Sharif University of Technology - CEDRA By: Professor Ali Meghdari
|
|
- Shannon Mason
- 5 years ago
- Views:
Transcription
1 Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai
2 Pupose: o exen he Enegy appoach in eiving euaions of oion i.e. Lagange s Meho fo Mechanical Syses. opics: Genealize Cooinaes Lagangian Euaion of Moion fo Inepenen Se of Genealize Cooinaes Lagangian Euaion of Moion fo Depenen Se of Genealize Cooinaes Hailonian Pinciple Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai
3 Hailon s Pinciples I is an inegal pinciple an consies he configuaion of a syse beween he ie ineval,. Avanages; Dynaics Foulaion is:. Reuce o he evaluaion of a scala efinie inegal,. Cooinae syse inepenen in expessing he inegan. Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai
4 By: Pofesso Ali Meghai Le us consie a syse of paicles. Using D Alebes s Pinciple an he Pinciple of Viual Wok we have:.. Can be wien as: i i i f U o x x f U ] [ ] [ Shaif Univesiy of echnology - CEDRA
5 By: Pofesso Ali Meghai Recall ha he Kineic Enegy fo a Syse of Paicles is:.3 Vaiaion in Kineic Enegy Subsiue in euaion. Shaif Univesiy of echnology - CEDRA
6 By: Pofesso Ali Meghai On he ohe han, since Viual Wok is efine as: Subsiuing Euaions.3 an.4 ino euaion., we obain: Inegaing Euaions.5 ove he ie ineval o esuls in:.6 ] [ bu U U f U.4.5 Shaif Univesiy of echnology - CEDRA
7 U [ bu ].6 U.7 x i x i Vaie Pah Geneal Fo of Hailon s Pinciple x i x i ue Pah Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai
8 U.7 Geneal Fo of Hailon s Pinciple: I saes ha he ue pah followe by he ynaic syse o go fo o is such ha he ie inegal of he su of he viual kineic enegy change an viual wok vanishes when subjece o viual isplaceens fo he ue pah. Hailon s Pinciple can be applie o boh on-holonoic an on-consevaive syses. x i x i Vaie Pah x i x i Shaif Univesiy of echnology - CEDRA ue Pah By: Pofesso Ali Meghai
9 Special Cases: when foces ae consevaive an he viual wok is elae o he change in poenial enegy V by U = -V, we have: L V Lagangian a scala whee : hen; Euaion.7 L,,, an V V becoes funcion V :.8, If he syse is Holonoic, hen euaion.8 becoes: I L Shaif Univesiy of echnology - CEDRA I L.9 By: Pofesso Ali Meghai
10 I L I.9 Euaion.9 saes ha he ue pah followe by a consevaive holonoic syse o go fo o is such ha he ie inegal I is exeize. L Poof of Lagange s Euaion fo Hailon s Pinciple: Hailon s Pinciple U.7 Fo Holonoic Syse of -Paicles wih egees of feeo we have: Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai
11 By: Pofesso Ali Meghai,,, = Genealize Cooinaes = { } Space =,,,, = Veco Cooinaes of Paicles hen, he oal Kineic Enegy fo he syse is:,,...,,,,...,,.3 Bu, viual wok one by genealize foces ae: aking he vaiaion of using euaion.3, an noing ha =, we have: Q Q U.3 subsiue in.7 Shaif Univesiy of echnology - CEDRA
12 By: Pofesso Ali Meghai subsiuing in euaion.3, we have: Inegaing he las e of E..3 by pas, we have: ] [ ] [ Q.3 subsiue in.3 [ = = Shaif Univesiy of echnology - CEDRA
13 [ Q ].33 Since in Holonoic Syses, he genealize cooinaes fo an inepenen se, heefoe, he coefficiens of each in euaion.33 us be zeo. heefoe: Q,,..., M.34 Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai
14 Exaple: A bea of ass is fee o slie on a hoop of aius R as shown. he hoop is oaing wih he consan angula velociy Ω. Fin he euaion of oion using Hailon s pinciple?. Moion: Le x, x, x 3 be aache o he hoop. g R x 3 Ω x Rsin e Rcos e 3 x v Rsin e R cos e R sin e 3 Dau Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai
15 . Kineic Enegy: v v [ Rsin R R sin R cos R sin 3. Poenial Enegy: aking θ= as he au, we have; V gr cos 4. Lagangian: L V R sin R gr cos Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai
