# a) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points.

Save this PDF as:

Size: px
Start display at page:

Download "a) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points."

## Transcription

1 Prole 3: Crnot Cyle of n Idel Gs In this prole, the strting pressure P nd volue of n idel gs in stte, re given he rtio R = / > of the volues of the sttes nd is given Finlly onstnt γ = 5/3 is given You do not know how ny oles of the gs re present Red over steps (- (4 elow nd sketh the pth of the yle on P plot on the grph elow Lel ll pproprite points ( In the first of four steps, to, n idel gs is opressed fro to while no het is llowed to flow into or out of the syste he opression of the gs rises the teperture fro n initil teperture nd to finl teperture During this proess the quntity γ = onstnt, where γ = 5/3 Wht is the pressure P nd volue of the stte of the gs fter the opression is finished? Wht is the hnge in internl energy of the gs during this hnge of stte? Wht is the work done y the gs during this opression? ( he gs is now llowed to expnd isotherlly fro to, fro volue to volue d Express the work done y the gs in this proess W nd the ount of het Q tht ust e dded fro the het soure t in ters of P,,,, nd

2 Is this het positive or negtive? Explin whether it is dded to the syste or reoved e Wht is the pressure P of the gs fter the expnsion is finished? (3 When the gs hs rehed point is expnds fro to d while no het is llowed to flow into or out of the syste he expnsion of the gs lowers the teperture nd pressure fro n initil teperture to finl teperture During this proess the quntity γ = onstnt f Wht is the pressure P nd volue d of the stte d of the gs fter the d expnsion is finished? g Wht is the hnge in internl energy of the gs during this hnge of stte? h Wht is the work done y the gs during this expnsion? (4 he gs is now opressed isotherlly fro d to t onstnt fro volue d k to i Find the work done y the syste on the surroundings W d nd the ount of het Qd tht flows etween the syste nd the surroundings Are these quntities positive or negtive? Explin whether het is dded to the syste or reoved fro the het soure t otl Cyle: j Wht is the totl work W yle done y the gs during this yle? k Wht is the totl het Q yle ( fro drwn fro the higher teperture het soure during this yle? l Wht is the effiieny of this yle ε x = W yle / Q yle ( fro? Prole 4 Het pup A reversile het engine n e run in the other diretion, in whih se it does negtive work W yle on the world while puping het Q yle (into into reservoir t n upper teperture,, fro lower teperture, he het gin of this yle, defined to e

3 g Q yle (into /W yle = (/ ε x where ε = ( x / is the xiu therodyni effiieny of het engine he refrigertor perforne is defined to e K Q yle ( fro / W yle = /( Consider tht you hve lrge swiing pool nd pln to het your house with het pup tht pups het fro the pool into your house A lrge plte in the wter will rein t 0 o C due to the fortion of ie You pik to e 50 o C, whih will e the teperture of the (lrge rditors used to het your house Assue tht your het pup hs the xiu effiieny llowed y therodynis Wht is the het gin nd the refrigertor perforne for this yle? Be reful to use units of Kelvin for teperture If your house forerly urned 00 gllons of oil in winter (t \$00/gllon, how uh will the eletriity ost (t \$00 per kilowtt-hour to reple this het using 8 the het pup? A gllon of oil hs ss 34 kg nd ontins 4 0 J gl - he ie ue tht ppers in your pool over the winter will e how ny eters on 6 eh side? (It tkes J to elt one kg of ie; it tkes up this uh het when 3 freezing he density of ie is kg -3 his would e gret for ooling your house in the suer even if the pool wred up enough to swi in it, you ould still ool your house y running the het pup in reverse s n ir onditioner! More prtilly, you ight e le to use ground wter (nd the dirt round it s the het sink

4 Prole 3: Crnot Cyle of n Idel Gs In this prole, the strting pressure P nd volue of n idel gs in stte, re given he rtio R = / > of the volues of the sttes nd is given Finlly onstnt γ = 5/3 is given You do not know how ny oles of the gs re present Red over steps (- (4 elow nd sketh the pth of the yle on P plot on the grph elow Lel ll pproprite points ( In the first of four steps, to, n idel gs is opressed fro to while no het is llowed to flow into or out of the syste he opression of the gs rises the teperture fro n initil teperture nd to finl teperture During this proess the quntity γ = onstnt, where γ = 5/3 Wht is the pressure P nd volue of the gs fter the opression is finished? Answer: Aording to the idel gs lw, = n R nd = n R so So the pressure = P = P he opression stisfies γ = γ so using the ove result for pressure P, we get 6

