ScheduleLecture Exp Date Lecture Topics Assignment

1 July 3

Course Overview

Discussion of Exp 1 – Goals, setup

(Deduce mean density of the

earth)

Lab:

Taylor:

-Prepare for Quiz #1

-Read chapters 1-3, HW 1

2

1

A July 5

Measurements, uncertainties.

Statistical Analysis

Intro to error propagation

Lab:

Taylor:

-Analyze data for Exp #1

-Read chapter 4, HW 2

3 B July 10

Discussion of Exp 2 – goals, setup

(Deduction of mass distribution)

Histograms & distributions

Lab:

Taylor:

-Prepare for quiz #2

-Read chapter 5, HW 3

4 A July 12

The Gaussian Distribution,

Maximum likelihood,Lab:

Taylor:

-Analyze data for Exp #2

-Read chapters 6-7, HW 42 Rejected data, Weighted mean

Taylor: -Read chapters 6-7, HW 4

5 B July 17Discussion of Exp 3 – goals, setup

(Tune a shock absorber)

Lab:

Taylor:

-Prepare for quiz #3

-Read chapter 8, HW 5

6

3

A July 19Fitting

Chi-squared test of distribution

Lab:

Taylor:

-Analyze data for Exp #3

-Read chapters 9 & 12

7 B July 24Discussion of Exp 4 – goals, setup

(Calibrate a voltmeter)

Lab:

Taylor:

-Prepare for quiz #4

-HW 6

84

A July 26Chi-squared

Covariance and correlation

Lab:

Taylor:

-Analyze data for Exp #4

-

9 B July 31 Final Exam Review Lab: -Prepare for final exam

10 August 2Final Exam8PM, York 2722

-Pick up graded work from

TAs

-Pick up final from LTAC

4th Lab Due!

1Physics 2BL Summer I 2012

Exp 4 Write-up, Weighted fits, Chi-squared

Review

Lecture # 8Physics 2BLSpring 2012

2Physics 2BL Summer I 2012

Lecture #8:• Issues from experiment 4?

– Tuesday you will need to turn in your lab notebook/report before the end of lab

– Start it at home! (may need to retake data)

• End of Session I logistics• Experiment 4 writeup• Experiment 4 writeup• Recap:

– Chi-Squared

• Homework – Review old homework/quizzes– No more homework!

3Physics 2BL Summer I 2012

End of session I• Tuesday – last lecture; Thursday – Final!• Office hours

– Chris – Monday 10 am – 12 pm– Me – Tues, Thur 6 – 7 pm

• EXTRA office hours: Chris – Wednesday (Aug 1) 4-5 pm in MHA 2722– Final questions– Pick up 4th lab before final

• CAPE evaluations:– Important for fine tuning of the course– Making changes– Giving feedback

4Physics 2BL Summer I 2012

Announcements1. Prepare for labs, seek help if needed as

resources are available2. In lieu of final, will have extended quiz

that may include questions not previously assigned

5Physics 2BL Summer I 2012

1. Understand basic concepts in error analysis

a. Significant figuresb. Propagation of errors – simple forms, general

formc. Gaussian distributions – mean, standard

deviation, standard deviation of the mean

Expectations - Review

deviation, standard deviation of the meand. Extract probabilities from t-valuese. Rejection of dataf. Weighted averagesg. Linear least squaresh. χ2 analysis

Concepts mentioned in this brief review are not all inclusive6Physics 2BL Summer I 2012

Expectations - Review2. Apply ideas to physics lab situation

a. Presentation of measurements and errors using proper number of significant figures

b. Propagation of errors through calculations (radius and density of earth)

c. Plot of histogramsd. Gaussian fits of data – mean, d. Gaussian fits of data – mean,

standard deviation, standard deviation of the mean

e. Extract probabilities from real data – used to determine variation in thickness of racket balls

f. Testing of a model with measurements – t-score analysis

g. Answer questions about the physics of the labs7Physics 2BL Summer I 2012

Experiment 4 - Measurements• Weigh separately: coat

hanger, circular capacitor plate (w/ rubber stopper), circular counterweight

• How did you measure l1, l2?• Radius of disks

FrFr ⊥=×=vvτ

d dδ• Radius of disks• Period measurements

– measure N periods (don’t forget to divide by N!)