16 5. he Vaiaion of Lagangian: L L L R g [ sin cos sin ] R R o apply Hailon s Pinciple, we nee o expess he n e in above euaion in es of δθ. Inegaing he n e by pas esuls: he inegae e in he above euaion vanishes by efiniion of he vaiaion a he beginning an en of he pah. heefoe, applying Hailon s Pinciple esuls: L [ R R sin cos g R sin ] Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai
17 6. Applying he Hailon s Pinciple: L [ R R sin cos g R sin ] Fo he eualiy o hol, he inegan us vanish a all ies. Because δθ is abiay, fo he inegan o be zeo, he coefficien of δθ us be zeo. heefoe, he Euaion of Moion will esuls as: sin g R cos Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai
18 Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai
ÖRNEK 1: THE LINEAR IMPULSE-MOMENTUM RELATION Calculate the linear momentum of a particle of mass m=10 kg which has a. kg m s
MÜHENDİSLİK MEKANİĞİ. HAFTA İMPULS- MMENTUM-ÇARPIŞMA Linea oenu of a paicle: The sybol L denoes he linea oenu and is defined as he ass ies he elociy of a paicle. L ÖRNEK : THE LINEAR IMPULSE-MMENTUM RELATIN
More informationENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 9 Solutions [Theorems of Gauss and Stokes]
ENGI 44 Avance alculus fo Engineeing Faculy of Engineeing an Applie cience Poblem e 9 oluions [Theoems of Gauss an okes]. A fla aea A is boune by he iangle whose veices ae he poins P(,, ), Q(,, ) an R(,,
More informationCombinatorial Approach to M/M/1 Queues. Using Hypergeometric Functions
Inenaional Mahemaical Foum, Vol 8, 03, no 0, 463-47 HIKARI Ld, wwwm-hikaicom Combinaoial Appoach o M/M/ Queues Using Hypegeomeic Funcions Jagdish Saan and Kamal Nain Depamen of Saisics, Univesiy of Delhi,
More informationTwo-dimensional Effects on the CSR Interaction Forces for an Energy-Chirped Bunch. Rui Li, J. Bisognano, R. Legg, and R. Bosch
Two-dimensional Effecs on he CS Ineacion Foces fo an Enegy-Chiped Bunch ui Li, J. Bisognano,. Legg, and. Bosch Ouline 1. Inoducion 2. Pevious 1D and 2D esuls fo Effecive CS Foce 3. Bunch Disibuion Vaiaion
More informationCircular Motion. Radians. One revolution is equivalent to which is also equivalent to 2π radians. Therefore we can.
1 Cicula Moion Radians One evoluion is equivalen o 360 0 which is also equivalen o 2π adians. Theefoe we can say ha 360 = 2π adians, 180 = π adians, 90 = π 2 adians. Hence 1 adian = 360 2π Convesions Rule
More information11. HAFAT İş-Enerji Power of a force: Power in the ability of a force to do work
MÜHENDİSLİK MEKNİĞİ. HFT İş-Eneji Pwe f a fce: Pwe in he abiliy f a fce d wk F: The fce applied n paicle Q P = F v = Fv cs( θ ) F Q v θ Pah f Q v: The velciy f Q ÖRNEK: İŞ-ENERJİ ω µ k v Calculae he pwe
More informationLecture-V Stochastic Processes and the Basic Term-Structure Equation 1 Stochastic Processes Any variable whose value changes over time in an uncertain
Lecue-V Sochasic Pocesses and he Basic Tem-Sucue Equaion 1 Sochasic Pocesses Any vaiable whose value changes ove ime in an unceain way is called a Sochasic Pocess. Sochasic Pocesses can be classied as
More informationEN221 - Fall HW # 7 Solutions
EN221 - Fall2008 - HW # 7 Soluions Pof. Vivek Shenoy 1.) Show ha he fomulae φ v ( φ + φ L)v (1) u v ( u + u L)v (2) can be pu ino he alenaive foms φ φ v v + φv na (3) u u v v + u(v n)a (4) (a) Using v
More informationLecture 23 Damped Motion
Differenial Equaions (MTH40) Lecure Daped Moion In he previous lecure, we discussed he free haronic oion ha assues no rearding forces acing on he oving ass. However No rearding forces acing on he oving
More informationr r r r r EE334 Electromagnetic Theory I Todd Kaiser
334 lecoagneic Theoy I Todd Kaise Maxwell s quaions: Maxwell s equaions wee developed on expeienal evidence and have been found o goven all classical elecoagneic phenoena. They can be wien in diffeenial
More informationWORK POWER AND ENERGY Consevaive foce a) A foce is said o be consevaive if he wok done by i is independen of pah followed by he body b) Wok done by a consevaive foce fo a closed pah is zeo c) Wok done
More information7 Wave Equation in Higher Dimensions