5 γ = P γ = γ his eoes using γ = 5/3 / 3 = /3 he volue is then 3/ = hus the rtio of the volues is 3/ = So the pressure P is 5/ P = P Wht is the hnge in internl energy of the gs during this hnge of stte? Answer: he hnge in internl energy is 3 3 ( U U = n R = Wht is the work done y the gs during this opression? Answer: Sine no het is exhnged Q = 0 U U = W + Q = W = 3 ( So 3 ( W = < 0 7

6 he surroundings do work opressing the gs ( he gs is now llowed to expnd isotherlly fro to, fro volue to volue d Express the work done y the gs in this proess W nd the ount of het Q tht ust e dded fro the het soure t in ters of P,,,, nd Is this het positive or negtive? Explin whether it is dded to the syste or reoved Answer: his is n isotherl expnsion so the teperture does not hnge the internl energy is onstnt, =0 hus 3 U U = n R = 0 he gs does work on the surroundings euse it is expnding he pressure is not onstnt during this expnsion Sine the gs is expnding y n isotherl proess, the Idel Gs Lw reltes the pressure nd volue vrition ording to nr P = herefore the work done y the gs on the surroundings is the integrl d W = n R = n R ln( / Using the result for the volue fro prt the work is 3/ =, = n R d = n R ln( 3/ W / Rell tht the volues re relted ording to R = / > 0 nd nr = P / so the work done is positive nd given y 8

7 W = n R ln( / = ln( 3/ R > 0 Fro he First Lw of herodynis, 0 = U U = W + Q, hus the het tht flows into the syste fro the het soure t teperture is equl to the work done y the expnding gs Q = W = ln( 3/ R > 0, Note tht this het flow ust flow fro the higher teperture het soure into the syste euse s the gs expnds it should lose internl energy nd would derese its teperture unless het flows into the syste keeping the internl energy nd hene the teperture onstnt e Wht is the pressure P of the gs fter the expnsion is finished? Answer: = n R = hus P = = P R (3 When the gs hs rehed point is expnds fro to d while no het is llowed to flow into or out of the syste he expnsion of the gs lowers the teperture nd pressure fro n initil teperture to finl teperture During this proess the quntity γ = onstnt f Wht is the pressure P d nd the volue d of the stte d of the gs fter the expnsion is finished? Answer: his lultion is identil to prt, with stte d repling stte, nd stte repling stte So the volue is then 9

8 3/ = d hus the rtio of the volues is So the pressure P is hene d 3/ = 5/ P = P d 5/ P d = P g Wht is the hnge in internl energy of the gs during this hnge of stte? Answer: he derese in the internl energy is due to the teperture derese of the idel gs during expnsion 3 ( U d U = P h Wht is the work done y the gs during this expnsion? Answer: Sine no het is exhnged Q d = 0 U d U = W d + Q = W d = 3 ( d P So 3 ( W d = P > 0 he gs does work on the surroundings sine the gs is expnding (4 he gs is now opressed isotherlly fro d to t onstnt fro volue d k to i Find the work done y the syste on the surroundings W d nd the ount of het Q tht flows etween the syste nd the surroundings Are these quntities d 0

9 positive or negtive? Explin whether het is dded to the syste or reoved fro the het soure t Answer: When the gs undergoes opression it will inrese its internl energy ut het flows out of the syste intining onstnt internl energy, U = 0 nd hene the opression is isotherl he lultion of the work nd het is siilr to step ( exept the teperture is held t he work done y the syste on the surroundings is negtive nd is given y the integrl W d d = n R = n R ln( / = P ln( / = P ln(r d d d 3/ Fro prt f the volue d = so the work done is d 3/ 3/ W d = n R = n R ln( / = P ln( / = P ln( R d d Aording to the First Lw this is equl to the het tht flows into the syste whih is lso negtive whih ens tht it tully flows out of the syste into the surroundings t teperture, 3/ Q d = W d = P ln( R otl Cyle: j Wht is the totl work W yle done y the gs during this yle? Answer: he work done y the het engine on the surroundings during the yle is positive nd given y W yle = P ln( 3/ R P 3/ 3/ ln( R = P ln( R k Wht is the totl het Q yle ( fro drwn fro the higher teperture het soure during this yle? Answer: he het tht flowed fro the higher teperture het soure ourred during step ( isotherl expnsion,

