Schedule
Lecture Exp
Date
A
July 5
3
B
July 10
4
A
July 12
5
B
July 17
6
A
July 19
B
July 24
A
July 26
B
July 31
Lecture Topics
Course Overview
Discussion of Exp 1 – Goals, setup
(Deduce mean density of the
earth)
Measurements, uncertainties.
Statistical Analysis
Intro to error propagation
Discussion of Exp 2 – goals, setup
(Deduction of mass distribution)
Histograms & distributions
The Gaussian Distribution,
Maximum likelihood,
Rejected data, Weighted mean
Discussion of Exp 3 – goals, setup
(Tune a shock absorber)
Fitting
Chi-squared test of distribution
Discussion of Exp 4 – goals, setup
(Calibrate a voltmeter)
Chi-squared
Covariance and correlation
Final Exam Review
August 2
Final Exam
1
July 3
2
1
2
3
7
8
9
10
4
Assignment
Lab: -Prepare for Quiz #1
Taylor: -Read chapters 1-3, HW 1
Lab: -Analyze data for Exp #1
Taylor: -Read chapter 4, HW 2
Lab: -Prepare for quiz #2
Taylor: -Read chapter 5, HW 3
Lab: -Analyze data for Exp #2
Taylor: -Read chapters 6-7, HW 4
Lab:
Taylor:
Lab:
Taylor:
Lab:
Taylor:
Lab:
Taylor:
Lab:
8PM, York 2722
Physics 2BL Summer I 2012
-Prepare for quiz #3
-Read chapter 8, HW 5
-Analyze data for Exp #3
-Read chapters 9 & 12
-Prepare for quiz #4
-HW 6
-Analyze data for Exp #4
-Prepare for final exam
-Pick up graded work from
TAs
-Pick up final from LTAC
4th Lab
Due!
1
Exp 4 Write-up, Weighted fits,
Chi-squared
Review
Lecture # 8
Physics 2BL
Spring 2012
Physics 2BL Summer I 2012
2
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
• Recap:
– Chi-Squared
• Homework – Review old homework/quizzes
– No more homework!
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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
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Announcements
1. Prepare for labs, seek help if needed as
resources are available
2. In lieu of final, will have extended quiz
that may include questions not
previously assigned
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5
Expectations - Review
1. Understand basic concepts in error
analysis
a. Significant figures
b. Propagation of errors – simple forms, general
form
c. Gaussian distributions – mean, standard
deviation, standard deviation of the mean
d. Extract probabilities from t-values
e. Rejection of data
f. Weighted averages
g. Linear least squares
h. χ2 analysis
Concepts mentioned in this brief review are not all inclusive
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Expectations - Review
2. 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 histograms
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 labs
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Experiment 4 - Measurements
• Weigh separately: coat
hanger, circular capacitor
plate (w/ rubber stopper),
circular counterweight
• How did you measure l1, l2?
• Radius of disks
• Period measurements
– measure N periods (don’t
forget to divide by N!)
v v
τ = r × F = r⊥ F
d
R=
2
δR =
δd
2
Top View
• θ (and θ0)
θ
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Experiment 4 - Calculations
• Kappa and its uncertainty
δκ δI
δT
2π 2 I
= ⊕2
κ= 2
κ
I
T
T
∂I
∂I
∂I
∂I
∂I
∂I
δI = δl1 ⊕ δl2 ⊕
δm1 ⊕
δm2 ⊕
δm ⊕ δR
∂l1
∂l2
∂m1
∂m2
∂m
∂R
• Vcalc(θ) and its uncertainty
2κθ
V =d
lAε 0
Check that
δV =
∂V
∂V
∂V
∂V
∂V
δd ⊕
δκ ⊕
δθ ⊕
δl ⊕
δA
∂d
∂κ
∂θ
∂l
∂A
δV = σ V =
1 N
2
(
V
−
A
−
Bx
)
∑ i
i
N − 2 i =1
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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!
A=
B=
2
x
∑ i ∑ yi − ∑ xi ∑ xi yi
∆
N ∑ xi yi − ∑ xi ∑ yi
∆
∆ = N ∑ xi − (∑ xi )
2
2
2
x
∑ i =?
∑x
i
=?
∑y =?
∑x y = ?
i
i
i
∆=?
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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?
– Dominant source of error
– Sources of systematic error?
• How did/would you improve?
(not an all inclusive list)
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Weighted Linear least squares fit
x ∑y −∑x ∑x y
∑
A=
2
i
• Non-weighted fit
i
i
i
∆
– Negligible δxi
– Assume δyi ~ σy
B=
N ∑ xi yi − ∑ xi ∑ yi
∆
∆ = N ∑ xi −
2
• Weighted fit
– Different δyi
– wi = 1/(δyi)2
i
A=
B=
(∑ x )
2
i
2
w
x
∑ i i ∑ wi yi − ∑ wi xi ∑ wi xi yi
∆
N ∑ wi ∑ wi xi yi − ∑ wi xi ∑ wi yi
∆
∆ = ∑ wi ∑ wi xi − (∑ wi xi )
2
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13
χ2 Test
Functional fit (i.e. linear)
 y j − f (x j )
2
χ = ∑