• θ (and θ0)

2

dR =

2

dR

δδ =

θ

Top View

8Physics 2BL Summer I 2012

Experiment 4 - Calculations

• Kappa and its uncertainty

2

22

T

Iπκ =T

T

I

I δδκδκ

2⊕=

RR

Im

m

Im

m

Im

m

Il

l

Il

l

II δδδδδδδ

∂∂⊕

∂∂⊕

∂∂⊕

∂∂⊕

∂∂⊕

∂∂= 2

21

12

21

1

• Vcalc(θ) and its uncertainty2121

0

2

εκθ

lAdV = A

A

Vl

l

VVVd

d

VV δδδθ

θδκ

κδδ

∂∂⊕

∂∂⊕

∂∂⊕

∂∂⊕

∂∂=

Check that ∑=

−−−

==N

iiiV BxAV

NV

1

2)(2

1σδ

9Physics 2BL Summer I 2012

Experiment 4 - Graph• (Don’t forget title, axis labels with

units, error bars, legend)• What is your expected distribution?

– y = A + Bx A = ?? B = ??

• What kind of relationship do you see?

• Show work for least squares!• Show work for least squares!

( )22

2

∑∑

∑∑∑

∑ ∑∑∑

−=∆∆−

=

∆−

=

ii

iiii

iiiii

xxN

yxyxNB

yxxyxA

?

??

??2

=∆

==

==

∑∑∑∑

iii

ii

yxx

yx

11Physics 2BL Summer I 2012

Experiment 4 - Conclusion• Calculated χ2 and reduced χ2

• P(χ2 > χ02) (from chart)

• With what significance do you reject? (100% - confidence level)

• Sources of error?• Sources of error?– Dominant source of error– Sources of systematic error?

• How did/would you improve?(not an all inclusive list)

12Physics 2BL Summer I 2012

Weighted Linear least squares fit• Non-weighted fit

– Negligible δxi

– Assume δyi ~ σy

• Weighted fit

A =x i

2∑ y i∑ − x i x iy i∑∑∆

B =N x iy i∑ − x i∑ y i∑

∆

∆ = N x i2∑ − x i∑( )2

• Weighted fit– Different δyi

– wi = 1/(δyi)2

i∑ i∑( )

( )22

2

∑∑∑

∑∑∑∑

∑ ∑∑∑

−=∆∆−

=

∆−

=

iiiii

iiiiiiii

iiiiiiiii

xwxww

ywxwyxwwNB

yxwxwywxwA

13Physics 2BL Summer I 2012

Distribution fitFunctional fit (i.e. linear)

χ2 Test

( )∑

=

−=

N

j y

jj xfy

1

2

2

σχ

χ 2 =

Ok

− Ek( )2

Eki=1

n

∑

d = N - c d = n - cd = N - c d = n - c

d

22~ χχ =

( )20

2 ~~ χχ ≥dP

14Physics 2BL Summer I 2012

Example: χ2 test• Die is tossed 600 times• Expectation: each face equally likely• Verification of expectation by computing the χ2

• Bins (n) = 6• Constraints (c) = 1 (N tosses)

1 2 3 4 5 6

χ 2 =

Ok

− Ek( )2

Eki=1

n

∑

d = 6 – 1 = 5

1 2 3 4 5 6

Ok 91 137 111 87 80 94

Ek 100 100 100 100 100 100

∆k2

χk2

χk2 for distribution is ∆k

2 divided by σk

2 = Ek

Total χ2 21.76

d 5

reduced χ2 4.35

81 1369 121 169 400 36

0.81 13.7 1.21 1.69 4.0 0.36

16Physics 2BL Summer I 2012

Application of χ2 – Use of Table D

~°

Agrees to 0.1% confidence

Reject at 99.9% significance

Prob that χ0

2 > 4by chance

~

17Physics 2BL Summer I 2012

ReviewDetermination of errors from measurements

Two types – random (statistical) and systematicRandom errors – intrinsic uncertainty (limitations)