10 totl Q tken fro het soure t = P ln( 3/ R l Wht is the effiieny of this yle ε x = W yle / Q yle ( fro? Answer: he effiieny is given y rtio of the work done divided y the eht flowing into the syste fro the higher teperture het soure 3/ 3/ ε x = W yle / Q yle ( fro = P ln( R / P ln( R ε = x / = = le : Sury of Het Engine Proess U f U i W f,i Q f,i 3 ( 0 diti 3 ( opression isotherl expnsion d diti expnsion d isotherl opression otl positive 0 3 ( negtive 0 0 negtive 3/ ln( R positive 3 ( positive 3/ ln( R negtive 3/ R 3/ ln( R ln( positive 3/ R ln( positive fro 0 3/ ln( R negtive, into 3/ R 3/ ln( R ln( positive, fro into

### Thermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R

/10/010 Question 1 1 mole of idel gs is rought to finl stte F y one of three proesses tht hve different initil sttes s shown in the figure. Wht is true for the temperture hnge etween initil nd finl sttes?

### U Q W The First Law of Thermodynamics. Efficiency. Closed cycle steam power plant. First page of S. Carnot s paper. Sadi Carnot ( )

0-9-0 he First Lw of hermoynmis Effiieny When severl lterntive proesses involving het n work re ville to hnge system from n initil stte hrterize y given vlues of the mrosopi prmeters (pressure p i, temperture

### Solutions to Assignment 1

MTHE 237 Fll 2015 Solutions to Assignment 1 Problem 1 Find the order of the differentil eqution: t d3 y dt 3 +t2 y = os(t. Is the differentil eqution liner? Is the eqution homogeneous? b Repet the bove

### Chapter 19: The Second Law of Thermodynamics

hpter 9: he Seon Lw of hermoynmis Diretions of thermoynmi proesses Irreversile n reversile proesses hermoynmi proesses tht our in nture re ll irreversile proesses whih proee spontneously in one iretion

### Non Right Angled Triangles

Non Right ngled Tringles Non Right ngled Tringles urriulum Redy www.mthletis.om Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in non-right ngled tringles. This unit

### (a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.

Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time

### Project 6: Minigoals Towards Simplifying and Rewriting Expressions

MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy

### NON-DETERMINISTIC FSA

Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is

### Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)

Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte single-vrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion

### STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

### Nondeterministic Finite Automata

Nondeterministi Finite utomt The Power of Guessing Tuesdy, Otoer 4, 2 Reding: Sipser.2 (first prt); Stoughton 3.3 3.5 S235 Lnguges nd utomt eprtment of omputer Siene Wellesley ollege Finite utomton (F)

### Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.

27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we

### Physics 9 Fall 2011 Homework 2 - Solutions Friday September 2, 2011

Physics 9 Fll 0 Homework - s Fridy September, 0 Mke sure your nme is on your homework, nd plese box your finl nswer. Becuse we will be giving prtil credit, be sure to ttempt ll the problems, even if you

### Unit #10 De+inite Integration & The Fundamental Theorem Of Calculus

Unit # De+inite Integrtion & The Fundmentl Theorem Of Clculus. Find the re of the shded region ove nd explin the mening of your nswer. (squres re y units) ) The grph to the right is f(x) = -x + 8x )Use

### Chapter 4 The second law of thermodynamics

hpter 4 he second lw of thermodynmics Directions of thermodynmic processes et engines Internl-combustion engines Refrigertors he second lw of thermodynmics he rnotcycle Entropy Directions of thermodynmic

### CIT 596 Theory of Computation 1. Graphs and Digraphs

CIT 596 Theory of Computtion 1 A grph G = (V (G), E(G)) onsists of two finite sets: V (G), the vertex set of the grph, often enote y just V, whih is nonempty set of elements lle verties, n E(G), the ege

