Rigid Bodies: Equivalent Systems of Forces

Size: px
Start display at page:

Download "Rigid Bodies: Equivalent Systems of Forces"

Transcription

1 Engneeng Statcs, ENGR 2301 Chapte 3 Rgd Bodes: Equvalent Sstems of oces

2 Intoducton Teatment of a bod as a sngle patcle s not alwas possble. In geneal, the se of the bod and the specfc ponts of applcaton of the foces must be consdeed. ost bodes n elementa mechancs ae assumed to be gd,.e., the actual defomatons ae small and do not affect the condtons of equlbum o moton of the bod. Cuent chapte descbes the effect of foces eeted on a gd bod and how to eplace a gven sstem of foces wth a smple equvalent sstem. moment of a foce about a pont moment of a foce about an as moment due to a couple An sstem of foces actng on a gd bod can be eplaced b an equvalent sstem consstng of one foce actng at a gven pont and one couple. 3-2

3 Etenal and Intenal oces oces actng on gd bodes ae dvded nto two goups: - Etenal foces - Intenal foces Etenal foces epesents the acton of othe bodes on the gd bod unde consdeaton. The ae entel esponsble fo the etenal behavo of the gd bod. If unopposed, each etenal foce can mpat a moton of tanslaton o otaton, o both. The ntenal foces ae the foces whch hold togethe the patcles fomng the gd bod. If the gd bod s stuctuall composed of seveal pats, the foce holdng the components pats togethe ae also defned as ntenal foces. 3-3

4 ncple of Tansmssblt: Equvalent oces ncple of Tansmssblt - Condtons of equlbum o moton ae not affected b tansmttng a foce along ts lne of acton. NOTE: and ae equvalent foces. ovng the pont of applcaton of the foce to the ea bumpe does not affect the moton o the othe foces actng on the tuc. ncple of tansmssblt ma not alwas appl n detemnng ntenal foces and defomatons. 3-4

5 ultplng vectos ultplng a vecto wth a vecto: Vecto (Coss) oduct The vecto poduct between two vectos a and b can be wtten as: The ght-hand ule allows us to fnd the decton of vecto c. The esult s a new vecto c, whch s: Hee a and b ae the magntudes of vectos a and b espectvel, and f s the smalle of the two angles between a and b vectos.

6 Vecto oduct of Two Vectos Concept of the moment of a foce about a pont s moe easl undestood though applcatons of the vecto poduct o coss poduct. Vecto poduct of two vectos and s defned as the vecto V whch satsfes the followng condtons: 1. Lne of acton of V s pependcula to plane contanng and. 2. agntude of V s V sn 3. Decton of V s obtaned fom the ght-hand ule. Vecto poducts: - ae not commutatve, - ae dstbutve, - ae not assocatve, S S 3-6

7 Vecto oducts: Rectangula Components 3-7 Vecto poducts of Catesan unt vectos, Vecto poducts n tems of ectangula coodnates V Refe to bottom of page 82 to smplf t futhe

8 oment of a oce About a ont A foce vecto s defned b ts magntude and decton. Its effect on the gd bod also depends on t pont of applcaton. The moment of about O s defned as O The moment vecto O s pependcula to the plane contanng O and the foce. agntude of O measues the tendenc of the foce to cause otaton of the bod about an as along O. O sn d The sense of the moment ma be detemned b the ght-hand ule. An foce that has the same magntude and decton as, s equvalent f t also has the same lne of acton and theefoe, poduces the same moment. 3-8

9 oment of a oce About a ont Two-dmensonal stuctues have length and beadth but neglgble depth and ae subected to foces contaned n the plane of the stuctue. The plane of the stuctue contans the pont O and the foce. O, the moment of the foce about O s pependcula to the plane. If the foce tends to otate the stuctue counteclocwse, the sense of the moment vecto s out of the plane of the stuctue and the magntude of the moment s postve. If the foce tends to otate the stuctue clocwse, the sense of the moment vecto s nto the plane of the stuctue and the magntude of the moment s negatve. 3-9

10 Vagnon s Theoem The moment about a gve pont O of the esultant of seveal concuent foces s equal to the sum of the moments of the vaous moments about the same pont O Vagon s Theoem maes t possble to eplace the dect detemnaton of the moment of a foce b the moments of two o moe component foces of. 3-10

11 Rectangula Components of the oment of a oce 3-11 O The moment of about O, O,

12 Sample oblem 3.1 A 100-lb vetcal foce s appled to the end of a leve whch s attached to a shaft at O. Detemne: a) moment about O, b) hoontal foce at A whch ceates the same moment, c) smallest foce at A whch poduces the same moment, d) locaton fo a 240-lb vetcal foce to poduce the same moment, e) whethe an of the foces fom b, c, and d s equvalent to the ognal foce. 3-12

13 Sample oblem 3.1 a) oment about O s equal to the poduct of the foce and the pependcula dstance between the lne of acton of the foce and O. Snce the foce tends to otate the leve clocwse, the moment vecto s nto the plane of the pape. O d O d 24n. cos lb12 n. 12 n. O 1200 lbn 3-13

14 Sample oblem 3.1 c) Hoontal foce at A that poduces the same moment, d O 24 n. d 1200 lbn n lbn n. sn n lb 3-14

15 Sample oblem 3.1 c) The smallest foce A to poduce the same moment occus when the pependcula dstance s a mamum o when s pependcula to OA. O 1200 lbn. d 24 n lbn. 24 n. 50 lb 3-15

16 Sample oblem 3.1 d) To detemne the pont of applcaton of a 240 lb foce to poduce the same moment, O 1200 lbn. d OB cos60 d 240 lb d 1200 lbn. 240 lb 5 n. 5 n. OB 10 n. 3-16

17 Sample oblem 3.1 e) Although each of the foces n pats b), c), and d) poduces the same moment as the 100 lb foce, none ae of the same magntude and sense, o on the same lne of acton. None of the foces s equvalent to the 100 lb foce. 3-17

