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
1
Final Review
Lecture # 9
Physics 2BL
Spring 2012
Physics 2BL Summer I 2012
2
Lecture #9:
•
•
•
•
End of Session I logistics
Recap
Questions
Homework
– Make Final cheat sheet!
– Review old homework/quizzes
Physics 2BL Summer I 2012
3
End of session I
• Thursday – Final!
• Office hours
– Chris – Wednesday 4-5 pm in MHA 2722
• Pick up your graded lab 4!
• Otherwise e-mail your lab TA (don’t email me!)
– Me –Thur 6-7 pm
• Final Exam pickup:
– From Chris Murphy Monday August 6th 10-11am MHA 2722
– Afterwards email Chris!
– Grades are due August 7th, so pick up Monday!!
• CAPE evaluations:
– Important for fine tuning of the course
– Making changes
– Giving feedback
Physics 2BL Summer I 2012
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Grading Policy
• ABSOLUTE grading scale
≥95%
≥90% & <95%
≥85% & <90%
A+
A
A-
≥80% & <85%
≥75% & <80%
≥65% & <75%
B+
B
B-
≥60% & <65%
≥55% & <60%
≥50% & <55%
C+
C
C-
≥40% & <50%
D
<40%
F
Grade scale may be
adjusted, but only in a
way that benefits you
(If everyone gets above
85%, everyone will get an
A!)
Physics 2BL Summer I 2012
5
Proper Sig Figs
Be sure you understand the rules! You WILL miss points on lab reports, quizzes,
& the final for not using proper sig figs!
Experimental uncertainties should almost always be rounded to ONE
significant figure!
NOTE: The exception is when that sig fig is equal to 1, then keep two sig figs
Measure l = 13.4 cm, estimate
uncertainty to be ¼ cm…
l = 13.4 ± 0.25 cm – WRONG
l = 13.4 ± 0.3 cm – RIGHT!
Calculate g = 9.85 m/s2,
uncertainty to be 0.095 m/s2 …
g = 9.85 ± 0.1 cm – WRONG
g = 9.85 ± 0.10 cm – RIGHT!
The last significant figure in the best estimate should be in the
same decimal position as the last (or only) decimal position
of the uncertainty
Measure θ = 25.75°, estimate uncertainty to be 2°…
θ = 25.75 ± 2° – WRONG
θ = 26 ± 2° – RIGHT!
Physics 2BL Summer I 2012
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Uncertainties
in Counting:
Error Propagation Summary
q = N (integer #)
δq = N
Uncertainties in Products
and Ratios:
q = xy
q = x/ y
δq = (δx )2 + (δy )2
δq ≤ δx + δy
Uncertainties
in Measured
Value and
exact constant:
q = xn
Uncertainties in Sums
and Differences:
q= x+ y
q= x− y
Uncertainties
in nth order
polynomial:
δq
 δx 
 δy 
=   +  
q
 x  y
δq δx δy
≤
+
q
x
y
2
q = Ax
2
δq
q
=n
δx
δq = A δx
x
 ∂q   ∂q 
δq =  δx  +  δy 
General Rule:
 ∂x   ∂y 
For independent random errors
(≤upper bound)
∂q
∂q
δq ≤ δx + δy
∂x
∂y
*always use radians when calculating the errors on trig functions
2
Physics 2BL Summer I 2012
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2
How to combine random and
systematic error?
δxtot =
(δxrandom ) + (δxsystematic )
2
Physics 2BL Summer I 2012
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10
Repeated Measurements
x1 , x2 ,..., xN
xbest = x
N measurements of the quantity x
Best estimate
the average or mean
x1 + x2 + ... + xN
1
x=
=
N
N
N
∑x
i =1
i
xi ± δx = xi ± σ x Standard deviation: uncertainty in any single
measurement of x
σx =
1
2
(
)
x
x
−
∑ i
N −1
xbest ± δxbest = x ± σ x Uncertainty in mean (best guess) is the
standard deviation of the mean
σx =
σx
N
Physics 2BL Summer I 2012
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Gaussian (Normal) Distribution
1
− ( x − X )2
G X ,σ ( x ) =
e
σ 2π
2σ 2
Width (& height) parameter
(~ standard deviation)
Peak position
(~ mean value)
Normalization factor
• Probability of measuring within a t-value of true value
t=
x A − xB
σA +σB
2
2
=
xk − x
σx
erf (t ) = ∫
X + tσ
X − tσ
G X ,σ ( x )dx
• Rejection of Data
x1 ,..., x N
tsus =
xsus − x
σx
n = N * Prob(|t| ≥ tsus)
If n < 0.5, the reject xsus
Physics 2BL Summer I 2012
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How to draw a Histogram
• Determine the range of your data
(largest value - smallest value)
• Choose number of bins ≥ 4 ≈ N
• Width of bins, ∆k, is range divided by #
• List bin boundaries, count number of data
points, nk, in each bin
• Draw histogram, y-scale, fk, may be the #
measurements in each bin
(
Physics 2BL Summer I 2012
)
14
Different Uncertainties
• Principle of maximum likelihood
1
− ( x − X )2
PX ,σ ( x ) =
e
σ 2π
2σ 2
– L = P(x1)P(x2)…P(xN)
– Prove that the mean maximizes the
X = x δX = σ x
likelihood when errors are equal
– Prove that the weighted mean
maximizes the likelihood when errors
xi wi
are different
1
∑
X = xwav =
δX = σ wav =
∑ wi
∑ wi
• Minimize chi-squared
χ
2
N
=
∑
i =1
 xi − X

