The Pendulum. Purpose

Size: px
Start display at page:

Download "The Pendulum. Purpose"

Transcription

1 The Pedulum Purpose To carry out a example illustratig how physics approaches ad solves problems. The example used here is to explore the differet factors that determie the period of motio of a pedulum. Apparatus Meter stick, stopwatch, strig, weights, ad a stad. I. Prelimiary Discussio The fist step is to ask the questio, i this case what are the quatities which affect the period (or frequecy) of a simple pedulum, ad determie quatitatively how they effect the period. May systems, both atural ad mamade, udergo periodic motio. Some examples iclude the beatig heart, the electros i a radio atea, metroomes, ad atoms i solid materials. Other simple systems also exhibit harmoic motio, such as the motio of a pedulum ad that of a mass attached to a sprig. The motio of all such systems has a period, T, which is determied by the elemets that make up the system. The period is simply the amout of time that the motio takes to complete oe cycle. Sometimes we describe a periodic motio by its frequecy, f, which is simply 1/T. For example, Old Faithful (Yellowstoe Natioal Forest) erupts oce every hour*. Old Faithful s period is oe hour, ad its frequecy is oce per hour. Period is usually measured i the most coveiet uit of time (secods, miutes, hours, etc.), while frequecy is ofte measured i Hertz (1 Hz = 1 cycle/secod) or Mega Hertz (1 MHz = 1000 Hz). I this experimet, we will test the differet elemets of a pedulum to determie which oes affect the period ad frequecy of the system. * Old Faithful's actual period actually varies betwee 35 miutes ad 2 hours. The iterval betwee successive eruptios however ca be quite accurately predicted based upo the duratio of the last eruptio.

2 II. Experimet A. The Pedulum Experimetal Procedure There are three pedulum elemets that we ca easily chage: the suspeded mass, the iitial amplitude, ad the strig's legth. We will test each of the elemets idividually to see which oe(s) affect the period of the pedulum. Record your data directly i a spreadsheet to save writig. Set up your pedulum by hagig a 200g mass from your stad with approximately 60cm of strig. Costruct a table usig your spreadsheet to record your data. Now, accurately measure the legth of the strig. NOTE: The legth is the distace from where the strig first touches the support stad to the ceter of gravity of the suspeded mass. Be very coscious of the chage i ceter of gravity height as you add or subtract weights from the pedulum. Record this legth ad the amout of suspeded mass i your table. We will first test the amplitude's effect o the period. Start with a small iitial amplitude (aroud 5-10 cm). Measure the amplitude horizotally from where the strig would hag aturally to where you are releasig the mass. Drop the mass from this amplitude ad measure how log it takes for the pedulum to make 20 cycles. HINT: If you release the mass from the left side of the stad, start your stop watch whe the mass completes its first full swig. The time 20 cycles. This will reduce much of the error iheret i tryig to start the stopwatch at the same time as the mass is released. Measure the time for 20 cycles two times ad record the average of the two readigs. Now icrease your amplitude by at least 5cm ad take 2 readigs of the time for 20 cycles agai. IMPORTANT NOTE: If you do ot otice a chage i the measured time, icrease the amplitude agai ad take oe more set of data, completig data for 3 differet amplitudes. If there is a differece i the time readigs, cotiue takig measuremets util you have data for 5 differet amplitudes, otherwise cotiue to the ext step below. Do ot take uecessary data. Now, test how the mass affects the period. This time do the etire experimet, chagig oly the mass. Keep the pedulum at the legth of 60 cm (ote that you may have to adjust your strig legth whe you chage the mass) ad a amplitude of your choosig. Choose weights betwee 20g ad 500g. Record your data for at least 3 differet masses. If you fid that mass chages the period, cotiue util you have data for 5 differet masses; otherwise proceed to the ext step. Next, test how the legth affects the period. For this part, use a pedulum

