Week Exp Date Lecture topic(s) Assignment. Course Overview. Intro: Measurements, uncertainties. Lab: Discussion of Exp 3 goals, setup

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1 Schedule for Physics BL - Fall 01 Find relevan Lab Guides on he course web sie o prepare for Quizzes. Reading maerial and pracice problems are from Taylor, Error Analysis. Week Exp Dae Lecure opic(s) Assignmen Oc 8-Oc 3 15-Oc 4 -Oc 5 9-Oc Nov Course Overview. nro: Measuremens, uncerainies. Discussion of Exp 1 goals, seup (Deduce mean densiy of earh) Saisical Analysis Saisical Analysis Hisograms & Disribuions The Gaussian Disribuion Discussion of Exp goals, seup (Deducion of mass disribuions) Maximum Likelihood, Rejecion of Daa, Weighed Mean Discussion of Exp 3 goals, seup (Tune a shock absorber) 7 1-Nov No Lecure Veerans Day Nov 9 6-Nov 10 3-Dec Discussion of Exp 4 goals, seup (Calibrae a volmeer) Covariance and Correlaion, es of a disribuion Final Exam Monday evening, 7PM, Peer 110 Taylor: -Read Chaps 1-3. Lab: Taylor: Lab: Taylor: Lab: Taylor: Lab: Taylor: Lab: Taylor: Lab: Taylor: Lab: Taylor: Lab: -Be prepared for Quiz #1, -Read Guidelines for Exp -Sudy probs. 3.10, 3.8, 3.36, 3.41 (HW1 pracice) -Hand in Exp. 1 lab repor -Read Chap 4. -Sudy probs. 4.18, 4.6, 4.6, 4.14 (HW pracice) -Be prepared for Quiz # -Read Chap 5 -Sudy probs. 5.0, 5.36, 5., 5.6 (HW3 pracice) -Hand in Exp. lab repor -Read Chap 6 & 7 -Sudy probs. 7., 6.4 (HW4 pracice) -Be prepared for Quiz #3 -Read Chap 8 -Sudy probs. 8.10, 8.6, 8.4 (HW5 pracice) -Hand in Exp. 3 lab repor -Read Chap 9 & 1. -Be prepared for Quiz #4 -Sudy probs. 9.14, 1.3, 1.14 (HW6 pracice) -Hand in Exp. 4 lab repor -Prepare for Final Exam -Aend review session nd Pre-Lab Quiz 3 rd Pre-Lab Quiz 4 h Pre-Lab Quiz 1

2 Today s Lecure: ssues from las week lab secions? Logisics nroducion o experimen # Sraegy Non-desrucive measuremen of variaion Discussion of daa nerpreaion & Error Analysis Compaibiliy of measured resuls T-score Confidence Level

3 Logisics During Thanksgiving week A04,A06,A07 need o be rescheduled: Sign-up shees for make-up sessions are already posed in he labs 3

4 The Four Experimens Deermine he average densiy of he earh Weigh he Earh, Measure is volume Measure simple hings like lenghs and imes Learn o esimae and propagae errors Non-Desrucive measuremens of densiies, inner srucure of objecs Absolue measuremens Vs. Measuremens of variabiliy Measure momens of ineria Use repeaed measuremens o reduce random errors Consruc, and une a shock absorber Adjus performance of a mechanical sysem Demonsrae criical damping of your shock absorber Measure coulomb force and calibrae a volmeer. Reduce sysemaic errors in a precise measuremen. 4

5 Noe Changes: The Rubric for his Experimen changed. Please download i again from he course websie. New Rubric since Sunday 11/1 evening. There is also a minor change in he experimen guidelines. Please download hem again from he course websie as well. New experimen guidelines since Monday 11/ lae afernoon. 5

6 Experimen Sufficien accuracy High speed Perform a simple, fas, and non-desrucive experimen o measure he variaion in hickness of he shell of a large number of racque balls in shipmens arriving a a number of sores, o deermine if he variaion in hickness is much less han 10%. The problem can be solved by measuring he mass and momen of ineria of he balls. Noe, we only need o measure he variaion in hickness 6

7 Racque Balls R R r R r = M 5 R r d Counerfeiers make balls wih he same mass and he same average momen of ineria,, bu have worse qualiy conrol on he hickness, d, and hence on. We are looking for a larger spread in d implying a spread in. 7