 σi
wi =



2
Physics 2BL Summer I 2012
1
σ i2
=
1
(δxi )2
15
Linear Least Squares
A=
B=
σy =
2
x
∑ i ∑ yi − ∑ xi ∑ xi yi
∆
N ∑ xi yi − ∑ xi ∑ yi
y = A + Bx
∆
1 N
2 uncertainty in the measurement of y
( yi − A − Bxi ) (If we already have an independent
∑
N − 2 i =1
estimate of the uncertainty in y , …, y we
1
σA =σy
σB = σ y
expect this estimate to compare with σy)
2
x
∑
uncertainties in the constants A and B
given by error propagation in terms
of uncertainties in y1, … , yN
∆
N
∆
Where
∆ = N ∑ x2 −
N
 ∂A 
σ A = ∑  σ y 
i =1  ∂yi

N
(∑ x)
2
Physics 2BL Summer I 2012
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16
χ2 Test
Functional fit (i.e. linear)
(Measured)
Distribution fit
(Predicted from fit)
 y j − f (x j )
χ = ∑

σy
j =1 


N
2
2
n
χ =∑
2
i =1
(O
k
− Ek
)
2
Ek
d=n-c
d=N-c
χ~ 2 =
(
χ2
d
2
Pd χ~ 2 ≥ χ~0
)
(Larger table in Taylor)
Physics 2BL Summer I 2012
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Experiment 1: Density of the Earth: Basics
Assuming a sphere with uniform density…
Me
3g
ρ=
=
4πGRe
4
3
 πRe 
3

Determine using
pendulum
What is the value of g?
What is the radius of the earth?
Determine by
walking to cliffs
Height of cliff
Height of person
The formula for error
analysis:
 h1 − h2 

Re = 2C 
 ω∆t 


tperson - tcliff
2
g = 4π 2l T 2
Physics 2BL Summer I 2012
18
Experiment 2: Variation in thickness due to
Manufacturing
5
5
−
2
R
r
I hollow sphere = M 3 3
I = ∫ r 2dm
5 R −r
1
1
2
2
Conservation of energy: Mgh = Mv f + Iω f
2
2
Rolling without slipping: v = R′ω
2


2 ght
I = MR′  2 − 1
 2x

Variation in thickness σ ≈ 27 σ (man )
d
d
t
t
σd
 d
δ
1 σd

=

N d

σ t (man ) = σ t (total )2 − σ t (meas )2
Physics 2BL Summer I 2012
19
Experiment 3: Construct and Tune a Shock Absorber
Terminal velocity
mg
mg − bvt = 0, vt =
b
Equation of motion for damped oscillator:
b
d 2x
dx
−(
± iω ) t
m 2 + b + kx = 0 x = x e 2 m
0
dt
dt
x0
k
b2
ω=
−
m 4m2
(a) Under-damped
(b) Critically damped
(c) Over damped
bcrit = 2 mk
− x0
Physics 2BL Summer I 2012
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Experiment 4: Construct a Voltmeter
F=
Electrostatic attraction:
1 Aε 0 2
V
2
2 d
Set equal and solve for V
2κθ
V =d
lAε 0
Force by torsion balance: F = κθ l
4π 2 I
κ= 2
T
m
m
R2
2
2
2
2
I = (l1 + l2 ) + (l2 − l1 ) + m1l1 + m2 l 2 + (m1 + m2 )
12
4
4
Make a graph of Vcalc vs VPS:
– x-axis is the voltage read from the power supply (600-1000 V)
– y-axis is the voltage calculated from the angle of capacitor plate separation
Fit to straight line
Calculate χ2
Discuss goodness of fit
Calculate probability of result
=V(θ)
=A + Bx
 yi − f ( xi )
χ = ∑

σ
yi


2
2
SDOM of V(θ)
Physics 2BL Summer I 2012
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Homework
• Study for the final
• Create final cheat sheet (hand written, 2
sides)
• Bring student ID to exam!!! Don’t be
late!
Physics 2BL Summer I 2012
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