### Section 6.1 Definite Integral

Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

### 1.3 SCALARS AND VECTORS

Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd

### Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point

GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

QUADRATIC EQUATION EXERCISE - 0 CHECK YOUR GRASP. Sine sum of oeffiients 0. Hint : It's one root is nd other root is 8 nd 5 5. tn other root 9. q 4p 0 q p q p, q 4 p,,, 4 Hene 7 vlues of (p, q) 7 equtions

### Prefix-Free Regular-Expression Matching

Prefix-Free Regulr-Expression Mthing Yo-Su Hn, Yjun Wng nd Derik Wood Deprtment of Computer Siene HKUST Prefix-Free Regulr-Expression Mthing p.1/15 Pttern Mthing Given pttern P nd text T, find ll sustrings

QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,

### Department of Mechanical Engineering ME 322 Mechanical Engineering Thermodynamics. Lecture 33. Psychrometric Properties of Moist Air

Deprtment of Mechnicl Engineering ME 3 Mechnicl Engineering hermodynmics Lecture 33 sychrometric roperties of Moist Air Air-Wter Vpor Mixtures Atmospheric ir A binry mixture of dry ir () + ter vpor ()

### Chem Homework 11 due Monday, Apr. 28, 2014, 2 PM

Chem 44 - Homework due ondy, pr. 8, 4, P.. . Put this in eq 8.4 terms: E m = m h /m e L for L=d The degenery in the ring system nd the inresed sping per level (4x bigger) mkes the sping between the HOO

### CS 310 (sec 20) - Winter Final Exam (solutions) SOLUTIONS

CS 310 (sec 20) - Winter 2003 - Finl Exm (solutions) SOLUTIONS 1. (Logic) Use truth tles to prove the following logicl equivlences: () p q (p p) (q q) () p q (p q) (p q) () p q p q p p q q (q q) (p p)

### Graph Theory. Simple Graph G = (V, E). V={a,b,c,d,e,f,g,h,k} E={(a,b),(a,g),( a,h),(a,k),(b,c),(b,k),...,(h,k)}

Grph Theory Simple Grph G = (V, E). V ={verties}, E={eges}. h k g f e V={,,,,e,f,g,h,k} E={(,),(,g),(,h),(,k),(,),(,k),...,(h,k)} E =16. 1 Grph or Multi-Grph We llow loops n multiple eges. G = (V, E.ψ)

### Geometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272.

Geometry of the irle - hords nd ngles Geometry of the irle hord nd ngles urriulum Redy MMG: 272 www.mthletis.om hords nd ngles HRS N NGLES The irle is si shpe nd so it n e found lmost nywhere. This setion

### ChE 548 Final Exam Spring, 2004

. Keffer, eprtment of Chemil Engineering, University of ennessee ChE 58 Finl Em Spring, Problem. Consider single-omponent, inompressible flid moving down n ninslted fnnel. erive the energy blne for this

### Homework Solution - Set 5 Due: Friday 10/03/08

CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.

### for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx

Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion

### Line Integrals and Entire Functions

Line Integrls nd Entire Funtions Defining n Integrl for omplex Vlued Funtions In the following setions, our min gol is to show tht every entire funtion n be represented s n everywhere onvergent power series

### 10. AREAS BETWEEN CURVES

. AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

### The Emission-Absorption of Energy analyzed by Quantum-Relativity. Abstract

The mission-absorption of nergy nlyzed by Quntum-Reltivity Alfred Bennun* & Néstor Ledesm** Abstrt The uslity horizon llows progressive quntifition, from n initil nk prtile, whih yields its energy s blk

### Math 8 Winter 2015 Applications of Integration

Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

### Homework Assignment 3 Solution Set

Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.

### m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r

CO-ORDINTE GEOMETR II I Qudrnt Qudrnt (-.+) (++) X X - - - 0 - III IV Qudrnt - Qudrnt (--) - (+-) Region CRTESIN CO-ORDINTE SSTEM : Retngulr Co-ordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr

### Data Compression Techniques (Spring 2012) Model Solutions for Exercise 4

58487 Dt Compressio Tehiques (Sprig 0) Moel Solutios for Exerise 4 If you hve y fee or orretios, plese ott jro.lo t s.helsii.fi.. Prolem: Let T = Σ = {,,, }. Eoe T usig ptive Huffm oig. Solutio: R 4 U

### Chapter Gauss Quadrature Rule of Integration

Chpter 7. Guss Qudrture Rule o Integrtion Ater reding this hpter, you should e le to:. derive the Guss qudrture method or integrtion nd e le to use it to solve prolems, nd. use Guss qudrture method to

### In the next lecture... Tutorial on ideal cycles and component performance.

In the next lecture... utoril on idel cycles nd component performnce. rof. Bhskr Roy, rof. A M rdeep, Deprtment of Aerospce, II Bomby roblem # Lect-9 A Bryton cycle opertes with regenertor of 75% effectiveness.