3 made up of a strig ad a small lead weight. Do the etire experimet, chagig oly the legth of the pedulum, usig legths from the smallest you ca achieve startig with these three legths: 4cm, 7cm ad 10cm. Record data for up to 5 differet legths if you otice a chage i the period of the pedulum. You ca use legths up to 100 cm for the other legths, if you eed more tha three legths. B. Experimetal Aalysis First, for each of the average time measuremet you have recorded, calculate the correspodig period, Timefor20cycles T = 20 Geerate a table with the amplitude i oe colum ad the correspodig average period i the ext colum to the right. Similarly, costruct a table for mass ad the correspodig average period, ad a table for legth ad the correspodig average period. Usig the spreadsheet s graphig fuctios, costruct graphs of period vs. amplitude, period vs. mass, ad period vs. strig legth. To graph your data easily make sure the legth, or mass, or amplitude are i a sigle colum, ad your correspodig period T, i the adjacet colum just to the right of it, with o spaces i the colums. If you did ot record your data this way you may easily geerate these adjacet colums by copyig ad pastig the data. Now select (highlight) these two colums. Now click o the chart wizard, third ico from the right o the tool bar which appears as various colored vertical bars. The chart type meu will appear Click o the x-y scatter plot. This is the oly optio which will produce a x-y plot i Excel. Click o ext. The graph should appear ad if it satisfactory click ext agai ad a meu to allow you to label axis ad title the graph will appear. Label your axes ad title the graph ad click fiish. You may adjust the size ad shape by clickig aywhere iside the border ad draggig the outlie with the poiter at oe of the small squares o the outlie. Make graphs of T (the period) versus each of the three variables, mass, legth ad amplitude. NOTE: I the graphs of period vs. amplitude ad the period vs. mass, you eed to adjust the scale o the y axis (set miimum to zero, maximum to 2) to get a better picture of the relatioship betwee these two variables. Make a pritout of these graphs for your report. It should be quite obvious ow which elemets affect the period of the pedulum. You should ow fid that oe of your graphs does ot give a straight horizotal lie plot ad looks like a curve. From this iformatio, ca we ow develop a algebraic expressio which relates the period to the quatities which cause it to chage? There are a ifiite umber of relatioships that are possible, but fortuately, may simple physical systems obey fairly simple relatioships. For example, the relatioship could be liear ( y = mx + b) where y could be the

4 period ad x the mass. Aother possibility would be a expoetial relatioship, kx y = e. You might recogize this as the radioactive decay law. Eve more commo is the power law behavior, y = Ax. The parameters that affect the period of the pedulum obey the power law relatioship. Power laws are difficult to work with, but there is a easy trick to simplify the procedures. By usig logarithms, we ca tur y = Ax ito log y = log x + log A. This looks very similar to the equatio of a straight lie ( y = mx + b). Therefore, if we costruct a plot of logy vs. log x, the slope of this lie will give us the expoet we are lookig for. (We will ot cocer ourselves with the costat A i this lab. It is a simple exercise to fid oce we have foud the power expoet). Select the graph for the elemet, or elemets, which affected the period of the pedulum. For this elemet which affects the period, we ow say that this elemet obeys a power law relatioship with the period of the pedulum, ad hece ca be aalyzed usig logarithms as show the previous paragraph. So ow, create a plot of Log T vs. log of the period affectig elemet. This will covert the curved plot to a straight lie log/log plot. Iclude a pritout of this graph i your report. Get the slope of the straight lie log/log plot. This slope gives us the expoet i the power law equatio relatig the period T ad the period affectig elemet. Usig the power law equatio y = Ax, get the equatio for the period of the pedulum by substitutig the period T for y, your slope for, ad replace x with a, m, or l (depedig o whether the amplitude, mass, or legth was the

5 elemet that affected the period). Cogratulatios, you have deduced the equatio for the period of a pedulum! III. Discussio Questios 1. Which of the three elemets (iitial amplitude, mass, or strig legth) affected the period of the pedulum? 2. Cosider two pedulums: Pedulum A has strig legth l, Pedulum B has strig legth 2l. Which pedulum has the logest period? Is it twice the period of the other pedulum? 3. Pedulum A has twice the mass of pedulum B. The force due to gravity actig o Pedulum A (2mg) is twice the magitude of the force due to gravity actig o Pedulum B (mg). Why the does't Pedulum A have twice the frequecy (half the period) of Pedulum B? 4. What is the actual equatio for the period of the pedulum (recall from lecture or search o the Web). What does it predict for the expoet ad how does it compare to the value of obtaied i your experimet? What is the value of the costat A i the actual equatio for the period of the pedulum? Origial Versio by Dr. Bill Moulto Modified for PHY2048 by B. Reyes, 4/2/2010

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A)

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A) REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data

More information

Problem 1. Problem Engineering Dynamics Problem Set 9--Solution. Find the equation of motion for the system shown with respect to:

Problem 1. Problem Engineering Dynamics Problem Set 9--Solution. Find the equation of motion for the system shown with respect to: 2.003 Egieerig Dyamics Problem Set 9--Solutio Problem 1 Fid the equatio of motio for the system show with respect to: a) Zero sprig force positio. Draw the appropriate free body diagram. b) Static equilibrium

More information

Chapter 4. Fourier Series

Chapter 4. Fourier Series Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials

3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered

More information

4.3 Growth Rates of Solutions to Recurrences

4.3 Growth Rates of Solutions to Recurrences 4.3. GROWTH RATES OF SOLUTIONS TO RECURRENCES 81 4.3 Growth Rates of Solutios to Recurreces 4.3.1 Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer.

