Schedule for Physics 2BL - Fall 2012
•
•
Find relevant Lab Guides on the course web site to prepare for Quizzes.
Reading material and practice problems are from Taylor, Error Analysis.
Week Exp
1
2
Date
Lecture topic(s)
1-Oct
Course Overview.
Intro: Measurements, uncertainties.
8-Oct
Discussion of Exp 1 – goals, setup
(Deduce mean density of earth)
Statistical Analysis
15-Oct
Statistical Analysis
Histograms & Distributions
The Gaussian Distribution
_______
Taylor:
22-Oct
Discussion of Exp 2 – goals, setup
(Deduction of mass distributions)
Lab:
_______
Taylor:
29-Oct
Maximum Likelihood,
Rejection of Data, Weighted Mean
Lab:
_______
Taylor:
1
3
4
6
-Read Chaps 1-3.
Lab:
_______
Taylor:
-Be prepared for Quiz #1,
-Read Guidelines for Exp
______________________
-Study probs. 3.10, 3.28,
3.36, 3.41 (HW1 practice)
-Hand in Exp. 1 lab report
______
-Read Chap 4.
-Study probs. 4.18, 4.26,
4.6, 4.14 (HW2 practice)
-Be prepared for Quiz #2
______________________
-Read Chap 5
-Study probs. 5.20, 5.36,
5.2, 5.6 (HW3 practice)
-Hand in Exp. 2 lab report
______________________
-Read Chap 6 & 7
-Study probs. 7.2, 6.4
(HW4 practice)
-Be prepared for Quiz #3
______________________
-Read Chap 8
-Study probs. 8.10, 8.6,
8.24 (HW5 practice)
-Hand in Exp. 3 lab report
______________________
-Read Chap 9 & 12.
-Be prepared for Quiz #4
______________________
-Study probs. 9.14, 12.3,
12.14 (HW6 practice)
-Hand in Exp. 4 lab report
______________________
-Prepare for Final Exam
-Attend review session
Lab:
2
5
Assignment
Taylor:
5-Nov
Discussion of Exp 3 – goals, setup
(Tune a shock absorber)
7
12-Nov
No Lecture – Veterans Day
8
19-Nov
Discussion of Exp 4 – goals, setup
(Calibrate a voltmeter)
9
26-Nov
Covariance and Correlation,
!2 test of a distribution
10
3-Dec
Final Exam
Monday evening, 7PM, Peter 110
3
4
Lab:
_______
Taylor:
Lab:
_______
Taylor:
Lab:
_______
Taylor:
Lab:
_______
2nd Pre-Lab Quiz
3rd Pre-Lab Quiz
4th Pre-Lab Quiz
1
Todayʼs Lecture:
• Issues from last week lab sections ?
• Logistics
• Introduction to experiment #2
– Strategy
– Non-destructive measurement of variation
– Discussion of data Interpretation & Error Analysis
• Compatibility of measured results
– T-score
–
Confidence Level
2
Logistics
• During Thanksgiving week A04,A06,A07 need to be
rescheduled:
Sign-up sheets for make-up sessions are already posted in the labs!
3
The Four Experiments
•
Determine the average density of the earth
•
Non-Destructive measurements of densities, inner structure of objects
•
Construct, and tune a shock absorber
•
Measure coulomb force and calibrate a voltmeter.
Weigh the Earth, Measure its volume
– Measure simple things like lengths and times
– Learn to estimate and propagate errors
– Absolute measurements Vs. Measurements of variability
– Measure moments of inertia
– Use repeated measurements to reduce random errors
– Adjust performance of a mechanical system
– Demonstrate critical damping of your shock absorber
– Reduce systematic errors in a precise measurement.
4
Note Changes:
• The Rubric for this Experiment changed. Please download
it again from the course website.
– New Rubric since Sunday 11/21 evening.
• There is also a minor change in the experiment guidelines.
