Unformatted text preview: Physics 121, Spring 2008 Mechanics
Frank L. H. Wolfs Department of Physics and Astronomy University of Rochester Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester Physics 121, Spring 2008. What are we going to talk about today?
• Goals of the course • Who am I? • Who are you? • Course information:
• • • • • • Text books Lectures Workshops Homework Exams Quizzes • Units and Measurements • Error Analysis (replaces the Physics 121 lab lecture). Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester Physics 121, Spring 2008. Goal of the course.
• Physics 121 is a survey course for physics and engineering majors. • Course topics include motion (linear, rotational, and harmonic), forces, work, energy, conservation laws, and thermodynamics. • I assume that you have some knowledge of calculus, but techniques will be reviewed when needed. • I do not assume you have any prior knowledge of physics.
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester 1 Physics 121, Spring 2008. Who am I?
• I am Frank Wolfs! • I am a professor in Physics in the Department of Physics and Astronomy. • I am an experimental nuclear physicist who is looking for dark matter in a deep mine in South Dakota. Did you know that the most dominant form of matter in our Universe is dark matter? We have never directly detected dark matter! • I consider teaching a very component of my job, and will do whatever I can to ensure you succeed in this course.
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester Physics 121, Spring 2008. Who are you? ece makes up 7% of the class woot Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester Physics 121, Spring 2008. Who are you? Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester 2 Physics 121, Spring 2008. Why are you here?
• Most of you will say:
• It is a requirement of my major! • I have no clue! I want to be an engineer, and computers do all the engineering calculations. • Some you may say:
• I was excited about Physics in high school and I like to learn more about the subject. • I like to Prof. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester Physics 121, Spring 2008. Why should you be here?
• All engineering calculations and models are based on physics. • A basic understanding of the principles of mechanics and the capability to determine whether solutions to problems make sense is a skill that any engineer needs to have. • Remember ….. A computer is only as smart as the person who programmed it (although some computers are smarter than others).
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester Physics 121, Spring 2008. Course Information.
• Text Book:
• Giancoli, Physics for Scientists and Engineers. The material covered in this course is covered in Volume 1 (Physics 122 will cover the material covered in Volume 2). • PRS:
• We will be using a Personal Response System in this course for inclass quizzes and concept tests. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester 3 Physics 121, Spring 2008. Course Components.
• Lecture:
• Focus on the concepts of the material, and its connections to areas outside physics. • Not a recital of the text book! • The lecture presentation is interspersed with conceptual questions and quizzes, solved with and without help from your neighbors. be sure to read the textbook, because not all information in the class is covered in the text
or there is additional information that is needed from the text that isn't covered in the class • Workshops:
• Small group meetings with a trained workshop leader. • Institutionalize the “study group”. • You discover how much you can learn from you fellow students. • Consistent attendance of workshops correlates with better grades.
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester Physics 121, Spring 2008. Course Components.
• Homework assignments:
• Homework is assigned to practice the material covered in this course and to enhance your analytical problem solving skills. • You will need to struggle with the assignments to do well in this course. • You will need to make sure you fully understand the solution to these problems! • Labs:
• Give you handson experience with making measurements and interpreting data. • Labs are pretty much separated from the course (not controlled by me), but are a required component.
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester starting the last week of January? Physics 121, Spring 2008. Course Components.
• Exams:
• Test you on your basic understanding of the material and your quantitative problem solving skills. • There will be 3 midterm exams and 1 ﬁnal exam. • There is no need to memorize formulas; you will be given an equation sheet with all important equations for the material covered on the exam. three midterms, one final cannot be moved • Final grades:
• Calculated in 4 different ways: the highest grade counts. • No grading on a curve: grade scale is ﬁxed and known to you! calculates grade in four ways, gives you the highest never grades on a curve
pretest and post test, used to judge range of students testes do not impact grade Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester 4 Physics 121, Spring 2008. Course Components.
• I am here to help you learn this material, but it is up to you to actually master it:
• If there is something you do not understand you need to ask for help …….. (come and talk, email, after class, etc.) • It is our job to teach you …… you are paying my salary ……… • In large lecture courses it is difﬁcult to see who needs help. You need to ask for the help you need before you fall behind.
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester Chapter 1. Making measurements. Using units.
• Theories in physics are developed on the basis of experimental observations, or are tested by comparing predictions with the results of experiments. • Being able to carry out experiments and understand their limitations is a critical part of physics or any experimental science. • In every experiment you make errors; understanding what to do with these errors is required if you want to compare experiments and theories. physics is an experimental science based entirely on
measurement. error is important, no measurement is perfect
need to know about the units, they are important for how that number is interpreted units need to be agreed on, Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester Making measurements. Using units.
• In order to report the results of experiments, we need to agree on a system of units to be used. • Only if all equipment is calibrated with respect to the same standard can we compare the results of different experiments. • Although different units can be used to report different measurements, we need to know what units are used and how to do unit conversions. • Using the wrong units can lead to http://science.ksc.nasa.gov/mars/msp98/images.html expensive mistakes.
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester 5 Making measurements. Using units.
