between the blocks, using the case of friction and no friction. 3 kg 8 kg 29. A device called Atwood’s Machine, shown in the drawing to the right, can be used to determine g, the gravitation field strength. Mass A = 5.1 kg, mass B = 2.7 kg, and the pulley C is massless. If a student measures the acceleration of the system as 3.0 m/s2, what is the student’s measured value for g? FT 25. A 500 kg racecar travels at a constant speed around a circular track whose radius is 1.6 km. (a) If the car travels once around the track in 2.0 minutes, what is the magnitude of the centripetal acceleration of the car? (b) If the force of static friction from the road on the tires is 1200 N, how fast can the car travel around the track without slipping? 26. A lunch tray is being held in one hand, as shown. The mass of the tray itself is 0.28 kg, and its center of gravity is located at its geometrical center. On the tray is a 1.0-kg plate of food and a 0.295-kg cup of coffee. Find the force T exerted by the thumb and the force F exerted by the four fingers. 30. HONORS ONLY: A 1220 N uniform beam is attached to a vertical wall at one end and is supported by a cable at the other end. A 1960 N crate hangs from the far end of the beam. Use the data shown in the drawing to the right. (a) Find the magnitude of the tension in the wire. (b) Find the magnitude of the horizontal and vertical components of the force that the wall exerts on the left end of the beam. Answers 1 E 2 F 3 H 4 D 5 B 6 G 7 C 8 A 9 D 10 B 11 D 12 D 13 C 14 D 15 C 16 B 17 C 18 A 19 2506 N 20 58.2 N 21 (a) 98 kg (b) 160 N (c) 98 kg 22 1.50 kg 23 (a) 12.9 N, 2 2 2 10.3 m/s right (b) 26.1 N, 3.17 m/s right (c) 10.3 N, 4.88 m/s up 24 (a) 5.27 m/s2 (b) 2.04 m/s2 (c) 15.8 N for both 25 (a) 4.39 m/s2 (b) 62.0 m/s 26 76.8 N, 61.4 N 27 7.0 x 1028 N 28 5.66 x 1026 kg 29 9.75 m/s2 30 2260, 1453, 1449 N 61 Notes 62 Notes 63 Notes 64 Notes 65 Notes 66 Notes 67 Energy, Momentum, and Relativity “For those who want some proof that physicists are human, the proof is in the idiocy of all the different units which they use for measuring energy” – RICHARD FEYNMAN “A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.” – MAX PLANCK 68 Energy, Momentum, Relativity Assignment Sheet Online: www.physicsclassroom.com/class/energy (Assignments 1-6) www.physicsclassroom.com/class/momentum (Assignments 7-11) No. Topics Reading Homework 1 work and energy, work done by a constant force 168-171 Ch 5: 1,7,9,10a 2 kinetic energy; work-energy theorem 172-175 Ch 5: 12,14,15,19,20 3 potential energy: gravitational and elastic 177-179 Ch 5: 13,23,24,25 4 conservation of mechanical energy, conservative forces 181-184 Ch 5: 28,33,37,39,43 5 conservation of total energy, non-conservative forces 185-186 Ch 5: 41,47,48,49 6 power versus energy 187-189 Ch 5: pg 189: SR 1-3 page 195, #35 7 linear momentum, impulse, and the impulse-momentum theorem 208-214 Ch 6: 5,7,13,15,16 8 more problems on impulse and momentum 9 conservation of momentum in one dimension 215-220 Ch 6: 17,23,24a,25, 10 collisions in one dimension: inelastic collisions 222-225 Ch 6: 31,35,48,49 11 collisions in one dimension: elastic collisions 226-230 Ch 6: 28,29,36,38,39 12 space-time, postulates of relativity, time dilation 282-289 Q 2,6 P 2,3b,d,f,11 13 twin paradox, space and time travel, length contraction, correspondence principle 290-295 Q 8,10 P 1,4,8 14 relativistic inertia & momentum, mass/energy equivalence, correspondence principle 302-308 Q 13,15 P 15,18,20,22 15 general relativity, gravity/space/time/geometry, test of general theory of relativity 308-316 Q 16,17,18,19,20 Ch 6: 6,10,12,14,56 69 Energy, Momentum, Relativity Assignment Sheet - Honors Online: www.physicsclassroom.com/class/energy (Assignments 1-6) www.physicsclassroom.com/class/momentum (Assignments 7-11) No. Topics Reading Homework 1 work and energy, work done by a constant force 6-1 Q2 P 2,3,4,9,10 2 kinetic energy; work-energy principle 6-3 Q 19 P 15,18,20,22,25 3 potential energy: gravitational and elastic 6-4 Q 14 P 26,28,30,31,32 4 conservation of mechanical energy, conservative forces 6-5 to 6-7 Q 10 P 36,37,39,40,43 5 conservation of total energy, nonconservative forces 6-8 6-9 Q 15 P 48,49,52,53,55 6 power vs. energy 6-10 P 58,59,66,67,69 7 linear momentum, impulse, and the impulse-momentum theorem 7-1 7-3 Q6 P 2,3,15,17,19 8 conservation of momentum in one dimension 7-2 7-4 Q8 P 4,6,7,8,12 9 collisions in one dimension: inelastic collisions 7-6 Q 11 P 31,32,34,35,36 10 collisions in one dimension: elastic collisions 7-5 P 22,24,27,28 11 conservation of momentum in two dimensions 7-7 P 40,41,42,43 12 space-time, postulates of relativity, time dilation 26-1 to 26-4 13 twin paradox, space and time travel, length contraction, correspondence principle 26-5 26-6 14 relativistic inertia & momentum, mass/energy equivalence, correspondence principle 26-7 to 26-9 Q 14,15 P 15,17,22,24 15 general relativity, gravity/space/time/geometry, test of general theory of relativity online: 308-316 Q 16,17,18,19,20 70 Q 2,6 P 3a,c,e,5,11,12 Q 8,9 P 1,4,7 Personal Power PURPOSE To measure your personal power while performing three exercises. EQUIPMENT - stopwatch - meter stick - various weights PROCEDURE 1. There are three exercises each person must do: A. Stair climb. Use the stairs by the Little Theater or towards the math classrooms. B. Sit ups. Use the carpet and pillow for comfort. Do 10 sit-ups. C. Arm curls. Use the 5, 7, or 10 pound weights at a lab table. Do 10 arm curls. d 2. Work in groups of two or three, but record data for each person so that everyone calculates their own personal power. 3. Use a meter stick to measure the displacement over which work is done. Measure only the displacement in the direction of the force so that work can be calculated as W = Fd. d d • Stair climb: measure the vertical displacement (the height of all stairs you climb). • Sit up: the vertical displacement of your armpit from start to finish • Arm curl: the vertical displacement of the mass (from elbow to fist) 4. Use a stopwatch to time the exercise. If you want, repeat trials for your personal best! DATA & ANALYSIS 1. Calculate the force used in the exercise. Use the conversion: 1 pound = 4.45 newtons • Stair climb: the force is your weight • Sit up: the force is half your weight • Arm curl: the force is the weight you lifted, plus 2% of your body weight to account for lifting your own forearm! 