8 Reminder- Wha is momen of ineria? Momen of ineria (), also called mass momen of ineria or he angular mass, (S unis kg m), is he roaional analog of mass. Tha is, i is he ineria of a rigid roaing body wih respec o is roaion. Momen of neria is a measure of resisance o change of angular velociy. = r dm Where m is he mass, and r is he (perpendicular) disance of he poin mass o he axis of roaion. Slide solen from Mike Riley 8

9 momen of ineria for spheres and cylinders Based on dimensional analysis alone, he momen of ineria of a non poin objec mus ake he form: = kim ir Where: M is he mass R is he radius of he objec from he cener of mass (in some cases, he lengh of he objec is used insead.) k is a dimensionless consan ha varies wih he objec in consideraion. nerial consans are used o accoun for he differences in he placemen of he mass from he cener of roaion. Examples include: k = 1, hin ring or hin walled cylinder around is cener k = /5, solid sphere around is cener k = 1/, solid cylinder or disk around is cener Slide solen from Mike Riley 9

10 Wha s he message? This is an experimen abou using repeaed measuremens o deermine he accuracy of a measuremen echnique. Experimenal mehods can be modified and improved in ligh of he resul of repeaed measuremens. We should learn o use averages o improve he accuracy of our resuls. Learn o disinguish beween sources of uncerainies. Error Propagaion is no a mahemaical exercise need o look, analyze and undersand he daa o do hese 10

11 Basic Sraegy - Overview Measure a ime. Repea measuremens o deermine error on. Calculae he Momen of neria,, from. Propagae error on o ge error on. Calculae he desired quaniy q (hickness) from. Propagae error on o ge error on q. The acual value of q is no ineresing in is own righ i maers only for comparison beween racke balls. >> Deermine how many repeaed measuremens are needed o ge he desired accuracy. 11

12 x 1 Rolling ball rackeball x h x 1 phoogae imer disance before saring imer Parameers you need o measure once and use for all ess: - disance, x, beween he saring poin and he nd phoo-gae; - disance, x 1, beween he saring poin and he 1s phoo-gae; - heigh difference, h, beween he wo phoo-gaes: h rolling = xsin radius R - widh of he groove, w, on he rail. Rolling radius of he ball, R, is differen from he acual radius of he ball, R. R ' = R ( w/ ) R w R 1

13 x 1 x h x 1 rackeball phoogae imer disance before saring imer v v/ How o use he measured parameers (ime and geomery) o calculae? 1 1 Mgh = Mv + Energy conservaion. v = R" Rolling radius. v = x 1 # $ % & ' ( Mgh = v M + R " x # $ gh = 1 % + & ' MR" ( MR # gh $ = ) 1 " % x & ' ( Newonian Mechanics for uniform acceleraion. Plug i all ino equaion. Solve for, we ge: = MR' gh " $ # x 1 % ' & 13

14 Repeaing Measuremens The errors on rolling ime will likely be bigger han he smalles division on your imer. You will need o measure repeaedly o find ou wha he error is. You may hen use he repeaed measuremens o reduce he random error on mean ime. # = = n 1 Mean ime for one ball afer n measuremens, 1,,, n, - i n i= 1 1 n n " 1 i= 1 our bes esimae of he acual ime. ( i " ) How many imes should you measure o ge he SDOM o 30% (0.3) of he SD? Sandard deviaion, SD, a measure of scaer of he individual daa poins, 1,,, n, and our esimae of error of an individual measuremen = n Sandard deviaion (error) of he mean (SDOM) our esimae of error of he calculaed mean value, based on he random uncerainies of a large number of individual measuremens. can be reduced o minimum by repeaing he measuremens. Key is reproducibiliy 14

15 Measuring Variaion in Balls 1. Measure rolling ime of one ball many imes o deermine he measuremen error in, σ measuremen N Nex week Chauvene. Measure rolling ime of many, differen balls o deermine he oal spread in, σ oal N one_ball calculae measured RMS variaion of average ime for Penn balls: ball 3. Calculae he spread in ime due o ball manufacure, σ manufacure, by subracing he measuremen error ball 4. Propagae error on ino error on and hen ino error on hickness d oal = manufacure measuremen oal > measuremen σ σ σ d variaion in variaion in variaion in d 15