18 Scala oduct of Two Vectos 3-18 The scala poduct o dot poduct between two vectos and s defned as esult scala cos Scala poducts: - ae commutatve, - ae dstbutve, - ae not assocatve, undefned S Scala poducts wth Catesan unt components,

19 Scala oduct of Two Vectos: Applcatons Angle between two vectos: cos cos oecton of a vecto on a gven as: OL cos poecton of along OL cos cos OL o an as defned b a unt vecto: OL cos cos cos 3-19

20 ed Tple oduct of Thee Vectos ed tple poduct of thee vectos, S scala esult The s med tple poducts fomed fom S,, and have equal magntudes but not the same sgn, S S S S S S Evaluatng S the med tple poduct, S S S S S S 3-20

21 oment of a Couple Two foces and - havng the same magntude, paallel lnes of acton, and opposte sense ae sad to fom a couple. oment of the couple, A B A B sn d The moment vecto of the couple s ndependent of the choce of the ogn of the coodnate aes,.e., t s a fee vecto that can be appled at an pont wth the same effect. 3-21

22 oment of a Couple Two couples wll have equal moments f 1 d1 2d 2 the two couples le n paallel planes, and the two couples have the same sense o the tendenc to cause otaton n the same decton. 3-22

23 Addton of Couples 3-23 Consde two ntesectng planes 1 and 2 wth each contanng a couple n plane n plane Resultants of the vectos also fom a couple 1 2 R B Vagon s theoem Sum of two couples s also a couple that s equal to the vecto sum of the two couples

24 Couples Can Be Repesented b Vectos A couple can be epesented b a vecto wth magntude and decton equal to the moment of the couple. Couple vectos obe the law of addton of vectos. Couple vectos ae fee vectos,.e., the pont of applcaton s not sgnfcant. Couple vectos ma be esolved nto component vectos. 3-24

25 Resoluton of a oce Into a oce at O and a Couple oce vecto can not be smpl moved to O wthout modfng ts acton on the bod. Attachng equal and opposte foce vectos at O poduces no net effect on the bod. The thee foces ma be eplaced b an equvalent foce vecto and couple vecto,.e, a foce-couple sstem. 3-25

Engineering Mechanics. Force resultants, Torques, Scalar Products, Equivalent Force systems

Engineering Mechanics. Force resultants, Torques, Scalar Products, Equivalent Force systems Engneeng echancs oce esultants, Toques, Scala oducts, Equvalent oce sstems Tata cgaw-hll Companes, 008 Resultant of Two oces foce: acton of one bod on anothe; chaacteed b ts pont of applcaton, magntude,

More information

Scalars and Vectors Scalar

Scalars and Vectors Scalar Scalas and ectos Scala A phscal quantt that s completel chaacteed b a eal numbe (o b ts numecal value) s called a scala. In othe wods a scala possesses onl a magntude. Mass denst volume tempeatue tme eneg

More information

Chapter I Matrices, Vectors, & Vector Calculus 1-1, 1-9, 1-10, 1-11, 1-17, 1-18, 1-25, 1-27, 1-36, 1-37, 1-41.

Chapter I Matrices, Vectors, & Vector Calculus 1-1, 1-9, 1-10, 1-11, 1-17, 1-18, 1-25, 1-27, 1-36, 1-37, 1-41. Chapte I Matces, Vectos, & Vecto Calculus -, -9, -0, -, -7, -8, -5, -7, -36, -37, -4. . Concept of a Scala Consde the aa of patcles shown n the fgue. he mass of the patcle at (,) can be epessed as. M (,

More information

Capítulo. Three Dimensions

Capítulo. Three Dimensions Capítulo Knematcs of Rgd Bodes n Thee Dmensons Mecánca Contents ntoducton Rgd Bod Angula Momentum n Thee Dmensons Pncple of mpulse and Momentum Knetc Eneg Sample Poblem 8. Sample Poblem 8. Moton of a Rgd

More information

DYNAMICS VECTOR MECHANICS FOR ENGINEERS: Kinematics of Rigid Bodies in Three Dimensions. Seventh Edition CHAPTER

DYNAMICS VECTOR MECHANICS FOR ENGINEERS: Kinematics of Rigid Bodies in Three Dimensions. Seventh Edition CHAPTER Edton CAPTER 8 VECTOR MECANCS FOR ENGNEERS: DYNAMCS Fednand P. Bee E. Russell Johnston, J. Lectue Notes: J. Walt Ole Teas Tech Unvest Knematcs of Rgd Bodes n Thee Dmensons 003 The McGaw-ll Companes, nc.

More information

1. A body will remain in a state of rest, or of uniform motion in a straight line unless it

1. A body will remain in a state of rest, or of uniform motion in a straight line unless it Pncples of Dnamcs: Newton's Laws of moton. : Foce Analss 1. A bod wll eman n a state of est, o of unfom moton n a staght lne unless t s acted b etenal foces to change ts state.. The ate of change of momentum

More information

Review of Vector Algebra and Vector Calculus Operations

Review of Vector Algebra and Vector Calculus Operations Revew of Vecto Algeba and Vecto Calculus Opeatons Tpes of vaables n Flud Mechancs Repesentaton of vectos Dffeent coodnate sstems Base vecto elatons Scala and vecto poducts Stess Newton s law of vscost

More information

PHY126 Summer Session I, 2008

PHY126 Summer Session I, 2008 PHY6 Summe Sesson I, 8 Most of nfomaton s avalable at: http://nngoup.phscs.sunsb.edu/~chak/phy6-8 ncludng the sllabus and lectue sldes. Read sllabus and watch fo mpotant announcements. Homewok assgnment

More information

Integral Vector Operations and Related Theorems Applications in Mechanics and E&M

