PHYSICS 2DL SPRING 2010 MODERN PHYSICS LABORATORY Monday May 3, - - PowerPoint PPT Presentation

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 R0+ΔR

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)

Purpose: 1) Agreement with theory? 2) Parameters

y(x) = Bx

χ2 TEST for FIT (Ch 12)

# of degrees of freedom

Least Sq. Fits : Derived from:

LEAST SQUARES FITTING

y=A+Bx+Cx2+Dx3+…+ ZxN

minimize … A,B,C…

LEAST SQUARES FITTING

1.

• 2. Minimize χ2:

y = f(x)

LINEAR FIT

y(x) = A +Bx : velocity at const acceleration Ohm’s law many other…

x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6

A B

?

LINEAR FIT

y(x) = A +Bx

x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6

y=-2+2x y=9+0.8x

LINEAR FIT

y(x) = A +Bx y=-2+2x y=9+0.8x Assumptions: 1) δxj << δyj ; δxj = 0 2) yj – normally distributed 3) σj: same for all yj

x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6

LINEAR FIT: y(x) = A + Bx

Assumptions: δxj << δyj ; δxj = 0 yj – normally distributed σj: same for all yj y3-(A+Bx3) y4-(A+Bx4) true value [yj-(A+Bxj)]

Σ

2 minimize

LINEAR FIT

y(x) = A +Bx y=-2+2x y=9+0.8x Assumptions: 1) δxj << δyj ; δxj = 0 2) yj – normally distributed 3) σj: same for all yj

x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6

true value

• f y

LINEAR FIT: y(x) = A + Bx

Best estimates of A&B  max Prob(y1…yN)  min

[yj-(A+Bxj)]

Σ

2

LINEAR FIT: y(x) = A + Bx

Best estimates of A&B  max Prob(y1…yN)  min

[yj-(A+Bxj)]

Σ

2 In Taylor p. 197

LEAST SQUARES FITTING

ln V=ln(Vo) – x/td Better to put in linear form: This is just ok, but hides details of fit

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 q = Bx II q = x + y II general case

prob(x) normally distributed prob(q)= prob(x=q-A)

q = x + A II q = Bx II q = x + y II general case

prob(q) = prob(x=q/B) normally distributed prob(x) X _ σx Bx _ Bσx

q = x + A II q = Bx II q = x + y II general case

x=y=0 _ _ prob(x) prob(y) prob(x and y) = prob(x)*prob(y) prob(x&y) prob(x&y) prob(x+y and z)

q = x + A II q = Bx II q = x + y II general case

prob(x+y and z) prob(x+y) =(2π)0.5 σx σy

q = x + A II q = Bx II q = x + y II general case

q(x,y) fixed number fixed number normally distributed with σx

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