7 Wave Equaion in Highe Dimensions We now conside he iniial-value poblem fo he wave equaion in n dimensions, u c u x R n u(x, φ(x u (x, ψ(x whee u n i u x i x i. (7. 7. Mehod of Spheical Means Ref: Evans,
More informationGeneral Non-Arbitrage Model. I. Partial Differential Equation for Pricing A. Traded Underlying Security
1 Geneal Non-Abiage Model I. Paial Diffeenial Equaion fo Picing A. aded Undelying Secuiy 1. Dynamics of he Asse Given by: a. ds = µ (S, )d + σ (S, )dz b. he asse can be eihe a sock, o a cuency, an index,
More informationMEEN 617 Handout #11 MODAL ANALYSIS OF MDOF Systems with VISCOUS DAMPING
MEEN 67 Handou # MODAL ANALYSIS OF MDOF Sysems wih VISCOS DAMPING ^ Symmeic Moion of a n-dof linea sysem is descibed by he second ode diffeenial equaions M+C+K=F whee () and F () ae n ows vecos of displacemens
More informationToday - Lecture 13. Today s lecture continue with rotations, torque, Note that chapters 11, 12, 13 all involve rotations
Today - Lecue 13 Today s lecue coninue wih oaions, oque, Noe ha chapes 11, 1, 13 all inole oaions slide 1 eiew Roaions Chapes 11 & 1 Viewed fom aboe (+z) Roaional, o angula elociy, gies angenial elociy
More informationLecture 18: Kinetics of Phase Growth in a Two-component System: general kinetics analysis based on the dilute-solution approximation
Lecue 8: Kineics of Phase Gowh in a Two-componen Sysem: geneal kineics analysis based on he dilue-soluion appoximaion Today s opics: In he las Lecues, we leaned hee diffeen ways o descibe he diffusion
More informationLecture 22 Electromagnetic Waves
Lecue Elecomagneic Waves Pogam: 1. Enegy caied by he wave (Poyning veco).. Maxwell s equaions and Bounday condiions a inefaces. 3. Maeials boundaies: eflecion and efacion. Snell s Law. Quesions you should
More informationKINEMATICS OF RIGID BODIES
KINEMTICS OF RIGID ODIES In igid body kinemaics, we use he elaionships govening he displacemen, velociy and acceleaion, bu mus also accoun fo he oaional moion of he body. Descipion of he moion of igid
More informationPractice Problems - Week #4 Higher-Order DEs, Applications Solutions
Pracice Probles - Wee #4 Higher-Orer DEs, Applicaions Soluions 1. Solve he iniial value proble where y y = 0, y0 = 0, y 0 = 1, an y 0 =. r r = rr 1 = rr 1r + 1, so he general soluion is C 1 + C e x + C
More informationPHYS GENERAL RELATIVITY AND COSMOLOGY PROBLEM SET 7 - SOLUTIONS
PHYS 54 - GENERAL RELATIVITY AND COSMOLOGY - 07 - PROBLEM SET 7 - SOLUTIONS TA: Jeome Quinin Mach, 07 Noe ha houghou hee oluion, we wok in uni whee c, and we chooe he meic ignaue (,,, ) a ou convenion..
More informationSTUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE WEIBULL DISTRIBUTION
Inenaional Jounal of Science, Technology & Managemen Volume No 04, Special Issue No. 0, Mach 205 ISSN (online): 2394-537 STUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE
More informationMATHEMATICAL FOUNDATIONS FOR APPROXIMATING PARTICLE BEHAVIOUR AT RADIUS OF THE PLANCK LENGTH
Fundamenal Jounal of Mahemaical Phsics Vol 3 Issue 013 Pages 55-6 Published online a hp://wwwfdincom/ MATHEMATICAL FOUNDATIONS FOR APPROXIMATING PARTICLE BEHAVIOUR AT RADIUS OF THE PLANCK LENGTH Univesias
More informationHomework 2 Solutions
Mah 308 Differenial Equaions Fall 2002 & 2. See he las page. Hoework 2 Soluions 3a). Newon s secon law of oion says ha a = F, an we know a =, so we have = F. One par of he force is graviy, g. However,
More informationThe Production of Polarization
Physics 36: Waves Lecue 13 3/31/211 The Poducion of Polaizaion Today we will alk abou he poducion of polaized ligh. We aleady inoduced he concep of he polaizaion of ligh, a ansvese EM wave. To biefly eview
More informationChapter 7. Interference
Chape 7 Inefeence Pa I Geneal Consideaions Pinciple of Supeposiion Pinciple of Supeposiion When wo o moe opical waves mee in he same locaion, hey follow supeposiion pinciple Mos opical sensos deec opical
