Class 7 Preclass Quiz on MasteringPhysics

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1 PHY131H1F Class 7 Today: Uncertainty Analysis Normal Distribution Standard Deviation Reading uncertainty Propagation of uncertainties Uncertainty in the Mean From The Lahman Baseball Database V Test tomorrow at 6:00pm. Material: Chapters 1-4, up to last week. This week s material will not be on the test tomorrow. (It will be on test Nov.17) Class 7 Preclass Quiz on MasteringPhysics This was due this morning at 8:00am 71% correct: In an equation for the normal distribution (Gaussian) sigma is related to the width of the distribution. 97% correct: Mean is the same as average. 56% correct: If a digital thermometer reads 1.8 C and the uncertainty specified in the manual is half the last digit, in this case it would be ±0.05 C. 64% correct: Every time you measure a certain quantity, x, in your laboratory, the uncertainty in this measurement is u x. If you make N measurements, and then take the mean of all N measurements, the uncertainty in the mean is u x / N. 1

2 Class 7 Preclass Quiz Student Comments Can you explain more on the standard deviation, it is a bit confusing on how to do the calculation. Harlow comment: I got lots of questions like these! I will try! Propagation of error was a little confusing because it's a new concept. Harlow comment: I got lots of questions like these! I will try! Are standard deviation and uncertainty on the first midterm? Harlow comment: I got lots of questions like these! The answer is NO. I guarantee it! I can't find the context in the book? it is just me or do I need sleep? stressing about this upcoming midterm and life... Harlow comment: You re right this is not in Wolfson, unfortunately. I have provided supplemental reading on portal. Last Wednesday I asked at the end of class: If your height is 150 cm, is there an uncertainty in that number? ANSWER: YES! Almost every measured number has an uncertainty, even if it is not stated. If you told me your height was 150 cm, I would guess the uncertainty is probably between 1 and 5 cm. [But there is no way to know this, unless you investigate how the 150 was measured.]

3 Clicker Question Demo and Example: What is the Period of a Swinging Ball? (Pendulum) Procedure: Measure the time for 5 oscillations, t 5. The period is calculated as T = t 5 / 5. t 5 data: Here were Harlow s measurements of t 5 : 7.53 s 7.38 s 7.47 s 7.43 s Which of the following might be a good estimate for the uncertainty in Harlow s first measurement of 7.53 seconds? A s B s C. 0.5 s D. 5 s E. Impossible to determine 3

4 Histogram: 4 Measurements Number of Measurements s 7.38 s 7.47 s 7.43 s Measured Time (half second bins) Number of Measurements Histogram: 8 Measurements s s s s s 7.56 s 7.48 s s Measured Time (half second bins) 4

5 Histogram: 1 Measurements Number of Measurements Measured Time (hafl second bins) Number of Measurements Histogram: 16 Measurements Measured Time (half second bins) 5

6 Number of Measurements Histogram: 16 Measurents Gaussian Curve (best fit) Stopwatch Measurements Measured Time (half second bins) The Gaussian: (xx ) N(x) Ae When you integrate this curve, you find that 68% of area lies between the dotted lines on the graph. 6

7 Gaussian Distributions turn up everywhere! Heights of some People (London, 1886) inches Random Walk Where does an object end up, if it takes N steps randomly left or right? The final distribution is described by a Gaussian function! 7

8 7.53 s 7.38 s 7.47 s 7.43 s x est The t 5 data: What are the individual uncertainties? 1 N N i1 1 x i N est ( x i xest ) N 1 i s 7.38 s 7.47 s 7.43 s The t 5 data: What are the individual uncertainties? N 1 est ( x i xest ) 1 N i1 8

9 Reading Uncertainty (Analog) Look at the speedometer below. How fast are you going? I think it s about 6 MPH, but what is the reading uncertainty? There is no fixed rule that will allow us to answer this question We must use our intuition and common sense! Reading Uncertainty (Analog) Could you be going 64 MPH? no, I don t think so Could it be 63 MPH? maybe And it could be as low as 61 MPH, but, in my opinion, no slower So the range is about 61 to 63 MPH So let s say we re going 6 ± 1 MPH. Meaning: I am 68% certain that my true speed is between 61 and 63 MPH. 9

10 Clicker Question Reading Uncertainty (Digital) For a measurement with an instrument with a digital readout, the reading uncertainty is usually ± one-half of the last digit. This means one-half of the power of ten represented in the last digit. With the digital thermometer shown, the last digit represents values of a tenth of a degree, so the reading uncertainty is ½ 0.1 = 0.05 C You should write the temperature as 1.80 ± 0.05 C. The digital speedometer on your Toyota reads 63 MPH. What is the uncertainty in this measurement? A MPH B MPH C. 0.5 MPH D. 5 MPH E. Impossible to determine 10

11 Choosing between Statistical and Reading Uncertainty In most cases, when you have both a standard deviation and a reading uncertainty, one is much larger than the other, and then you should choose the larger to be the uncertainty For example, if every time you measure something you always get the same numerical answer, this indicates that the reading uncertainty is dominant However, if every time you measure something you get different answers which differ more than the reading uncertainty you might estimate, then the standard deviation is dominant Propagation of uncertainties Rule #1 (sum or difference rule): If z = x + y or z = x y then u Rule # (product or division rule): If z = xy or z = x/y x then z u u y u z z ux x uy y 11