More information

Math 2784 (or 2794W) University of Connecticut

Math 2784 (or 2794W) University of Connecticut ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really

More information

1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable

More information

mx bx kx F t. dt IR I LI V t, Q LQ RQ V t,

mx bx kx F t. dt IR I LI V t, Q LQ RQ V t, Lecture 5 omplex Variables II (Applicatios i Physics) (See hapter i Boas) To see why complex variables are so useful cosider first the (liear) mechaics of a sigle particle described by Newto s equatio

More information

Essential Question How can you recognize an arithmetic sequence from its graph?

Essential Question How can you recognize an arithmetic sequence from its graph? . Aalyzig Arithmetic Sequeces ad Series COMMON CORE Learig Stadards HSF-IF.A.3 HSF-BF.A. HSF-LE.A. Essetial Questio How ca you recogize a arithmetic sequece from its graph? I a arithmetic sequece, the

More information

CHAPTER 10 INFINITE SEQUENCES AND SERIES

CHAPTER 10 INFINITE SEQUENCES AND SERIES CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece

More information

Section 6.4: Series. Section 6.4 Series 413

Section 6.4: Series. Section 6.4 Series 413 ectio 64 eries 4 ectio 64: eries A couple decides to start a college fud for their daughter They pla to ivest $50 i the fud each moth The fud pays 6% aual iterest, compouded mothly How much moey will they

More information

EXPERIMENT OF SIMPLE VIBRATION

EXPERIMENT OF SIMPLE VIBRATION EXPERIMENT OF SIMPLE VIBRATION. PURPOSE The purpose of the experimet is to show free vibratio ad damped vibratio o a system havig oe degree of freedom ad to ivestigate the relatioship betwee the basic

More information

SEQUENCE AND SERIES NCERT

SEQUENCE AND SERIES NCERT 9. Overview By a sequece, we mea a arragemet of umbers i a defiite order accordig to some rule. We deote the terms of a sequece by a, a,..., etc., the subscript deotes the positio of the term. I view of

More information

Section 11.8: Power Series

Section 11.8: Power Series Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i

More information

INTEGRATION BY PARTS (TABLE METHOD)

INTEGRATION BY PARTS (TABLE METHOD) INTEGRATION BY PARTS (TABLE METHOD) Suppose you wat to evaluate cos d usig itegratio by parts. Usig the u dv otatio, we get So, u dv d cos du d v si cos d si si d or si si d We see that it is ecessary

More information

14.2 Simplifying Expressions with Rational Exponents and Radicals

14.2 Simplifying Expressions with Rational Exponents and Radicals Name Class Date 14. Simplifyig Expressios with Ratioal Expoets ad Radicals Essetial Questio: How ca you write a radical expressio as a expressio with a ratioal expoet? Resource Locker Explore Explorig

More information

Analysis of Experimental Data

Analysis of Experimental Data Aalysis of Experimetal Data 6544597.0479 ± 0.000005 g Quatitative Ucertaity Accuracy vs. Precisio Whe we make a measuremet i the laboratory, we eed to kow how good it is. We wat our measuremets to be both

More information

Signals & Systems Chapter3

Signals & Systems Chapter3 Sigals & Systems Chapter3 1.2 Discrete-Time (D-T) Sigals Electroic systems do most of the processig of a sigal usig a computer. A computer ca t directly process a C-T sigal but istead eeds a stream of

More information

Response Variable denoted by y it is the variable that is to be predicted measure of the outcome of an experiment also called the dependent variable

Response Variable denoted by y it is the variable that is to be predicted measure of the outcome of an experiment also called the dependent variable Statistics Chapter 4 Correlatio ad Regressio If we have two (or more) variables we are usually iterested i the relatioship betwee the variables. Associatio betwee Variables Two variables are associated

More information

P.3 Polynomials and Special products

P.3 Polynomials and Special products Precalc Fall 2016 Sectios P.3, 1.2, 1.3, P.4, 1.4, P.2 (radicals/ratioal expoets), 1.5, 1.6, 1.7, 1.8, 1.1, 2.1, 2.2 I Polyomial defiitio (p. 28) a x + a x +... + a x + a x 1 1 0 1 1 0 a x + a x +... +