Can be determined from multiple measurementsMean and standard deviation, standard deviation of the mean

Propagation or uncertainties through formulas

deviation of the mean

Simple formula for adding two terms (a=b+c)Simple formula for multiplying two terms (a=b*c)General formula for g(x,y,z)

Determine total uncertainty from random ⊕ systematic

19Physics 2BL Summer I 2012

Overview

May be given basic physics equationsNeed to know how to use them (labs)

Understand significant figures and how to quotevalues properly

Need to know basic error propagation formulas

Need to know Gaussian distributionsmean, standard deviation, standard deviation of the mean

values properly

20Physics 2BL Summer I 2012

Overview

Understand rejection of data – Chauvenet’s principle

Know how to determine t-values extract probability information from those values

principle

Know how to calculate weighted averages

Let’s do an example

21Physics 2BL Summer I 2012

Example Exam Question

You want to determine the torsional constantfor the wire you used in the last experiment.You do this by measuring the period of oscillation. You make 5 measurements of 15.1 s, 13.2s, 14.4 s,15.4 s and 14.6 s. What is the best value for the15.4 s and 14.6 s. What is the best value for thetorsional constant κ with the proper number ofsignificant figures and uncertainty. You also determined the moment of inertia to be(2420 ± 120) g cm2.

{ Ti (s)} = 15.1, 13.2, 14.4, 15.4, 14.6I = 2420 ± 120 g cm2κ = ?

22Physics 2BL Summer I 2012

Example Solution(1) Identify given parameters

(3) Write the equation(s) necessaryto calculate κ

Given T measurements and I ± δI

{ Ti (s)} = 15.1, 13.2, 14.4, 15.4, 14.6I = 2420 ± 120 g cm2κ = ?

(2) Identify objective Want κ ± δκ

to calculate κ

(4) Calculate best value for T

Tbest = Tave = 14.54 s

κπ I

T 2= 2

24

T

Iπκ =

23Physics 2BL Summer I 2012

Example Solution(5) Calculate uncertainty in T

σΤ = 0.847 s

σΤ = 0.424 s = 0.4 s

Τ = (14.5 ± 0.4) s

{ Ti (s)} = 15.1, 13.2, 14.4, 15.4, 14.6Tave = 14.54 ± ?? sI = 2420 ± 120 g cm2κ = ?

(6) Calculate κ from best values

κ = 4π2I/T2 = 454.4 unitsg cm2/s2

Τbest = (14.5 ± 0.4) s

2

22

T

Iπκ =

24Physics 2BL Summer I 2012

Example Solution(7) Calculate uncertainty for κ

{ Ti (s)} = 15.1, 13.2, 14.4, 15.4, 14.6<T> = 14.5 ± 0.4 sI = 2420 ± 120 g cm2κ = 454.4 g cm2/s2 ± ??

22

2

+

=T

T

I

I δδκδκ

2

22

T

Iπκ =

22

5.14

4.02

2420

120

+

=κδκ ( ) ( )22 0552.00496.0 +=

κδκ

σκ = κ * (0.07) = 30 g cm2/s2

Most significant source of uncertainty?

Thus, κ = (450 ± 30) g cm2/s2

5.142420 κ κ

07.0=κδκ

25Physics 2BL Summer I 2012

Homework

• Finish Experiment #4! • If you need to retake data, visit Chris’s

office hours (M 10am-12pm)• Start analysis so you finish lab 4 on time• Start analysis so you finish lab 4 on time• Study for the final, bring questions to

Tuesday’s lecture• Create final cheat sheet (hand written, 2

sides)

26Physics 2BL Summer I 2012