2. Calculate the TOTAL vertical displacement for each exercise. If the exercise requires and up and down motion (sit ups and arm curl), technically the net displacement is zero, so instead calculate the absolute value of all the displacements to get a non-zero answer. For example, since you do 10 arm curl repetitions, this total vertical displacement is 20 times the length of your forearm, from fist to elbow. 3. Calculate the TOTAL work done in each exercise. 4. Calculate your personal power for each exercise in watts. 5. Convert personal power for each exercise into horsepower using the conversion 1 horsepower = 746 watts 6. Show calculations (below the table) for the arm curl showing how you calculated the total displacement, force, work, power in watts, and power in horsepower. 71 type of work TOTAL force displacement (newtons) (meters) time (seconds) work (joules) power (watts) power (hp) stair climb xsit up arm curl QUESTIONS & CALCULATIONS 1. Show calculations below for the arm curl, showing how you calculated the total displacement, force, work, power in watts, and power in horsepower. 2. A rock climber wears a 7.5 kg backpack while scaling a cliff. After 30 minutes, the climber is 8.2 m above the starting point. a. How much work does the climber do on the backpack? b. If the climber weighs 645 newtons, how much work does she do lifting herself and the backpack? c. What is the average power developed by the climber? 3. You have an after school job carrying cartons of new copy paper up a flight of stairs, and then carrying recycled paper back down the stairs. The mass of the paper does not change. Your physics teacher says that you do not work all day, so you should not be paid. In what sense is the physics teacher correct? What arrangement of payments might you make to ensure that you are properly compensated? 72 Work and Energy PURPOSE The purpose of this activity is to compare the work done on a cart to the change in kinetic energy of the cart. Determine the relationship of work done to the change in energy. EQUIPMENT - PASPORT Xplorer GL - Dynamics Track - Mass Set - PASPORT Motion Sensor - Dynamics Car - String - PASPORT Force Sensor - Super Pulley with Clamp BACKGROUND For an object with mass m that experiences a net force Fnet over a distance d that W = Fnet d is parallel to the net force, the equation on the right shows the work done, W. If the work changes the object's vertical position, the object's gravitational potential energy changes. However, if the work changes only the object's speed, the object's kinetic energy, KE, changes as shown in the next W = ΔKE = KE − KE = 1 mv 2 − 1 mv 2 f i f i equation where W is the work, vf is the final speed of the 2 2 object and vi is the initial speed of the object. PREVIEW Use a Force Sensor to measure the force applied to a cart by a string attached to a descending mass. Use the Motion Sensor to measure the movement of the cart as it is pulled by the string. Use the Xplorer GLX to record and display the force and the motion. Determine the work done on the system and the final kinetic energy of the system. Compare the work done to the final kinetic energy. PREDICTION (record answers to questions in the Questions & Calculations section) 1. As work is done to accelerate a cart, what will happen to its kinetic energy? 2. How would the work done on the cart compare to its final kinetic energy? PROCEDURE GLX Setup 1. Turn on the GLX ( ) and open the GLX setup file labeled work energy. To open a specific GLX file, go to the Home Screen ( ). In the Home Screen, select Data Files and press to activate your choice. In the Data Files screen, use the arrow keys to navigate to the file you want. Press to open the file. Press the Home button to return to the Home Screen. Press to open the Graph. • The file is set up to measure force 50 times per second (50 Hz) and to measure motion 20 times per second (20 Hz). The Graph screen opens with a graph (Graph 1) of Position (m) and Time (s). The file also has a second graph (Graph 2) of Force (N) versus Position (m). (go to to switch between graphs). 2. Connect the Motion Sensor to sensor port 1 on the GLX and connect the Force Sensor to sensor port 2. Be sure the pins are aligned properly, and then press firmly. 3. Set the range selection switch on top of the Motion Sensor to the ‘near’ (cart) setting, as shown in Fig 1. 4. Check that the Dynamics Track is level and adjust the leveling screw on the left side as needed. 5. Be sure the Super Pulley with Clamp at the right end of the track is secure, and the Motion Sensor at the left end of the track is aimed directly at the pulley. 73 6. Add two 100 g of mass to the cart. The cart is 500 g. The Force Sensor is 100 g of mass. 7. Press the ZERO button on the Force Sensor to reset the sensor 8. Attach a string to the cart and put the string over the pulley. Adjust the length of the string so that when the cart is almost to the pulley, the end of the string almost reaches the floor. (Note: if you ZERO the sensor again later, be sure you remove the string first so there is no force!) 9. Put a 50 g mass on the end of the string. The mass hanger is another 5 g. Adjust the pulley up or down so the string is parallel to the track. 10. The kinetic energy of the system depends on all the mass that is in motion. Add up the total mass (cart & mass, sensor, hanger & mass) and record in the Lab Report. Fig. 2: Equipment setup • NOTE: The procedure is easier if one person handles the cart and a second person handles the Xplorer GLX. 11. Pull the cart away from the pulley until the cart is about 15 cm from the Motion Sensor. (The sensor doesn’t work well with a distance less than 15 cm). 12. Support the Force Sensor’s cable so the cart can move freely. 13. Press Start 14. Press • • to start recording data. Release the cart so it moves toward the pulley. to stop data recording just before the cart reaches the pulley. NOTE: Catch the cart before it hits the end stop on the track. NOTE: You may need to try several Runs before you get good data. Be sure to look at the correct Run (see Fig 3 and information below). DATA & ANALYSIS Use the Graph screen to examine the Position versus Time and the Velocity versus Time data. Use the second graph (Graph 2) to examine the Force versus Position. 