Schedule. Lab: Taylor: - earth) Measurements, uncertainties. Statistical Analysis. Lab: Taylor:
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1 Schedule Lecture Ep Date Lecture Topics Assignment Course Overview 1 Jul 3 Discussion of Ep 1 Goals, setup Lab: -Prepare for Quiz #1 (Deduce mean densit of the Talor: -Read chapters 1-3, HW 1 earth) A Jul 5 Measurements, uncertainties. Statistical Analsis 1 Intro to error propagation Discussion of Ep goals, setup 3 B Jul 10 (Deduction of mass distribution) Histograms & distributions The Gaussian Distribution, 4 A Jul 1 Maimum likelihood, Rejected data, Weighted mean 5 B Jul 17 Discussion of Ep 3 goals, setup (Tune a shock absorber) Fitting 6 A Jul 19 Chi-suared test of distribution 3 Discussion of Ep 4 goals, setup 7 B Jul 4 (Calibrate a voltmeter) 8 Chi-suared A Jul 6 4 Covariance and correlation Lab: Talor: Lab: Talor: -Analze data for Ep #1 -Read chapter 4, HW -Prepare for uiz # -Read chapter 5, HW 3 Lab: -Analze data for Ep # Talor: -Read chapters 6-7, HW 4 Lab: -Prepare for uiz #3 Talor: -Read chapter 8, HW 5 Lab: -Analze data for Ep #3 Talor: -Read chapters 9 & 1 Lab: -Prepare for uiz #4 Talor: -HW 6 Lab: -Analze data for Ep #4 Talor: - 9 B Jul 31 Final Eam Review Lab: -Prepare for final eam -Pick up graded work from TAs -Pick up final from LTAC 10 August Final Eam 8PM, York 7 Phsics BL Summer I 01 Lab 1 Due!! nd Quiz 3 rd Quiz 4 th Quiz 1
2 Measurement, Uncertainties, Statistical Analsis, Intro to Error Propagation Lecture # Phsics BL Summer Session I 01 Phsics BL Summer I 01
3 Lecture #: Issues from eperiment 1? Tuesda ou will need to turn in our lab notebook/report before the end of lab Start it at home! Errors random & sstematic Statistical analsis (uncertainties for datasets) Error propagation general formula and special cases Homework Consult the course web site regularl! Our main wa to communicate with ou Phsics BL Summer I 01 3
4 Measurement and Observation Measurement: deciding the amount of a given propert b observation Empirical Not logical deduction Not all measurements are created eual Phsics BL Summer I 01 4
5 Reproducibilit Same results under similar circumstances Reliable/precise Similar - a slipper thing Measure resistance of metal need same sample purit for repeatable measurement same size and shape need same people in room? same potential difference? Measure outcome of treatment on patients Can t repeat on same patient Patients not the same Phsics BL Summer I 01 5
6 Precision and Accurac Not the same thing! Precise - reproducible Accurate - close to true value Eample - temperature measurement thermometer with fine divisions (high precision) or with coarse divisions (low precision) and that reads 0 C in ice water (highl accurate) or 5 C in ice water (not so accurate) Phsics BL Summer I 01 6
7 Accurac Accurac vs. Precision Distance from true value Spread between trials P r e c i s i o n Accurac Phsics BL Summer I 01 7
8 What do we mean when we sa error(s)? Errors (not mistakes) Uncertainties Inevitable and intrinsic part of an eperiment How do we estimate uncertainties? Phsics BL Summer I 01 8
9 How to estimate uncertaint on a single measurement E: Length For lengths with well-defined (eas to reach) edges: use precision of measurement tool E: measuring distance L from beach to cliff in Ep 1 For things with unclear edges: Estimate the range within which the length is likel to fall and use the center of that range as the best guess E: measuring l from pivot point to the center of mass of the bob Phsics BL Summer I 01 9
10 Tpes of Measurement Errors: Random Errors: Unavoidable Can be estimated b repeating measurements Can also be reduced b repeating measurements Sstematic Errors: Hard to estimate, hard to reduce Can be due to calibration errors, neglecting small corrections, or procedural mistakes Get those clickers read if ou want etra credit Phsics BL Summer I 01 10
11 Statistical Analsis How to arrive at the the best estimate and uncertaint for a collection of repeated measurements as opposed to a single measurement We use repeated measurements to Improve accurac Estimate uncertaint best ± How to determine best? How to estimate?? Phsics BL Summer I 01 1
