One-dimensional kinematics
|
|
- Emory Harrell
- 5 years ago
- Views:
Transcription
1 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 d v t dv d nstantaneous acceleaton: a lm One-dmensonal moton wth constant acceleaton: () v v + at () + ( v + v)t (3) + v t + at v v + a (4) ( ) Fee all (postve decton o taken to be upwad) and a g n the above 4 equatons o knematcs: () v v gt () + ( v + v)t (3) + v t gt v v g (4) ( ) Vectos n -D I a vecto A s wtten n component om as A A ˆ + A ˆ A, A then: Gettng magntude and decton o A om the components: A A + A (magntude o A ) A θ actan (decton o A ) A
2 Fomula Sheet Eam 3 Page Gettng components om magntude and decton: A A cosθ ( component o A ) A snθ ( component o A θ undestood to be the angle that ) } A makes wth the postve as A Vectos n 3-D I A A ˆ + A ˆ + A ˆ A, A, A then: A A + A + A -D Knematcs poston vecto: () t () t ˆ + ( t) ˆ ( t), ( t) Δ Δ Δ aveage veloct: v av, d d d nstantaneous veloct: v, aveage acceleaton: aav, nstantaneous acceleaton: dv dv dv a, d d d a, nstantaneous speed v (magntude o the nstantaneous veloct) 3-D knematcs poston vecto: () t () t ˆ + ( t) ˆ + ( t) ˆ ( t), ( t), ( t) Δ Δ Δ Δ aveage veloct: v av,, d d d d nstantaneous veloct: v,, aveage acceleaton: aav,, nstantaneous acceleaton: dv dv dv dv a,, d d d d a,, nstantaneous speed v (magntude o the nstantaneous veloct)
3 Fomula Sheet Eam 3 Page 3 Pojectle Moton decton (moton wth constant veloct): a v v + v t decton (ee all... postve decton o taken to be upwad): () v v gt () + ( v + v) t (3) + vt gt v v g Relatve Moton (4) ( ) vpa vpb + vba Newton s Laws o Moton Two boad categoes o oces: contact oces (objects n contact wth one anothe) and eld oces (objects not n contact wth one anothe). Gavt s the onl eld oce we wll deal wth n ths couse. Weght: w mg st law: v s constant unless object (o sstem) epeences net etenal oce. nd law : F ma mples thee statements, n geneal: F ma, F ma, and F ma *o sstems o objects, F m a et ss ss 3 d law: Wheneve one object eets a oce on a second object, the second eets a oce on the st; these two oces ae equal n magntude and opposte n decton: F F Equlbum An object s sad to be n equlbum (eall n tanslatonal equlbum) : F a *eall mples thee sepaate equements o tanslatonal equlbum: ) F a ) F a 3) F a Fcton oces s μs k μk n n
4 Fomula Sheet Eam 3 Page 4 Ccula Moton nd law (centpetal decton): ( F ) maad Radal (centpetal) acceleaton: a ad ad v a I thee s a tangental acceleaton a also, then: dv a Dot Poduct (Scala Poduct) Fo an two vectos A A, A, A A B AB cosθ, n whch: and B B, B, B : A A + A + A Wok B B + B + B Altenatve (equvalent) denton o dot poduct: A B A B + A B + A B Wok Vaable Foces: W F d (geneal denton o wok) I F has onl an component and ths component depends on, then: W F( ) d Constant Foces: W F Δ (-D o 3-D path) W FΔ (-D path) Spngs Hooke s law: F k. k the spng constant o the oce constant. Potental eneg stoed n a spng (elastc potental eneg): Wok-Eneg Theoem ΔK W net Uel k Knetc eneg: K mv Powe Instantaneous Powe:
5 Fomula Sheet Eam 3 Page 5 de Geneal Denton: Rate at whch eneg beng suppled b o to a sstem: P I eneg comes om wok beng done, then the powe s the ate at whch wok s done: dw P P F v (altenatve denton) Aveage Powe: ΔE ΔW Pav Consevatve Foces, Potental Eneg, and Consevaton o Eneg Consevatve Foces: Wok done b a consevatve oce: W c ΔU, o some potental eneg U Wok done s ndependent o path. Wok done gong once aound closed path s eo. Potental Eneg: Gavtatonal potental eneg (nea suace o Eath): U gav mg Elastc Potental Eneg: Uel Total mechancal eneg: E K + U Nonconsevatve Foces: Wnc Δ E I W nc, E conseved. Foce and Potental Eneg F F du d du d du F d Lnea Momentum Lnea momentum: k p mv p mv p mv p mv dp Δp Fnet Fnet av Δ t Newton s second law: ( ) net F s net (etenal) oce ( Fnet F ) Impulse Vang oce: t J F t Constant oce: J F (geneal denton o mpulse)