More informationOn Control Problem Described by Infinite System of First-Order Differential Equations
Ausalian Jounal of Basic and Applied Sciences 5(): 736-74 ISS 99-878 On Conol Poblem Descibed by Infinie Sysem of Fis-Ode Diffeenial Equaions Gafujan Ibagimov and Abbas Badaaya J'afau Insiue fo Mahemaical
More informationInternational Journal of Pure and Applied Sciences and Technology
In. J. Pue Appl. Sci. Technol., 4 (211, pp. 23-29 Inenaional Jounal of Pue and Applied Sciences and Technology ISS 2229-617 Available online a www.ijopaasa.in eseach Pape Opizaion of he Uiliy of a Sucual
More informationPhysics 207 Lecture 13
Physics 07 Lecue 3 Physics 07, Lecue 3, Oc. 8 Agenda: Chape 9, finish, Chape 0 Sa Chape 9: Moenu and Collision Ipulse Cene of ass Chape 0: oaional Kineaics oaional Enegy Moens of Ineia Paallel axis heoe
More informationControl Volume Derivation
School of eospace Engineeing Conol Volume -1 Copyigh 1 by Jey M. Seizman. ll ighs esee. Conol Volume Deiaion How o cone ou elaionships fo a close sysem (conol mass) o an open sysem (conol olume) Fo mass
More informationPhysics 2001/2051 Moments of Inertia Experiment 1
Physics 001/051 Momens o Ineia Expeimen 1 Pelab 1 Read he ollowing backgound/seup and ensue you ae amilia wih he heoy equied o he expeimen. Please also ill in he missing equaions 5, 7 and 9. Backgound/Seup
More informationThe shortest path between two truths in the real domain passes through the complex domain. J. Hadamard
Complex Analysis R.G. Halbud R.Halbud@ucl.ac.uk Depamen of Mahemaics Univesiy College London 202 The shoes pah beween wo uhs in he eal domain passes hough he complex domain. J. Hadamad Chape The fis fundamenal
More informationThus the force is proportional but opposite to the displacement away from equilibrium.
Chaper 3 : Siple Haronic Moion Hooe s law saes ha he force (F) eered by an ideal spring is proporional o is elongaion l F= l where is he spring consan. Consider a ass hanging on a he spring. In equilibriu
More informationA GENERAL METHOD TO STUDY THE MOTION IN A NON-INERTIAL REFERENCE FRAME
The Inenaional onfeence on opuaional Mechanics an Viual Engineeing OME 9 9 OTOBER 9, Basov, Roania A GENERAL METHOD TO STUDY THE MOTION IN A NON-INERTIAL REFERENE FRAME Daniel onuache, Vlaii Mainusi Technical
More informationTwo-Pion Exchange Currents in Photodisintegration of the Deuteron
Two-Pion Exchange Cuens in Phoodisinegaion of he Deueon Dagaa Rozędzik and Jacek Goak Jagieonian Univesiy Kaków MENU00 3 May 00 Wiiasbug Conen Chia Effecive Fied Theoy ChEFT Eecoagneic cuen oeaos wihin
More informationSPHERICAL WINDS SPHERICAL ACCRETION
SPHERICAL WINDS SPHERICAL ACCRETION Spheical wins. Many sas ae known o loose mass. The sola win caies away abou 10 14 M y 1 of vey ho plasma. This ae is insignifican. In fac, sola aiaion caies away 4 10
More informationLecture 17: Kinetics of Phase Growth in a Two-component System:
Lecue 17: Kineics of Phase Gowh in a Two-componen Sysem: descipion of diffusion flux acoss he α/ ineface Today s opics Majo asks of oday s Lecue: how o deive he diffusion flux of aoms. Once an incipien
More informationMATH 31B: MIDTERM 2 REVIEW. x 2 e x2 2x dx = 1. ue u du 2. x 2 e x2 e x2] + C 2. dx = x ln(x) 2 2. ln x dx = x ln x x + C. 2, or dx = 2u du.
MATH 3B: MIDTERM REVIEW JOE HUGHES. Inegraion by Pars. Evaluae 3 e. Soluion: Firs make he subsiuion u =. Then =, hence 3 e = e = ue u Now inegrae by pars o ge ue u = ue u e u + C and subsiue he definiion
More informationψ(t) = V x (0)V x (t)
.93 Home Work Se No. (Professor Sow-Hsin Chen Spring Term 5. Due March 7, 5. This problem concerns calculaions of analyical expressions for he self-inermediae scaering funcion (ISF of he es paricle in
More informationSHM SHM. T is the period or time it takes to complete 1 cycle. T = = 2π. f is the frequency or the number of cycles completed per unit time.
