Welcome to Physics 2BL: A Laboratory on measurements of physical

quantities, error analysis and interpretation of data

Physics 2BL •  In this course you will:

– Learn how scientists form models of nature’s phenomena •  How science is actually done (not the simplified version you

read in textbooks) – Learn how to assess the accuracy of measurements – Extend your understanding of Mechanics & Static

electricity through a hands-on exposure – Learn how to report scientific results

•  Pre-reqs: – PHYS 2A, 4A or equivalent – Concurrent enrollment in PHYS 2B/4C – Basic course in calculus (Differentiation & Integration)

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Physics 2BL: Spring 2013

Instructor Vivek Sharma, 5561 MHA Lab coordinator (LTAC) Tera Austrum, 2722 MHA Lab location 2722 MHA Course web page http://2bl.ucsd.edu

all course related material will be posted there

2BL team email address 2bl@physics.ucsd.edu Instructor Office hour Mondays, 10 -11 am Lab coordinator office hours Mondays, 2-4 pm Final Exam Monday June 3rd during 2BL

lecture hour (in RBC Auditorium) Text book An Introduction to error analysis,

2nd edition by J.R Taylor. Available at UCSD bookstore 3

Course Components & Your Tasks •  Attend all lectures in RBC Auditorium

– Provides context of each lab; Procedure for error analysis •  Read Lab descriptions (available at course website) before

coming to assigned section of lab •  Perform 4 Lab experiments in MHA 2722

– First lab starting Tue. April 9th (check your section time) – Two 3 hour sessions (A & B) to complete each lab – Learn how to compute measurement uncertainties

•  Estimate measurement errors •  Propagate errors

•  Read assigned chapters from the textbook •  Do weekly Homework based on textbook (self graded) 4

2BL Labs •  3 hours per week

–  the class is divided into several sections which do labs on different days of the week

•  Organized around different aspects of scientific methods (observation, forming and testing models etc)

•  Must read lab description (available at 2bl website) and do pre-lab home work before coming to the lab session – Will be tested on your knowledge of the lab by a short

quiz at the beginning of Session A of each lab •  Buy TWO quad-ruled notebook for lab report •  Record contact information for your lab TA; Listen to their

instructions carefully; TA is the best friend you will have in this course ! 5

The Four 2BL Experiments (Labs) •  Determine the average density of the earth; Weigh the Earth,

Measure its volume – Measure simple things like lengths and times – Learn to estimate and propagate errors

• Non-Destructive measurements of densities, inner structure of objects – Absolute measurements vs. measurements of variability – Measure moments of inertia – Use repeated measurements to reduce random errors

• Construct and tune a shock absorber – Adjust performance of a mechanical system – Demonstrate critical damping of your shock absorber

• Measure the Coulomb force and calibrate a voltmeter – Reduce systematic errors in a precise measurement

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In-Lab Quizzes •  Held during the first 15 minutes of each “A” session

– Come to the lab promptly ! –  If you show up after the quiz has already been collected,

you will receive a “0” •  All questions are directly from HW/pre-lab reading

–  2 pre-lab questions –  2 HW questions – Open book/open notes

•  No Makeup quizzes or final exam

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Lab Reports & Notebooks

•  Notebooks: –  Use a PEN & write clearly !

•  Report (10 pts per lab, 10% each on the following) –  Introduction (Objective, Essential Physics, Procedure) –  Methodology (Apparatus, Equations) –  Data/Observations (HOW did you estimate uncertainty?) –  Calculations/Error Analysis/Results –  Conclusions (dominant error source, discrepancy with published

results, possible improvements)

See 2bl web site (2bl.ucsd.edu) for detailed Lab notebook guidelines

-2.5 points for every day that the report is late!

Text Book & Homework Introduction to Error Analysis, by Taylor No information on experiments, but easy to

read and contains very good info on how to handle data. Helpful with error analysis, conclusion writing…

Source of HW problems (similarly in finals) Many helpful examples!

Brush up on Important 2A/4A physics you learnt. Review you Mechanics text book or consult

Fundamentals of Physics, By Halliday, Resnick & Walker 7th Ed.

