The Nature of Scientific Progress
+
Error Analysis
Session 3
Physics 2CL
Summer 2009
Outline
• How scientific knowledge progresses:
– Replacing models
– Restricting models
• What you should know about error analysis (so
far) and more
• Limiting Gaussian distribution
• Back to series and parallel circuits
• Preview of Exp. 1
• Reminder
Lab Objectives, This Week
• Models
– Extend knowledge of resistors, capacitors and inductors
to a circuit
– Testing/limiting
• Models of circuit analysis
• Models for circuit responses
• Physics
– Oscillation and Damping
– Resonance
• More complicated analysis
Models
• Invented
• Properties correspond closely to real world
• Must be testable
How Models Fit Into Process of
Doing Science
• Science is a process that studies the world by:
–
–
–
–
–
–
Limiting the focus to a specific topic (making a choice)
Observing (making a measurement)
Refining Intuitions (making sense) Creating
Extending (seeking implications) Predicting
Demanding consistency (making it fit) Refining or Replacing
Community evaluation and critique
How Models Change
• If models disagree with observation,
we change the model
– Refine - add to existing structure
– Restrict - limit scope of utility
– Replace - start over
Refining
• Original model consistent w/ new
observations, but not complete
• Extend model to account for new
observations
• May include new concepts
e.g. Model of interaction between charged
objects; to include interactions between
charged & uncharged add concept of
induced charge
Restriction
• New model correct in situations where
old isn’t
• New model agrees w/ old over some
range
Ö Old still useful in limited range
e.g. General relativity vs. classical
gravitational theory
Replacement
• Old model can’t be extended
consistently
• Replace entire model
Ö Earlier observations provide limits for
new model
e.g. Geocentric vs heliocentric models
for solar system
Random and independent?
•
•
•
o
o
Yes
Estimating between
marks on ruler or meter
Releasing object from
‘rest’
Mechanical vibration
Judgment
Problems of definition
•
•
•
o
o
No
End of ruler screwy
Reading meter from
the side (speedometer
effect)
Scale not zeroed
Reaction time delay
Calibration
Zero
Random & independent errors:
q = x+ y−z
q = Bx
δq = B δx
δq = (δx) 2 + (δy ) 2 + (δz ) 2
δq ≤ δx + δy + δz
δq
q
=
δx
x
q = q(x, y,z)
q= x×y÷z
⎛ δx ⎞ ⎛ δy ⎞ ⎛ δz ⎞ 2
δq
= ⎜ ⎟ +⎜ ⎟ +⎜ ⎟
⎝x⎠ ⎝y⎠ ⎝z⎠
q
⎛ ∂q ⎞ ⎛ ∂q ⎞ ⎛ ∂q ⎞ 2
δq = ⎜ δx ⎟ + ⎜ δy ⎟ + ⎜ δz⎟
⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎝ ∂z ⎠
δq
δz
δq
z
q
2
2
q
≤
δx
x
+
δy
y
+
2
≤
2
∂q
∂q
∂q
δx + δy + δz
∂x
∂y
∂z
Propagation in formulas
Independent
Propagate error in steps
For example:
x
q=
y−z
• Then
• First find
p= y−z
δp = (δy ) 2 + (δz ) 2
x
q=
p
δq
⎛ δx ⎞ ⎛ δp ⎞
= ⎜ ⎟ + ⎜⎜ ⎟⎟
q
⎝ x⎠ ⎝ p⎠
2
2
Propagation in formulas
Dependent
Variable(s) appear more Use general propagation
formula:
than once
ÖErrors compensate
q = q(x, y,z)
For example:
2
2
2
x
⎛
⎞
⎛
⎛
⎞
⎞
∂q
∂q
∂q
q=
δq = ⎜ δx ⎟ + ⎜ δy ⎟ + ⎜ δz⎟
x−z
⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎝ ∂z ⎠
An Important Simplifying Point
1 2
h = gt
2
g = 2h t 2 , δh h = 5%, δt / t = 0.1% Requires random & ind. errors!
δg
⎛ δh ⎞ ⎛ δt ⎞
= ⎜ ⎟ + ⎜2 ⎟
g
⎝ h⎠ ⎝ t ⎠
2
2
δg g = 5% 2 + (2 × 0.1% )2
δg g = 0.050039984 = 5%
ÖSimplifies calc.