6 Fomula Sheet Eam 3 Page 6 Impulse-momentum theoem: Jnet Δp, n whch: Collsons t Jnet Fnet (vang oce) o Jnet Fnet (constant oce) t Two boad categoes: head-on and glancng Fo each catego, thee classes:. elastc: p and K conseved. nelastc: p conseved, K not 3. completel nelastc: p conseved, K not, objects stck togethe Head-on collsons:. elastc: v + m v m v m v (p-consevaton) m + v v v + v + ( othe eq deved n class). nelastc: m v + mv mv + mv (p-consevaton) 3. completel nelastc: v + m v m m v (p-consevaton) Glancng (-D) collsons: p p p p ( ) m + Cente o Mass and Sstems o Patcles m + m+ + mnn m + m+ + mnn X CM m+ m + + mn Mtot m + m + + mnn m + m + + mnn YCM m+ m + + mn Mtot nd law o sstem: dp ( F ), n whch P mv + mv+ + mnvn s total momentum o sstem. et P MtotvCM I mass o sstem s constant, then: ( F ) MtotaCM Rotatonal Knematcs et s angle θ n adans: θ (s length o ac swept out) angula dsplacement: Δ θ θ θ Δθ θ θ aveage angula veloct: ω av t t dθ nstantaneous angula veloct: ω Δω ω aveage angula acceleaton: α av Δ t t ω t
7 Fomula Sheet Eam 3 Page 7 nstantaneous angula acceleaton: dω d θ α *θ undestood to be n adans n all o the above Rotatonal moton wth constant angula acceleaton: () ω ω + α t () θ θ + ( ω + ω )t (3) θ θ + ω t + α t ω ω + α θ θ (4) ( ) Rotatonal and Lnea Quanttes tan tan v v ω a a α Radal (centpetal) acceleaton: a a ω ad Rollng Wthout Slppng vcm Rω ( v tanslatonal speed o cente o mass o ollng object) Rotatonal Knetc Eneg and Moment o Ineta Geneal Fomula o Moment o Ineta: I dm Moment o Ineta o Collecton o N Pont Masses: m Moments o Ineta o Dstbuted Objects: N I Rotatonal Knetc Eneg: K ot Iω Total Knetc Eneg: K total K tans + K ot MvCM + I CM ω
8 Fomula Sheet Eam 3 Page 8 (Note: Hee v CM s the tanslatonal veloct o the cente o mass and I CM s the moment o neta about the cente o mass.) Paallel-as Theoem I I + Md P CM Pependcula-as Theoem I I + I Coss Poduct (Vecto Poduct) A B ( A ) ˆ ( ) ˆ B AB + AB AB + ( AB AB) ˆ A B ABsnφ Toque and Angula Acceleaton Toque: τ F τ F Relaton Between Toque and Angula Acceleaton (Newton s Second Law o Rotatonal Moton): τ I α Angula Momentum net Fo pont patcle: L p. Fo gd bod otatng about a ed smmet as: L Iω. dl Newton s Second Law o Rotatonal Moton: τ net. (Note hee that τ net s the net etenal toque.) Equlbum Thee condtons equed o tue equlbum (tanslatonal and otatonal equlbum):. F (.e., a ). F (.e., a ) 3. τ o an as (.e., α about an as) net Gavt mm Fgav G (Newton s law o unvesal gavtaton) N m G 6.67 (gavtatonal constant) kg g GM (acceleaton due to gavt above planet, moon, etc., o mass M)
9 Fomula Sheet Eam 3 Page 9 mm U G (gavtatonal potental eneg o an two masses m and m ) gav GM vesc (escape speed) R Smple Hamonc Moton (SHM) Relatonshps among equenc, angula equenc and peod o snusods: T ω π π T ω Mass on Spng Acos ωt+ φ (most geneal om o dsplacement as a uncton o tme) ( ) ω k m m T π k v A + ω v φ tan ω Smple Pendulum L T π g Phscal Pendulum T π I mgd Sola Sstem Data adus o Eath: adus o Moon: mass o Eath: mass o Moon: mass o Sun: R 6.37 m E R.74 m M M 5.97 kg E 7.35 kg 3. kg Eath-Moon dstance (cente-to-cente): 3.84 m Eath-Sun dstance (cente-to-cente):.5 m 8
One-dimensional kinematics
Phsics 45 Formula Sheet Eam One-dimensional kinematics Vectors displacement: Δ f i total distance traveled average speed total time Δ f i average velocit: vav t f ti Δ instantaneous velocit: v lim Δ t