Integral Vector Operations and Related Theorems Applications in Mechanics and E&M Dola Bagayoko (0) Integal Vecto Opeatons and elated Theoems Applcatons n Mechancs and E&M Ι Basc Defnton Please efe to you calculus evewed below. Ι, ΙΙ, andιιι notes and textbooks fo detals on the concepts

More information

Rotary motion

Rotary motion ectue 8 RTARY TN F THE RGD BDY Notes: ectue 8 - Rgd bod Rgd bod: j const numbe of degees of feedom 6 3 tanslatonal + 3 ota motons m j m j Constants educe numbe of degees of feedom non-fee object: 6-p

More information

Set of square-integrable function 2 L : function space F

Set of square-integrable function 2 L : function space F Set of squae-ntegable functon L : functon space F Motvaton: In ou pevous dscussons we have seen that fo fee patcles wave equatons (Helmholt o Schödnge) can be expessed n tems of egenvalue equatons. H E,

More information

24-2: Electric Potential Energy. 24-1: What is physics

24-2: Electric Potential Energy. 24-1: What is physics D. Iyad SAADEDDIN Chapte 4: Electc Potental Electc potental Enegy and Electc potental Calculatng the E-potental fom E-feld fo dffeent chage dstbutons Calculatng the E-feld fom E-potental Potental of a

More information

One-dimensional kinematics

One-dimensional kinematics Phscs 45 Fomula Sheet Eam 3 One-dmensonal knematcs Vectos dsplacement: Δ total dstance taveled aveage speed total tme Δ aveage veloct: vav t t Δ nstantaneous veloct: v lm Δ t v aveage acceleaton: aav t

More information

2/24/2014. The point mass. Impulse for a single collision The impulse of a force is a vector. The Center of Mass. System of particles

2/24/2014. The point mass. Impulse for a single collision The impulse of a force is a vector. The Center of Mass. System of particles /4/04 Chapte 7 Lnea oentu Lnea oentu of a Sngle Patcle Lnea oentu: p υ It s a easue of the patcle s oton It s a vecto, sla to the veloct p υ p υ p υ z z p It also depends on the ass of the object, sla

More information

LINEAR MOMENTUM. product of the mass m and the velocity v r of an object r r

LINEAR MOMENTUM. product of the mass m and the velocity v r of an object r r LINEAR MOMENTUM Imagne beng on a skateboad, at est that can move wthout cton on a smooth suace You catch a heavy, slow-movng ball that has been thown to you you begn to move Altenatvely you catch a lght,

More information

Dynamics of Rigid Bodies

Dynamics of Rigid Bodies Dynamcs of Rgd Bodes A gd body s one n whch the dstances between consttuent patcles s constant thoughout the moton of the body,.e. t keeps ts shape. Thee ae two knds of gd body moton: 1. Tanslatonal Rectlnea

More information

Physics 1: Mechanics

Physics 1: Mechanics Physcs : Mechancs Đào Ngọc Hạnh Tâm Offce: A.503, Emal: dnhtam@hcmu.edu.vn HCMIU, Vetnam Natonal Unvesty Acknowledgment: Sldes ae suppoted by Pof. Phan Bao Ngoc Contents of Physcs Pat A: Dynamcs of Mass

More information

Physics 202, Lecture 2. Announcements

Physics 202, Lecture 2. Announcements Physcs 202, Lectue 2 Today s Topcs Announcements Electc Felds Moe on the Electc Foce (Coulomb s Law The Electc Feld Moton of Chaged Patcles n an Electc Feld Announcements Homewok Assgnment #1: WebAssgn

More information

SCALARS AND VECTORS All physical quantities in engineering mechanics are measured using either scalars or vectors.

SCALARS AND VECTORS All physical quantities in engineering mechanics are measured using either scalars or vectors. SCALARS AND ECTORS All phscal uanttes n engneerng mechancs are measured usng ether scalars or vectors. Scalar. A scalar s an postve or negatve phscal uantt that can be completel specfed b ts magntude.

More information

UNIVERSITÀ DI PISA. Math thbackground

UNIVERSITÀ DI PISA. Math thbackground UNIVERSITÀ DI ISA Electomagnetc Radatons and Bologcal l Inteactons Lauea Magstale n Bomedcal Engneeng Fst semeste (6 cedts), academc ea 2011/12 of. aolo Nepa p.nepa@et.unp.t Math thbackgound Edted b D.

More information

COLLEGE OF FOUNDATION AND GENERAL STUDIES PUTRAJAYA CAMPUS FINAL EXAMINATION TRIMESTER /2017

COLLEGE OF FOUNDATION AND GENERAL STUDIES PUTRAJAYA CAMPUS FINAL EXAMINATION TRIMESTER /2017 COLLEGE OF FOUNDATION AND GENERAL STUDIES PUTRAJAYA CAMPUS FINAL EXAMINATION TRIMESTER 1 016/017 PROGRAMME SUBJECT CODE : Foundaton n Engneeng : PHYF115 SUBJECT : Phscs 1 DATE : Septembe 016 DURATION :

More information

Rotational Kinematics. Rigid Object about a Fixed Axis Western HS AP Physics 1

Rotational Kinematics. Rigid Object about a Fixed Axis Western HS AP Physics 1 Rotatonal Knematcs Rgd Object about a Fxed Axs Westen HS AP Physcs 1 Leanng Objectes What we know Unfom Ccula Moton q s Centpetal Acceleaton : Centpetal Foce: Non-unfom a F c c m F F F t m ma t What we

More information

PHYS 705: Classical Mechanics. Derivation of Lagrange Equations from D Alembert s Principle

PHYS 705: Classical Mechanics. Derivation of Lagrange Equations from D Alembert s Principle 1 PHYS 705: Classcal Mechancs Devaton of Lagange Equatons fom D Alembet s Pncple 2 D Alembet s Pncple Followng a smla agument fo the vtual dsplacement to be consstent wth constants,.e, (no vtual wok fo

