Chapter 8. Potential Energy and Conservation of Energy


 Randell Chambers
 6 years ago
 Views:
Transcription
1 Chapter 8 Potental Energy and Conservaton of Energy In ths chapter we wll ntroduce the followng concepts: Potental Energy Conservatve and nonconservatve forces Mechancal Energy Conservaton of Mechancal Energy The conservaton of energy theorem wll be used to solve a varety of problems As was done n Chapter 7 we use scalars such as work,knetc energy, and mechancal energy rather than vectors. Therefore the approach s mathematcally smpler. (81)
2 h v o B A v o g Work and Potental Energy: Consder the tomato of mass m shown n the fgure. The tomato s taken together wth the earth as the system we wsh to study. The tomato s thrown upwards wth ntal speed v o at pont A. Under the acton of the gravtatonal force t slows down and stops completely at pont B. Then the tomato falls back and by the tme t reaches pont A ts speed has reached the orgnal value v o. Below we analyze n detal what happens to the tomatoearth system. Durng the trp from A to B the gravtatonal force F g does negatve work W 1 = mgh. Energy s transferred by F g from the knetc energy of the tomato to the gravtatonal potental energy U of the tomatoearth system. Durng the trp from B to A the transfer s reversed. The work W 2 done by F g s postve ( W 2 = mgh ). The gravtatonal force transfers energy from the gravtatonal potental energy U of the tomatoearth system to the knetc energy of the tomato. The change n the potental energy U s defned as: U = W (82)
3 A A m B k B Consder the mass m attached to a sprng of sprng constant k as shown n the fgure. The mass s taken together wth the sprng as the system we wsh to study. The mass s gven an ntal speed v o at pont A. Under the acton of the sprng force t slows down and stops completely at pont B whch corresponds to a sprng compresson. Then the mass reverses the drecton of ts moton and by the tme t reaches pont A ts speed has reached the orgnal value v o. As n the prevous eample we analyze n detal what happens to the masssprng system. Durng the trp from A to B the sprng force F s does negatve work W 1 = k 2 /2. Energy s transferred by F s from the knetc energy of the mass to the potental energy U of the masssprng system. Durng the trp from B to A the transfer s reversed. The work W 2 done by F s s postve ( W 2 = k 2 /2 ). The sprng force transfers energy from the potental energy U of the masssprng system to the knetc energy of the mass. The change n the potental energy U s defned as: U = W (83)
4 f k A m v o f k d B m Conservatve and nonconservatve forces. The gravtatonal force as the sprng force are called conservatve because the can transfer energy from the knetc energy of part of the system to potental energy and vce versa. Frctonal and drag forces on the other hand are called nonconservatve for reasons that are eplaned below. Consder a system that conssts of a block of mass m and the floor on whch t rests. The block starts to move on a horzontal floor wth ntal speed v o at pont A. The coeffcent of knetc frcton between the floor and the block s μ k. The block wll slow down by the knetc frcton f k and wll stop at pont B after t has traveled a dstance d. Durng the trp from pont A to pont B the frctonal force has done work W f =  μ k mgd. The frctonal force transfers energy from the knetc energy of the block to a type of energy called thermal energy. Ths energy transfer cannot be reversed. Thermal energy cannot be transferred back to knetc energy of the block by the knetc frcton. Ths s the hallmark of nonconservatve forces. (84)
5 (85) Path Independence of Conservatve Forces In ths secton we wll gve a test that wll help us decde whether a force s conservatve or nonconservatve. A force s conservatve f the net work done on a partcle durng a round trp s always equal to zero (see fg.b). W net = Such a round trp along a closed path s shown n fg.b. In the eamples of the tomatoearth and masssprng system W net = W ab,1 + W ba,2 = 0 We shall prove that f a force s conservatve then the work done on a partcle between two ponts a and b does not depend on the path. From fg. b we have: W net = W ab,1 + W ba,2 = 0 W ab,1 =  W ba,2 (eqs.1) 0 From fg.a we have: W ab,2 =  W ba,2 (eqs.2) If we compare eqs.1 and eqs.2 we get: W = W ab,1 ab,2