• In this course we will use the SI System of units:
• Length: meter • Time: second • Mass: kg • The SI units are related to the units you use in your daily life:
• Length: 1” = 2.54 cm = 0.0254 m • Conversion factors can be found in the front cover of the book. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester The base units. The unit of length: changes over time!
• One tenmillionth of the meridian line from the north pole to the equator that passes though Paris. • Distance between 2 ﬁne lines engraved near the ends of a PlatinumIridium bar kept at the International Bureau of Weights and Measures in Paris. • 1,650,763.73 Wavelengths of a particular orangered light emitted by Krypton86 in a gas discharge tube. • Path length traveled by light in vacuum during a time interval of 1/299,792,458 of a second.
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester The base units. Their current deﬁnitions.
• TIME  UNIT: SECOND (s)
• One second is the time occupied by 919,263,170 vibrations of the light (of a speciﬁed wavelength) emitted by a Cesium133 atom. • LENGTH  UNIT: METER (m)
• Path length traveled by light in vacuum during a time interval of 1/299,792,458 of a second. • MASS  UNIT: KILOGRAM (kg)
• One kilogram is the mass of a PlatinumIridium cylinder kept at the International Bureau of Weights and Measures in Paris. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester 6 still the measurement of a kilogram, even though this The base SI units. system is not particularly good The current standard of the kg and the old standard of the m.
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester Error Analysis. Some (but certainly not all) important facts.
• Why should we care? • Types of errors. • The Gaussian distribution  not all results can be described in terms of such distribution, but most of them can. • Estimate the parameters of the Gaussian distribution (the mean and the width). • Error propagation. • The weighted mean. • Note: Some of the following slides are based on the slides for a lab lecture, prepared by Prof. Manly of the Department of Physics and Astronomy.
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester Error Analysis. Is statistics relevant to you personally?
Month 1 Bush Dukakis Undecided 42% 40% 18% Month 2 41% 43% 16%
±4% Headline (1988): Dukakis surges past Bush in polls! Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester 7 Error Analysis. Is statistics relevant to you personally? Global Warming Analytical medical diagnostics Effect of EM radiation Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester Error Analysis. Type of Errors.
• Statistical errors:
• Results from a random ﬂuctuation in the process of measurement. Often quantiﬁable in terms of “number of measurements or trials”. Tends to make measurements less precise. through repetition statistical errors will disappear errors based on stupidity are systematic errors more difficult to treat • Systematic errors:
• Results from a bias in the observation due to observing conditions or apparatus or technique or analysis. Tend to make measurements less accurate. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester The Gaussian distribution: the most common error distribution. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester 8 The Gaussian Distribution: its mean and its standard deviation.
1σ is roughly the halfwidth at halfmaximum of the distribution. 2σ gx=
Frank L. H. Wolfs () 1 2! " # x# µ ( )2
Department of Physics and Astronomy, University of Rochester e 2" 2 Making measurements: increasing the number of measurements increases the accuracy. Length = 10 m, σ = 1 m. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester Probability of a single measurement falling within ±1σ of the mean is 0.683. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester 9 Probability of a single measurement falling within ±2σ of the mean is 0.954. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester Probability of a single measurement falling within ±3σ of the mean is 0.997. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester Do you agree?
Month 1 Bush Dukakis Undecided 42% 40% 18% Month 2 41% 43% 16%
±4% Headline: Dukakis surges past Bush in polls! Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester 10 How to determine the mean µ and width σ of a distribution based on N measurements? Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester How to determine the mean µ and width σ of a distribution based on N measurements?
x= x1 + x2 + ! + x N !1 + x N 1 = N N " xi = µ
i =1 N ? Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester How to determine the mean µ and width σ of a distribution based on N measurements?
The “standard deviation” is a measure of the error in each of the N measurements: #= ! (x " µ)
i i =1 N 2 N Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester 11 How to determine the mean µ and width σ of a distribution based on N measurements? • The standard deviation is equal to • But ….. µ is unknown. So we will use the mean (which is your best estimate of µ). We also change the denominator to increase the error slightly due to using the mean. • This is the form of the standard deviation you use in practice: • Note: This quantity cannot be determined from a single measurement.
Frank L. H. Wolfs #= ! (x " µ)
i i =1 N 2 N != # ( xi " x ) 2
i =1 N N "1 Department of Physics and Astronomy, University of Rochester What matters? The standard deviation or the error in the mean?
• The standard deviation is a measure of the error made in each individual measurement. • Often you want to measure the mean and the error in the mean. • Which should have a smaller error, an individual measurement or the mean? • The answer ….. the mean, if you do more than one measurement: !m = !
N
Department of Physics and Astronomy, University of Rochester Frank L. H. Wolfs Applying this in the laboratory. Measuring g.
Student 1: 9.0 m/s2 Student 2: 8.8 m/s2 Student 3: 9.1 m/s2 Student 4: 8.9 m/s2 Student 5: 9.1 m/s2 What is the best estimate of the gravitational acceleration measured by these students? a=
Frank L. H. Wolfs 9.0 + 8.8 + 9.1 + 8.9 + 9.1 m = 9.0 2 5 s
Department of Physics and Astronomy, University of Rochester 12 Applying this in the laboratory. Measuring g.