Please download them again from the course website as
well.
– New experiment guidelines since Monday 11/22 late afternoon.
5
Experiment 2
•
Sufficient accuracy
High speed
Perform a simple, fast, and non-destructive experiment to measure the
variation in thickness of the shell of a large number of racquet balls in
shipments arriving at a number of stores, to determine if the variation in
thickness is much less than 10%.
The problem can be solved by
measuring the mass and
moment of inertia of the balls.
Note, we only need to measure
the variation in thickness
6
Racquet Balls
R
r
2
R5 ! r5
I= M 3 3
5 R !r
R
d
Counterfeiters make balls with the same mass and the same
average moment of inertia, I, but have worse quality control on
the thickness, d, and hence on I. We are looking for a larger
spread in d implying a spread in I.
7
Reminder- What is “moment of inertia?”
• Moment of inertia (I), also called mass moment of inertia or
the angular mass, (SI units kg m2), is the rotational analog
of mass. That is, it is the inertia of a rigid rotating body
with respect to its rotation.
• Moment of Inertia is a measure of resistance to change of
angular velocity.
2
I = ! r dm
• Where
– m is the mass,
– and r is the (perpendicular) distance of the point mass to the axis
of rotation.
Slide stolen from Mike Riley
8
moment of inertia for spheres and
cylinders
• Based on dimensional analysis alone, the moment of inertia of a non‐point
object must take the form:
Where:
I = kiM iR
2
– M is the mass
– R is the radius of the object from the center of mass (in some cases, the
length of the object is used instead.)
– k is a dimensionless constant that varies with the object in consideration.
• Inertial constants are used to account for the differences in the
placement of the mass from the center of rotation. Examples include:
– k = 1, thin ring or thin‐walled cylinder around its center
– k = 2/5, solid sphere around its center
– k = 1/2, solid cylinder or disk around its center
Slide stolen from Mike Riley
9
Whatʼs the message?
• This is an experiment about using repeated measurements to
determine the accuracy of a measurement technique.
• Experimental methods can be modified and improved in light of the
result of repeated measurements.
• We should learn to use averages to improve the accuracy of our
results.
• Learn to distinguish between sources of uncertainties.
Error Propagation is not a mathematical exercise!
need to look, analyze and understand the data to do these
10
Basic Strategy - Overview
• Measure a “time” t.
– Repeat measurements to determine error on t.
• Calculate the Moment of Inertia, I, from t.
– Propagate error on t to get error on I.
• Calculate the desired quantity q (thickness) from I.
– Propagate error on I to get error on q.
– The actual value of q is not interesting in its own right …
… it matters only for comparison between racket balls.
>> Determine how many repeated measurements are needed
to get the desired accuracy.
11
t
Rolling ball
x1
racketball
photogate timer
x
h
!
x1 distance before
starting timer
Parameters you need to measure once and use for all tests:
- distance, x, between the starting point and the 2nd photo-gate;
- distance, x1, between the starting point and the 1st photo-gate;
- height difference, h, between the two photo-gates: hrolling
= x sinradius
! R’
- width of the groove, w, on the rail.
“Rolling radius” of the ball, Rʼ, is different from
the actual radius of the ball, R.
R ' = R 2 ! ( w / 2) 2
R
R’
w
12
t
x1
racketball
v
photogate timer
x
h
v/2
x1 distance before
starting timer
How to use the measured parameters (time and geometry) to calculate I?
1
1 2
2
Mgh = Mv + I!
Energy conservation.
2
2
v = R"!
Rolling radius.
v=
2x
t
Newtonian Mechanics for
uniform acceleration.
1 2#
I $
Mgh = v % M + 2 &
2 '
R" (
2 x2 #
I $
gh = 2 % 1 +
&
t '
MR"2 (
2
# ght
$
I
= % 2 ) 1&
2
MR" ' 2 x
(
Plug it all into equation.