More information

CSJM University Class: B.Sc.-II Sub:Physics Paper-II Title: Electromagnetics Unit-1: Electrostatics Lecture: 1 to 4

CSJM University Class: B.Sc.-II Sub:Physics Paper-II Title: Electromagnetics Unit-1: Electrostatics Lecture: 1 to 4 CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 Electostatcs: It deals the study of behavo of statc o statonay Chages. Electc Chage: It s popety by

More information

Chapter 8. Linear Momentum, Impulse, and Collisions

Chapter 8. Linear Momentum, Impulse, and Collisions Chapte 8 Lnea oentu, Ipulse, and Collsons 8. Lnea oentu and Ipulse The lnea oentu p of a patcle of ass ovng wth velocty v s defned as: p " v ote that p s a vecto that ponts n the sae decton as the velocty

More information

Chapter 12 Equilibrium and Elasticity

Chapter 12 Equilibrium and Elasticity Chapte 12 Equlbum and Elastcty In ths chapte we wll defne equlbum and fnd the condtons needed so that an object s at equlbum. We wll then apply these condtons to a vaety of pactcal engneeng poblems of

More information

Physics Exam II Chapters 25-29

Physics Exam II Chapters 25-29 Physcs 114 1 Exam II Chaptes 5-9 Answe 8 of the followng 9 questons o poblems. Each one s weghted equally. Clealy mak on you blue book whch numbe you do not want gaded. If you ae not sue whch one you do

More information

Chapter 10 and elements of 11, 12 Rotation of Rigid Bodies

Chapter 10 and elements of 11, 12 Rotation of Rigid Bodies Chapte 10 and elements of 11, 1 Rotaton of Rgd Bodes What s a Rgd Body? Rotatonal Knematcs Angula Velocty ω and Acceleaton α Rotaton wth Constant Acceleaton Angula vs. Lnea Knematcs Enegy n Rotatonal Moton:

More information

iclicker Quiz a) True b) False Theoretical physics: the eternal quest for a missing minus sign and/or a factor of two. Which will be an issue today?

iclicker Quiz a) True b) False Theoretical physics: the eternal quest for a missing minus sign and/or a factor of two. Which will be an issue today? Clce Quz I egsteed my quz tansmtte va the couse webste (not on the clce.com webste. I ealze that untl I do so, my quz scoes wll not be ecoded. a Tue b False Theoetcal hyscs: the etenal quest fo a mssng

More information

Physics 11b Lecture #2. Electric Field Electric Flux Gauss s Law

Physics 11b Lecture #2. Electric Field Electric Flux Gauss s Law Physcs 11b Lectue # Electc Feld Electc Flux Gauss s Law What We Dd Last Tme Electc chage = How object esponds to electc foce Comes n postve and negatve flavos Conseved Electc foce Coulomb s Law F Same

More information

CSU ATS601 Fall Other reading: Vallis 2.1, 2.2; Marshall and Plumb Ch. 6; Holton Ch. 2; Schubert Ch r or v i = v r + r (3.

CSU ATS601 Fall Other reading: Vallis 2.1, 2.2; Marshall and Plumb Ch. 6; Holton Ch. 2; Schubert Ch r or v i = v r + r (3. 3 Eath s Rotaton 3.1 Rotatng Famewok Othe eadng: Valls 2.1, 2.2; Mashall and Plumb Ch. 6; Holton Ch. 2; Schubet Ch. 3 Consde the poston vecto (the same as C n the fgue above) otatng at angula velocty.

More information

Chapter Fifiteen. Surfaces Revisited

Chapter Fifiteen. Surfaces Revisited Chapte Ffteen ufaces Revsted 15.1 Vecto Descpton of ufaces We look now at the vey specal case of functons : D R 3, whee D R s a nce subset of the plane. We suppose s a nce functon. As the pont ( s, t)

More information

Physics 111 Lecture 11

Physics 111 Lecture 11 Physcs 111 ectue 11 Angula Momentum SJ 8th Ed.: Chap 11.1 11.4 Recap and Ovevew Coss Poduct Revsted Toque Revsted Angula Momentum Angula Fom o Newton s Second aw Angula Momentum o a System o Patcles Angula

More information

ALL QUESTIONS ARE WORTH 20 POINTS. WORK OUT FIVE PROBLEMS.

ALL QUESTIONS ARE WORTH 20 POINTS. WORK OUT FIVE PROBLEMS. GNRAL PHYSICS PH -3A (D. S. Mov) Test (/3/) key STUDNT NAM: STUDNT d #: -------------------------------------------------------------------------------------------------------------------------------------------

More information

1. Starting with the local version of the first law of thermodynamics q. derive the statement of the first law of thermodynamics for a control volume

1. Starting with the local version of the first law of thermodynamics q. derive the statement of the first law of thermodynamics for a control volume EN10: Contnuum Mechancs Homewok 5: Alcaton of contnuum mechancs to fluds Due 1:00 noon Fda Febua 4th chool of Engneeng Bown Unvest 1. tatng wth the local veson of the fst law of themodnamcs q jdj q t and

More information

Chapter 23: Electric Potential

Chapter 23: Electric Potential Chapte 23: Electc Potental Electc Potental Enegy It tuns out (won t show ths) that the tostatc foce, qq 1 2 F ˆ = k, s consevatve. 2 Recall, fo any consevatve foce, t s always possble to wte the wok done

More information

Remember: When an object falls due to gravity its potential energy decreases.