### Chapter 4 Regular Grammar and Regular Sets. (Solutions / Hints)

C K Ngpl Forml Lnguges nd utomt Theory Chpter 4 Regulr Grmmr nd Regulr ets (olutions / Hints) ol. (),,,,,,,,,,,,,,,,,,,,,,,,,, (),, (c) c c, c c, c, c, c c, c, c, c, c, c, c, c c,c, c, c, c, c, c, c, c,

### Physics 505 Homework No. 11 Solutions S11-1

Physis 55 Homework No 11 s S11-1 1 This problem is from the My, 24 Prelims Hydrogen moleule Consider the neutrl hydrogen moleule, H 2 Write down the Hmiltonin keeping only the kineti energy terms nd the

### PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.

PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic

### Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

### Section 4.4. Green s Theorem

The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higher-dimensionl nlogues) with the definite integrls

### dy ky, dt where proportionality constant k may be positive or negative

Section 1.2 Autonomous DEs of the form 0 The DE y is mthemticl model for wide vriety of pplictions. Some of the pplictions re descried y sying the rte of chnge of y(t) is proportionl to the mount present.

### CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)

CS 373, Spring 29. Solutions to Mock midterm (sed on first midterm in CS 273, Fll 28.) Prolem : Short nswer (8 points) The nswers to these prolems should e short nd not complicted. () If n NF M ccepts

### Triangles The following examples explore aspects of triangles:

Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x - 4 +x xmple 3: ltitude of the

### MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

### Homework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama

CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 4 1. UsetheproceduredescriedinLemm1.55toconverttheregulrexpression(((00) (11)) 01) into n NFA. Answer: 0 0 1 1 00 0 0 11 1 1 01 0 1 (00)

### Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero

### Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.

1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples

### 2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).

AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following

### Linear Systems with Constant Coefficients

Liner Systems with Constnt Coefficients 4-3-05 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system

### The Thermodynamics of Aqueous Electrolyte Solutions

18 The Thermodynmics of Aqueous Electrolyte Solutions As discussed in Chpter 10, when slt is dissolved in wter or in other pproprite solvent, the molecules dissocite into ions. In queous solutions, strong

### 1 From NFA to regular expression

Note 1: How to convert DFA/NFA to regulr expression Version: 1.0 S/EE 374, Fll 2017 Septemer 11, 2017 In this note, we show tht ny DFA cn e converted into regulr expression. Our construction would work

### Naming the sides of a right-angled triangle

6.2 Wht is trigonometry? The word trigonometry is derived from the Greek words trigonon (tringle) nd metron (mesurement). Thus, it literlly mens to mesure tringle. Trigonometry dels with the reltionship

### CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010

CS 311 Homework 3 due 16:30, Thursdy, 14 th Octoer 2010 Homework must e sumitted on pper, in clss. Question 1. [15 pts.; 5 pts. ech] Drw stte digrms for NFAs recognizing the following lnguges:. L = {w

9/6/7 Mustf Jrrr: Leture Notes in Disrete Mthemtis. Birzeit University Plestine 05 Funtions 7.. Introdution to Funtions 7. One-to-One Onto Inverse funtions mjrrr 05 Wth this leture nd downlod the slides

### Pre-Lie algebras, rooted trees and related algebraic structures

Pre-Lie lgers, rooted trees nd relted lgeri strutures Mrh 23, 2004 Definition 1 A pre-lie lger is vetor spe W with mp : W W W suh tht (x y) z x (y z) = (x z) y x (z y). (1) Exmple 2 All ssoitive lgers

### PHYSICS 211 MIDTERM I 21 April 2004

PHYSICS MIDERM I April 004 Exm is closed book, closed notes. Use only your formul sheet. Write ll work nd nswers in exm booklets. he bcks of pges will not be grded unless you so request on the front of