More information

a is some real number (called the coefficient) other

a is some real number (called the coefficient) other Precalculus Notes for Sectio.1 Liear/Quadratic Fuctios ad Modelig http://www.schooltube.com/video/77e0a939a3344194bb4f Defiitios: A moomial is a term of the form tha zero ad is a oegative iteger. a where

More information

Polynomial Functions and Their Graphs

Polynomial Functions and Their Graphs Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively

More information

ARITHMETIC PROGRESSIONS

ARITHMETIC PROGRESSIONS CHAPTER 5 ARITHMETIC PROGRESSIONS (A) Mai Cocepts ad Results A arithmetic progressio (AP) is a list of umbers i which each term is obtaied by addig a fixed umber d to the precedig term, except the first

More information

Algebra II Notes Unit Seven: Powers, Roots, and Radicals

Algebra II Notes Unit Seven: Powers, Roots, and Radicals Syllabus Objectives: 7. The studets will use properties of ratioal epoets to simplify ad evaluate epressios. 7.8 The studet will solve equatios cotaiig radicals or ratioal epoets. b a, the b is the radical.

More information

Section 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations

Section 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?

More information

Synopsis of Euler s paper. E Memoire sur la plus grande equation des planetes. (Memoir on the Maximum value of an Equation of the Planets)

Synopsis of Euler s paper. E Memoire sur la plus grande equation des planetes. (Memoir on the Maximum value of an Equation of the Planets) 1 Syopsis of Euler s paper E105 -- Memoire sur la plus grade equatio des plaetes (Memoir o the Maximum value of a Equatio of the Plaets) Compiled by Thomas J Osler ad Jase Adrew Scaramazza Mathematics

More information

PH 425 Quantum Measurement and Spin Winter SPINS Lab 1

PH 425 Quantum Measurement and Spin Winter SPINS Lab 1 PH 425 Quatum Measuremet ad Spi Witer 23 SPIS Lab Measure the spi projectio S z alog the z-axis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured

More information

Zeros of Polynomials

Zeros of Polynomials Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree

More information

Position Time Graphs 12.1

Position Time Graphs 12.1 12.1 Positio Time Graphs Figure 3 Motio with fairly costat speed Chapter 12 Distace (m) A Crae Flyig Figure 1 Distace time graph showig motio with costat speed A Crae Flyig Positio (m [E] of pod) We kow

More information

FREE VIBRATION RESPONSE OF A SYSTEM WITH COULOMB DAMPING

FREE VIBRATION RESPONSE OF A SYSTEM WITH COULOMB DAMPING Mechaical Vibratios FREE VIBRATION RESPONSE OF A SYSTEM WITH COULOMB DAMPING A commo dampig mechaism occurrig i machies is caused by slidig frictio or dry frictio ad is called Coulomb dampig. Coulomb dampig

More information

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense, 3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [

More information

Summary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram.

Summary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram. Key Cocepts: 1) Sketchig of scatter diagram The scatter diagram of bivariate (i.e. cotaiig two variables) data ca be easily obtaied usig GC. Studets are advised to refer to lecture otes for the GC operatios

More information

Solutions to Final Exam

Solutions to Final Exam Solutios to Fial Exam 1. Three married couples are seated together at the couter at Moty s Blue Plate Dier, occupyig six cosecutive seats. How may arragemets are there with o wife sittig ext to her ow

More information

EDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 4 - CALCULUS

EDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 4 - CALCULUS EDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 4 - CALCULUS TUTORIAL 1 - DIFFERENTIATION Use the elemetary rules of calculus arithmetic to solve problems that ivolve differetiatio

More information

September 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1

September 2012 C1 Note. C1 Notes (Edexcel) Copyright   - For AS, A2 notes and IGCSE / GCSE worksheets 1 September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright

More information

NUMERICAL METHODS FOR SOLVING EQUATIONS

NUMERICAL METHODS FOR SOLVING EQUATIONS Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 1 of 11 M.K. HOME TUITION Mathematics Revisio Guides Level: GCSE Higher Tier NUMERICAL METHODS FOR SOLVING EQUATIONS Versio:. Date:

More information

FINALTERM EXAMINATION Fall 9 Calculus & Aalytical Geometry-I Questio No: ( Mars: ) - Please choose oe Let f ( x) is a fuctio such that as x approaches a real umber a, either from left or right-had-side,