SHM A ω = k d x x = Acos ( ω +) dx v = = ω Asin( ω + ) vax = ± ωa dv a = = ω Acos + k + x Apliude ( ω ) = 0 a ax = ± ω A SHM x = Acos is he period or ie i akes o coplee cycle. ω = π ( ω +) π = = π ω k
More informationPHYS PRACTICE EXAM 2
PHYS 1800 PRACTICE EXAM Pa I Muliple Choice Quesions [ ps each] Diecions: Cicle he one alenaive ha bes complees he saemen o answes he quesion. Unless ohewise saed, assume ideal condiions (no ai esisance,
More informationFundamental Vehicle Loads & Their Estimation
Fundaenal Vehicle Loads & Thei Esiaion The silified loads can only be alied in he eliinay design sage when he absence of es o siulaion daa They should always be qualified and udaed as oe infoaion becoes
More informationr P + '% 2 r v(r) End pressures P 1 (high) and P 2 (low) P 1 , which must be independent of z, so # dz dz = P 2 " P 1 = " #P L L,
Lecue 36 Pipe Flow and Low-eynolds numbe hydodynamics 36.1 eading fo Lecues 34-35: PKT Chape 12. Will y fo Monday?: new daa shee and daf fomula shee fo final exam. Ou saing poin fo hydodynamics ae wo equaions:
More informationwhere the coordinate X (t) describes the system motion. X has its origin at the system static equilibrium position (SEP).
Appendix A: Conservaion of Mechanical Energy = Conservaion of Linear Momenum Consider he moion of a nd order mechanical sysem comprised of he fundamenal mechanical elemens: ineria or mass (M), siffness
More informationAn Open cycle and Closed cycle Gas Turbine Engines. Methods to improve the performance of simple gas turbine plants
An Open cycle and losed cycle Gas ubine Engines Mehods o impove he pefomance of simple gas ubine plans I egeneaive Gas ubine ycle: he empeaue of he exhaus gases in a simple gas ubine is highe han he empeaue
More information4.5 Constant Acceleration
4.5 Consan Acceleraion v() v() = v 0 + a a() a a() = a v 0 Area = a (a) (b) Figure 4.8 Consan acceleraion: (a) velociy, (b) acceleraion When he x -componen of he velociy is a linear funcion (Figure 4.8(a)),
More informationA note on characterization related to distributional properties of random translation, contraction and dilation of generalized order statistics
PobSa Foum, Volume 6, July 213, Pages 35 41 ISSN 974-3235 PobSa Foum is an e-jounal. Fo eails please visi www.pobsa.og.in A noe on chaaceizaion elae o isibuional popeies of anom anslaion, conacion an ilaion
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationField due to a collection of N discrete point charges: r is in the direction from
Physcs 46 Fomula Shee Exam Coulomb s Law qq Felec = k ˆ (Fo example, f F s he elecc foce ha q exes on q, hen ˆ s a un veco n he decon fom q o q.) Elecc Feld elaed o he elecc foce by: Felec = qe (elecc
More informationLecture 23: Central Force Motion
Lectue 3: Cental Foce Motion Many of the foces we encounte in natue act between two paticles along the line connecting the Gavity, electicity, and the stong nuclea foce ae exaples These types of foces
More informationExam #2 PHYSICS 211 Monday July 6 th, 2009 Please write down your name also on the back page of this exam
Exa #2 PHYSICS 211 Monday July 6 h, 29 NME Please wrie down your nae also on he back pae of his exa 1. The fiure ives how he force varies as a funcion of he posiion. Such force is acin on a paricle, which
More informationFig. 1S. The antenna construction: (a) main geometrical parameters, (b) the wire support pillar and (c) the console link between wire and coaxial
a b c Fig. S. The anenna consucion: (a) ain geoeical paaees, (b) he wie suppo pilla and (c) he console link beween wie and coaial pobe. Fig. S. The anenna coss-secion in he y-z plane. Accoding o [], he
More informationThe sudden release of a large amount of energy E into a background fluid of density
10 Poin explosion The sudden elease of a lage amoun of enegy E ino a backgound fluid of densiy ceaes a song explosion, chaaceized by a song shock wave (a blas wave ) emanaing fom he poin whee he enegy
More informationCHAPTER 5: Circular Motion; Gravitation
CHAPER 5: Cicula Motion; Gavitation Solution Guide to WebAssign Pobles 5.1 [1] (a) Find the centipetal acceleation fo Eq. 5-1.. a R v ( 1.5 s) 1.10 1.4 s (b) he net hoizontal foce is causing the centipetal