### Homework 3 Solutions

CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

### Theoretische Physik 2: Elektrodynamik (Prof. A.-S. Smith) Home assignment 4

WiSe 1 8.1.1 Prof. Dr. A.-S. Smith Dipl.-Phys. Ellen Fischermeier Dipl.-Phys. Mtthis Sb m Lehrstuhl für Theoretische Physik I Deprtment für Physik Friedrich-Alexnder-Universität Erlngen-Nürnberg Theoretische

### 6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 CH 3. CH 3 C a. NMR spectroscopy. Different types of NMR

6.. Spetrosopy NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the lowenergy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht

### ON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs

ON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs A.I. KECHRINIOTIS AND N.D. ASSIMAKIS Deprtment of Eletronis Tehnologil Edutionl Institute of Lmi, Greee EMil: {kehrin,

### The Ellipse. is larger than the other.

The Ellipse Appolonius of Perg (5 B.C.) disovered tht interseting right irulr one ll the w through with plne slnted ut is not perpendiulr to the is, the intersetion provides resulting urve (oni setion)

### Lecture 3. Limits of Functions and Continuity

Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live

### Math RE - Calculus II Area Page 1 of 12

Mth --RE - Clculus II re Pge of re nd the Riemnn Sum Let f) be continuous function nd = f) f) > on closed intervl,b] s shown on the grph. The Riemnn Sum theor shows tht the re of R the region R hs re=

### This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2

1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion

### f (x)dx = f(b) f(a). a b f (x)dx is the limit of sums

Green s Theorem If f is funtion of one vrible x with derivtive f x) or df dx to the Fundmentl Theorem of lulus, nd [, b] is given intervl then, ording This is not trivil result, onsidering tht b b f x)dx

### The Riemann-Stieltjes Integral

Chpter 6 The Riemnn-Stieltjes Integrl 6.1. Definition nd Eistene of the Integrl Definition 6.1. Let, b R nd < b. ( A prtition P of intervl [, b] is finite set of points P = { 0, 1,..., n } suh tht = 0

### Data Structures LECTURE 10. Huffman coding. Example. Coding: problem definition

Dt Strutures, Spring 24 L. Joskowiz Dt Strutures LEURE Humn oing Motivtion Uniquel eipherle oes Prei oes Humn oe onstrution Etensions n pplitions hpter 6.3 pp 385 392 in tetook Motivtion Suppose we wnt

### Test , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes

Test 2 8., 8.2, 8.4 (density only), 8.5 (work only), 9., 9.2 nd 9.3 relted test mteril nd mteril from prior clsses Locl to Globl Perspectives Anlyze smll pieces to understnd the big picture. Exmples: numericl

### Particle Lifetime. Subatomic Physics: Particle Physics Lecture 3. Measuring Decays, Scatterings and Collisions. N(t) = N 0 exp( t/τ) = N 0 exp( Γt/)

Sutomic Physics: Prticle Physics Lecture 3 Mesuring Decys, Sctterings n Collisions Prticle lifetime n with Prticle ecy moes Prticle ecy kinemtics Scttering cross sections Collision centre of mss energy

### 38.2. The Uniform Distribution. Introduction. Prerequisites. Learning Outcomes

The Uniform Distribution 8. Introduction This Section introduces the simplest type of continuous probbility distribution which fetures continuous rndom vrible X with probbility density function f(x) which

### Solutions to assignment 3

D Sruure n Algorihm FR 6. Informik Sner, Telikeplli WS 03/04 hp://www.mpi-.mpg.e/~ner/oure/lg03/inex.hml Soluion o ignmen 3 Exerie Arirge i he ue of irepnie in urreny exhnge re o rnform one uni of urreny

### x ) dx dx x sec x over the interval (, ).

Curve on 6 For -, () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor

### PHYS 705: Classical Mechanics. Small Oscillations: Example A Linear Triatomic Molecule

PHYS 75: Clssicl echnics Sll Oscilltions: Exple A Liner Tritoic olecule A Liner Tritoic olecule x b b x x3 x Experientlly, one ight be interested in the rdition resulted fro the intrinsic oscilltion odes

### 5: The Definite Integral

5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

### Physics Dynamics: Atwood Machine

plce of ind F A C U L Y O F E D U C A I O N Deprtent of Curriculu nd Pedoy Physics Dynics: Atwood Mchine Science nd Mthetics Eduction Reserch Group Supported by UBC echin nd Lernin Enhnceent Fund 0-04

### Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs

Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:

### PDE Notes. Paul Carnig. January ODE s vs PDE s 1

PDE Notes Pul Crnig Jnury 2014 Contents 1 ODE s vs PDE s 1 2 Section 1.2 Het diffusion Eqution 1 2.1 Fourier s w of Het Conduction............................. 2 2.2 Energy Conservtion.....................................