Remember: When an object falls due to gravity its potential energy decreases. Chapte 5: lectc Potental As mentoned seveal tmes dung the uate Newton s law o gavty and Coulomb s law ae dentcal n the mathematcal om. So, most thngs that ae tue o gavty ae also tue o electostatcs! Hee

More information

Units, Physical Quantities and Vectors

Units, Physical Quantities and Vectors What s Phscs? Unts, Phscal Quanttes and Vectos Natual Phlosoph scence of matte and eneg fundamental pncples of engneeng and technolog an epemental scence: theo epement smplfed (dealed) models ange of valdt

More information

BALANCING OF ROTATING MASSES

BALANCING OF ROTATING MASSES www.getyun.co YIS OF HIES IG OF ROTTIG SSES www.getyun.co Rotatng centelne: The otatng centelne beng defned as the axs about whch the oto would otate f not constaned by ts beangs. (lso called the Pncple

More information

gravity r2,1 r2 r1 by m 2,1

gravity r2,1 r2 r1 by m 2,1 Gavtaton Many of the foundatons of classcal echancs wee fst dscoveed when phlosophes (ealy scentsts and atheatcans) ted to explan the oton of planets and stas. Newton s ost faous fo unfyng the oton of

More information

Thermodynamics of solids 4. Statistical thermodynamics and the 3 rd law. Kwangheon Park Kyung Hee University Department of Nuclear Engineering

Thermodynamics of solids 4. Statistical thermodynamics and the 3 rd law. Kwangheon Park Kyung Hee University Department of Nuclear Engineering Themodynamcs of solds 4. Statstcal themodynamcs and the 3 d law Kwangheon Pak Kyung Hee Unvesty Depatment of Nuclea Engneeng 4.1. Intoducton to statstcal themodynamcs Classcal themodynamcs Statstcal themodynamcs

More information

Energy in Closed Systems

Energy in Closed Systems Enegy n Closed Systems Anamta Palt palt.anamta@gmal.com Abstact The wtng ndcates a beakdown of the classcal laws. We consde consevaton of enegy wth a many body system n elaton to the nvese squae law and

More information

Physics 201 Lecture 4

Physics 201 Lecture 4 Phscs 1 Lectue 4 ltoda: hapte 3 Lectue 4 v Intoduce scalas and vectos v Peom basc vecto aleba (addton and subtacton) v Inteconvet between atesan & Pola coodnates Stat n nteestn 1D moton poblem: ace 9.8

More information

V. Principles of Irreversible Thermodynamics. s = S - S 0 (7.3) s = = - g i, k. "Flux": = da i. "Force": = -Â g a ik k = X i. Â J i X i (7.

V. Principles of Irreversible Thermodynamics. s = S - S 0 (7.3) s = = - g i, k. Flux: = da i. Force: = -Â g a ik k = X i. Â J i X i (7. Themodynamcs and Knetcs of Solds 71 V. Pncples of Ievesble Themodynamcs 5. Onsage s Teatment s = S - S 0 = s( a 1, a 2,...) a n = A g - A n (7.6) Equlbum themodynamcs detemnes the paametes of an equlbum

More information

Tensor. Syllabus: x x

Tensor. Syllabus: x x Tenso Sllabus: Tenso Calculus : Catesan tensos. Smmetc and antsmmetc tensos. Lev Vvta tenso denst. Pseudo tensos. Dual tensos. Dect poduct and contacton. Dads and dadc. Covaant, Contavaant and med tensos.

More information

UNIT10 PLANE OF REGRESSION

UNIT10 PLANE OF REGRESSION UIT0 PLAE OF REGRESSIO Plane of Regesson Stuctue 0. Intoducton Ojectves 0. Yule s otaton 0. Plane of Regesson fo thee Vaales 0.4 Popetes of Resduals 0.5 Vaance of the Resduals 0.6 Summay 0.7 Solutons /

More information

If there are k binding constraints at x then re-label these constraints so that they are the first k constraints.

If there are k binding constraints at x then re-label these constraints so that they are the first k constraints. Mathematcal Foundatons -1- Constaned Optmzaton Constaned Optmzaton Ma{ f ( ) X} whee X {, h ( ), 1,, m} Necessay condtons fo to be a soluton to ths mamzaton poblem Mathematcally, f ag Ma{ f ( ) X}, then

More information

Physics 207 Lecture 16

Physics 207 Lecture 16 Physcs 07 Lectue 6 Goals: Lectue 6 Chapte Extend the patcle odel to gd-bodes Undestand the equlbu of an extended object. Analyze ollng oton Undestand otaton about a fxed axs. Eploy consevaton of angula

More information

VECTOR MECHANICS FOR ENGINEERS: STATICS

VECTOR MECHANICS FOR ENGINEERS: STATICS 4 Equilibium CHAPTER VECTOR MECHANICS FOR ENGINEERS: STATICS Fedinand P. Bee E. Russell Johnston, J. of Rigid Bodies Lectue Notes: J. Walt Ole Texas Tech Univesity Contents Intoduction Fee-Body Diagam

More information

COMPLEMENTARY ENERGY METHOD FOR CURVED COMPOSITE BEAMS

COMPLEMENTARY ENERGY METHOD FOR CURVED COMPOSITE BEAMS ultscence - XXX. mcocd Intenatonal ultdscplnay Scentfc Confeence Unvesty of skolc Hungay - pl 06 ISBN 978-963-358-3- COPLEENTRY ENERGY ETHOD FOR CURVED COPOSITE BES Ákos József Lengyel István Ecsed ssstant

More information

PHYS Week 5. Reading Journals today from tables. WebAssign due Wed nite

PHYS Week 5. Reading Journals today from tables. WebAssign due Wed nite PHYS 015 -- Week 5 Readng Jounals today fom tables WebAssgn due Wed nte Fo exclusve use n PHYS 015. Not fo e-dstbuton. Some mateals Copyght Unvesty of Coloado, Cengage,, Peason J. Maps. Fundamental Tools

More information

Test 1 phy What mass of a material with density ρ is required to make a hollow spherical shell having inner radius r i and outer radius r o?

Test 1 phy What mass of a material with density ρ is required to make a hollow spherical shell having inner radius r i and outer radius r o? Test 1 phy 0 1. a) What s the pupose of measuement? b) Wte all fou condtons, whch must be satsfed by a scala poduct. (Use dffeent symbols to dstngush opeatons on ectos fom opeatons on numbes.) c) What

More information

Physics 2A Chapter 10 - Moment of Inertia Fall 2018

Physics 2A Chapter 10 - Moment of Inertia Fall 2018 Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.

