PHYSICS 2DL – SPRING 2010
MODERN PHYSICS
LABORATORY
Monday May 3, 2010
Course Week 6
Lab Week
Prof. Brian Keating
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2Day in 2DL
• Questions/Announcements
• Error Analysis: Review Ch 6 (rejecting data) Ch 7
weighted averages; NEW! Ch8 least squares fitting.
• HW due in lab this week
• Special Topic: Franck-Hertz: Using Data Acquisition to
improve your life.
New Today Ch 8
• Ch 6,7 Review
• Ch 8 = Least Squares fitting
Final example
A student makes several measurements of a resistance R and determines
_
R=3.5Ω and σR=0.2 Ω.
Write an expression for the probability of obtaining a
measured value between R0 and R0+ΔR.
What is the probability of obtaining a measured value
between 3.35Ω and 3.8Ω?
R0+ΔR
R0
t=1.5
Then divide by 2
and add to
previous
(t=0.75 sigma)
probability.
Chapter 7
New Today Ch 8
• Ch 6,7 Review
• Ch 8 = Least Squares fitting
LEAST SQUARES FITTING (Ch.8)
y(x) = Bx
Purpose:
1) Agreement with theory?
2) Parameters
Least Sq. Fits : Derived from:
χ2 TEST for FIT (Ch 12)
# of degrees of
freedom
LEAST SQUARES FITTING
y=A+Bx+Cx2+Dx3+…+ ZxN
minimize
…
A,B,C…
LEAST SQUARES FITTING
y = f(x)
1.
2. Minimize χ2:
LINEAR FIT
y(x) = A +Bx :
velocity at const acceleration
Ohm’s law
many other…
B
A
x1
x2
x3
x4
x5
x6
y1
y2
y3
y4
y5
y6
LINEAR FIT
?
y(x) = A +Bx
y=-2+2x
y=9+0.8x
x1
x2
x3
x4
x5
x6
y1
y2
y3
y4
y5
y6
LINEAR FIT
y(x) = A +Bx
y=-2+2x
y=9+0.8x
x1
x2
x3
x4
x5
x6
y1
y2
y3
y4
y5
y6
Assumptions:
1) δxj << δyj ; δxj = 0
2) yj – normally distributed
3) σj: same for all yj
LINEAR FIT: y(x) = A + Bx
minimize
Σ [y -(A+Bx )] 2
j
e
tru
j
e
lu
a
v
y4-(A+Bx4)
y3-(A+Bx3)
Assumptions:
δxj << δyj ; δxj = 0
yj – normally distributed
σj: same for all yj
LINEAR FIT
y(x) = A +Bx
true value
of y
y=-2+2x
y=9+0.8x
x1
x2
x3
x4
x5
x6
y1
y2
y3
y4
y5
y6
Assumptions:
1) δxj << δyj ; δxj = 0
2) yj – normally distributed
3) σj: same for all yj
LINEAR FIT: y(x) = A + Bx
Best estimates of A&B  max Prob(y1…yN)  min
Σ [y -(A+Bx )] 2
j
j
LINEAR FIT: y(x) = A + Bx
In Taylor p. 197
Best estimates of A&B  max Prob(y1…yN)  min
Σ [y -(A+Bx )] 2
j
j
LEAST SQUARES FITTING
This is just ok, but
hides details of fit
Better to put in linear form:
ln V=ln(Vo) – x/td
Here’s our “final” example of the
general technique when fitting for
See Taylor
Problem 8.18
Don’t Use Linear fit with A=0!
LEAST SQUARES FITTING
y=eAx
y=A+Bx+Cx2+Dx3+…+ ZxN
y=f(x)
minimize
…
A,B,C…
q=x+A
II
normally
distributed
q = Bx
II
q=x+y
prob(x)
prob(q)= prob(x=q-A)
II
general case
q=x+A
II
q = Bx
II
q=x+y
II
normally
distributed
prob(x)
prob(q) = prob(x=q/B)
σx
_
X
Bσx
_
Bx
general case
q=x+A
II
q = Bx
II
q=x+y
prob(x and y) = prob(x)*prob(y)
II
general case
_ _
x=y=0
prob(x)
prob(x&y)
prob(y)
prob(x&y)
prob(x+y and z)
q=x+A
II
q = Bx
II
q=x+y
II
general case
σx
=(
2π
) 0.5
σy
prob(x+y)
prob(x+y and z)
q=x+A
II
q = Bx
II
q=x+y
q(x,y)
fixed
number
fixed
number
normally distributed
with σx
II
general case
How DAQ can simplify your
(experimental) life.
e--Atom collisions overhead!
Electronic Measurement using
Digital to Analog Conversion
Franck Hertz DAQ
•Program
(called a “.vi”
file) is on
Floppy drive.
•Save data to
hard disk, on
desktop.
•Email
yourself the
data from IE
•Save channel
1 data
(acceleration
voltage)
•Channel 2 is
current,
measured as
a voltage