ÖSuggests improvements
in experiment
• Often the error is
dominated by error in
least accurate
measurement
More Complicated Example
What is error in
q = Bx – yn
Start with
δq2 = (δ(Βx))2 + (δ(yn))2
δ(Bx) =
B δ(x)
Example continued
δ(yn) =
= ε(yn) * yn
= n ε(y) yn = n yn-1 δy
Then,
δq2 = (B δx)2 – (n yn-1 δy)2
Analyzing Multiple Measurements
• Repeat measurement of x many times
• Best estimate of x is average (mean)
x1 , x2 ,⋅ ⋅ ⋅, x N
xbest
x1 + x2 + ⋅ ⋅ ⋅ + x N
=x=
N
∑ xi
x=
N
Repeated Measurements
Number measurements
16
14
12
10
8
6
4
2
0
69.4
69.6
69.8
70.0
70.2
height (inches)
70.4
• If errors are random
and independent:
– Expect most values
near true value
– Expect few values far
from true value
ÖAssume values are
distributed normally
Number measurements
How are Measured Values
Distributed?
69.88
4
3
70.04
69.72
2
1
0
69.4
69.6
69.8
70.0
70.2
height (inches)
1
G X ,σ ( x) =
exp(− ( x − x) 2 2σ 2 )
σ 2π
70.4
Normal Distribution
Number measurements
x
69.88
4
x−σ
x+σ
3
70.04
69.72
2
1
0
69.4
69.6
69.8
70.0
70.2
70.4
height (inches)
1
2
2
G X ,σ ( x) =
exp(− ( x − x) 2σ )
σ 2π
Error of an Individual Measurement
• How precise are
measurements of x?
• Start with each value’s
deviations from mean
• Deviations average to
zero, so square, then
average, then take
square root
• ~68% of time, xi will
be w/in σx of true value
d i ≡ xi − x
d =0
σx ≡
=
(d i )2
1 N
2
( xi − x )
∑
N − 1 i =1
Take σx as error in
individual measurement called standard deviation
Number measurements
Standard Deviation
69.88
4
3
70.04
69.72
2
1
0
69.4
69.6
69.8
70.0
70.2
70.4
height (inches)
Measurement
16
12
8
4
0
69.4
69.6
69.8
70.0
height (inches)
70.2
70.4
Error of the Mean
• Expect error of mean
to be lower than error
of the measurements
it’s calculated from
• Divide SD by square
root of number of
measurements
• Decreases slowly with
more measurements
Standard Deviation of
the Mean (SDOM)
or
Standard Error
or
Standard Error of the
Mean
σx = σx
N
Summary
∑ xi
x=
N
• Average
• Standard deviation
σ =
x
1 N
2
(
x
−
x
)
∑
i
N − 1 i =1
• Standard deviation of the mean
σ x= σ x / N
Series and Parallel
R
Simple circuit
ε
R1
R2
series
R2
parallel
R1
Uncertainties - Series and
Parallel
R2
R1
RTOTAL = R1 + R2
δRTOTAL = ??
series
δRTOTAL = δR12 + δR22
R2
1/RTOTAL = 1/R1 + 1/R2
RTOTAL = R2*R1/(R1 + R2)
parallel
R1
εRTOTAL = ??
εRTOTAL = ε(R2*R1)2 + ε(R2+R1)2
Quoting Uncertainties
R2
R1
series
500 ± 1 Ω
100 ± 1 Ω
RTOTAL = R1 + R2
δRTOTAL = δR12 + δR22
RTOTAL = (600 ± 1) Ω
R2
RTOTAL = R2*R1/(R1 + R2)
parallel
R1
RTOTAL = 83 Ω
εRTOTAL = ε(R2*R1)2 + ε(R2+R1)2
Quoting Uncertainties – cont’d
100 ± 1 Ω R2
500 ± 1 Ω
parallel
εRTOTAL = ε(R2*R1)2 + ε(R2+R1)2
ε(R2*R1)2 = ε(R1)2 + ε(R2)2
R1
ε(R2*R1)2 = (.002)2 + (.01)2
ε(R2*R1)2 = .000104 = 1.04 x 10-4
ε(R2+R1)2 = δ(R2+R1)2/(R2+R1)2
ε(R2+R1)2 = 2 Ω2/360,000 Ω2 = 5.6 x 10-6
εRTOTAL = (1.04 x 10-4 + 5.6 x 10-6)1/2
εRTOTAL = (.01)
δRTOTAL = (.01) * 83.3 Ω = 0.8 Ω
RTOTAL = (83.3 ± 0.8) Ω
Circuit Analysis
Kirchoff’s Rules
Circuit Exp. 1
Charge
Discharge
Charge Capacitor
Discharge Capacitor
Discharge Capacitor
Remember
•
•
•
•
Experiment writeups
Prelab questions
Read Taylor through Chapter 4
Homework for lab 2 (Problems 4.2, 4.18)