More informationPHY121 Formula Sheet
HY Foula Sheet One Denson t t Equatons o oton l Δ t Δ d d d d a d + at t + at a + t + ½at² + a( - ) ojectle oton y cos θ sn θ gt ( cos θ) t y ( sn θ) t ½ gt y a a sn θ g sn θ g otatonal a a a + a t Ccula
More informationRotary 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 informationPHY126 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 informationCOLLEGE 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 informationReview. Physics 231 fall 2007
Reew Physcs 3 all 7 Man ssues Knematcs - moton wth constant acceleaton D moton, D pojectle moton, otatonal moton Dynamcs (oces) Enegy (knetc and potental) (tanslatonal o otatonal moton when detals ae not
More informationDynamics 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 information1. 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 informationCapítulo. 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 informationRotational 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 informationScalars 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 informationDYNAMICS 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 informationPhysics 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 information2/24/2014. 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 informationPhysics 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 informationRigid Bodies: Equivalent Systems of Forces
Engneeng Statcs, ENGR 2301 Chapte 3 Rgd Bodes: Equvalent Sstems of oces 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
More information10/23/2003 PHY Lecture 14R 1
Announcements. Remember -- Tuesday, Oct. 8 th, 9:30 AM Second exam (coverng Chapters 9-4 of HRW) Brng the followng: a) equaton sheet b) Calculator c) Pencl d) Clear head e) Note: If you have kept up wth
More informationChapter 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 informationPhysics 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 informationStudy Guide For Exam Two
Study Gude For Exam Two Physcs 2210 Albretsen Updated: 08/02/2018 All Other Prevous Study Gudes Modules 01-06 Module 07 Work Work done by a constant force F over a dstance s : Work done by varyng force
More informationPhysics 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 informationa v2 r a' (4v) 2 16 v2 mg mg (2.4kg)(9.8m / s 2 ) 23.52N 23.52N N
Conceptual ewton s Law Applcaton Test Revew 1. What s the decton o centpetal acceleaton? see unom ccula moton notes 2. What aects the magntude o a ctonal oce? see cton notes 3. What s the deence between
More informationPhysics 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 informationChapter 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 informationDescription Linear Angular position x displacement x rate of change of position v x x v average rate of change of position
Chapte 5 Ccula Moton The language used to descbe otatonal moton s ey smla to the language used to descbe lnea moton. The symbols ae deent. Descpton Lnea Angula poston dsplacement ate o change o poston
More informationChapter 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 informationExam 3: Equation Summary
MAACHUETT INTITUTE OF TECHNOLOGY Depatment of Physics Physics 8. TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t = Exam 3: Equation ummay = Impulse: I F( t ) = p Toque: τ =,P dp F P τ =,P
More information24-2: 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 informationReview 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 informationFrom Newton to Einstein. Mid-Term Test, 12a.m. Thur. 13 th Nov Duration: 50 minutes. There are 20 marks in Section A and 30 in Section B.