1. To change the Graph screen to show a specific run of data, press to activate the vertical axis menu. Press the arrow keys ( ) to move to ‘Run #_’ in the upper left hand corner. Press to open the menu, select the data run in the menu, and press to activate your choice. 2. Change the Graph screen to show Velocity versus Time. Press to activate the vertical axis. Press to open the vertical axis menu. Use the arrow keys to select ‘Velocity’ and press again to activate your choice. 3. Move the cursor to the maximum value of velocity and record the value in the Data Table. 74 Fig. 3: Select data run Fig. 4: Select ‘Velocity’ 4. Switch to Graph 2. Press F4 ( activate your choice. ) to open the ‘Graphs’ menu. Select ‘Graph 2’ and press to 5. Find the area under the curve. Move the cursor to the beginning of the data. Press F3 ( ) to open the ‘Tools’ menu. Select ‘Area Tool’ and press to activate your choice. 6. The area under the curve is shown above the X-axis. Record the value as the work done. 7. Use the maximum velocity and the mass of the system (cart, sensor, string, hanging mass) to calculate the final kinetic energy of the system. 8. Please Delete All Runs when you are finished with the lab so the next class starts with No Data. QUESTIONS & CALCULATIONS Prediction 1. As work is done to accelerate a cart, what will happen to its kinetic energy? 2. How would the work done on the cart compare to its final kinetic energy? Data Sketch a graph of velocity versus time and a graph of force versus position for one run of data. Include units and labels for your axes. 75 Data Table Item Value Mass of system, total Velocity, maximum Work done Kinetic energy, final Percent difference Calculations Use the mass of the system and the final (maximum) velocity to calculate the final kinetic energy of 1 the system. Kinetic energy is KE = mv 2 where m is the mass and v is the velocity. 2 Calculate the percent difference between the work done (area under force-position curve) and the final kinetic energy. %diff = W − KE × 100% . W + KE 2 Questions 1. What happens to the kinetic energy as work is done on the system? 2. How does the final kinetic energy compare to the work done? Refer to the calculation for the percent difference between the work done and the final kinetic energy. 3. The kinetic energy is measured in joules and the work done is measured in newton•meters (N m). What is the relationship between a joule and a newton•meter? 4. Do your results support your predictions? What sources of error account for the difference between work and energy? 76 Conservation of Energy PURPOSE To use the principle of conservation of mechanical energy (elastic potential, gravitation potential, and kinetic energy), comparing the initial and final energy of a system. EQUIPMENT - dynamics carts - mass set and hanger - block mass - pulley & string - photogate timers - meter stick & ruler PROCEDURE PART A – ELASTIC (SPRING) POTENTIAL TO KINETIC ENERGY dynamics cart with block and flag m1 end stop photogate timer 1. Begin by using the dynamics cart with the built-in plunger to store elastic potential energy. The cart is 500 grams, the block is 500 grams and the flag is 17 grams. Record the mass m1 in kilograms below. 2. Measure the depth of the plunger; that is, measure how far the plunger protrudes from the surface of the dynamics cart, in meters. This is the amount of spring displacement, x. 3. Now measure the width of the black flag on the cart, in meters. This distance will be used to calculate the velocity of the cart. 4. Push the plunger in (and slightly up) until it is even with the end of the cart. Do not push it any farther in or it will jam. Set the cart on the dynamics track until the plunger lightly touches the rubber bumper. Move the photogate timer back so that the plastic flag on the cart will trigger the timer just after the released plunger loses contact with the bumper. 5. Turn the photogate timer to Gate Mode, which measures the time for the plastic flag to pass through the gate. Release the plunger by tapping it with another black block. Record the time in the table below. Repeat four more times. Ignore obvious outliers. DATA &ANALYSIS mass of cart w/ flag (m1): ________ plunger depth (x): ________ black flag width (d): _________ t1 (s) t2 (s) t3 (s) t4 (s) t5 (s) tave (s) 1. Calculate the cart’s velocity, in meters per second, by dividing the black flag width by the average time. 77 2. Set the elastic potential energy equal to the kinetic energy, and solve for the spring constant of the plunger. PROCEDURE PART B – GRAVITATIONAL POTENTIAL TO KINETIC ENERGY dynamics cart with block and flag m1 end stop pulley accessory photogate photogate timer mass hanger m2 1. Attach the string to the dynamics cart and the mass hanger as shown above. Use the method shown in class to make it easier to take the string off at the end of the lab. 2. Place 45 grams of mass on the 5-gram mass hanger. Recall that the dynamics cart is 500 grams, the block is 500 grams, and the photogate flag is 17 grams. Record the cart/block/flag mass m1 and the hanger/mass m2 in kilograms, in the space below. 3. Pull the cart back until it touches the rubber bumper on the dynamics track. This will be the release point for the cart for the entire lab. Move the photogate timer back so that the flag on the cart will immediately trigger the timer when the cart is released. The initial velocity of the cart will now be zero. Place the other photogate about 0.70 m away from the photogate timer. With a meter stick, carefully measure the distance between the photogates. Record the distance, in meters below. 4. Turn the photogate timer to Pulse Mode, which measures the time between gates. Also set it to the 1 ms setting. Release the cart with someone ready to catch it after it passes through the second photogate. Record the time in your lab book below. Repeat four more times. Ignore obvious outliers. DATA &ANALYSIS Distance between photogates (Δx): ____________ m1: ____________ t1 (s) t2 (s) t3 (s) t4 (s) t5 (s) m2: ____________ tave (s) 1. Use an appropriate kinematic equation to calculate the final speed of the system. 78 2. Calculate the final kinetic energy of the system. Remember to include both masses. 3. Calculate the initial potential energy of the system. Careful! Which mass has a change in gravitational potential energy? (Recognize that the change in vertical position of the hanging mass is equal to the photogate distance measured earlier.) 