More information

subcaptionfont+=small,labelformat=parens,labelsep=space,skip=6pt,list=0,hypcap=0 subcaption ALGEBRAIC COMBINATORICS LECTURE 8 TUESDAY, 2/16/2016

subcaptionfont+=small,labelformat=parens,labelsep=space,skip=6pt,list=0,hypcap=0 subcaption ALGEBRAIC COMBINATORICS LECTURE 8 TUESDAY, 2/16/2016 subcaptiofot+=small,labelformat=pares,labelsep=space,skip=6pt,list=0,hypcap=0 subcaptio ALGEBRAIC COMBINATORICS LECTURE 8 TUESDAY, /6/06. Self-cojugate Partitios Recall that, give a partitio λ, we may

More information

Solutions to Final Exam Review Problems

Solutions to Final Exam Review Problems . Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the

More information

( ) = p and P( i = b) = q.

( ) = p and P( i = b) = q. MATH 540 Radom Walks Part 1 A radom walk X is special stochastic process that measures the height (or value) of a particle that radomly moves upward or dowward certai fixed amouts o each uit icremet of

More information

11.1 Radical Expressions and Rational Exponents

11.1 Radical Expressions and Rational Exponents Name Class Date 11.1 Radical Expressios ad Ratioal Expoets Essetial Questio: How are ratioal expoets related to radicals ad roots? Resource Locker Explore Defiig Ratioal Expoets i Terms of Roots Remember

More information

A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence

A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as

More information

Introduction to Signals and Systems, Part V: Lecture Summary

Introduction to Signals and Systems, Part V: Lecture Summary EEL33: Discrete-Time Sigals ad Systems Itroductio to Sigals ad Systems, Part V: Lecture Summary Itroductio to Sigals ad Systems, Part V: Lecture Summary So far we have oly looked at examples of o-recursive

More information

Machine Learning for Data Science (CS 4786)

Machine Learning for Data Science (CS 4786) Machie Learig for Data Sciece CS 4786) Lecture & 3: Pricipal Compoet Aalysis The text i black outlies high level ideas. The text i blue provides simple mathematical details to derive or get to the algorithm

More information

Analytic Theory of Probabilities

Analytic Theory of Probabilities Aalytic Theory of Probabilities PS Laplace Book II Chapter II, 4 pp 94 03 4 A lottery beig composed of umbered tickets of which r exit at each drawig, oe requires the probability that after i drawigs all

More information

Math 312 Lecture Notes One Dimensional Maps

Math 312 Lecture Notes One Dimensional Maps Math 312 Lecture Notes Oe Dimesioal Maps Warre Weckesser Departmet of Mathematics Colgate Uiversity 21-23 February 25 A Example We begi with the simplest model of populatio growth. Suppose, for example,

More information

On a Smarandache problem concerning the prime gaps

On a Smarandache problem concerning the prime gaps O a Smaradache problem cocerig the prime gaps Felice Russo Via A. Ifate 7 6705 Avezzao (Aq) Italy felice.russo@katamail.com Abstract I this paper, a problem posed i [] by Smaradache cocerig the prime gaps

More information

Dotting The Dot Map, Revisited. A. Jon Kimerling Dept. of Geosciences Oregon State University

Dotting The Dot Map, Revisited. A. Jon Kimerling Dept. of Geosciences Oregon State University Dottig The Dot Map, Revisited A. Jo Kimerlig Dept. of Geoscieces Orego State Uiversity Dot maps show the geographic distributio of features i a area by placig dots represetig a certai quatity of features

More information

Math 116 Second Midterm November 13, 2017

Math 116 Second Midterm November 13, 2017 Math 6 Secod Midterm November 3, 7 EXAM SOLUTIONS. Do ot ope this exam util you are told to do so.. Do ot write your ame aywhere o this exam. 3. This exam has pages icludig this cover. There are problems.

More information

The z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j

The z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.

More information

Principle Of Superposition

Principle Of Superposition ecture 5: PREIMINRY CONCEP O RUCUR NYI Priciple Of uperpositio Mathematically, the priciple of superpositio is stated as ( a ) G( a ) G( ) G a a or for a liear structural system, the respose at a give

More information

Quadratic Functions. Before we start looking at polynomials, we should know some common terminology.

Quadratic Functions. Before we start looking at polynomials, we should know some common terminology. Quadratic Fuctios I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively i mathematical

More information

Statistics 511 Additional Materials

Statistics 511 Additional Materials Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability

More information

Regression, Part I. A) Correlation describes the relationship between two variables, where neither is independent or a predictor.

Regression, Part I. A) Correlation describes the relationship between two variables, where neither is independent or a predictor. Regressio, Part I I. Differece from correlatio. II. Basic idea: A) Correlatio describes the relatioship betwee two variables, where either is idepedet or a predictor. - I correlatio, it would be irrelevat

More information

SOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1.

SOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1. SOLUTIONS TO EXAM 3 Problem Fid the sum of the followig series 2 + ( ) 5 5 2 5 3 25 2 2 This series diverges Solutio: Note that this defies two coverget geometric series with respective radii r 2/5 < ad

More information

x a x a Lecture 2 Series (See Chapter 1 in Boas)

x a x a Lecture 2 Series (See Chapter 1 in Boas) Lecture Series (See Chapter i Boas) A basic ad very powerful (if pedestria, recall we are lazy AD smart) way to solve ay differetial (or itegral) equatio is via a series expasio of the correspodig solutio

More information

Math 113, Calculus II Winter 2007 Final Exam Solutions

Math 113, Calculus II Winter 2007 Final Exam Solutions Math, Calculus II Witer 7 Fial Exam Solutios (5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute x x + dx The check your aswer usig the Evaluatio Theorem Solutio: I this

More information

. (24) If we consider the geometry of Figure 13 the signal returned from the n th scatterer located at x, y is

. (24) If we consider the geometry of Figure 13 the signal returned from the n th scatterer located at x, y is .5 SAR SIGNA CHARACTERIZATION I order to formulate a SAR processor we first eed to characterize the sigal that the SAR processor will operate upo. Although our previous discussios treated SAR cross-rage

More information

Riemann Sums y = f (x)

Riemann Sums y = f (x) Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, o-egative fuctio o the closed iterval [a, b] Fid

More information

multiplies all measures of center and the standard deviation and range by k, while the variance is multiplied by k 2.

multiplies all measures of center and the standard deviation and range by k, while the variance is multiplied by k 2. Lesso 3- Lesso 3- Scale Chages of Data Vocabulary scale chage of a data set scale factor scale image BIG IDEA Multiplyig every umber i a data set by k multiplies all measures of ceter ad the stadard deviatio

More information

The multiplicative structure of finite field and a construction of LRC

The multiplicative structure of finite field and a construction of LRC IERG6120 Codig for Distributed Storage Systems Lecture 8-06/10/2016 The multiplicative structure of fiite field ad a costructio of LRC Lecturer: Keeth Shum Scribe: Zhouyi Hu Notatios: We use the otatio

More information

SPEC/4/PHYSI/SPM/ENG/TZ0/XX PHYSICS PAPER 1 SPECIMEN PAPER. 45 minutes INSTRUCTIONS TO CANDIDATES

SPEC/4/PHYSI/SPM/ENG/TZ0/XX PHYSICS PAPER 1 SPECIMEN PAPER. 45 minutes INSTRUCTIONS TO CANDIDATES SPEC/4/PHYSI/SPM/ENG/TZ0/XX PHYSICS STANDARD LEVEL PAPER 1 SPECIMEN PAPER 45 miutes INSTRUCTIONS TO CANDIDATES Do ot ope this examiatio paper util istructed to do so. Aswer all the questios. For each questio,

More information

Recurrence Relations

Recurrence Relations Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The

More information

Linear Regression Demystified

Linear Regression Demystified Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to

More information

Activity 3: Length Measurements with the Four-Sided Meter Stick

Activity 3: Length Measurements with the Four-Sided Meter Stick Activity 3: Legth Measuremets with the Four-Sided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a four-sided meter

More information

17 Phonons and conduction electrons in solids (Hiroshi Matsuoka)

17 Phonons and conduction electrons in solids (Hiroshi Matsuoka) 7 Phoos ad coductio electros i solids Hiroshi Matsuoa I this chapter we will discuss a miimal microscopic model for phoos i a solid ad a miimal microscopic model for coductio electros i a simple metal.

More information

ECON 3150/4150, Spring term Lecture 3

ECON 3150/4150, Spring term Lecture 3 Itroductio Fidig the best fit by regressio Residuals ad R-sq Regressio ad causality Summary ad ext step ECON 3150/4150, Sprig term 2014. Lecture 3 Ragar Nymoe Uiversity of Oslo 21 Jauary 2014 1 / 30 Itroductio

More information

18th Bay Area Mathematical Olympiad. Problems and Solutions. February 23, 2016

18th Bay Area Mathematical Olympiad. Problems and Solutions. February 23, 2016 18th Bay Area Mathematical Olympiad February 3, 016 Problems ad Solutios BAMO-8 ad BAMO-1 are each 5-questio essay-proof exams, for middle- ad high-school studets, respectively. The problems i each exam