Fom Newton to Einstein Mid-Tem Test, a.m. Thu. 3 th Nov. 008 Duation: 50 minutes. Thee ae 0 maks in Section A and 30 in Section B. Use g = 0 ms in numeical calculations. You ma use the following epessions
More informationEnergy 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 informationChapter 5 Circular Motion
Chapte 5 Ccula Moton In a gd body, the dstances between the pats o the body eman constant. We begn nestgatng the otaton o a gd body. We conclude ou nestgaton n Chapte 8. The language used to descbe otatonal
More informationEngineering 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 informationPhysics 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 informationLINEAR 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 information10/15/2013. PHY 113 C General Physics I 11 AM-12:15 PM MWF Olin 101
10/15/01 PHY 11 C Geneal Physcs I 11 AM-1:15 PM MWF Oln 101 Plan fo Lectue 14: Chapte 1 Statc equlbu 1. Balancng foces and toques; stablty. Cente of gavty. Wll dscuss elastcty n Lectue 15 (Chapte 15) 10/14/01
More informationProblem While being compressed, A) What is the work done on it by gravity? B) What is the work done on it by the spring force?
Problem 07-50 A 0.25 kg block s dropped on a relaed sprng that has a sprng constant o k 250.0 N/m (2.5 N/cm). The block becomes attached to the sprng and compresses t 0.12 m beore momentarl stoppng. Whle
More information10/24/2013. PHY 113 C General Physics I 11 AM 12:15 PM TR Olin 101. Plan for Lecture 17: Review of Chapters 9-13, 15-16
0/4/03 PHY 3 C General Physcs I AM :5 PM T Oln 0 Plan or Lecture 7: evew o Chapters 9-3, 5-6. Comment on exam and advce or preparaton. evew 3. Example problems 0/4/03 PHY 3 C Fall 03 -- Lecture 7 0/4/03
More informationPhysics 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 informationRemember: 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 informationExam 3: Equation Summary
MAACHUETT INTITUTE OF TECHNOLOGY Depatment of Physics Physics 8. TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t = Exam 3: Equation ummay = Impulse: I F( t ) = p Toque: τ =,P dp F P τ =,P
More informationPHYS 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 informationMark answers in spaces on the answer sheet
Mak answes in spaces 31-43 on the answe sheet PHYSICS 1 Summe 005 EXAM 3: July 5 005 9:50pm 10:50pm Name (pinted): ID Numbe: Section Numbe: INSTRUCTIONS: Some questions ae one point, othes ae two points,
More informationChapter 3 and Chapter 4
Chapter 3 and Chapter 4 Chapter 3 Energy 3. Introducton:Work Work W s energy transerred to or rom an object by means o a orce actng on the object. Energy transerred to the object s postve work, and energy
More informationPart 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 informationPHYS 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 informationKepler's 1 st Law by Newton
Astonom 10 Section 1 MWF 1500-1550 134 Astonom Building This Class (Lectue 7): Gavitation Net Class: Theo of Planeta Motion HW # Due Fida! Missed nd planetaium date. (onl 5 left), including tonight Stadial
More informationChapter 1: Mathematical Concepts and Vectors
Chapte : Mathematical Concepts and Vectos giga G 9 mega M 6 kilo k 3 centi c - milli m -3 mico μ -6 nano n -9 in =.54 cm m = cm = 3.8 t mi = 58 t = 69 m h = 36 s da = 86,4 s ea = 365.5 das You must know
More informationALL QUESTIONS ARE WORTH 20 POINTS. WORK OUT FIVE PROBLEMS.