4. Calculate the percent difference (not error) between the initial and final energy of the system. difference % = QUESTIONS & CALCULATIONS (Show all work and evidence for answer) 1. There is some error in this lab that comes from not investigating all possible energies of the system. What energies did we ignore? 2. Why is it generally easier to investigate mechanics using energy conservation than to use Newton’s laws? 3. Honors only: If the dynamics track is set at an angle of 15˚ to the horizontal, how far up the track would the cart go if it is launched using the spring plunger from Part I? 79 Conservation of Momentum PURPOSE To investigate the principle of conservation of momentum in separations (explosions), elastic collisions, and perfectly inelastic collisions. EQUIPMENT - dynamics track - photogate timer with accessory photogate - dynamics carts and flags - meter stick d negative direction photogate flag positive direction cart 2 accessory photogate photogate timer PROCEDURE 1. You will be measuring photogate times which allows you to calculate cart velocity and momentum (remember, both are vectors so +/- direction is important.) Each cart has a mass of 500 g, and the photogate flags are 17 g. 2. Begin by using the dynamics cart with the built-in plunger to generate an explosion between carts. Push the plunger in and slightly up until it is even with the end of the cart. Do not push it any farther in or it will jam. Place the carts face-to-face on the track. 3. Place the carts together between the photogate timers, but off to the left side so that the left gate triggers before the right gate triggers. You will measure the velocity of each cart with the timer in GATE mode, by calculating v = d/t where d is the flag width (measured to the tenth of a millimeter!). flag width, d, (in meters): _____________ 4. You will use one photogate system to measure two times by setting the MEMORY on. Only the first time displays on the timer. To get the second time, toggle to READ, and then subtract the first time from the READ time. For example if the first time displays 0.0685 s and the READ time is 0.136 s, then the second time is 0.136 - 0.0685 = 0.0675 s. 6. Be sure that one photogate is triggered on and off before the other is triggered on. The proper placement of both photogates is very important for collecting good data with little error! 7. Use a metal block to tap and release the plunger. Don't let the carts rebound & retrigger the timers. Make three trials and record the times in the tables. Repeat each trial from the same position on the track and leave the photogates in the same position. 8. Place a 500 g block in cart 2, repeat three trials, and record all data in the table. DATA & ANALYSIS 1. Complete the data tables, which include calculating initial and final velocities, initial and final momentum, and initial and final kinetic energy of each cart in the system. 80 PART A - CARTS SEPARATING plunger catch cart here cart 1 cart 2 accessory photogate photogate timer mass (kg) time #1 (s) time #2 (s) time #3 time ave (s) (s) catch cart here vi (m/s) vf (m/s) pi (kgm/s) pf (kgm/s) KEi (J) cart 1 0 0 0 cart 2 0 0 0 mass (kg) time #1 (s) time #2 (s) time #3 time ave (s) (s) vi (m/s) vf (m/s) pi (kgm/s) pf (kgm/s) KEi (J) cart 1 0 0 0 cart 2 +500g 0 0 0 KEf (J) KEf (J) QUESTIONS & CALCULATIONS 1. For part A, calculate the percent difference between the magnitude (absolute value) of the final momentum of one cart with the other cart. Do this for the equal mass trials, and then again for the unequal mass trials. equal mass unequal mass 2. In each separation of the carts, where did the kinetic energy come from? 3. In the second set of trials, why does the smaller mass (cart 1) have more kinetic energy than the larger mass (cart 2 + 500 g)? 81 PART B - PERFECTLY INELASTIC COLLISION plunger velcro cart 1 end stop cart 2 photogate timer accessory photogate PROCEDURE 1. Arrange the carts with the Velcro tabs aligned so that the carts will stick together after the collision. The stationary cart must have the side with no magnets facing the plunger cart. 2. Set up the photogate timers so that the velocity of one cart before the collision and the velocity of both carts after the collision can be measured, as shown above. Triggering should occur as soon as possible for best results. (Note: it’s important that cart 2 is set up beyond both photogates initially, as shown in the diagram. 3. Use the plunger for launching, set at the second click in, not all the way. Send the plunger cart into the stationary cart, so that they stick together and pass through the second photogate. Record the times in the table below. Remember to subtract from read times! 4. Place a 500 g block on the stationary cart and repeat. Record all times in the table. mass (kg) time #1 (s) time #2 (s) time #3 time ave (s) (s) vi (m/s) vf (m/s) pi (kgm/s) pf (kgm/s) KEi (J) KEf (J) cart 1 cart 2 0 mass (kg) time #1 (s) time #2 (s) time #3 time ave (s) (s) vi (m/s) 0 vf (m/s) pi (kgm/s) 0 pf (kgm/s) KEi (J) KEf (J) cart 1 cart 2 +500g 0 0 0 QUESTIONS & CALCULATIONS 1. For part B, calculate the percent difference between the initial momentum of the system and the final momentum of the system. Do this for the equal mass trials, and then again for the unequal mass trials. equal mass unequal mass 2. For these perfectly inelastic collisions, why isn’t kinetic energy conserved? 82 PART C - ELASTIC COLLISION plunger magnets end stop cart 1 cart 2 photogate timer photogate timer PROCEDURE 1. Turn the stationary cart around so the repelling magnets in both carts face each other to create an elastic collision. Put the photogate flag back in the stationary cart. 2. Again use the plunger for launching, set at the second click in, not all the way. Send the plunger cart into the stationary cart. With equal masses the carts will switch velocities (why?) so only one final velocity is measured. Record all times in the table below. 