" ght 2 %
Solve for I, we get: I = MR' $ 2 ! 1'
# 2x
&
2
13
Repeating Measurements
•
•
•
The errors on rolling time will likely be bigger than the smallest division on your timer.
You will need to measure repeatedly to find out what the error is.
You may then use the repeated measurements to reduce the random error on mean time.
1 n
t = ! ti
n i=1
How many times should you measure to get
the SDOM to 30% (0.3) of the SD?
Mean time for one ball after n measurements, t1, t2, …, tn, our best estimate of the actual time.
1 n
2
#t =
(
t
"
t
)
!
i
n " 1 i=1
!t
!t =
n
Standard deviation, SD, – a measure of scatter of
the individual data points, t1, t2, …, tn, and our
estimate of error of an individual measurement
Standard deviation (error) of the mean (SDOM) – our estimate
of error of the calculated mean value, based on the random
uncertainties of a large number of individual measurements.
It can be reduced to minimum by repeating the measurements.
Key is reproducibility!!!
14
Measuring Variation in Balls
N
•
•
•
•
1. Measure rolling time of one ball
many times to determine the
measurement error in t, σmeasurement
Next week
Chauvenet
tone_ball
2. Measure rolling time of many,
different balls to determine the
total spread in t, σtotal
N calculate measured RMS variation of
average time for Penn balls: !tball
3. Calculate the spread in time due
to ball manufacture, σmanufacture, by
subtracting the measurement error
4. Propagate error on t into error
on I and then into error on
thickness d
tball
! total = ! manufacture ! ! measurement
! total > ! measurement
σt
variation in t
σI
variation in I
σd
variation in d
15
Relating <t> to the thickness
• We now know how to analyze the timing of the balls.
• We next need to relate this to the thickness of the balls.
• We need this in order to understand how a 10% variation in
thickness (as claimed by the manufacturer) turns into a X%
variation in <t>. I.e. we need to determine X here.
• We will find that the math involved is too painful to do
analytically.
• We will thus learn how to do this numerically.
16
“Normalized” moment of inertia
We measured the moment of inertia of the ball(s),
5
5
2
R
!
r
I, Defined as:
I= M 3
;
3
5 R !r
Side-Bar:
• We measure a time and compute the moment of inertia.
• The Moment of Inertia is a useful intermediate quantity.
• We have computed I, but have specified accuracy needed on thickness
• We need to compute thickness and its associated error.
• Do it numerically for the ball
• Donʼt try to solve the 5th order equation…
We define “normalized” moment
of inertia:
And finally:
2
$ ght 2 '
I
R
"
I! !
= 2 & 2 # 1)
2
MR
R % 2x
(
5
I
2
1
!
(r
/
R)
I! =
=
2
MR
5 1 ! (r / R)3
17
Propagate Error from I to d - Numerically
~
~
We measure I and want to know, how much the error of I
influences the error of d = R ! r
Typical value for the balls are d = 4.5 mm and R = 28.25 mm, r = 23.75 mm:
~ 2 1 ! (23.75 / 28.25) 5
I =
= 0.5717
3
5 1 ! (23.75 / 28.25)
Make a small
“perturbation”…
Letʼs take d = 4.4 mm; we obtain for it r = 23.85 mm and
~ 2 1 ! (23.85 / 28.25) 5
I =
= 0.5736
3
5 1 ! (23.85 / 28.25)
~
!d 0.1
"I
0.5736 ! 0.5717
=
= 0.0222 and
= 0.00331
We have
~ =
d
4. 5
0.5736
I
~
~
!d
0.0222 !I
!I
That is
=
" ~ = 6.7 ~ for the fractional errors
d
0.00331 I
I
~
To get the error of d down to 5%, we need to know I with a precision of 0.75%
18
Propagate Error from I to d - Summary
We needed to propagate errors for a complex function
2 1 ! [( R ! d ) / R ]5
~
I (d ) =
5 1 ! [( R ! d ) / R ]3
The numerical method we used was to perturb the argument near a
known value, to calculate the change in the function and the relative
values of the two. It can be applied in general and is exactly the same
as calculating:
~
~ "I
!I =
!d
"d
We needed to know and plug in the actual approximate values of d and R.