GNRAL PHYSICS PH -3A (D. S. Mov) Test (/3/) key STUDNT NAM: STUDNT d #: -------------------------------------------------------------------------------------------------------------------------------------------
More informationc) (6) Assuming the tires do not skid, what coefficient of static friction between tires and pavement is needed?
Geneal Physics I Exam 2 - Chs. 4,5,6 - Foces, Cicula Motion, Enegy Oct. 10, 2012 Name Rec. Inst. Rec. Time Fo full cedit, make you wok clea to the gade. Show fomulas used, essential steps, and esults with
More informationCSU 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 informationIntegral 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 informationChapter 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 informationRE 6.d Electric and Rest Energy RE 6.e EP6, HW6: Ch 6 Pr s 58, 59, 91, 99(a-c), 105(a-c)
ed. Lab., Mon. Tues. ed. Lab. Mon. Tues. 6.1-.4 (.1) Intoducng Enegy & ok Quz 5 L5: Buoyancy, Ccles & Pendulums 6.5-.7 (.) Rest Mass,ok by Changng oces tudy Day tudy Day 6.8-.9(.18,.19) Intoducng Potental
More informationAP-C WEP. h. Students should be able to recognize and solve problems that call for application both of conservation of energy and Newton s Laws.
AP-C WEP 1. Wok a. Calculate the wok done by a specified constant foce on an object that undegoes a specified displacement. b. Relate the wok done by a foce to the aea unde a gaph of foce as a function
More informationPeriod & Frequency. Work and Energy. Methods of Energy Transfer: Energy. Work-KE Theorem 3/4/16. Ranking: Which has the greatest kinetic energy?
Perod & Frequency Perod (T): Tme to complete one ull rotaton Frequency (): Number o rotatons completed per second. = 1/T, T = 1/ v = πr/t Work and Energy Work: W = F!d (pcks out parallel components) F
More informationSpring 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 information7/1/2008. 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( ) ( ) Review of Force. Review of Force. r = =... Example 1. What is the dot product for F r. Solution: Example 2 ( )
: PHYS 55 (Pat, Topic ) Eample Solutions p. Review of Foce Eample ( ) ( ) What is the dot poduct fo F =,,3 and G = 4,5,6? F G = F G + F G + F G = 4 +... = 3 z z Phs55 -: Foce Fields Review of Foce Eample
More informationPhysics for Scientists and Engineers. Chapter 9 Impulse and Momentum
Physcs or Scentsts and Engneers Chapter 9 Impulse and Momentum Sprng, 008 Ho Jung Pak Lnear Momentum Lnear momentum o an object o mass m movng wth a velocty v s dened to be p mv Momentum and lnear momentum
More information1. 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 informationFundamental 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= 4 3 π( m) 3 (5480 kg m 3 ) = kg.