3. Place a 500 g block in the stationary cart and repeat the elastic collision. This time the plunger cart will rebound, so a third time must be measured. This is not possible with one photogate and its accessory gate, so join another lab group and use two photogate timers without the accessory gates. Record all times in the table below. (Initial time for cart 1 goes above the dotted line, final time for cart 1 goes below the dotted line.) mass (kg) time #1 (s) time #2 (s) time #3 time ave (s) (s) vi (m/s) cart 1 vf (m/s) pi (kgm/s) 0 cart 2 0 mass (kg) time #1 (s) time #2 (s) time #3 time ave (s) (s) vi (m/s) pf (kgm/s) 0 0 vf (m/s) KEi (J) pi (kgm/s) KEf (J) 0 0 pf (kgm/s) KEi (J) KEf (J) cart 1 cart 2 +500g 0 0 0 QUESTIONS & CALCULATIONS 1. For part C, calculate the percent difference between the initial momentum of the system and the final momentum of the system. Do this for the equal mass trials, and then again for the unequal mass trials. equal mass unequal mass 2. Has kinetic energy been conserved for these elastic collision? Calculate the ratio of KEfinal to KEinitial to best answer this question. 83 Model Rockets PURPOSE To investigate the principles of rocket flight. To build and launch a toy rocket, and to apply laws of mechanics to analyze the vertical motion of the launched rocket. DATA Rocket model: ____________________ Rocket mass (in kg) with recovery wadding, but without the engine: ____________________ The table below includes data that is common to all groups, plus your group’s angles for launch: Air Resistance on rocket when using A8-3 engine Engine Masses Engine Stage Mass (kg) Initial 0.0163 End Thrust 0.0131 End Coast 0.0105 Rocket Type Awesome Alpha Wizard Viking Air Resist. (N) 0.44 0.47 0.54 0.56 Angle Measurements Angle a 1. 2. 3. Angle b 4. 5. 6. DATA & ANALYSIS MAXIMUM MEASURED ALTITUDE 1. Calculate the rocket’s measured altitude using the data collected during the launch and the equation shown below. Be sure to show all data and average angle calculations. h A a b h= B ABsin(a)sin(b) sin(a + b) ROCKET MASS 2. Calculate the rocket’s mass at each engine stage listed above: Average Thrust Stage m1, End Thrust Stage m2, and Average Coast Stage m3. 84 THRUST STAGE 3. Calculate the impulse from each of the forces (lift from engine, gravity, air resistance) acting on the rocket during the Thrust Stage. 4. Use the impulse-momentum theorem to calculate the rocket’s velocity at the end of the Thrust Stage. 5. Use an appropriate kinematic equation to calculate the upward vertical displacement of the rocket during the Thrust Stage. COAST STAGE 6. Calculate the rocket’s kinetic energy at the beginning of the Coast Stage (this is the same as the end of the Thrust Stage.) 7. Use the conservation of energy equation to calculate the rocket’s vertical displacement during the Coast Stage. Set the zero level for potential energy at the height where the Coast Stage begins. MAXIMUM PREDICTED ALTITUDE 8. Add the Thrust Stage and the Coast Stage displacements to determine the maximum predicted altitude of the rocket above the ground level where it was launched. 85 QUESTIONS & CALCULATIONS 1. Calculate the percent difference between the maximum measured altitude and the maximum predicted altitude. 2. Give a detailed account for any discrepancy between the actual and predicted altitudes. 3. Use a kinematic equation to calculate the rocket’s acceleration during the thrust stage. 4. Use a kinematic equation to calculate the rocket’s acceleration during the coast stage. 5. Calculate what the maximum altitude would be without air resistance. Use separate paper to show all your work for this problem. 6. Use your altitude from the previous question to calculate the speed of the rocket as it hit the ground if there were no air resistance. Use separate paper to show all your work for this problem. 7. Honors only: Calculate the actual drag force for the rocket. Show that this answer is accurate by recalculating the predicted altitude and comparing to the actual altitude. Use separate paper to show all your work for this problem. 86 Energy, Momentum, and Relativity Review Multiple choice. Circle the best answer. 1 A librarian picks a 2.0-kilogram book up 1.5 meters from the floor. She then carries it 15 meters to a bookshelf. Raising it an additional 0.5-meter, she places it on the bookshelf. How much work has she done on the book? (A) 3.92 J (C) 39.2 J (B) 30 J (D) 333 J 8 A 3-gram bullet traveling horizontally at 400 m/s hits a 3-kilogram wooden block, which is initially at rest on a frictionless horizontal table. The bullet buries itself in the block without passing through. The speed of the block after the collision is (A) 1.33 m/s (C) 12.0 m/s (B) 0.40 m/s (D) 40.0 m/s 2 A spring has a spring constant of 120 newtons per meter. How much potential energy is stored in the spring as it is stretched 0.20 meter? (A) 2.4 J (C) 12 J (B) 4.8 J (D) 24 J 9 The diagram below shows two carts on a horizontal, frictionless surface being pushed apart when a compressed spring attached to one of the carts is released Cart A has a mass of 3.0 kilograms and cart B has a mass of 5.0 kilograms. The speed of cart A is 0.33 meters per second after the spring is released. 3 A 10-newton force is required to move a 3.0-kilogram box at constant speed. How much power is required to move the box 8.0 meters in 2.0 seconds? (A) 40 W (C) 15 W (B) 20 W (D) 12 W 4 A bullet traveling at 5.0 × 102 meters per second is brought to rest by an impulse of 50 newton-seconds. What is the mass of the bullet? (A) 2.5 × 104 kg (C) 1.0 × 10-1 kg (B) 1.0 × 101 kg (D) 1.0 × 10-2 kg If the carts are initially at rest, what is the approximate speed of cart B after the spring is released? (A) 0.12 m/s (C) 0.33 m/s (B) 0.20 m/s (D) 0.55 m/s 10 A 2.5-kilogram stone is released from rest and falls toward the Earth. After 4.0 seconds, its momentum is (A) 98 kg·m/s (C) 39 kg·m/s (B) 78 kg·m/s (D) 24 kg·m/s 5 A box weighing 1.0 × 102 newtons is dragged to the top of an incline, as shown below. The gravitational potential energy of the box at the top of the incline is approximately (A) 1.0 × 102 J (C) 8.0 × 102 J (B) 6.0 × 102 J (D) 1.0 × 103 J 6 A cart of mass M on a frictionless track starts from rest at the top of a hill having height h1, as shown in the diagram below. Mg(h1 − h2 ) 12 A 2-kilogram mass moving with a speed of 5 meters per second to the right strikes a wall. It rebounds, moving with a speed of 3 meters per second to the left. The magnitude of the change in the momentum of the mass is: (A) 4 kg·m/s (C) 16 kg·m/s (B) 8 kg·m/s (D) 32 kg·m/s 13 A student drops two eggs of equal mass simultaneously from the same height. Egg A lands on the tile floor