More information

3.1 Electrostatic Potential Energy and Potential Difference

3.1 Electrostatic Potential Energy and Potential Difference 3. lectostatc Potental negy and Potental Dffeence RMMR fom mechancs: - The potental enegy can be defned fo a system only f consevatve foces act between ts consttuents. - Consevatve foces may depend only

More information

Machine Learning 4771

Machine Learning 4771 Machne Leanng 4771 Instucto: Tony Jebaa Topc 6 Revew: Suppot Vecto Machnes Pmal & Dual Soluton Non-sepaable SVMs Kenels SVM Demo Revew: SVM Suppot vecto machnes ae (n the smplest case) lnea classfes that

More information

Mechanics Physics 151

Mechanics Physics 151 Mechancs Physcs 151 Lectue 18 Hamltonan Equatons of Moton (Chapte 8) What s Ahead We ae statng Hamltonan fomalsm Hamltonan equaton Today and 11/6 Canoncal tansfomaton 1/3, 1/5, 1/10 Close lnk to non-elatvstc

More information

COORDINATE SYSTEMS, COORDINATE TRANSFORMS, AND APPLICATIONS

COORDINATE SYSTEMS, COORDINATE TRANSFORMS, AND APPLICATIONS Dola Bagaoo 0 COORDINTE SYSTEMS COORDINTE TRNSFORMS ND PPLICTIONS I. INTRODUCTION Smmet coce of coodnate sstem. In solvng Pscs poblems one cooses a coodnate sstem tat fts te poblem at and.e. a coodnate

More information

Part V: Velocity and Acceleration Analysis of Mechanisms

Part V: Velocity and Acceleration Analysis of Mechanisms Pat V: Velocty an Acceleaton Analyss of Mechansms Ths secton wll evew the most common an cuently pactce methos fo completng the knematcs analyss of mechansms; escbng moton though velocty an acceleaton.

More information

BALANCING OF ROTATING MASSES

BALANCING OF ROTATING MASSES VTU EUST PROGRE - 7 YIS OF HIES Subject ode - E 54 IG OF ROTTIG SSES otes opled by: VIJYVITH OGE SSOITE PROFESSOR EPRTET OF EHI EGIEERIG OEGE OF EGIEERIG HSS -57. KRTK oble:94488954 E-al:vvb@cehassan.ac.n

More information

Physics 5153 Classical Mechanics. Principle of Virtual Work-1

Physics 5153 Classical Mechanics. Principle of Virtual Work-1 P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal

More information

Physics 2A Chapter 11 - Universal Gravitation Fall 2017

Physics 2A Chapter 11 - Universal Gravitation Fall 2017 Physcs A Chapte - Unvesal Gavtaton Fall 07 hese notes ae ve pages. A quck summay: he text boxes n the notes contan the esults that wll compse the toolbox o Chapte. hee ae thee sectons: the law o gavtaton,

More information

1. The tie-rod AB exerts the 250 N force on the steering knuckle AO as shown. Replace this force by an equivalent force-couple system at O.

1. The tie-rod AB exerts the 250 N force on the steering knuckle AO as shown. Replace this force by an equivalent force-couple system at O. 1. The terod AB exerts the 50 N force on the steerng knuckle AO as shown. Replace ths force by an equvalent forcecouple system at O. . The devce shown s part of an automoble seatbackrelease mechansm. The

More information

Amplifier Constant Gain and Noise

Amplifier Constant Gain and Noise Amplfe Constant Gan and ose by Manfed Thumm and Wene Wesbeck Foschungszentum Kalsuhe n de Helmholtz - Gemenschaft Unvestät Kalsuhe (TH) Reseach Unvesty founded 85 Ccles of Constant Gan (I) If s taken to

More information

7.2.1 Basic relations for Torsion of Circular Members

7.2.1 Basic relations for Torsion of Circular Members Section 7. 7. osion In this section, the geomety to be consideed is that of a long slende cicula ba and the load is one which twists the ba. Such poblems ae impotant in the analysis of twisting components,

More information

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 29,

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 29, hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe 9, 08 47 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe 9, 08 48 I. ETOR AND TENOR ANALYI I... Tenso functon th Let A

More information

2 dependence in the electrostatic force means that it is also

2 dependence in the electrostatic force means that it is also lectc Potental negy an lectc Potental A scala el, nvolvng magntues only, s oten ease to wo wth when compae to a vecto el. Fo electc els not havng to begn wth vecto ssues woul be nce. To aange ths a scala

More information

One Dimensional Axial Deformations

One Dimensional Axial Deformations One Dmensonal al Deformatons In ths secton, a specfc smple geometr s consdered, that of a long and thn straght component loaded n such a wa that t deforms n the aal drecton onl. The -as s taken as the

More information

Moment. F r F r d. Magnitude of moment depends on magnitude of F and the length d

Moment. F r F r d. Magnitude of moment depends on magnitude of F and the length d Moment Tanslation Tanslation + Rotation This otation tenency is known as moment M of foce (toque) xis of otation may be any line which neithe intesects no paallel to the line of action of foce Magnitue

More information

Physics Exam 3

Physics Exam 3 Physcs 114 1 Exam 3 The numbe of ponts fo each secton s noted n backets, []. Choose a total of 35 ponts that wll be gaded that s you may dop (not answe) a total of 5 ponts. Clealy mak on the cove of you

More information

TEST-03 TOPIC: MAGNETISM AND MAGNETIC EFFECT OF CURRENT Q.1 Find the magnetic field intensity due to a thin wire carrying current I in the Fig.