More information

f(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim

f(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =

More information

SEQUENCES AND SERIES

SEQUENCES AND SERIES Sequeces ad 6 Sequeces Ad SEQUENCES AND SERIES Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives rise to what is called a sequece. Sequeces

More information

t distribution [34] : used to test a mean against an hypothesized value (H 0 : µ = µ 0 ) or the difference

t distribution [34] : used to test a mean against an hypothesized value (H 0 : µ = µ 0 ) or the difference EXST30 Backgroud material Page From the textbook The Statistical Sleuth Mea [0]: I your text the word mea deotes a populatio mea (µ) while the work average deotes a sample average ( ). Variace [0]: The

More information

Z ß cos x + si x R du We start with the substitutio u = si(x), so du = cos(x). The itegral becomes but +u we should chage the limits to go with the ew

Z ß cos x + si x R du We start with the substitutio u = si(x), so du = cos(x). The itegral becomes but +u we should chage the limits to go with the ew Problem ( poits) Evaluate the itegrals Z p x 9 x We ca draw a right triagle labeled this way x p x 9 From this we ca read off x = sec, so = sec ta, ad p x 9 = R ta. Puttig those pieces ito the itegralrwe

More information

Practice Problems: Taylor and Maclaurin Series

Practice Problems: Taylor and Maclaurin Series Practice Problems: Taylor ad Maclauri Series Aswers. a) Start by takig derivatives util a patter develops that lets you to write a geeral formula for the -th derivative. Do t simplify as you go, because

More information

Upper bound for ropelength of pretzel knots

Upper bound for ropelength of pretzel knots Upper boud for ropelegth of pretzel kots Safiya Mora August 25, 2006 Abstract A model of the pretzel kot is described. A method for predictig the ropelegth of pretzel kots is give. A upper boud for the

More information

Assignment 2 Solutions SOLUTION. ϕ 1 Â = 3 ϕ 1 4i ϕ 2. The other case can be dealt with in a similar way. { ϕ 2 Â} χ = { 4i ϕ 1 3 ϕ 2 } χ.

Assignment 2 Solutions SOLUTION. ϕ 1  = 3 ϕ 1 4i ϕ 2. The other case can be dealt with in a similar way. { ϕ 2 Â} χ = { 4i ϕ 1 3 ϕ 2 } χ. PHYSICS 34 QUANTUM PHYSICS II (25) Assigmet 2 Solutios 1. With respect to a pair of orthoormal vectors ϕ 1 ad ϕ 2 that spa the Hilbert space H of a certai system, the operator  is defied by its actio

More information

Substitute these values into the first equation to get ( z + 6) + ( z + 3) + z = 27. Then solve to get

Substitute these values into the first equation to get ( z + 6) + ( z + 3) + z = 27. Then solve to get Problem ) The sum of three umbers is 7. The largest mius the smallest is 6. The secod largest mius the smallest is. What are the three umbers? [Problem submitted by Vi Lee, LCC Professor of Mathematics.

More information

ENGI 4421 Probability and Statistics Faculty of Engineering and Applied Science Problem Set 1 Solutions Descriptive Statistics. None at all!

ENGI 4421 Probability and Statistics Faculty of Engineering and Applied Science Problem Set 1 Solutions Descriptive Statistics. None at all! ENGI 44 Probability ad Statistics Faculty of Egieerig ad Applied Sciece Problem Set Solutios Descriptive Statistics. If, i the set of values {,, 3, 4, 5, 6, 7 } a error causes the value 5 to be replaced

More information

is also known as the general term of the sequence

is also known as the general term of the sequence Lesso : Sequeces ad Series Outlie Objectives: I ca determie whether a sequece has a patter. I ca determie whether a sequece ca be geeralized to fid a formula for the geeral term i the sequece. I ca determie

More information

Bertrand s Postulate

Bertrand s Postulate Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a

More information

Mathematics for Queensland, Year 12 Worked Solutions to Exercise 5L

Mathematics for Queensland, Year 12 Worked Solutions to Exercise 5L Mathematics for Queeslad, Year 12 Worked Solutios to Exercise 5L Hits Mathematics for Queeslad, Year 12 Mathematics B, A Graphics Calculator Approach http://mathematics-for-queeslad.com 23. Kiddy Bolger,

More information

Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 19

Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 19 CS 70 Discrete Mathematics ad Probability Theory Sprig 2016 Rao ad Walrad Note 19 Some Importat Distributios Recall our basic probabilistic experimet of tossig a biased coi times. This is a very simple

More information

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0.