Homework Exercises

•  Homework problems will be listed on 2bl web site – Self graded: solutions will be posted on web site after the

quiz – All problems are from Textbook (Taylor) – They will help you understand and practice new concepts

Solving them on your own will help you prepare for the quizzes & final!

Course Grading

•  Quizzes 20% •  Lab Reports 40% •  Final Exam 40%

Details: Quizzes (4) 5% each

(graded out of 10 points) Quiz questions come directly from

pre-lab & homework questions Lab Reports (4) 10% each Description of the experiment,

diagram of apparatus, presentation of the data, error analysis, calculations, discussion and conclusions

Final Exam 40% Final exam will ask questions about

the labs and will be similar to assigned homework

•  Preparation is key for labs! •  Time is limited, so be prepared

to perform the experiment •  If you run out of time, plan to

attend LTAC office hours •  Mondays 2-4 pm

Grading Policy

Absolute grading scale

Score Grade

> 95 % A+ > 85 % A > 70 % B > 50 % C > 40 % D < 40 % F

How To Do Well In This Course •  Complete recommended readings before Monday lecture

and definitely before your lab session –  2bl site will have all information starting Tuesday 2nd

•  Show up to all lectures and labs (ON TIME!) •  Before each lab:

–  Review lecture slides –  Read experiment guidelines –  Answer all quiz questions within –  Do the homework

– Analyze data before the second lab session •  Ask questions during lecture & Lab ! •  For the final exam, feel free to make your own 2-page cheat

sheet 13

Academic Integrity •  Please read the section on UCSD Policy on Integrity and

Scholarship in: http://www.ucsd.edu/catalog/front/AcadRegu.html •  Acknowledge your partner’s contributions (write their name

on your lab report) •  Do not plagiarize your partner’s work •  Collaboration in important

–  Work on experiment in teams –  Lab report must be written individually!

•  If in doubt, ask your lab TAs/LTAC/me

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Know The Error Of Thy Ways (and learn how to report them correctly)

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Measurement Error As Uncertainty •  Measurement uncertainty is inevitable ! •  Uncertainty (or error) in a measurement is not the same as a

mistake •  Uncertainty results from:

– Limits of instruments •  finite spacing of markings on ruler

– Design of the measurement •  using stopwatch instead of a photogate

– Less-well defined quantities •  Such as composition of earth

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Report measurement of a quantity = x ± δx

Uncertainty In Measuring Devices

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At other times you have to estimate it…

Reporting Result Of Measurement: Significant Figures

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Experimental uncertainties should almost always be rounded to ONE significant figure!

NOTE: The exception is when that sig fig is equal to 1, then keep two sig figs

The last significant figure in the best estimate should be in the same decimal position as the last (or only) decimal position of the uncertainty

Measure l = 13.4 cm, estimate uncertainty to be ¼ cm… l = 13.4 ± 0.25 cm – WRONG l = 13.4 ± 0.3 cm – RIGHT!

Calculate g = 9.85 m/s2, uncertainty to be 0.095 m/s2 … g = 9.85 ± 0.1 cm – WRONG g = 9.85 ± 0.10 cm – RIGHT!

Measure θ = 25.75°, estimate uncertainty to be 2°… θ = 25.75 ± 2° – WRONG θ = 26 ± 2° – RIGHT!

Be sure to follow the rules! You WILL miss points on lab reports, quizzes, & the final for not using proper sig figs !

Measurement Accuracy & Precision •  Accuracy: Close to target ; Precision: Close to each other

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Precision

Accuracy

Distance from true value Spread between trials

Types Of Measurement Errors •  Random Errors:

– Unavoidable – Can be estimated by repeating measurements – Can also be reduced by repeating measurements

•  Systematic Errors: – Harder to estimate, hard to reduce – Can be due to calibration errors, neglecting small

corrections, or procedural mistake

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Statistical Analysis Of data

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•  How to arrive at the the best estimate and uncertainty for a collection of repeated measurements as opposed to a single measurement –  We use repeated measurements to

•  Improve accuracy •  Estimate uncertainty xbest ±δ x

How to determine xbest? How to estimate δx ?

Repeated Measurements

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Make N measurements of the quantity x Best estimate à the average or mean Standard deviation: uncertainty in any single

measurement of x Uncertainty in mean (best guess) is the

standard deviation of the mean

xxbest = ∑=

=+++

=N

ii

N xNN

xxxx1

21 1...