and breaks. Egg B lands intact, without bouncing, on a foam pad lying on the floor. Compared to the magnitude of the impulse on egg A as it lands, the magnitude of the impulse on egg B as it lands is (A) less (B) greater (C) the same What is the kinetic energy of the cart when it reaches the top of the next hill, having height h2? (A) Mgh1 (C) Mg(h2 − h3 ) (B) 11 A cart is moving at 4.0 meters per second at the top of a hill 6 meter high. It rolls down the hill and up an adjoining hill 5 meters high. What is the cart’s speed when it reaches the top of the second hill? (Assume there is no friction. (A) 2 m/s (C) 6 m/s (B) 4 m/s (D) 8 m/s 14 A horizontal force of 500 newtons is applied to a 200kilogram cart for a distance of 10 meters. The kinetic energy gained by the cart is (A) 25 J (C) 5000 J (B) 2000 J (D) 10000 J (D) 0 7 A cart of mass m traveling at speed v has kinetic energy KE. If the mass of the cart is doubled and its speed is halved, the kinetic energy of the cart will be (A) halved (C) quartered (B) doubled (D) quadrupled 87 15 If the speed of a moving object is doubled, which quantity must also double? (A) momentum (C) acceleration (B) kinetic energy (D) potential energy HONORS ONLY – QUESTIONS #27-#32 27 The unstretched spring in the diagram below has a length of 0.40 meter and spring constant k. A weight is hung from the spring, causing it to stretch to a length of 0.60 meter. 16 In an elastic collision between two particles (A) neither particle loses any of its kinetic energy (B) neither particle loses any of its momentum (C) the velocity gained by one particle is equal to that lost by the other (D) the total kinetic energy before and after the collision remains constant 17 A 5-newton force causes a spring to stretch 0.2 meter. What is the potential energy stored in the stretched spring? (A) 1 J (B) 0.5 J (C) 0.2 J (D) 0.1 J 18 An object of mass 1.0 kilogram is whirled in a horizontal circle of radius 0.5 meter at a constant speed of 2 meters per second. The work done on the object during one revolution is (A) 0 J (B) 2.0 J (C) 4.0 J (D) 8.0 J 19 A postulate of Einstein’s theory of relativity is: (A) moving clocks appear to run slower than when at rest (B) moving rods appear longer than when at rest (C) light has both wave and particle properties (D) the laws of physics must be the same for observers moving with uniform velocity relative to each other. 20 Repeat previous question, but replace the word “postulate” with the word “consequence”. How many joules of elastic potential energy are stored in this stretched spring? (A) 0.020 × k (C) 0.18 × k (B) 0.080 × k (D) 2.0 × k 28 Block A has mass 2 kg and moves to the right at 10 m/s, block B has mass 3 kg and moves to the left at 5 m/s. After they collide head-on elastically their velocities are, respectively: (use relative velocity!) (A) -10 m/s, +5 m/s (C) -8 m/s, +7 m/s (B) -9 m/s, +6 m/s (D) -5 m/s, +10 m/s 29 Camping equipment weighing 6000 newtons is pulled across a frozen lake by means of a horizontal rope. The coefficient of kinetic friction is 0.05. The work done by the campers in pulling the equipment 1000 meters at constant velocity is (A) 1.5 × 105 J (C) 6.0 × 105 J 5 (B) 3.0 × 10 J (D) 6.0 × 106 J 30 A man pushes an 80-newton crate a distance of 5.0 meters upward along a frictionless slope at an angle of 30˚ above horizontal. The force he exerts is parallel to the slope. If the speed of the crate is constant, then the work done by the man is (A) -200 J (C) 200 J (B) 140 J (D) 260 J 21 A millionaire was told in 1992 that he had exactly 15 years to live. However, if he travels away from the Earth at 0.8c and then returns at the same speed, he will live until the year (A) 2001 (B) 2007 (C) 2010 (D) 2017 22 An observer notices a moving clock runs slow by a factor of exactly 10. The speed of the clock is (A) 0.995c (B) 0.900c (C) 0.990c (D) 0.100c 31 A 0.3 kg puck, initially at rest, is struck by a 0.2 kg puck moving along the x-axis at 2 m/s. After the collision, the 0.2 kg puck has a speed of 1 m/s at an angle of 53˚ above the x-axis. After the collision, the velocity of the 0.3 kg puck is now: (A) 0.896 m/s, -26.6˚ (C) 0.269 m/s, 63.4˚ (B) 0.067 m/s, 3.81˚ (D) 1.07 m/s, -29.7˚ 23 A meter stick moves at 0.95c in the direction of its length through a laboratory. According to the measurements taken in the laboratory, its length is (A) 0.098 m (B) 0.31 m (C) 3.2 m (D) 1.0 m 32 According to relativity theory a particle of mass m with a momentum of 2mc has a speed of (A) 2c (B) 4c (C) c (D) 0.89c 24 A particle with rest mass m moves with speed 0.6c. Its kinetic energy is (A) mc2 (B) 0.18mc2 (C) 0.25mc2 (D) 1.25mc2 25 A 3-gram bullet is fired horizontally into a 2kilogram block of wood suspended by a rope from the ceiling. The block swings in an arc, rising 3 millimeters above its lowest position. The kinetic energy of the block at the bottom of its swing is (A) 0.0589 J (C) 589 J (B) 0.147 J (D) 147 J 26 Referring to the previous question, the velocity of the bullet before it struck the block was (A) 0.242 m/s (C) 242 m/s (B) 0.162 m/s (D) 162 m/s 88 ANSWERS 1 C 7 2 A 8 3 A 9 4 C 10 5 B 11 6 B 12 A B B A C C 13 14 15 16 17 18 C C A D B A 19 20 21 22 23 24 D A D A B C 25 26 27 28 29 30 A D A C B C 31 D 32 D Mousetrap Racecar Project OBJECTIVE Build a vehicle powered solely by the energy of one standard sized mousetrap that will travel the greatest linear distance. RULES 1. The car must be powered by a single mouse trap (size is about 1 3/4" × 3 7/8"). This is not a rat trap! 2. The mousetrap cannot be physically altered except for the following: a. holes can be drilled in the wood base to mount the mousetrap to the car frame. b. the spring can have a lever arm attached to it. Note: the spring from cannot be altered in any other way (like adding more coils). 3. The device cannot have any additional potential or kinetic energy at the start other than what can be stored in the mousetrap's spring itself. No extra springs, rubber bands, elastic, ramps, etc. are allowed. 4. The mousetrap must be contained in the car and must propel the car by means of a wheel or wheels in contact with the ground. A launcher or device that pushes the car from a stationary object is prohibited. 5. The spring cannot be wound more than its normal rotation angle of 180 degrees. 