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0. THE SOLUTION OF NONLINEAR EQUATIONS f( ) = 0. Noliear Equatio Solvers Bracketig. Graphical. Aalytical Ope Methods Bisectio False Positio (Regula-Falsi) Fied poit iteratio Newto Raphso Secat The root of

More information

The Method of Least Squares. To understand least squares fitting of data.

The Method of Least Squares. To understand least squares fitting of data. The Method of Least Squares KEY WORDS Curve fittig, least square GOAL To uderstad least squares fittig of data To uderstad the least squares solutio of icosistet systems of liear equatios 1 Motivatio Curve

More information

Chapter 7 z-transform

Chapter 7 z-transform Chapter 7 -Trasform Itroductio Trasform Uilateral Trasform Properties Uilateral Trasform Iversio of Uilateral Trasform Determiig the Frequecy Respose from Poles ad Zeros Itroductio Role i Discrete-Time

More information

7 Sequences of real numbers

7 Sequences of real numbers 40 7 Sequeces of real umbers 7. Defiitios ad examples Defiitio 7... A sequece of real umbers is a real fuctio whose domai is the set N of atural umbers. Let s : N R be a sequece. The the values of s are

More information

CALCULUS BASIC SUMMER REVIEW

CALCULUS BASIC SUMMER REVIEW CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=

More information

The Random Walk For Dummies

The Random Walk For Dummies The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli

More information

Linear Regression Analysis. Analysis of paired data and using a given value of one variable to predict the value of the other

Linear Regression Analysis. Analysis of paired data and using a given value of one variable to predict the value of the other Liear Regressio Aalysis Aalysis of paired data ad usig a give value of oe variable to predict the value of the other 5 5 15 15 1 1 5 5 1 3 4 5 6 7 8 1 3 4 5 6 7 8 Liear Regressio Aalysis E: The chirp rate

More information

UNIT #5. Lesson #2 Arithmetic and Geometric Sequences. Lesson #3 Summation Notation. Lesson #4 Arithmetic Series. Lesson #5 Geometric Series

UNIT #5. Lesson #2 Arithmetic and Geometric Sequences. Lesson #3 Summation Notation. Lesson #4 Arithmetic Series. Lesson #5 Geometric Series UNIT #5 SEQUENCES AND SERIES Lesso # Sequeces Lesso # Arithmetic ad Geometric Sequeces Lesso #3 Summatio Notatio Lesso #4 Arithmetic Series Lesso #5 Geometric Series Lesso #6 Mortgage Paymets COMMON CORE

More information

The Minimum Distance Energy for Polygonal Unknots

The Minimum Distance Energy for Polygonal Unknots The Miimum Distace Eergy for Polygoal Ukots By:Johaa Tam Advisor: Rollad Trapp Abstract This paper ivestigates the eergy U MD of polygoal ukots It provides equatios for fidig the eergy for ay plaar regular

More information

Appendix: The Laplace Transform

Appendix: The Laplace Transform Appedix: The Laplace Trasform The Laplace trasform is a powerful method that ca be used to solve differetial equatio, ad other mathematical problems. Its stregth lies i the fact that it allows the trasformatio

More information

Math 155 (Lecture 3)

Math 155 (Lecture 3) Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,

More information

Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day

Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the

More information

Alternating Series. 1 n 0 2 n n THEOREM 9.14 Alternating Series Test Let a n > 0. The alternating series. 1 n a n.

Alternating Series. 1 n 0 2 n n THEOREM 9.14 Alternating Series Test Let a n > 0. The alternating series. 1 n a n. 0_0905.qxd //0 :7 PM Page SECTION 9.5 Alteratig Series Sectio 9.5 Alteratig Series Use the Alteratig Series Test to determie whether a ifiite series coverges. Use the Alteratig Series Remaider to approximate

More information

11 Correlation and Regression

11 Correlation and Regression 11 Correlatio Regressio 11.1 Multivariate Data Ofte we look at data where several variables are recorded for the same idividuals or samplig uits. For example, at a coastal weather statio, we might record

More information

TEACHER CERTIFICATION STUDY GUIDE

TEACHER CERTIFICATION STUDY GUIDE COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra

More information

Probability, Expectation Value and Uncertainty

Probability, Expectation Value and Uncertainty Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such

More information

Sequences, Series, and All That

Sequences, Series, and All That Chapter Te Sequeces, Series, ad All That. Itroductio Suppose we wat to compute a approximatio of the umber e by usig the Taylor polyomial p for f ( x) = e x at a =. This polyomial is easily see to be 3

More information

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3 MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special

More information