6 ... O F() f Determnng Potental Energy Values: In ths secton we wll dscuss a method that can be used to determne the dfference n potental energy U of a conservatve force F between ponts f and on the as f we know F( ) A conservatve force F moves an object along the as from an ntal pont to a fnal pont. The work W that the force F does on the object s gven by : W f = F( ) d The correspondng change n potental energy U was defned as: f U = W Therefore the epresson for U becomes: U f = F( ) d (86)
7 dy mg Gravtatonal Potental Consder a partcle of mass m movng vertcally along the yas ( ) from pont y to pont y. At the same tme the gravtatonal force does work W energy: f on the partcle whch changes the potental energy of the partcleearth system. We use the result of the prevous secton to calculate U ( ) [ ] U = F( ) dy F = mg U = mg dy = mg dy = mg y U = mg y y = mg y We assgn the fnal pont y to be the "generc" ( ) pont y on the yas whose potental s U ( y). U ( y) U = mg y y Snce y y y f f f y y y f f only changes n the potental are physcally menangful, ths allows us to defne arbtrarly y and U The most convenent choce s: y y... y f m y y O = 0, U = 0 Ths partcular choce gves: y y f U ( y) = mgy (87)
8 O O O (a) (b) f (c) 2 f k U = k 2 = f k We assgn the fnal pont to be the "generc" 2 2 k k pont on the as whose potental s U( ). U( ) U = 2 2 Snce only changes n the potental are physcally menangful, ths allows us to defne arbtrarly and U The most convenent choce s: y = 0, U = 0 Ths partcular choce gves: Potental Energy of a sprng: Consder the blockmass system shown n the fgure. The block moves from pont to pont. At the same tme the sprng force does work W on the block whch change s the potental energy of the blocksprng system by an amount f f f W = F( ) d = kd = k d U = W U = f f k 2 2 (88)
9 Conservaton of Mechancal Energy: Mechancal energy of a system s defned as the sum of potental and knetc energes Emech = K + U We assume that the system s solated.e. no eternal forces change the energy of the system. We also assume that all the forces n the system are conservatve. When an nteral force does work W on an object of the system ths changes the knetc energy by K = W (eqs.1) Ths amount of work also changes the potental energy of the system by an amount If we compare equatons 1 and 2 we have: K ( ) 1 K = U U K + U = K + U K = U Ths U = W equaton s known as the prncple of conservaton of mechancal energy. It can be summarzed as: Emech = K + U = 0 For an solated system n whch the forces are a mture of conservatve and non conservatve forces the prncple takes the followng form E = W mech nc (89) (eqs.2) Here, s defned as the work of all the nonconsrvatve W nc force s of the system
10 (810) An eample of the prncple of conservaton of mechancal energy s gven n the fgure. It conssts of a pendulum bob of mass m movng under the acton of the gravtatonal force The total mechancal energy of the bobearth system remans constant. As the pendulum swngs, the total energy E s transferred back and forth between knetc energy K of the bob and potental energy U of the bobearth system We assume that U s zero at the lowest pont of the pendulum orbt. K s mamum n frame a, and e (U s mnmum there). U s mamum n frames c and g (K s mnmum there)
11 ... O A F B + Δ Fndng the Force F( ) analytcally from the potental energy U ( ) Consder an object that moves along the as under the nfluence of an unknown force F whose potental energy U() we know at all ponts of the as. The object moves from pont A (coordnate ) to a close by pont B (coordnate + ). The force does work W on the object gven by the equaton: W = F eqs.1 The work of the force changes the potental energy U of the system by the amount: U = W eqs.2 If we combne equatons 1 and 2 we get: U F = We take the lmt as 0 and we end up wth the equaton: F( ) = du ( ) d (811)
12 (812) The potental Energy Curve If we plot the potental energy U versus for a force F that acts along the as we can glean a wealth of nformaton about the moton of a partcle on whch F s actng. The frst parameter that we can determne s the force F() usng the equaton: F( ) = du ( ) d An eample s gven n the fgures below. In fg.a we plot U() versus. In fg.b we plot F() versus. For eample at 2, 3 and 4 the slope of the U() vs curve s zero, thus F = 0. The slope du/d between 3 and 4 s negatve; Thus F > 0 for the ths nterval. The slope du/d between 2 and 3 s postve; Thus F < 0 for the same nterval