More informationHow to Solve System Dynamic s Problems
How o Solve Sye Dynaic Proble A ye dynaic proble involve wo or ore bodie (objec) under he influence of everal exernal force. The objec ay uliaely re, ove wih conan velociy, conan acceleraion or oe cobinaion
More informationVariance and Covariance Processes
Vaiance and Covaiance Pocesses Pakash Balachandan Depamen of Mahemaics Duke Univesiy May 26, 2008 These noes ae based on Due s Sochasic Calculus, Revuz and Yo s Coninuous Maingales and Bownian Moion, Kaazas
More information1. VELOCITY AND ACCELERATION
1. VELOCITY AND ACCELERATION 1.1 Kinemaics Equaions s = u + 1 a and s = v 1 a s = 1 (u + v) v = u + as 1. Displacemen-Time Graph Gradien = speed 1.3 Velociy-Time Graph Gradien = acceleraion Area under
More informationChapter Three Systems of Linear Differential Equations
Chaper Three Sysems of Linear Differenial Equaions In his chaper we are going o consier sysems of firs orer orinary ifferenial equaions. These are sysems of he form x a x a x a n x n x a x a x a n x n
More informationIntegration of the constitutive equation
Inegaion of he consiive eqaion REMAINDER ON NUMERICAL INTEGRATION Analyical inegaion f ( x( ), x ( )) x x x f () Exac/close-fom solion (no always possible) Nmeical inegaion. i. N T i N [, T ] [ i, i ]
More informationSome Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
More informationRelative and Circular Motion
Relaie and Cicula Moion a) Relaie moion b) Cenipeal acceleaion Mechanics Lecue 3 Slide 1 Mechanics Lecue 3 Slide 2 Time on Video Pelecue Looks like mosly eeyone hee has iewed enie pelecue GOOD! Thank you
More informationPH126 Exam I Solutions
PH6 Exam I Solutions q Q Q q. Fou positively chage boies, two with chage Q an two with chage q, ae connecte by fou unstetchable stings of equal length. In the absence of extenal foces they assume the equilibium
More informationGravity and SHM Review Questions
Graviy an SHM Review Quesions 1. The mass of Plane X is one-enh ha of he Earh, an is iameer is one-half ha of he Earh. The acceleraion ue o raviy a he surface of Plane X is mos nearly m/s (B) 4m/s (C)
More informationRotational Motion and the Law of Gravity
Chape 7 7 Roaional Moion and he Law of Gaiy PROBLEM SOLUTIONS 7.1 (a) Eah oaes adians (360 ) on is axis in 1 day. Thus, ad 1 day 5 7.7 10 ad s 4 1 day 8.64 10 s Because of is oaion abou is axis, Eah bulges
More informationWall. x(t) f(t) x(t = 0) = x 0, t=0. which describes the motion of the mass in absence of any external forcing.
MECHANICS APPLICATIONS OF SECOND-ORDER ODES 7 Mechanics applicaions of second-order ODEs Second-order linear ODEs wih consan coefficiens arise in many physical applicaions. One physical sysems whose behaviour
More informationAnswers to test yourself questions
Answes to test youself questions opic. Cicula motion π π a he angula speed is just ω 5. 7 ad s. he linea speed is ω 5. 7 3. 5 7. 7 m s.. 4 b he fequency is f. 8 s.. 4 3 a f. 45 ( 3. 5). m s. 3 a he aeage
More informationGauge invariance and the vacuum state. Dan Solomon Rauland-Borg Corporation 3450 W. Oakton Skokie, IL Please send all correspondence to:
Gauge invaiance and he vacuum sae 1 Gauge invaiance and he vacuum sae by Dan Solomon Rauland-Bog Copoaion 345 W. Oakon Skokie, IL 676 Please send all coespondence o: Dan Solomon 164 Bummel Evanson, IL
More informationI. Define the Situation
I. efine he Siuaion This exam explores he relaionship beween he applied volage o a permanen magne C moond he linear velociy (speed) of he elecric vehicle (EV) he moor drives. The moor oupu is fed ino a
More information= + t ] can be used to calculate the angular
WEEK-7 Reciaion PHYS 3 Mar 8, 8 Ch. 8 FOC:, 4, 6,, 3 &5. () Using Equaion 8. (θ = Arc lengh / Raius) o calculae he angle (in raians) ha each objec subens a your eye shows ha θ Moon = 9. 3 ra, θ Pea = 7.
More information3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon
3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of
More information( ) = 0.43 kj = 430 J. Solutions 9 1. Solutions to Miscellaneous Exercise 9 1. Let W = work done then 0.