TEST-03 TOPIC: MAGNETISM AND MAGNETIC EFFECT OF CURRENT Q.1 Find the magnetic field intensity due to a thin wire carrying current I in the Fig. TEST-03 TPC: MAGNETSM AND MAGNETC EFFECT F CURRENT Q. Fnd the magnetc feld ntensty due to a thn we cayng cuent n the Fg. - R 0 ( + tan) R () 0 ( ) R 0 ( + ) R 0 ( + tan ) R Q. Electons emtted wth neglgble

More information

Rotating Disk Electrode -a hydrodynamic method

Rotating Disk Electrode -a hydrodynamic method Rotatng Dsk Electode -a hdodnamc method Fe Lu Ma 3, 0 ente fo Electochemcal Engneeng Reseach Depatment of hemcal and Bomolecula Engneeng Rotatng Dsk Electode A otatng dsk electode RDE s a hdodnamc wokng

More information

Physics 2020, Spring 2005 Lab 5 page 1 of 8. Lab 5. Magnetism

Physics 2020, Spring 2005 Lab 5 page 1 of 8. Lab 5. Magnetism Physics 2020, Sping 2005 Lab 5 page 1 of 8 Lab 5. Magnetism PART I: INTRODUCTION TO MAGNETS This week we will begin wok with magnets and the foces that they poduce. By now you ae an expet on setting up

More information

Electric Charge and Field

Electric Charge and Field lectic Chage and ield Chapte 6 (Giancoli) All sections ecept 6.0 (Gauss s law) Compaison between the lectic and the Gavitational foces Both have long ange, The electic chage of an object plas the same

More information

8 Baire Category Theorem and Uniform Boundedness

8 Baire Category Theorem and Uniform Boundedness 8 Bae Categoy Theoem and Unfom Boundedness Pncple 8.1 Bae s Categoy Theoem Valdty of many esults n analyss depends on the completeness popety. Ths popety addesses the nadequacy of the system of atonal

More information

Chapter 3 Vector Integral Calculus

Chapter 3 Vector Integral Calculus hapte Vecto Integal alculus I. Lne ntegals. Defnton A lne ntegal of a vecto functon F ove a cuve s F In tems of components F F F F If,, an ae functon of t, we have F F F F t t t t E.. Fn the value of the

More information

4 SingularValue Decomposition (SVD)

4 SingularValue Decomposition (SVD) /6/00 Z:\ jeh\self\boo Kannan\Jan-5-00\4 SVD 4 SngulaValue Decomposton (SVD) Chapte 4 Pat SVD he sngula value decomposton of a matx s the factozaton of nto the poduct of thee matces = UDV whee the columns

More information

So far: simple (planar) geometries

So far: simple (planar) geometries Physcs 06 ecture 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap. to 3 Rotatonal quanttes as vectors Cross product Torque epressed as a vector Angular momentum defned Angular momentum as a vector

More information

Summer Workshop on the Reaction Theory Exercise sheet 8. Classwork

Summer Workshop on the Reaction Theory Exercise sheet 8. Classwork Joned Physcs Analyss Cente Summe Wokshop on the Reacton Theoy Execse sheet 8 Vncent Matheu Contact: http://www.ndana.edu/~sst/ndex.html June June To be dscussed on Tuesday of Week-II. Classwok. Deve all

More information

Physics 201, Lecture 4. Vectors and Scalars. Chapters Covered q Chapter 1: Physics and Measurement.

Physics 201, Lecture 4. Vectors and Scalars. Chapters Covered q Chapter 1: Physics and Measurement. Phscs 01, Lecture 4 Toda s Topcs n Vectors chap 3) n Scalars and Vectors n Vector ddton ule n Vector n a Coordnator Sstem n Decomposton of a Vector n Epected from prevew: n Scalars and Vectors, Vector

More information

Correspondence Analysis & Related Methods

Correspondence Analysis & Related Methods Coespondence Analyss & Related Methods Ineta contbutons n weghted PCA PCA s a method of data vsualzaton whch epesents the tue postons of ponts n a map whch comes closest to all the ponts, closest n sense

More information

The Forming Theory and the NC Machining for The Rotary Burs with the Spectral Edge Distribution

The Forming Theory and the NC Machining for The Rotary Burs with the Spectral Edge Distribution oden Appled Scence The Fomn Theoy and the NC achnn fo The Rotay us wth the Spectal Ede Dstbuton Huan Lu Depatment of echancal Enneen, Zhejan Unvesty of Scence and Technoloy Hanzhou, c.y. chan, 310023,

More information

Vectors Serway and Jewett Chapter 3

Vectors Serway and Jewett Chapter 3 Vectos Sewa and Jewett Chapte 3 Scalas and Vectos Vecto Components and Aithmetic Vectos in 3 Dimensions Unit vectos i, j, k Pactice Poblems: Chapte 3, poblems 9, 19, 31, 45, 55, 61 Phsical quantities ae

More information

Spring 2002 Lecture #13

Spring 2002 Lecture #13 44-50 Sprng 00 ecture # Dr. Jaehoon Yu. Rotatonal Energy. Computaton of oments of nerta. Parallel-as Theorem 4. Torque & Angular Acceleraton 5. Work, Power, & Energy of Rotatonal otons Remember the md-term

More information

P 365. r r r )...(1 365

P 365. r r r )...(1 365 SCIENCE WORLD JOURNAL VOL (NO4) 008 www.scecncewoldounal.og ISSN 597-64 SHORT COMMUNICATION ANALYSING THE APPROXIMATION MODEL TO BIRTHDAY PROBLEM *CHOJI, D.N. & DEME, A.C. Depatment of Mathematcs Unvesty

More information

Unit_III Complex Numbers: Some Basic Results: 1. If z = x +iy is a complex number, then the complex number z = x iy is

Unit_III Complex Numbers: Some Basic Results: 1. If z = x +iy is a complex number, then the complex number z = x iy is Unt_III Comple Nmbes: In the sstem o eal nmbes R we can sole all qadatc eqatons o the om a b c, a, and the dscmnant b 4ac. When the dscmnant b 4ac