6. The car must be started from a standstill by releasing the mousetrap spring in a manner that imparts no additional energy to the vehicle (i.e., the vehicle may not be given a push start in the forward direction or side direction.). 7. The trigger on the mousetrap can be, but doesn’t have to be, used to release the mousetrap. 8. No purchased mousetrap car kits will be accepted. 9. No use of Legos or K-nex, or any part designed specifically for the purpose of making a car part. CONTEST 1. The racetrack will be on the smooth hallway floor (or in the gym if available). 2. Each group will have two attempts. The winner will be that car that has obtained the greatest distance on any one of the two attempts. Any ties will be decided by a single run off between the groups that tied. 3. Distance measurements will be determined by the forward displacement of the racecar. Distance will be measured from the front of the tape at the starting line to the front of the car when it comes to rest. 4. If the car curves to one side, only the forward distance at the point will be measured. GRADING See next page for grading rubric. 89 Mousetrap Racecar Grading Rubric Checkpoint Innovation Craftsmanship Assembly Competitio n Rank Disassembly Construction Materials Rules Design Energy Progress Each category will be graded on a 5-point rubric. Competition results count double. 5 4 3 2 • checkpoint includes all details: group names, racecar name, 1 photo and 1 movie, 3 or more results, and written summary. • race car uses spring energy very efficiently and completely. • checkpoint missing small details: group names, racecar name, 1 photo and 1 movie, 3 or more results, and written summary. • race car uses spring energy mostly efficiently and completely. • checkpoint missing some details: group names, racecar name, 1 photo and 1 movie, 3 or more results, and written summary. • race car uses spring energy somewhat efficiently and completely. • checkpoint missing most details: group names, racecar name, photo (or movie), results (3 or more) and written summary. • race car uses spring energy not efficiently and completely. • race car design uses • race car design uses • race car design uses • race car design does very creative and mostly creative and somewhat creative and not use creative and unique techniques. unique techniques. unique techniques. unique techniques. • race car design complies with all rules of competition. • race car design has minor issue that doesn’t comply with all rules. • race car design has major issue that doesn’t comply with all rules. (0 points) • race car made from very high quality materials with great • race car made from mostly high quality materials • race car made from mid quality materials • race car made from low quality materials • race car very well assembled and can perform many races. • race car fairly well assembled and can perform many races. • race car fairly well assembled and may perform some races. • race car poorly assembled and won’t perform many races. • race car is built with • race car is built with • race car is built with • race car is built with great skill and great good skill and good fair skill and fair poor skill and poor attention to detail. attention to detail. attention to detail. attention to detail. • groups disassembles the race car after the competition, recovering as much of the materials as possible. • groups disassembles the race car after the competition, recovering most of the materials as possible. • groups disassembles the race car after the competition, recovering some of the materials as possible. • groups does not disassembles the race car after the competition. • ranks in the 95th percentile compared with all groups in all classes. • ranks in the 75th percentile compared with all groups in all classes. • ranks in the 55th percentile compared with all groups in all classes. • ranks in the 35th percentile compared with all groups in all classes. 90 Notes 91 Notes 92 Notes 93 Notes 94 Notes 95 Notes 96 Appendix A: Unit Conversion, the Right Way A few years ago NASA launched a spacecraft called the Climate Orbiter designed to help relay information about the possibility of past life on Mars. Ten months after leaving Earth, the spacecraft began maneuvering into orbit around the Red Planet. Unfortunately the orbiter flew too close to the Martian atmosphere, and to the shock and great disappointment of many NASA engineers, it burned into pieces. The hopes and dreams of the mission were lost in an instant. A week after the disaster, NASA reported that the $125 million Climate Orbiter was lost because the engineering team that manufactured the spacecraft in Colorado used metric units while the mission navigation team in California used English units. They simply forgot to check for unit conversion. That’s a small mistake that wasted a large amount of taxpayers’ money. In physics, we often need to convert units either from one system to another, as in English to metric units, or even just from one power of ten to another, as in millimeters into meters. Surprisingly, many physics students do not know how to convert units, or have poor habits in attempting to convert units. Enough is enough - you should all get this straightened out immediately! A teacher might say “If I had a nickel for every time a student made a simple unit conversion mistake...” Well, this teacher wouldn’t retire early, but it’d be a lot of money. So let’s practice this technique. First, let’s look at a conversion from metric to English. For example, there are 1609 meters in a mile, so how many miles are there in a 10,000-meter race? (Common conversions and prefixes are listed in the back of the textbook and the lab manual.) 1 mile 10, 000 meters × = 6.215 miles 1609 meters Easy, right? Notice a few things. First, the fractions are written vertically. Never write fractions that slant (for example, 1 mile/1609 meters). Also note that the unit of meter cancels properly, like numbers in fractions do, because it appears in numerator of one fraction and the denominator of the other (10,000 meters doesn’t appear as a fraction, but division by 1 is implied). Now, let’s try a harder one. Let’s convert 65 miles per hour into meters per second. 