19
Propagate Error from Time to I
~
$
I
R ' 2 ' ght 2
~
!
d
!
I
We have got
= 2 %%
( 1"" ! 0.572
= 6.7 ~ and I =
2
2
MR
R & 2x
d
I
#
~
!t
!I
We want to evaluate ~ in terms of
(because it is t that we measure)
t
I
~
R ' 2 ght
~ "I
!I =
!t = 2 2 !t
Go straight to:
"t
R x
Multiply by t/t &
Rearrange…
' ~ R ' 2 $ !t
~ R ' 2 ght 2 !t
!I = 2
= 2 ( %% I + 2 ""
2
R x
t
R # t
&
~
~
!I
2 # ( I + R ' 2 / R 2 ) !t 2 # (0.572 + R ' 2 / R 2 ) !t
!t
"
"4
~ =
~
t
0.572
t
t
I
I
Finally, we obtain:
~
!d
!I
!t
!t
" 6.7 ~ " 6.7 # 4 " 27
d
t
t
I
The error analysis indicates that to obtain d with a precision of 10%, we need
20
a precision of ~0.4% in t.
Summary
We are only trying to find differences
between balls, therefore, many errors
can be ignored.
Only the measured rolling times are
important.
We must measure one ball many times
to determine the measurement
accuracy.
We must measure many balls of each
type experimentally to determine the
spread in thickness.
Propagate error on I into error on
thickness.
In practical terms:
as long as the setup and the
intended ball starting point do not
change, only random errors of
photo-gates time are important.
To make a conclusion, whether the
thicknesses of different balls are
uniform within 10%, it is desirable
to measure individual shell
thicknesses to ~1% error.
==> An error of 3% in ball
thickness, d, implies that error of
average time, t, as small as 0.04%!
21
Continuing from last lecture…
Quick reminder…
T-score
Compatibility of measurements
Confidence level
22
Gauss distribution:
the meaning of σ
G X ,! =
1
! 2"
! G X ," dx
GX,!
=
X - σ
= 0.68
The area under a segment from X -σ
to X+σ accounts for 68% of the
total area under the bell-shaped curve.
σ
σ
That is, 68% of the measured
points fall within σ from the
best estimate x = X
0
20
40
2! 2
!x =!
X + σ
x=X
e
#( x# X )2
60
80
100
120 23
GX,!
What about the probabilities to
find a point within 0.5σ from X,
1.7σ from X, or in general tσ
from X ?
G X ,! =
1
! 2"
e
#( x# X )2
2! 2
To find those probabilities
we need to calculate
X + t#
0.5σ 0.5σ
!G
X ,#
( x)
X "t#
0
20
1.7σ
1.7σ
tσ
tσ
40
60
Unfortunately, we cannot do
it analytically and have to
look it up in a table
80
100 24
120
t = 1.47
25
Compatibility of a measured result
t-score
• Best estimate of x: "
xbest ± ! X
• Compare with expected answer xexp and compute t-score: "
t!
xbest " xexp ected
#X
"
• This is the number of standard deviations that xbest differs from
xexp. "
• Therefore, the probability of obtaining an answer that differs
from xexp by t or more standard deviations is: "
Prob(outside tσ) = 1-Prob(within tσ))
26
“Acceptability” of a measured result
Conventions
• Large probability means likely outcome and hence reasonable
discrepancy.
• “reasonable” is a matter of convention…
• We define:
< 5 % - significant discrepancy, t > 1.96
< 1 % - highly significant discrepancy, t > 2.58
boundary for unreasonable improbability
erf(t) – error function
If the discrepancy is beyond the chosen
boundary for unreasonable improbability,
==> the theory and the measurement are
incompatible (at the stated level)
27
Example: Confidence Level
Two students measure the radius of a planet.