More information

7/1/2008. Adhi Harmoko S. a c = v 2 /r. F c = m x a c = m x v 2 /r. Ontang Anting Moment of Inertia. Energy

7/1/2008. Adhi Harmoko S. a c = v 2 /r. F c = m x a c = m x v 2 /r. Ontang Anting Moment of Inertia. Energy 7//008 Adh Haoko S Ontang Antng Moent of neta Enegy Passenge undego unfo ccula oton (ccula path at constant speed) Theefoe, thee ust be a: centpetal acceleaton, a c. Theefoe thee ust be a centpetal foce,

More information

Khintchine-Type Inequalities and Their Applications in Optimization

Khintchine-Type Inequalities and Their Applications in Optimization Khntchne-Type Inequaltes and The Applcatons n Optmzaton Anthony Man-Cho So Depatment of Systems Engneeng & Engneeng Management The Chnese Unvesty of Hong Kong ISDS-Kolloquum Unvestaet Wen 29 June 2009

More information

INTRODUCTION. consider the statements : I there exists x X. f x, such that. II there exists y Y. such that g y

INTRODUCTION. consider the statements : I there exists x X. f x, such that. II there exists y Y. such that g y INRODUCION hs dssetaton s the eadng of efeences [1], [] and [3]. Faas lemma s one of the theoems of the altenatve. hese theoems chaacteze the optmalt condtons of seveal mnmzaton poblems. It s nown that

More information

APPLICATIONS OF SEMIGENERALIZED -CLOSED SETS

APPLICATIONS OF SEMIGENERALIZED -CLOSED SETS Intenatonal Jounal of Mathematcal Engneeng Scence ISSN : 22776982 Volume Issue 4 (Apl 202) http://www.mes.com/ https://stes.google.com/ste/mesounal/ APPLICATIONS OF SEMIGENERALIZED CLOSED SETS G.SHANMUGAM,

More information

Chapter 13 - Universal Gravitation

Chapter 13 - Universal Gravitation Chapte 3 - Unesal Gataton In Chapte 5 we studed Newton s thee laws of moton. In addton to these laws, Newton fomulated the law of unesal gataton. Ths law states that two masses ae attacted by a foce gen

More information

19 The Born-Oppenheimer Approximation

19 The Born-Oppenheimer Approximation 9 The Bon-Oppenheme Appoxmaton The full nonelatvstc Hamltonan fo a molecule s gven by (n a.u.) Ĥ = A M A A A, Z A + A + >j j (883) Lets ewte the Hamltonan to emphasze the goal as Ĥ = + A A A, >j j M A

More information

Physics 1501 Lecture 19

Physics 1501 Lecture 19 Physcs 1501 ectue 19 Physcs 1501: ectue 19 Today s Agenda Announceents HW#7: due Oct. 1 Mdte 1: aveage 45 % Topcs otatonal Kneatcs otatonal Enegy Moents of Ineta Physcs 1501: ectue 19, Pg 1 Suay (wth copason

More information

Physics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding.

Physics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding. Physcs 53 Rotatonal Moton 3 Sr, I have found you an argument, but I am not oblged to fnd you an understandng. Samuel Johnson Angular momentum Wth respect to rotatonal moton of a body, moment of nerta plays

More information

Physics 2A Chapter 3 HW Solutions

Physics 2A Chapter 3 HW Solutions Phscs A Chapter 3 HW Solutons Chapter 3 Conceptual Queston: 4, 6, 8, Problems: 5,, 8, 7, 3, 44, 46, 69, 70, 73 Q3.4. Reason: (a) C = A+ B onl A and B are n the same drecton. Sze does not matter. (b) C

More information

Physics 201 Lecture 18

Physics 201 Lecture 18 Phsics 0 ectue 8 ectue 8 Goals: Define and anale toque ntoduce the coss poduct Relate otational dnamics to toque Discuss wok and wok eneg theoem with espect to otational motion Specif olling motion (cente

More information

Chapter I Vector Analysis

Chapter I Vector Analysis . Chpte I Vecto nlss . Vecto lgeb j It s well-nown tht n vecto cn be wtten s Vectos obe the followng lgebc ules: scl s ) ( j v v cos ) ( e Commuttv ) ( ssoctve C C ) ( ) ( v j ) ( ) ( ) ( ) ( (v) he lw

More information

Fundamental principles

Fundamental principles JU 07/HL Dnacs and contol of echancal sstes Date Da (0/08) Da (03/08) Da 3 (05/08) Da 4 (07/08) Da 5 (09/08) Da 6 (/08) Content Reve of the bascs of echancs. Kneatcs of gd bodes coodnate tansfoaton, angula

More information

Groupoid and Topological Quotient Group

Groupoid and Topological Quotient Group lobal Jounal of Pue and Appled Mathematcs SSN 0973-768 Volume 3 Numbe 7 07 pp 373-39 Reseach nda Publcatons http://wwwpublcatoncom oupod and Topolocal Quotent oup Mohammad Qasm Manna Depatment of Mathematcs

More information

Week 11: Chapter 11. The Vector Product. The Vector Product Defined. The Vector Product and Torque. More About the Vector Product

Week 11: Chapter 11. The Vector Product. The Vector Product Defined. The Vector Product and Torque. More About the Vector Product The Vector Product Week 11: Chapter 11 Angular Momentum There are nstances where the product of two vectors s another vector Earler we saw where the product of two vectors was a scalar Ths was called the

More information

Generating Functions, Weighted and Non-Weighted Sums for Powers of Second-Order Recurrence Sequences

Generating Functions, Weighted and Non-Weighted Sums for Powers of Second-Order Recurrence Sequences Geneatng Functons, Weghted and Non-Weghted Sums fo Powes of Second-Ode Recuence Sequences Pantelmon Stăncă Aubun Unvesty Montgomey, Depatment of Mathematcs Montgomey, AL 3614-403, USA e-mal: stanca@studel.aum.edu

More information