CHAPTER 11 THE GRAVITATIONAL FIELD Newton s Law of Gavitation m 1 m A foce of attaction occus between two masses given by Newton s Law of Gavitation Inetial mass and gavitational mass Gavitational potential
More informationCourse Outline. 1. MATLAB tutorial 2. Motion of systems that can be idealized as particles
Couse Outlne. MATLAB tutoal. Moton of systems that can be dealzed as patcles Descpton of moton, coodnate systems; Newton s laws; Calculatng foces equed to nduce pescbed moton; Devng and solvng equatons
More informationChapter 13: Gravitation
v m m F G Chapte 13: Gavitation The foce that makes an apple fall is the same foce that holds moon in obit. Newton s law of gavitation: Evey paticle attacts any othe paticle with a gavitation foce given
More informationA Tale of Friction Basic Rollercoaster Physics. Fahrenheit Rollercoaster, Hershey, PA max height = 121 ft max speed = 58 mph
A Tale o Frcton Basc Rollercoaster Physcs Fahrenhet Rollercoaster, Hershey, PA max heght = 11 t max speed = 58 mph PLAY PLAY PLAY PLAY Rotatonal Movement Knematcs Smlar to how lnear velocty s dened, angular
More informationPart C Dynamics and Statics of Rigid Body. Chapter 5 Rotation of a Rigid Body About a Fixed Axis
Part C Dynamcs and Statcs of Rgd Body Chapter 5 Rotaton of a Rgd Body About a Fxed Axs 5.. Rotatonal Varables 5.. Rotaton wth Constant Angular Acceleraton 5.3. Knetc Energy of Rotaton, Rotatonal Inerta
More informationPhysics C Rotational Motion Name: ANSWER KEY_ AP Review Packet
Linea and angula analogs Linea Rotation x position x displacement v velocity a T tangential acceleation Vectos in otational motion Use the ight hand ule to detemine diection of the vecto! Don t foget centipetal
More informationForce of gravity and its potential function
F. W. Phs0 E:\Ecel files\ch gavitational foce and potential.doc page of 6 0/0/005 8:9 PM Last pinted 0/0/005 8:9:00 PM Foce of gavit and its potential function (.) Let us calculate the potential function
More informationTEST-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 informationRigid body simulation
Rgd bod smulaton Rgd bod smulaton Once we consder an object wth spacal etent, partcle sstem smulaton s no longer suffcent Problems Problems Unconstraned sstem rotatonal moton torques and angular momentum
More informationPhysics 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 informationPrinciples of Physics I
Pinciples of Physics I J. M. Veal, Ph. D. vesion 8.05.24 Contents Linea Motion 3. Two scala equations........................ 3.2 Anothe scala equation...................... 3.3 Constant acceleation.......................
More informationb) (5) What is the magnitude of the force on the 6.0-kg block due to the contact with the 12.0-kg block?
Geneal Physics I Exam 2 - Chs. 4,5,6 - Foces, Cicula Motion, Enegy Oct. 13, 2010 Name Rec. Inst. Rec. Time Fo full cedit, make you wok clea to the gade. Show fomulas used, essential steps, and esults with
More informationPHYS 1443 Section 003 Lecture #17
PHYS 144 Secton 00 ecture #17 Wednesda, Oct. 9, 00 1. Rollng oton of a Rgd od. Torque. oment of Inerta 4. Rotatonal Knetc Energ 5. Torque and Vector Products Remember the nd term eam (ch 6 11), onda, Nov.!
More informationb) (5) What average force magnitude was applied by the students working together?
Geneal Physics I Exam 3 - Chs. 7,8,9 - Momentum, Rotation, Equilibium Nov. 3, 2010 Name Rec. Inst. Rec. Time Fo full cedit, make you wok clea to the gade. Show fomulas used, essential steps, and esults
More informationgravity 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 informationPHYSICS 231 Review problems for midterm 2
PHYSICS 31 Revew problems for mdterm Topc 5: Energy and Work and Power Topc 6: Momentum and Collsons Topc 7: Oscllatons (sprng and pendulum) Topc 8: Rotatonal Moton The nd exam wll be Wednesday October
More informationDynamics of Rotational Motion
Dynamics of Rotational Motion Toque: the otational analogue of foce Toque = foce x moment am τ = l moment am = pependicula distance though which the foce acts a.k.a. leve am l l l l τ = l = sin φ = tan
More informationF 12. = G m m 1 2 F 21 = F 12. = G m 1m 2. Review. Physics 201, Lecture 22. Newton s Law Of Universal Gravitation