16 Relaing <> o he hickness We now know how o analyze he iming of he balls. We nex need o relae his o he hickness of he balls. We need his in order o undersand how a 10% variaion in hickness (as claimed by he manufacurer) urns ino a X% variaion in <>..e. we need o deermine X here. We will find ha he mah involved is oo painful o do analyically. We will hus learn how o do his numerically. 16

17 Normalized momen of ineria We measured he momen of ineria of he ball(s),, Defined as: Side-Bar: We measure a ime and compue he momen of ineria. The Momen of neria is a useful inermediae quaniy. We have compued, bu have specified accuracy needed on hickness We need o compue hickness and is associaed error. Do i numerically for he ball Don ry o solve he 5h order equaion We define normalized momen of ineria: = 5 M R5 r 5 R 3 r 3 ; MR = R" $ gh R x # 1 ' & % ) ( And finally: = MR = 5 1 (r / R) 5 1 (r / R) 3 17

18 Propagae Error from o d - Numerically We measure ~ and wan o know, how much he error of ~ influences he error of d = R r Typical value for he balls are d = 4.5 mm and R = 8.5 mm, r = 3.75 mm: 5 ~ 1 (3.75/ 8.5) = = Make a small (3.75/ 8.5) perurbaion Le s ake d = 4.4 mm; we obain for i r = 3.85 mm and 5 ~ 1 (3.85/ 8.5) = = (3.85/ 8.5) ~ d 0.1 " We have = = 0.0 and ~ = = d d 0.0 ~ ~ Tha is = " ~ = 6.7 ~ for he fracional errors d To ge he error of d down o 5%, we need o know ~ wih a precision of 0.75% 18

19 Propagae Error from o d - Summary We needed o propagae errors for a complex funcion ~ ( d) = 5 1 [( R 1 [( R d) / R] d) / R] The numerical mehod we used was o perurb he argumen near a known value, o calculae he change in he funcion and he relaive values of he wo. can be applied in general and is exacly he same as calculaing: ~ ~ " = d " d We needed o know and plug in he acual approximae values of d and R

20 0 Propagae Error from Time o d d ~ ~.7 = 6 We have go and ' ~ " # $ % & ' ( = = x gh R R MR We wan o evaluae in erms of (because i is ha we measure) ~ ~ x gh R R ' ~ ~ = " " = Go sraigh o: R R x gh R R " # $ % & ' + ( = = ' ~ ' ~ R R R R ) / ' (0.57 ~ ) / ' ~ ( ~ ~ " + # " + # = Finally, we obain: d d ~ ~.7 6 " # " " The error analysis indicaes ha o obain d wih a precision of 10%, we need a precision of ~0.4% in. Muliply by / & Rearrange

21 Summary We are only rying o find differences beween balls, herefore, many errors can be ignored. Only he measured rolling imes are imporan. We mus measure one ball many imes o deermine he measuremen accuracy. We mus measure many balls of each ype experimenally o deermine he spread in hickness. Propagae error on ino error on hickness. n pracical erms: as long as he seup and he inended ball saring poin do no change, only random errors of phoo-gaes ime are imporan. To make a conclusion, wheher he hicknesses of differen balls are uniform wihin 10%, i is desirable o measure individual shell hicknesses o ~1% error. ==> An error of 3% in ball hickness, d, implies ha error of average ime,, as small as 0.04% 1

22 Coninuing from las lecure Quick reminder T-score Compaibiliy of measuremens Confidence level

23 GX, Gauss disribuion: he meaning of σ x = σ X σ G X, = e " X + σ G X," X - σ = dx #( x# X ) 1 = x = 0.68 The area under a segmen from X -σ o X+σ accouns for 68% of he oal area under he bell-shaped curve. Tha is, 68% of he measured poins fall wihin σ from he bes esimae x = X

24 GX, Wha abou he probabiliies o find a poin wihin 0.5σ from X, 1.7σ from X, or in general σ from X? 0.5σ 0.5σ 1.7σ 1.7σ σ σ G = e X, " G 1 #( x# X ) To find hose probabiliies we need o calculae X + # X " # X ( x),# Unforunaely, we canno do i analyically and have o look i up in a able