Soluions 9 Soluions o Miscellaneous Exercise 9. Le W work done hen.9 W PdV Using Simpson's rule (9.) we have. W { 96 + [ 58 + 6 + 77 + 5 ] + [ + 99 + 6 ]+ }. kj. Using Simpson's rule (9.) again: W.5.6
More informationx y θ = 31.8 = 48.0 N. a 3.00 m/s
4.5.IDENTIY: Vecor addiion. SET UP: Use a coordinae sse where he dog A. The forces are skeched in igure 4.5. EXECUTE: + -ais is in he direcion of, A he force applied b =+ 70 N, = 0 A B B A = cos60.0 =
More information336 ERIDANI kfk Lp = sup jf(y) ; f () jj j p p whee he supemum is aken ove all open balls = (a ) inr n, jj is he Lebesgue measue of in R n, () =(), f
TAMKANG JOURNAL OF MATHEMATIS Volume 33, Numbe 4, Wine 2002 ON THE OUNDEDNESS OF A GENERALIED FRATIONAL INTEGRAL ON GENERALIED MORREY SPAES ERIDANI Absac. In his pape we exend Nakai's esul on he boundedness
More informationChapter 31 Faraday s Law
Chapte 31 Faaday s Law Change oving --> cuent --> agnetic field (static cuent --> static agnetic field) The souce of agnetic fields is cuent. The souce of electic fields is chage (electic onopole). Altenating
More informationHomework Set 3 Physics 319 Classical Mechanics
Homewok Set 3 Phsics 319 lassical Mechanics Poblem 5.13 a) To fin the equilibium position (whee thee is no foce) set the eivative of the potential to zeo U 1 R U0 R U 0 at R R b) If R is much smalle than
More informationON 3-DIMENSIONAL CONTACT METRIC MANIFOLDS
Mem. Fac. Inegaed As and Sci., Hioshima Univ., Se. IV, Vol. 8 9-33, Dec. 00 ON 3-DIMENSIONAL CONTACT METRIC MANIFOLDS YOSHIO AGAOKA *, BYUNG HAK KIM ** AND JIN HYUK CHOI ** *Depamen of Mahemaics, Faculy
More informationMOMENTUM CONSERVATION LAW
1 AAST/AEDT AP PHYSICS B: Impulse and Momenum Le us run an experimen: The ball is moving wih a velociy of V o and a force of F is applied on i for he ime inerval of. As he resul he ball s velociy changes
More informationAST1100 Lecture Notes
AST00 Lecue Noes 5 6: Geneal Relaiviy Basic pinciples Schwazschild geomey The geneal heoy of elaiviy may be summaized in one equaion, he Einsein equaion G µν 8πT µν, whee G µν is he Einsein enso and T
More information( ) ( ) ( ) ( u) ( u) = are shown in Figure =, it is reasonable to speculate that. = cos u ) and the inside function ( ( t) du
Porlan Communiy College MTH 51 Lab Manual The Chain Rule Aciviy 38 The funcions f ( = sin ( an k( sin( 3 38.1. Since f ( cos( k ( = cos( 3. Bu his woul imply ha k ( f ( = are shown in Figure =, i is reasonable
More informationPracticing Problem Solving and Graphing
Pracicing Problem Solving and Graphing Tes 1: Jan 30, 7pm, Ming Hsieh G20 The Bes Ways To Pracice for Tes Bes If need more, ry suggesed problems from each new opic: Suden Response Examples A pas opic ha
More informationThe Fundamental Theorems of Calculus
FunamenalTheorems.nb 1 The Funamenal Theorems of Calculus You have now been inrouce o he wo main branches of calculus: ifferenial calculus (which we inrouce wih he angen line problem) an inegral calculus
More informationPhysics Courseware Physics II Electric Field and Force
Physics Cousewae Physics II lectic iel an oce Coulomb s law, whee k Nm /C test Definition of electic fiel. This is a vecto. test Q lectic fiel fo a point chage. This is a vecto. Poblem.- chage of µc is
More informationSection 7.4 Modeling Changing Amplitude and Midline
488 Chaper 7 Secion 7.4 Modeling Changing Ampliude and Midline While sinusoidal funcions can model a variey of behaviors, i is ofen necessary o combine sinusoidal funcions wih linear and exponenial curves
More informationTopics in Combinatorial Optimization May 11, Lecture 22
8.997 Topics in Combinaorial Opimizaion May, 004 Lecure Lecurer: Michel X. Goemans Scribe: Alanha Newman Muliflows an Disjoin Pahs Le G = (V,E) be a graph an le s,,s,,...s, V be erminals. Our goal is o
More informationLecture 4 Kinetics of a particle Part 3: Impulse and Momentum
MEE Engineering Mechanics II Lecure 4 Lecure 4 Kineics of a paricle Par 3: Impulse and Momenum Linear impulse and momenum Saring from he equaion of moion for a paricle of mass m which is subjeced o an
More informationAn Analytical Study of Strong Non Planer Shock. Waves in Magnetogasdynamics
Adv Theo Appl Mech Vol no 6 9-97 An Analyical Sdy of Song Non Plane Shock Waves in Magneogasdynaics L P Singh Depaen of Applied Maheaics Insie of Technology Banaas Hind Univesiy Vaanasi-5 India Akal Hsain
More information2. v = 3 4 c. 3. v = 4c. 5. v = 2 3 c. 6. v = 9. v = 4 3 c
Vesion 074 Exam Final Daf swinney (55185) 1 This pin-ou should have 30 quesions. Muliple-choice quesions may coninue on he nex column o page find all choices befoe answeing. 001 10.0 poins AballofmassM
More informationSuggested Practice Problems (set #2) for the Physics Placement Test
Deparmen of Physics College of Ars and Sciences American Universiy of Sharjah (AUS) Fall 014 Suggesed Pracice Problems (se #) for he Physics Placemen Tes This documen conains 5 suggesed problems ha are
More informationME 141. Engineering Mechanics
ME 141 Engineeing Mechnics Lecue 13: Kinemics of igid bodies hmd Shhedi Shkil Lecue, ep. of Mechnicl Engg, UET E-mil: sshkil@me.bue.c.bd, shkil6791@gmil.com Websie: eche.bue.c.bd/sshkil Couesy: Veco Mechnics
More informationAN EFFICIENT INTEGRAL METHOD FOR THE COMPUTATION OF THE BODIES MOTION IN ELECTROMAGNETIC FIELD
AN EFFICIENT INTEGRAL METHOD FOR THE COMPUTATION OF THE BODIES MOTION IN ELECTROMAGNETIC FIELD GEORGE-MARIAN VASILESCU, MIHAI MARICARU, BOGDAN DUMITRU VĂRĂTICEANU, MARIUS AUREL COSTEA Key wods: Eddy cuen
More informationOn The Estimation of Two Missing Values in Randomized Complete Block Designs
Mahemaical Theoy and Modeling ISSN 45804 (Pape ISSN 505 (Online Vol.6, No.7, 06 www.iise.og On The Esimaion of Two Missing Values in Randomized Complee Bloc Designs EFFANGA, EFFANGA OKON AND BASSE, E.
