Establishing Relationships Linear Least Squares Fitting. Lecture 5 Physics 2CL Summer 2010

Transcription

1 Establishing Relationships Linear Least Squares Fitting Lecture 5 Physics 2CL Summer 2010

2 Outline Determining the relationship between measured values Review of experiment # 2 Physics for experiment # 3 Oscillations & resonance

3 Important Reminder Entering the last 3 labs Need to sign-up for the setups Formal report on FIRST of the last set of labs CAPE evaluations

4 Schedule Meeting Experiment 1 (Aug. 3 or 4) none 2 (Aug. 5 or 6) 0 3 (Aug. 10 or 11) 1 4 (Aug. 12 or 13) 1 5 (Aug. 17 or 18) 2 6 (Aug. 19 or 20) 3 7 (Aug. 24 or 25) 4,5,6 8 (Aug. 26 or 27) 4,5,6 9 (Aug. 31 or Sept. 1) 4,5,6

5 Relationships So far, we ve talked about measuring a single quantity Often experiments measure two variables, both varying simultaneously Want to know mathematical relationship between them Want to compare to models How to analyze quantitatively?

6 Principle of Maximum Likelihood Best estimates of X and s from N measurements (x 1 - x N ) are those for which Prob X,s (x i ) is a maximum

7 Yagil

8 Yagil

9 Yagil

10 Weighted averages (Chapter 7) Yagil

11 Yagil

12 ± ±

13

14 b) Best estimate is the weighted mean:

15 Y value Linear Relationships: y = A + Bx Data would lie on a straight line, except for errors What is best line through the points? What is uncertainty in constants? How well does the relationship describe the data? Velocity vs. constant acceleration Ohms law Slope = X Value

16 Y value A Rough Cut Best means line close to all points Draw various lines that pass through data points Estimate error in constants from range of values Good fit if points within error bars of line slope = slope = 1.06 slope = X Value

17 Y value More Analytical Best means minimize the square of the deviations between line and points Can use error analysis to find constants, error Slope = 1.01 ± X Value

18 The Details of How to Do This (Chapter 8)

19 Finding the coefficients A and B Want to find A, B that minimize difference between data and line Since line above some data, below other, minimize sum of squares of deviations Find A, B that minimize this sum A B y y i y A Bx y i y y i A Bx i N i1 (y i A Bx i ) 2 deviation of y i y i AN Bx i 0 x iy i A x i B 2 x i 0

20 Finding A and B After minimization, solve equations for A and B Looks nasty, not so bad See Taylor, example 8.1 A B y i AN Bx i 0 x 2 iy i A x i Bx i 0 2 x i y i x i x i y i A B N x y i i x i N x i 2 2 x i y i

21 Uncertainty in Measurements of y Before, measure several times and take standard deviation as error in y Can t now, since y i s are different quantities Instead, find standard deviation of deviations s y s x 1 N 2 1 N N 1 i1 N i1 ( x i x) (y i A Bx i ) 2 2

22 Uncertainty in A and B A, B are calculated from x i, y i Know error in x i, y i ; use error propagation to find error in A, B A distant extrapolation will be subject to large uncertainty s A s y s B s y N N x i 2 2 x i 2 x i

23 Uncertainty in x So far, assumed negligible uncertainty in x If uncertainty in x, not y, just switch them If uncertainty in both, convert error in x to error in y, then add errors actual error in x y Bx s y (equiv) Bs x s y (equiv) equivalent error in y s 2 y Bs x 2

24 Other Functions y Ae Bx Convert to linear Can now use least squares fitting to get ln A and B y Ae Bx ln y ln A Bx

25 Experiment #2 Oscillations and Damping RLC Circuit DC response

26 RLC Circuit Response

27 Graph of RLC Circuit Response

28 Critical Damping Define critical damping time constant No oscillations observed

29 Three Regimes for Damping

30 Lab Objectives 1) Determine w and Q 2) Achieve Critical Damping 3) Determine unknown L

31 Lab 3 Resonance Sinusoidal Response Complete circuit Model circuit

32 Lab 3 Resonance Q w 0 w 2 w 1

33 Uncertainty in Q Q = w 0 /(w 2 - w 1 ) Q = w 0 /(w) where w = w 2 - w 1 eq) = {e(w 0 ) 2 + e(w) 2 } 1/2 e(w 0 ) = d(w 0 )/w 0 e(w) = d(w)/w e(w 2 - w 1 ) = d(w 2 - w 1 )/ w 2 - w 1 d(w 2 - w 1 ) = {d(w 2 ) 2 + d(w 1 ) 2 } 1/2

34 Voltage Response

35 Origin and Voltage Response Derived Equation Origin fit Equation V R I Z R V 0 Z R Z Total R 1 Q 2 V 0 R R w w 0 w 0 w 2 y y = VR x = w A 2 2 x C 1 B C x A = V 0 R R /R B = Q C = w 0

36 Phase Shifts

37 Phase Response

38 Q-Multiplier Maximum voltage across capacitor is Q times driving voltage V 0

39 Outline Lab # 3 1). Preliminary calculations of w 0 and Q 2). Measure w 0 and Q 3). Graph Frequency Response 4). Q dependence on R 5). Q-Multiplier 6). Analysis 7). Conclusions

40 Remember CAPE evaluations Lab Writeup Read next session s lab description, do prelab Homework 5 (Taylor 6.1, 6.4) Read Taylor through chapter 8