σy
j =1 


d=N-c
N
Distribution fit
2
n
χ =∑
2
i =1
(O
k
− Ek
)
2
Ek
d=n-c
χ
2
~
χ =
d
2
(
2
2
~
~
Pd χ ≥ χ 0
)
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Example:
•
•
•
•
•
2
χ
test
n
Die is tossed 600 times
Ok − Ek )
(
2
χ =∑
Expectation: each face equally likely
Ek
i =1
Verification of expectation by computing the χ2
Bins (n) = 6
d=6–1=5
Constraints (c) = 1 (N tosses)
2
1
2
3
4
5
6
Ok
91
137
111
87
80
94
Ek
100
100
100
100
100
100
∆k2
81
1369
121
169
400
36
χk2
0.81
13.7
1.21
1.69
4.0
0.36
χk2 for distribution is ∆k2 divided
by σk2 = Ek
Total χ2
d
reduced χ2
Physics 2BL Summer I 2012
21.76
5
4.35
16
Application of χ2 – Use of Table D
Agrees to 0.1% confidence
~
°
Reject at 99.9% significance
Prob that
χ~02 > 4
by chance
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Review
Determination of errors from measurements
Two types – random (statistical) and systematic
Random errors – intrinsic uncertainty (limitations)
Can be determined from multiple measurements
Mean and standard deviation, standard
deviation of the mean
Determine total uncertainty from random ⊕ systematic
Propagation or uncertainties through formulas
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)
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Overview
May be given basic physics equations
Need to know how to use them (labs)
Understand significant figures and how to quote
values properly
Need to know basic error propagation formulas
Need to know Gaussian distributions
mean, standard deviation, standard
deviation of the mean
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Overview
Know how to determine t-values
extract probability information from
those values
Understand rejection of data – Chauvenet’s
principle
Know how to calculate weighted averages
Let’s do an example
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Example Exam Question
You want to determine the torsional constant
for 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 the
torsional constant κ with the proper number of
significant 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, 14.6
I = 2420 ± 120 g cm2
κ=?
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{Ti (s)} = 15.1, 13.2,
14.4, 15.4, 14.6
I = 2420 ± 120 g cm2
κ=?
Example Solution
(1) Identify given parameters
Given T measurements and I ± δI
(2) Identify objective Want κ ± δκ
(3) Write the equation(s) necessary
to calculate κ
T = 2π
I
κ
4π 2 I
κ= 2
T
(4) Calculate best value for T
Tbest = Tave = 14.54 s
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{Ti (s)} = 15.1, 13.2,
14.4, 15.4, 14.6
Tave = 14.54 ± ?? s
I = 2420 ± 120 g cm2
κ=?
Example Solution
(5) Calculate uncertainty in T
σΤ = 0.847 s
σΤ = 0.424 s = 0.4 s
Τbest = (14.5 ± 0.4) s
(6) Calculate κ from best values
2π 2 I
κ= 2
T
g cm2/s2
κ = 4π2I/T2 = 454.4 units
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{Ti (s)} = 15.1, 13.2,
14.4, 15.4, 14.6
<T> = 14.5 ± 0.4 s
I = 2420 ± 120 g cm2
κ = 454.4 g cm2/s2 ± ??
Example Solution
2π 2 I
κ= 2
T
(7) Calculate uncertainty for κ
δκ
 δI   δT 
=   + 2 
κ
 I   T 
2
δκ
 120   0.4  δκ =
= 
 + 2

κ
 2420   14.5  κ
2
2
2
(0.0496)2 + (0.0552 )2
Most significant source of uncertainty?
δκ
= 0.07
κ
σκ = κ * (0.07) = 30 g cm2/s2
Thus, κ = (450 ± 30) g cm2/s2
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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
• Study for the final, bring questions to
Tuesday’s lecture
• Create final cheat sheet (hand written, 2
sides)
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