65 miles 1609 meters 1 hour 1 minute m × × × = 29.05 s hour 1 mile 60 minutes 60 seconds Notice that the conversion between miles and meters is used again, but this time the fraction is reciprocated so that proper cancellation occurs. Also, if you remember that there are 3600 (60 squared) seconds in an hour you can simplify this. And, if you look up the direct conversion from miles per hour to meters per second, this would simplify the calculation even more. Next, what about conversions that involve powers of ten, where prefixes are used to denote specific powers of ten? For example, let’s convert 10,000 kilometers into millimeters. Recall that kilo- means 103 and milli- means 10-3. 10 3 meters 1 millimeter 10, 000 kilometers × × −3 = 1010 millimeters 1 kilometer 10 meters Be aware that a prefix describes a particular power of ten. A mistake is often made with conversions that are usually well understood, like 100 centimeters in a meter. Yes, that is a true statement, but you might mistakenly think that centi- means 102 - it does not! The power of ten may then be changed incorrectly when fractions are done mentally, without care. It’s better to be careful and actually write out the unit analysis, than to just try to do it in your head. For example the length wavelength of red light is 660 nanometers (nm), so when converting to centimeters, set up the unit analysis as follows: 660 nanometer × 10 −9 meter 1 centimeter × −2 = 6.60 × 10 −5 centimeters 1 nanometer 10 meter 97 Practice Problems Show all work by setting up the proper fraction for each conversion, show cancellations, and then determine the final result. Express answers in scientific notation with 3 significant figures. Use the prefix table conversions table in the back of the textbook or lab manual as a reference. 1. Convert 2600 feet into meters. 2. Convert 1300 kilometers into inches. 3. Convert 220 centigrams into kilograms. 4. Convert 55 miles per hour into feet per second. 5. Convert 42.3 centimeters into nanometers. 6. Convert 350 megabytes into terabytes. 7. Convert 5 x 107 gigajoules into picojoules. 8. Convert 8.30 kilopounds into newtons. 2 7 -3 1 Answers: 1. 7.92 × 10 meters 2. 5.12 × 10 inches 3. 2.20 × 10 kilograms 4. 8.07 × 10 ft/sec 5. 4.23 × 108 nanometers 6. 3.50 × 10-4 terabytes 7. 5.00 × 1028 picojoules 8. 3.69 × 104 newtons 98 Appendix B: Useful Information 1024 1021 1018 1015 1012 109 106 103 102 101 Prefixes yotta (Y) 10-1 zetta (Z) 10-2 exa (E) 10-3 peta (P) 10-6 tera (T) 10-9 giga (G) 10-12 mega (M) 10-15 kilo (k) 10-18 hecto (h) 10-21 deca (da) 10-24 English - Metric Conversion Factors 1 inch = 2.54 centimeters 1 foot = 30.48 centimeters 1 yard = 91.44 centimeters 1 mile = 5280 feet = 1609 meters 1 meter = 39.37 inches = 3.281 feet 1 pound = 4.448 newtons 1 horsepower = 745.7 watts 1 gallon = 3.785 liters 1 calorie = 4.187 joules 1 btu = 1055 joules deci (d) centi (c) milli (m) micro (µ) nano (n) pico (p) femto (f) atto (a) zepto (z) yocto (y) Trigonometry r Constants mass of earth Me 5.97 × 1024 kg mass of sun Ms 1.99 × 1030 kg mass of moon Mm 7.36 × 1022 kg radius of earth Re 6.37 × 106 m radius of moon Rm 1.74 × 106 m radius of sun Rs 6.96 × 108 m Earth-moon distance 3.84 × 108 m Earth-sun distance 1.496 × 1011 m speed of light c 3.0 × 108 m/s grav. constant G 6.67 × 10-11 Nm2/kg2 Coulomb constant k 9.0 × 109 Nm2/C2 – electron charge e –1.60 × 10-19 C + proton charge e +1.60 × 10-19 C electron rest mass me 9.11 × 10-31 kg proton rest mass mp 1.673 × 10-27 kg neutron rest mass mn 1.675 × 10-27 kg y θ x sin θ = y r cosθ = r = x 2 + y2 x r tan θ = y x ⎛ y⎞ θ = tan −1 ⎜ ⎟ ⎝ x⎠ International System of Units (Metric) Base Units Physical quantity Name of Unit Symbol length meter m mass kilogram kg time second s electric current ampere A temperature Kelvin K amount of substance mole mol luminous intensity candela cd International System of Units (Metric) Derived Units Physical quantity Name of Unit Symbol frequency hertz Hz energy joule J force newton N power watt W electric charge coulomb C electric potential volt V electric resistance ohm Ω Percent Error Percent Difference Percent error is calculated to compare an experimental value to a known value. Percent Error = Percent difference is calculated to compare two experimental values to each other. Known - Experimental × 100% Known 99 Percent Diff. = Exp1 - Exp 2 × 100% ( Exp1 + Exp 2 ) / 2 Appendix C: Second Semester Equations Kinematics savg = average speed vavg = average velocity aavg = ave. acceleration t = time d = distance Δx, Δy = displacement Dynamics ΣF = resultant (net) force m, M = mass Fg = force of gravity (weight) Energy and Momentum W = work d = displacement t = time g = 9.8 P = power N kg = grav. field strength ⎫ ⎪ v f 2 = vi 2 + 2aΔx ⎬ OR Δy ⎪ Δx = 12 (vi + v f )t ⎭ Projectile motion k = spring constant Fn = normal force Fr = air resistance (drag) Fs = static friction force Fk = kinetic friction force µ s = coefficient of static friction µ k = coefficient of kinetic friction FT = tension Fsp = spring force ΣF = 0 (1st Law) ΣF = ma (2nd Law) F1,2 = − F2,1 (3rd Law) t = time Δx = horiz. displacement Δy = vert. displacement g = acceleration of gravity Fs = µ s Fn Fk = µ k Fn Circular Motion & Gravitation ac = centripetal acceleration a = inst. acceleration vi = initial velocity v f = final velocity d Δx Δv vavg = aavg = t t t v f = vi + at savg = Δx = vi t + 12 at 2 v = velocity vx = horizontal velocity vyi = initial vertical velocity vyf = final vertical velocity Components/Resultant: vx = v cosθ v = vx 2 + vy 2 vy = vsin θ θ = tan −1 (vy / vx ) Horizontal motion (ax = 0) Δx = vx t Vertical motion (g = −9.8 vyf = vyi + gt Δy = vyi t + 12 gt 2 vyf 2 = vyi 2 + 2gΔy Δy = 12 (vyi + vyf )t m s2 ) Fg = mg Fsp = kΔx Fc = centripetal force Fg = gravitational force vt = tangential speed T = period r, R = distance, orbital radius vt2 4π 2 r ac = = 2 r T 2π r vt = T mv 2 4π 2 mr Fc = t = r T2 Gm1m2 4π 2 R 3 2 Fg = T = r2 GM Torque τ = Fd sin θ = F⊥ d Στ = 0 (equilibrium) 100 KE = kinetic energy GPE = grav. potential energy EPE = elastic potential energy TE = thermal energy (heat) Fk = kinetic friction force I = impulse p = momentum W = Fd cosθ W P= = Fv t KE = 12 mv 2 GPE = mgh EPE = 12 kx 2 TE = Fk d W + KEi + PEi = KE f + PE f + TE I = Ft Ft = mΔv p = mv m1v1i + m2 v2i = m1v1 f + m2 v2 f Relativity γ = Lorentz factor v = relative velocity c = speed of light t 0 = time in observer's frame t = time in another frame L0 = length in observer's frame L = length in another frame v2 γ = 1/ 1− 2 c t = γ t0 E = γ mc 2 L= L0 γ p = γ mv KE = (γ − 1)mc 2