12 Propagation of uncertainties Rule #.1 (multiply by exact constant rule): If z = xy or z = x/y and x is an exact number, so that u x =0 then u x Rule #3 (exponent rule): z uy If z = x n then uz z ux n x What is the uncertainty in the mean of N measurements with the same uncertainty u x? 1

13 Clicker Question The Uncertainty in the Mean Many individual, independent measurements are repeated N times Each individual measurement has the same uncertainty u x Using uncertainty propagation you can show that the uncertainty in the estimated mean is: ux umean N You wish to know the time it takes to travel from Finch station to Yonge/Bloor by subway. You ask 10 people to take a stopwatch and time the trip. After analyzing all the data you find that it takes an average of 6 minutes and 40 seconds, with an uncertainty in this average of ± 100 seconds. If you expand your survey and ask 1000 people to time the trip, when you analyze the data, what would you expect to be the uncertainty in the average time? A.100 seconds B.50 seconds ux C.10 seconds umean N D.1 second E.0.05 seconds 13

14 Significant Figures The rules for significant figures when uncertainties are involved are: 1. Uncertainties should be specified to one or two significant figures..the most precise column in the number for the uncertainty should also be the most precise column in the number for the value. These are the actual sig.fig. rules used by scientists. They replace the sig.fig. rules from Ch.1 of your textbook, which no one really uses. Clicker Question Example: If a calculated result is d = / m, how should you report this? A. 7.1 ± 0.7 m B ± 0.71 m C ± m D. Any of the above E. Either A or B, but not C 14

15 Term Test 1 Info The first term test in PHY131H1F will be written tomorrow, Tuesday, Oct 6, from 6:10 to 7:30 pm. It will be based on Wolfson Chapters 1-4, plus the related material in classes 1-6, practicals 1 and, and MasteringPhysics problem sets 1 and. The test will have: 10 multiple-choice questions worth points each (total = 0) Two long-answer problems counting for a total of 1 points, which will be graded in detail; part marks may be awarded, but only if you show your work. Term Test 1 Info The room you will write in is based on the first few letters of your last name. You must attend the correct room, or you will not be allowed to write the test: A HN: EX100 HO N: EX00 O SA: EX300 SE TO: EX310 TR W: EX30 X YO: GB404 YU - Z: GB405 EX is the Exam Centre at 55 McCaul Street GB is the Galbraith Building at 35 St. George Street. Alternate sitting students please do NOT go to this room go to the room that was indicated in the sent to you last week. 15

16 Term Test 1 Info Please be sure to bring your T-Card, as invigilators will be collecting signatures and checking your photo-id. Allowed aids include a pocket calculator with no communication ability, a single hand-written aid-sheet prepared by you, no larger than 8.5 x11, written on both sides, a hard-copy English translation dictionary, and a ruler. If you wish to see the first page of the test and read the instructions, I have posted it at 15/test1FirstPages.pdf. (See the announcement today on portal.) Term Test 1 Info During the test, you must leave your other belongings at the designated location (front or back) of the room. You may bring a small sized paper or plastic bag to contain your valuables (wallets, powered-off electronic devices such as cell-phone and tablet, etc.) under your seat. At the end of the test at 7:30 pm, you must stop writing and put your pen/pencil down the very moment the invigilators declare the test over. Immediately put your answer sheet inside the question paper and have it ready to hand to the invigilator who will be picking it up. Please remain seated until you are told you can leave. 16

17 Midterm Test 1 - hints Don t be late. If you re very early, just wait outside the room. Spend the first or 3 minutes skimming over the entire test from front to back before you begin. Look for the easy problems that you have confidence to solve first. Before you answer anything, read the question very carefully. The most common mistake is misreading the question! Manage your time; if you own a watch, bring it. 1 problems over 80 minutes means an average of about 7 minutes per problem. You CANNOT HAVE YOUR PHONE with you or in your pocket at a test or exam at U of T you must store it in your backpack at the edge of the room, or in a special bag underneath your desk Midterm Test 1 more hints! Some of the multiple choice are conceptual and can be answered in less than minutes.. Maybe do these ones first? If you start a longer problem but can't finish it within about 10 minutes, leave it, make a mark on the edge of the paper beside it, and come back to it after you have solved all the easier problems. When you are in a hurry and your hand is not steady, you can make little mistakes; if there is time, do the calculation twice and obtain agreement. Bring a snack or drink. Don't leave a test early! You might spend the first half getting 95% of the marks you're going to get, and the second half getting the other 5%, but it's still worth it. 17

18 Little survey: How are you feeling about the test tomorrow? A. I feel confident about the test tomorrow; I believe I will get an A B. I m not too sure what to expect, but I m hopeful I ll do well C. I have no particularly positive or negative feelings about the test tomorrow.. D. I m not too sure what to expect, but I m worried it will be awful E. I am very worried about the test tomorrow; I m afraid I m going to fail! Before Class 8 on Wednesday Please read Wolfson Chapter 5, sections 5.1 through 5.3. Note there is no pre-class quiz on Wednesday morning. Something to think about: A ball is whirled on a string in a vertical circle. As it is going around, the tension in the string is A.constant. B.greatest at the top of the motion C.greatest at the bottom of the motion D.greatest somewhere in between the top and bottom. 18

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