• Student A gets R=9000 km and estimates an error of σ = 600 km
• Student B gets R=6000 km with an error of σ =1000 km
• What is the probability that the two measurements would disagree by
more than this (given the error estimates)?
==> Define the discrepancy q = RA-RB = 3000 km. The expected q is
zero. Use propagation of errors to determine the error on q.
! q = ! A2 + ! B2 = 1170 km
• Compute t the number of observed standard deviations from the
q 9000 " 6000
expected value of q:
t=
=
= 2.56
!q
1170
• Now we look at Table A ==> 2.56 σ corresponds to 98.95%
So, The probability to get a worse result is 1.05% (=100-98.95)
We call this the Confidence Level, and this is a bad one.
28
Example: Confidence Level
A student measures g, the acceleration of gravity, repeatedly and
carefully, and gets a final answer of 9.5 m/s2 with a standard deviation
of 0.1 m/s2.
If the measurements were normally distributed, with a center at the
accepted value of 9.8, what is the probability of getting an answer that
differs from 9.8 by as much as (or more than) his result.
9.8 ! 9.5
t=
=3
0.1
Its three standard deviations off the mean. Looking up the probability,
we see that 99.73% are within 3
sigma, so the required probability is
0.27%.
Slide stolen from Jim Branson 29
Example: Confidence Level
•
The Confidence Level is the probability to get a “worse”
result than you measured.
Average Re = 6191 ± 167 km
compared to 6370 true.
•
What is the probability to be further off the correct
radius of the earth than the measured value ?
6370 ! 6191
t=
= 1.07
167
What are the units of t?
• Looking in table A for 1.07, we read 71.54%.
• This is the probability to be less than 1.07 sigma away so the
C.L. is 100% - 71.54% =28.46%.
30
•
Example: Confidence Level
(from one of your colleagues)
The Confidence Level is the probability to get a
“worse” result than you measured.
Average Re = 3000 ± 500 km
compared to 6370 true.
•
What is the probability to be further off the correct
radius of the earth than the measured value ?
Try it with an uncertainty of 2000km
6370 ! 3000
t=
= 6.74
500
• Looking in table A for 6.74, we read 99.999%.
• This is the probability to be more than 6 sigma away so the
C.L. is 100% - 99.9999% = very very small number.
31
The Normal Distribution - Example
The grades of students in a course where found to be normally distributed with a mean of 80
points and standard deviation of 5 points. There were 275 students in the course. How many
students would you expect to have scores:
a)
b)
c)
d)
Between 75 and 85 points?
Higher than 90 points?
Between 70 and 90 points?
Bellow 65 Points?
Solution:
a) This grades range is: +/- 1σ therefore, 68% of the scores should be within this range
275*0.68 = 187 students.
b) Grades higher than 90 points are +2σ and above the average. Therefore:
N = 0.5 * (1 ! prob(2" )) * 275 = 0.5 * (1 ! 0.9545) * 275 = 6
c) Scores between 70-90 points are within +/-2σ of the mean. Therefore 95.45% of scores are expected to be
within this range:
N = 0.9545 * 275 = 262
d) Scores bellow 65 points are more than -3σ bellow the average score. We know that +/-3σ is expected to
contain 99.7% of the scores. The negative tail (outside this range) is expected to contain half as many:
N = 0.5 * (1 ! prob(3" )) * 275 = 0.5 * (1 ! 0.997) * 275 = 0.4
Note: one sided ---> 0.5
32
33
Next Weekʼs Lecture:
• Issues from last weekʼs lab-sections
• Rejection of Data
• The Principle of Maximum Likelihood
– The weighted average
– Introduction to fitting
Read Taylor Chap 7
34