### University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10

University of Wshington Deprtment of Chemistry Chemistry 45 Winter Qurter Homework Assignment 4; Due t 5p.m. on // We lerned tht the Hmiltonin for the quntized hrmonic oscilltor is ˆ d κ H. You cn obtin

### u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph.

nlyzing Dmped Oscilltions Prolem (Medor, exmple 2-18, pp 44-48): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + \$

### [ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves

Gols: 1. To find the re etween two curves Section 6.1 Are of Regions etween two Curves I. Are of Region Between Two Curves A. Grphicl Represention = _ B. Integrl Represention [ ( ) ( )] f x g x dx = C.

### 1 Nondeterministic Finite Automata

1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you

### The Dirichlet Problem in a Two Dimensional Rectangle. Section 13.5

The Dirichlet Prolem in Two Dimensionl Rectngle Section 13.5 1 Dirichlet Prolem in Rectngle In these notes we will pply the method of seprtion of vriles to otin solutions to elliptic prolems in rectngle

### 5.1 Estimating with Finite Sums Calculus

5.1 ESTIMATING WITH FINITE SUMS Emple: Suppose from the nd to 4 th hour of our rod trip, ou trvel with the cruise control set to ectl 70 miles per hour for tht two hour stretch. How fr hve ou trveled during

### Study Guide Final Exam Solutions. Part A: Kinetic Theory, First Law of Thermodynamics, Heat Engines

Massachusetts Institute of Technology Department of Physics 8.0T Fall 004 Study Guide Final Exam Solutions Part A: Kinetic Theory, First Law of Thermodynamics, Heat Engines Problem Energy Transformation,

### UNCORRECTED SAMPLE PAGES. Australian curriculum NUMBER AND ALGEBRA

7A 7B 7C 7D 7E 7F 7G 7H 7I 7J 7K Chpter Wht ou will lern 7Prols nd other grphs Eploring prols Skething prols with trnsformtions Skething prols using ftoristion Skething ompleting the squre Skething using

### Section 7.1 Area of a Region Between Two Curves

Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region

### Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

### Signal Flow Graphs. Consider a complex 3-port microwave network, constructed of 5 simpler microwave devices:

3/3/009 ignl Flow Grphs / ignl Flow Grphs Consider comple 3-port microwve network, constructed of 5 simpler microwve devices: 3 4 5 where n is the scttering mtri of ech device, nd is the overll scttering

### ( ) as a fraction. Determine location of the highest

AB/ Clulus Exm Review Sheet Solutions A Prelulus Type prolems A1 A A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f( x) Set funtion equl to Ftor or use qudrti eqution if qudrti Grph to

### 4.1. Probability Density Functions

STT 1 4.1-4. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile - vers - discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of

### H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a.

Chpter Review 89 IGURE ol hord GH of the prol 4. G u v H (, ) (A) Use the distne formul to show tht u. (B) Show tht G nd H lie on the line m, where m ( )/( ). (C) Solve m for nd sustitute in 4, otining

### APPROXIMATION AND ESTIMATION MATHEMATICAL LANGUAGE THE FUNDAMENTAL THEOREM OF ARITHMETIC LAWS OF ALGEBRA ORDER OF OPERATIONS

TOPIC 2: MATHEMATICAL LANGUAGE NUMBER AND ALGEBRA You shoul unerstn these mthemtil terms, n e le to use them ppropritely: ² ition, sutrtion, multiplition, ivision ² sum, ifferene, prout, quotient ² inex

### ENERGY AND PACKING. Outline: MATERIALS AND PACKING. Crystal Structure

EERGY AD PACKIG Outline: Crstlline versus morphous strutures Crstl struture - Unit ell - Coordintion numer - Atomi pking ftor Crstl sstems on dense, rndom pking Dense, regulr pking tpil neighor ond energ