( )∑ −−

= 2

11 xxN ixσ

xii xxx σδ ±=±

xbestbest xxx σδ ±=±

Nx

xσ

σ =

•  In Expt 1, with a simple pendulum you will measure acceleration due to gravity

•  If you measure

– Length – Period

•  What is the resulting uncertainty on g ? – Need to propagate measurement errors

Propagating Errors

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g = 4π 2l T 2

= T ±δT

Example : You measure xbest ± δx Quantity q = q(x) What are qbest & δqbest ?

= l ±δ l

General Formula For Error Propagation

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Single variable function q(x) For any function q(x, y,…, z)

xxqq δδ∂∂

=

δq = ∂q∂x

δ x⎛⎝⎜

⎞⎠⎟2

+ ∂q∂y

δ y⎛⎝⎜

⎞⎠⎟

2

+ ...+ ∂q∂z

δ z⎛⎝⎜

⎞⎠⎟2

Add independent & random errors in quadrature (similar to Pythagorean Theorem)

Partial derivative: Differentiate w.r.t one variable while treating the others constant

Back To The Simple Pendulum Experiment Acceleration due to gravity g = 4π 2 /T 2

Your data : T = 1.443± 0.018s , = 0.520 ± 0.003mImplies g = 9.8589 m/s2,δg = ??

Apply previous formula:δq = ∂q∂x

δ x⎛⎝⎜

⎞⎠⎟

2

+ ∂q∂y

δ y⎛⎝⎜

⎞⎠⎟

2

∂g∂l

δ l = 4π2

T 2 δ l = 4π 2

1.443s( )20.003m( ) = 0.0569m s2

∂g∂T

δT = 8π2l

T 3 δT =8π 2 0.520m( )1.443s( )3

0.018s( ) = 0.246m s2 Since ∂∂T

AT −2( ) = −2AT −3⎡⎣⎢

⎤⎦⎥

So

δg = 0.0569m s2( )2 + 0.246m s2( )2 = 0.25m s2 ≈ 0.3m s2

èAcceleration due to gravity: 23.09.9 smg ±=

Error Propagation : Some Examples

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Sum and differences:

( ) ( )yxq

yxq

yxqyxq

δδδ

δδδ

+≤

+=

−=

+=

22If measurement errors on x & y are independent then

Example: error on Δt = t1 – t2 Measure t1 ± δt1 t2 ± δt2

δ Δt( ) = δ t1( )2 + δ t2( )2

Error Propagation : Fractional Error

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For Products and quotients:

q = xyq = x yThen fractional error

δqq

= δ xx

⎛⎝⎜

⎞⎠⎟

2

+ δ yy

⎛⎝⎜

⎞⎠⎟

2

note : δqq

≤ δ xx+ δ yy

Error Propagation: Summary

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( ) ( )yxq

yxq

yxqyxq

δδδ

δδδ

+≤

+=

−=

+=

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Uncertainties in Sums and Differences:

yy

xx

qq

yy

xx

qq

yxqxyq

δδδ

δδδ

+≤

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠

⎞⎜⎝

⎛=

=

=

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/

Uncertainties in Products and Ratios:

General Rule: For independent random errors

(≤upper bound)

xxn

qq

xq n

δδ=

=

Uncertainties in nth order polynomial:

*always use radians when calculating the errors on trig functions

Nq

Nq

=

=

δ

Uncertainties in Counting:

(integer #)

xAq

Axq

δδ =

=

Uncertainties in Measured Value and exact constant:

yyqx

xqq

yyqx

xqq

δδδ

δδδ

∂∂

+∂∂

≤

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

+⎟⎠

⎞⎜⎝

⎛∂∂

=22

Schedule For This Week & Next •  This Week

– No labs this week – Read chapters 1- 4 of the Taylor book (easy reading!) – Work out Homework set #1

•  Next Week: – Read Experiment #1 information posted at website

•  Overview & guidelines •  Grading rubric

– Refresh your knowledge of physics involved – Lecture will discuss experiment # 1 – Labs will start Tuesday onwards, be prepared !

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