More information5-1. We apply Newton s second law (specifically, Eq. 5-2). F = ma = ma sin 20.0 = 1.0 kg 2.00 m/s sin 20.0 = 0.684N. ( ) ( )
5-1. We apply Newon s second law (specfcally, Eq. 5-). (a) We fnd he componen of he foce s ( ) ( ) F = ma = ma cos 0.0 = 1.00kg.00m/s cos 0.0 = 1.88N. (b) The y componen of he foce s ( ) ( ) F = ma = ma
More information( ) rad ( 2.0 s) = 168 rad
.) α 0.450 ω o 0 and ω 8.00 ω αt + ω o o t ω ω o α HO 9 Solution 8.00 0 0.450 7.8 b.) ω ω o + αδθ o Δθ ω 8.00 0 ω o α 0.450 7. o Δθ 7. ev.3 ev π.) ω o.50, α 0.300, Δθ 3.50 ev π 7π ev ω ω o + αδθ o ω ω
More information2.1 Harmonic excitation of undamped systems
2.1 Haronic exciaion of undaped syses Sanaoja 2_1.1 2.1 Haronic exciaion of undaped syses (Vaienaaoan syseein haroninen heräe) The following syse is sudied: y x F() Free-body diagra f x g x() N F() In
More informationChapter 3: Vectors and Two-Dimensional Motion
Chape 3: Vecos and Two-Dmensonal Moon Vecos: magnude and decon Negae o a eco: eese s decon Mulplng o ddng a eco b a scala Vecos n he same decon (eaed lke numbes) Geneal Veco Addon: Tangle mehod o addon
More informationThe Contradiction within Equations of Motion with Constant Acceleration
The Conradicion wihin Equaions of Moion wih Consan Acceleraion Louai Hassan Elzein Basheir (Daed: July 7, 0 This paper is prepared o demonsrae he violaion of rules of mahemaics in he algebraic derivaion
More informationDepartment of Chemical Engineering University of Tennessee Prof. David Keffer. Course Lecture Notes SIXTEEN
D. Keffe - ChE 40: Hea Tansfe and Fluid Flow Deamen of Chemical Enee Uniesi of Tennessee Pof. Daid Keffe Couse Lecue Noes SIXTEEN SECTION.6 DIFFERENTIL EQUTIONS OF CONTINUITY SECTION.7 DIFFERENTIL EQUTIONS
More informationQuantum Mechanics I - Session 5
Quantum Mechanics I - Session 5 Apil 7, 015 1 Commuting opeatos - an example Remine: You saw in class that Â, ˆB ae commuting opeatos iff they have a complete set of commuting obsevables. In aition you
More informationFerent equation of the Universe
Feen equaion of he Univese I discoveed a new Gaviaion heoy which beaks he wall of Planck scale! Absac My Nobel Pize - Discoveies Feen equaion of he Univese: i + ia = = (... N... N M m i= i ) i a M m j=
More informationP h y s i c s F a c t s h e e t
P h y s i c s F a c s h e e Sepembe 2001 Numbe 20 Simple Hamonic Moion Basic Conceps This Facshee will:! eplain wha is mean by simple hamonic moion! eplain how o use he equaions fo simple hamonic moion!
More informationTorque, Angular Momentum and Rotational Kinetic Energy
Toque, Angula Moentu and Rotational Kinetic Enegy In ou peious exaples that inoled a wheel, like fo exaple a pulley we wee always caeful to specify that fo the puposes of the poble it would be teated as
More information