Physics 201, Lectue 22 Review Today s Topics n Univesal Gavitation (Chapte 13.1-13.3) n Newton s Law of Univesal Gavitation n Popeties of Gavitational Foce n Planet Obits; Keple s Laws by Newton s Law
More informationPhysics 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 information10/9/2003 PHY Lecture 11 1
Announcements 1. Physc Colloquum today --The Physcs and Analyss of Non-nvasve Optcal Imagng. Today s lecture Bref revew of momentum & collsons Example HW problems Introducton to rotatons Defnton of angular
More informationChapter 3. r r. Position, Velocity, and Acceleration Revisited
Chapter 3 Poston, Velocty, and Acceleraton Revsted The poston vector of a partcle s a vector drawn from the orgn to the locaton of the partcle. In two dmensons: r = x ˆ+ yj ˆ (1) The dsplacement vector
More informationMechanics 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 informationWeek 9 Chapter 10 Section 1-5
Week 9 Chapter 10 Secton 1-5 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,
More informationChapter 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 informationCentral Force Motion
Cental Foce Motion Cental Foce Poblem Find the motion of two bodies inteacting via a cental foce. Examples: Gavitational foce (Keple poblem): m1m F 1, ( ) =! G ˆ Linea estoing foce: F 1, ( ) =! k ˆ Two
More informationChapter 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 informationGravitation. Chapter 12. PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman. Lectures by James Pazun
Chapte 12 Gavitation PowePoint Lectues fo Univesity Physics, Twelfth Edition Hugh D. Young and Roge A. Feedman Lectues by James Pazun Modified by P. Lam 5_31_2012 Goals fo Chapte 12 To study Newton s Law
More informationChapter 07: Kinetic Energy and Work
Chapter 07: Knetc Energy and Work Conservaton o Energy s one o Nature s undamental laws that s not volated. Energy can take on derent orms n a gven system. Ths chapter we wll dscuss work and knetc energy.
More informationPhysics 207 Lecture 13. Lecture 13
Physcs 07 Lecture 3 Goals: Lecture 3 Chapter 0 Understand the relatonshp between moton and energy Defne Potental Energy n a Hooke s Law sprng Develop and explot conservaton of energy prncple n problem
More informationChapter 13 Gravitation
Chapte 13 Gavitation In this chapte we will exploe the following topics: -Newton s law of gavitation, which descibes the attactive foce between two point masses and its application to extended objects
More informationLesson 8: Work, Energy, Power (Sections ) Chapter 6 Conservation of Energy
Lesson 8: Wok, negy, Powe (Sectons 6.-6.8) Chapte 6 Conseaton o negy Today we begn wth a ey useul concept negy. We wll encounte many amla tems that now hae ey specc dentons n physcs. Conseaton o enegy
More informationCentral Force Problem. Central Force Motion. Two Body Problem: Center of Mass Coordinates. Reduction of Two Body Problem 8.01 W14D1. + m 2. m 2.
Cental oce Poblem ind the motion of two bodies inteacting via a cental foce. Cental oce Motion 8.01 W14D1 Examples: Gavitational foce (Keple poblem): 1 1, ( ) G mm Linea estoing foce: ( ) k 1, Two Body
More information3.1 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 informationPHYS 1443 Section 002
PHYS 443 Secton 00 Lecture #6 Wednesday, Nov. 5, 008 Dr. Jae Yu Collsons Elastc and Inelastc Collsons Two Dmensonal Collsons Center o ass Fundamentals o Rotatonal otons Wednesday, Nov. 5, 008 PHYS PHYS
More informationω = θ θ o = θ θ = s r v = rω
Unifom Cicula Motion Unifom cicula motion is the motion of an object taveling at a constant(unifom) speed in a cicula path. Fist we must define the angula displacement and angula velocity The angula displacement
More informatione.g: If A = i 2 j + k then find A. A = Ax 2 + Ay 2 + Az 2 = ( 2) = 6
MOTION IN A PLANE 1. Scala Quantities Physical quantities that have only magnitude and no diection ae called scala quantities o scalas. e.g. Mass, time, speed etc. 2. Vecto Quantities Physical quantities
More informationUNIVERSITÀ 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 informationF(r) = r f (r) 4.8. Central forces The most interesting problems in classical mechanics are about central forces.
4.8. Cental foces The most inteesting poblems in classical mechanics ae about cental foces. Definition of a cental foce: (i) the diection of the foce F() is paallel o antipaallel to ; in othe wods, fo
More information