13 (813) Turnng Ponts: The total mechancal energy s E = K( ) + U ( ) Ths energy s constant (equal to 5 J n the fgure) and s thus represented by a horzontal lne. We can slolve ths equaton for mec K( ) and get: K( ) = E U ( ) At any pont on the as mec we can read the value of the equaton above and determne U ( ). Then we can solve 2 mv From the defnton of K = the knetc energy cannot be negatve. 2 Ths property of K allows us to determne whch regons of the as moton s allowed. K ( ) = K ( ) = E U ( ) If K > 0 E U ( ) > 0 U ( ) < mech If K < 0 E U ( ) < 0 U ( ) > mech The ponts at whch: E mec for the moton. For eample 1 mec E mec mec = U ( ) are known as turnng ponts E s the turnng K Moton s allowed Moton s forbdden pont U versus plot above. At the turnng pont K = 0 for the
14 (814) Gven the U() versus curve the turnng ponts and the regons for whch moton s allowed depends on the value of the mechancal energy E mec In the pcture to the left consder the stuaton when E mec = 4 J (purple lne) The turnng ponts (E mec = U ) occur at 1 and > 5. Moton s allowed for > 1 If we reduce E mec to 3 J or 1 J the turnng ponts and regons of allowed moton change accordngly. Equlbrum Ponts: A poston at whch the slope du/d = 0 and thus F = 0 s called an equlbrum pont. A regon for whch F = 0 such as the regon > 5 s called a regon of neutral equlbrum. If we set E mec = 4 J the knetc energy K = 0 and any partcle movng under the nfluence of U wll be statonary at any pont wth > 5 Mnma n the U versus curve are postons of stable equlbrum Mama n the U versus curve are postons of unstable equlbrum
15 (815) Note: The blue arrows n the fgure ndcate the drecton of the force F as determned from the equaton: F( ) = du ( ) d Postons of Stable Equlbrum. An eample s pont 4 where U has a mnmum. If we arrange E mec = 1 J then K = 0 at pont 4. A partcle wth E mec = 1 J s statonary at 4. If we dsplace slghtly the partcle ether to the rght or to the left of 4 the force tends to brng t back to the equlbrum poston. Ths equlbrum s stable. Postons of Unstable Equlbrum. An eample s pont 3 where U has a mamum. If we arrange E mec = 3 J then K = 0 at pont 3. A partcle wth E mec = 3 J s statonary at 3. If we dsplace slghtly the partcle ether to the rght or to the left of 3 the force tends to take t further away from the equlbrum poston. Ths equlbrum s unstable
16 Work done on a System by an Eternal Force Up to ths pont we have consdered only solated systems n whch no eternal forces were present. We wll now consder a system n whch there are forces eternal to the system The system under study s a bowlng ball beng hurled by a player. The system conssts of the ball and the earth taken together. The force eerted on the ball by the player s an eternal force. In ths case the mechancal energy E mec of the system s not constant. Instead t changes by an amount equal to the work W done by the eternal force accordng to the equaton: W = Emec = K + U (816)
PHYS 1443 Section 004 Lecture #12 Thursday, Oct. 2, 2014
PHYS 1443 Secton 004 Lecture #1 Thursday, Oct., 014 WorkKnetc Energy Theorem Work under rcton Potental Energy and the Conservatve Force Gravtatonal Potental Energy Elastc Potental Energy Conservaton o
More informationSpring Force and Power
Lecture 13 Chapter 9 Sprng Force and Power Yeah, energy s better than orces. What s net? Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi IN THIS CHAPTER, you wll learn how to solve problems
More informationChapter 8: Potential Energy and The Conservation of Total Energy
Chapter 8: Potental Energy and The Conservaton o Total Energy Work and knetc energy are energes o moton. K K K mv r v v F dr Potental energy s an energy that depends on locaton. Dmenson F x d U( x) dx
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 informationCHAPTER 8 Potential Energy and Conservation of Energy
CHAPTER 8 Potental Energy and Conservaton o Energy One orm o energy can be converted nto another orm o energy. Conservatve and nonconservatve orces Physcs 1 Knetc energy: Potental energy: Energy assocated
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 7. Potential Energy and Conservation of Energy
Chapter 7 Potental Energy and Conservaton o Energy 1 Forms o Energy There are many orms o energy, but they can all be put nto two categores Knetc Knetc energy s energy o moton Potental Potental energy
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 informationYou will analyze the motion of the block at different moments using the law of conservation of energy.