25 =

26 Compaibiliy of a measured resul -score Bes esimae of x: " x bes ± X Compare wih expeced answer x exp and compue -score: " x bes " x exp eced # X " This is he number of sandard deviaions ha x bes differs from x exp. " Therefore, he probabiliy of obaining an answer ha differs from x exp by or more sandard deviaions is: " Prob(ouside σ) = 1-Prob(wihin σ)) 6

27 Accepabiliy of a measured resul Convenions Large probabiliy means likely oucome and hence reasonable discrepancy. reasonable is a maer of convenion We define: < 5 % - significan discrepancy, > 1.96 < 1 % - highly significan discrepancy, >.58 boundary for unreasonable improbabiliy erf() error funcion f he discrepancy is beyond he chosen boundary for unreasonable improbabiliy, ==> he heory and he measuremen are incompaible (a he saed level) 7

28 Example: Confidence Level Two sudens measure he radius of a plane. Suden A ges R=9000 km and esimaes an error of σ = 600 km Suden B ges R=6000 km wih an error of σ =1000 km Wha is he probabiliy ha he wo measuremens would disagree by more han his (given he error esimaes)? ==> Define he discrepancy q = R A -R B = 3000 km. The expeced q is zero. Use propagaion of errors o deermine he error on q. = + = 1170 km q A B Compue he number of observed sandard deviaions from he expeced value of q: q 9000 " 6000 = = = q Now we look a Table A ==>.56 σ corresponds o 98.95% So, The probabiliy o ge a worse resul is 1.05% (= ) We call his he Confidence Level, and his is a bad one. 8

29 Example: Confidence Level A suden measures g, he acceleraion of graviy, repeaedly and carefully, and ges a final answer of 9.5 m/s wih a sandard deviaion of 0.1 m/s. f he measuremens were normally disribued, wih a cener a he acceped value of 9.8, wha is he probabiliy of geing an answer ha differs from 9.8 by as much as (or more han) his resul = = 0.1 s hree sandard deviaions off he mean. Looking up he probabiliy, 3 we see ha 99.73% are wihin 3 sigma, so he required probabiliy is 0.7%. Slide solen from Jim Branson 9

30 Example: Confidence Level The Confidence Level is he probabiliy o ge a worse resul han you measured. Average R e = 6191± 167 km compared o 6370 rue. Wha is he probabiliy o be furher off he correc radius of he earh han he measured value? = = Looking in able A for 1.07, we read 71.54%. Wha are he unis of? This is he probabiliy o be less han 1.07 sigma away so he C.L. is 100% % =8.46%. 30

31 Example: Confidence Level (from one of your colleagues) The Confidence Level is he probabiliy o ge a worse resul han you measured. Average R e = 3000 ± 500 km compared o 6370 rue. Wha is he probabiliy o be furher off he correc radius of he earh han he measured value? = = 6.74 Looking in able A for 6.74, we read %. Try i wih an uncerainy of 000km This is he probabiliy o be more han 6 sigma away so he C.L. is 100% % = very very small number. 31

32 The Normal Disribuion - Example The grades of sudens in a course where found o be normally disribued wih a mean of 80 poins and sandard deviaion of 5 poins. There were 75 sudens in he course. How many sudens would you expec o have scores: a) Beween 75 and 85 poins? b) Higher han 90 poins? c) Beween 70 and 90 poins? d) Bellow 65 Poins? Soluion: a) This grades range is: +/- 1σ herefore, 68% of he scores should be wihin his range 75*0.68 = 187 sudens. b) Grades higher han 90 poins are +σ and above he average. Therefore: N = 0.5 * (1 prob(" )) * 75 = 0.5 * ( ) * 75 = 6 c) Scores beween poins are wihin +/-σ of he mean. Therefore 95.45% of scores are expeced o be wihin his range: N = * 75 = 6 d) Scores bellow 65 poins are more han -3σ bellow he average score. We know ha +/-3σ is expeced o conain 99.7% of he scores. The negaive ail (ouside his range) is expeced o conain half as many: N = 0.5 * (1 prob(3" )) * 75 = 0.5 * ( ) * 75 = 0.4 Noe: one sided ---> 0.5 3

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34 Nex Week s Lecure: ssues from las week s lab-secions Rejecion of Daa The Principle of Maximum Likelihood The weighed average nroducion o fiing Read Taylor Chap 7 34