12 The mean, 1,..., N N measurements of the uantit best Best estimate the average or mean N N N N i 1 i N i 1 i i i i Phsics BL Summer I 01 13
13 Standard Deviation (RMS) d i 1 N i d i Deviation of i th measurement from the mean: Average sum of the suares of the uncertaint σ 1 i N 1 ( ) Standard deviation: uncertaint in an single measurement of ± ± σ i i 68% of all measurements will fall within the range of true ± σ How to estimate the uncertaint of best? Phsics BL Summer I 01 14
14 The Standard deviation of the Mean σ σ N (SDOM) Uncertaint in mean (best guess) is the standard deviation of the mean Based on the N measured values 1,, N we can state our final answer for the value of : (best value of ) best ± best σ σ N Phsics BL Summer I 01 15
15 Eample: lab 1 We measure the period of a pendulum 3 times and find the results: T 1.43 s, 1.48 s, 1.4 s. Give the best guess for T? Best guess? T s 3 Standard deviation (error for 1.43 s, 1.48 s, 1.4 s): Error for the best guess (SDOM): Best guess: 1 [( ) ( ) ( ) ] s σ T T ± s T 0.03 σ σ s T N 3 ALWAYS state values with uncertainties (and units), without which the value is meaningless Phsics BL Summer I 01 16
16 Propagation of error: E Lab 1 g 4π l T Several period measurements: T ± s Suppose ou estimate l 5.0 ± 0.3 cm From l and T, get best estimate of g ( 0.50m ) ( 1.443s) 4 m g π m What about uncertaint in g: g? Error analsis tells us how errors propagate through mathematical functions s Suppose ou measure ±, what is the uncertaint in () Phsics BL Summer I 01 17
17 Error propagation Phsics BL Summer I 01 18
18 Error Propagation Phsics BL Summer I 01 19
19 Error Propagation What if multiple dependent variables, i.e. g(l, T)? Phsics BL Summer I 01 0
20 General Formula for Error Propagation Single variable function () For an function (,,, z) Partial derivative: Differentiate w.r.t one variable while treating the others constant... z z Add independent & random errors in uadrature (similar to Pthagorean Theorem) Phsics BL Summer I 01 1
21 E: Error Propagation for Eperiment 1 Acceleration due to gravit: Lab data: Error in g: g l l g T T g g g g l T l T 4π 4π l 0569 T 8π l T 3 T g ( ) ( 0.003m ) m s 1.443s 8π ( 0.50m) ( 0.018s) 0.46m s 3 ( 1.443s) 4π l T T ± s l 0.50 ± m g m s ± g Dominant uncertaint Acceleration due to gravit: T ( m s ) ( 0.46m s ) 0.5m s 0.3m s Phsics BL Summer I 01 l ( Al ) A g 9.9 ± 0.3m s 3 ( AT ) AT You should show this much work when ou do error propagation!
22 Error Propagation Special cases Sum and differences: Independent errors ( ) ( ) Upper limit (just FYI) Eample: error for t t 1 t t 1 ± t 1 t ± t ( ) ( ) t t ( ) 1 t Phsics BL Summer I 01 3
23 Error Propagation Special cases Products and uotients: Independent errors (just FYI) E: error for ρ 3g/4πGR E g ± g R E ± R E E E R R g g ρ ρ 4 Phsics BL Summer I 01 Fractional errors: ratio of error to the true value
24 Error Propagation Special cases nth order polnomial: n A Measured value & eact constant: A n Eample: error for g 4πl/T l ± l T ± T g g l l T T A E: error for single period, T, from 10 periods, T (10T)/10: (10T) ± (10T) T ( 10T ) 10 Phsics BL Summer I 01 5
25 Error Propagation Summar Uncertainties in Sums and Differences: / Uncertainties in Products and Ratios: n Uncertainties in nth order polnomial: N N Uncertainties in Counting: (integer #) A Uncertainties in Measured Value and eact constant: ( ) ( ) General Rule: For independent random errors ( upper bound) n *alwas use radians when calculating the errors on trig functions A 6 Phsics BL Summer I 01
26 When all else fails, use general rule Eample: error for height of cliff h cliff h person Lcosθ for error propagation! h cliff h person ± h person L ± L θ ± θ h cliff hcliff hcliff h person L θ hperson L θ hcliff h 1 cliff h cliff cosθ Lsin θ hperson L θ Use radians!!!!! h ( h ) ( cosθl) ( L sin θθ ) cliff person Phsics BL Summer I 01 7
27 How to combine random and sstematic error? tot ( ) ( ) random sstematic Phsics BL Summer I 01 9
28 Homework Measure time between sunsets! If ou didn t finish pendulum, visit Chris s office hours (M 1-pm) Read Talor chapter 4 Start analsis so ou finish Lab 1 on time! HW: Talor problems 4.6, 4.14, 4.18, 4.6 Phsics BL Summer I 01 31
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