Student 1: 9.0 m/s2 Student 2: 8.8 m/s2 Student 3: 9.1 m/s2 Student 4: 8.9 m/s2 Student 5: 9.1 m/s2 What is the best estimate of the standard deviation of the gravitational acceleration measured by these students? "= (9.0 ! 9.0) 2 + (8.8 ! 9.0) 2 + (9.1 ! 9.0) 2 + (8.9 ! 9.0) 2 + (9.1 ! 9.0) 2 5 !1 m = 0.12 2 s
Department of Physics and Astronomy, University of Rochester Frank L. H. Wolfs Applying this in the laboratory. Measuring g.
Student 1: 9.0 m/s2 Student 2: 8.8 m/s2 Student 3: 9.1 m/s2 Student 4: 8.9 m/s2 Student 5: 9.1 m/s2 error analysis is critical !m = 0.12 5 = 0.054 m s2 Note: this procedure is valid if you can assume that all your measurements have the same measurement error. Final result: g = 9.0 ± 0.05 m/s2. Does this agree with the accepted value of 9.8 m/s2?
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester How does an error in one measurable affect the error in another measurable?
y y1 + δ y y1 y1  δ y x1  δ x x1 x1 + δ x y = F(x) x Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester 13 How does an error in one measurable affect the error in another measurable?
The degree to which an error in one measurable affects the error in another is driven by the functional dependence of the variables (or the slope: dy/dx) y y1 + δ y y1 y1  δ y x1  δ x x1
Frank L. H. Wolfs y = F(x) x1 + δ x x
Department of Physics and Astronomy, University of Rochester How does an error in one measurable affect the error in another measurable?
• But …… Most physical relationships involve multiple measurables! x = x o + vo t + F = Ma 12 at 2 • We must take into account the dependence of the parameter of interest, f, on each of the contributing quantities, x, y, z, ….: f = F(x, y, z,…) Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester Error Propagation. Partial Derivatives.
• The partial derivative with respect to a certain variable is the ordinary derivative of the function with respect to that variable where all the other variables are treated as constants. $F ( x, y, z ,...) dF ( x, y, z...) # = ! $x dx " y , z...const Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester 14 Error Propagation. Partial Derivatives: an example. F ( x, y, z ) = x 2 yz 3
!F = 2 xyz 3 !x !F = x2 z3 !y !F = x 2 y3z 2 !z
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester Error Propagation. The formula!
• Consider that a parameter of interest f = F(x, y, z, …) depends on the measured parameters x, y, z, …. • The error in f, σf, depends on the function F, measured parameters x, y, z, …, and their errors, σx, σy, σz, …, and can be calculated using the following formula: & 'F # 2 & 'F # 2 & 'F # 2 (f = $ ! (x +$ ! $ 'y ! ( y + $ 'z ! ( z + ... " % % 'x " " %
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester 2 2 2 The formula for error propagation. An Example.
A pitcher throws a baseball a distance of 30 ± 0.5 m at 40 ± 3 m/s (~ 90 mph). From this data, calculate the time of ﬂight of the baseball. t= d v !t " 1% 2 " d % 2 = $ ' !d + $ ( 2 ' !v = # v& # v& " 0.5 % " 30 % 2 + 3 = 0.058 ) =$ &# & # 40 ' $ 402 '
2 2 2 2 !F 1 = !d v d !F =" 2 !v v
Frank L. H. Wolfs t = 0.75 ± 0.058s
Department of Physics and Astronomy, University of Rochester 15 Another example of error propagation.
v = at: determine a and its error. Suppose we just had 1 data point, which data point would provide the best estimate of a? Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester Another example of error propagation.
• For each data point we can determine a (= v/t) and its error:
"1 % " v % = $ !v ' + $ 2 !t ' = & #t & #t = v " !v % " !t % + t $v' $t' #&#&
2 2 2 2 !a • We see that the error in a is different for different points. Simple averaging will not be the proper way to determine a and its error.
Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester The weighted mean.
• When the data have different errors, we need to use the weighted mean to estimate the mean value. • This procedure requires you to assign a weight to each data point:
wi = 1 ! i2 • Note: when the error decreases the weight increases. • The weighted mean and its error are deﬁned as:
y= !w y
i =1 N i N i !w
i =1 "y = 1 i !w
i =1 N i Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester 16 The end of my error analysis. The start of your learning curve.
• Certainly there is a lot more about statistical treatment of data than we can cover in part of one lecture. • A true understanding comes with practice, and this is what you will do in the laboratory. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester We are done (for today)!
• We are done for today. • Next week we will start discussing the material in Chapter 2 and start using the PRS. • If you have not received any email from me, you are not on my class list. Send me an email with your name and student id so that I can add you to our list server and to our homework server. • See you next week on Tuesday! Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester 17 ...
View
Full
Document
This note was uploaded on 10/20/2010 for the course PHY PHY 121 taught by Professor Wolfs during the Spring '08 term at Rochester.
 Spring '08
 Wolfs
 mechanics

Click to edit the document details