Physcs 00A Homework 7 Chapter 8 Where s the Energy? In ths problem, we wll consder the ollowng stuaton as depcted n the dagram: A block o mass m sldes at a speed v along a horzontal smooth table. It next
More informationConservation of Energy
Lecture 3 Chapter 8 Physcs I 0.3.03 Conservaton o Energy Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi Lecture Capture: http://echo360.uml.edu/danylov03/physcsall.html 95.4, Fall 03,
More informationFirst Law: A body at rest remains at rest, a body in motion continues to move at constant velocity, unless acted upon by an external force.
Secton 1. Dynamcs (Newton s Laws of Moton) Two approaches: 1) Gven all the forces actng on a body, predct the subsequent (changes n) moton. 2) Gven the (changes n) moton of a body, nfer what forces act
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1s tme nterval. The velocty of the partcle
More informationChapter 7: Conservation of Energy
Lecture 7: Conservaton o nergy Chapter 7: Conservaton o nergy Introucton I the quantty o a subject oes not change wth tme, t means that the quantty s conserve. The quantty o that subject remans constant
More informationPhysics 181. Particle Systems
Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system
More informationPhysics 207: Lecture 20. Today s Agenda Homework for Monday
Physcs 207: Lecture 20 Today s Agenda Homework for Monday Recap: Systems of Partcles Center of mass Velocty and acceleraton of the center of mass Dynamcs of the center of mass Lnear Momentum Example problems
More informationWeek 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 informationPhysics 5153 Classical Mechanics. Principle of Virtual Work1
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 informationPHYS 1441 Section 002 Lecture #15
PHYS 1441 Secton 00 Lecture #15 Monday, March 18, 013 Work wth rcton Potental Energy Gravtatonal Potental Energy Elastc Potental Energy Mechancal Energy Conservaton Announcements Mdterm comprehensve exam
More informationχ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body
Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown
More informationWork is the change in energy of a system (neglecting heat transfer). To examine what could
Work Work s the change n energy o a system (neglectng heat transer). To eamne what could cause work, let s look at the dmensons o energy: L ML E M L F L so T T dmensonally energy s equal to a orce tmes
More informationLecture 16. Chapter 11. Energy Dissipation Linear Momentum. Physics I. Department of Physics and Applied Physics
Lecture 16 Chapter 11 Physcs I Energy Dsspaton Lnear Momentum Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi Department o Physcs and Appled Physcs IN IN THIS CHAPTER, you wll learn
More informationWeek 8: Chapter 9. Linear Momentum. Newton Law and Momentum. Linear Momentum, cont. Conservation of Linear Momentum. Conservation of Momentum, 2
Lnear omentum Week 8: Chapter 9 Lnear omentum and Collsons The lnear momentum of a partcle, or an object that can be modeled as a partcle, of mass m movng wth a velocty v s defned to be the product of
More informationSpring 2002 Lecture #13
4450 Sprng 00 ecture # Dr. Jaehoon Yu. Rotatonal Energy. Computaton of oments of nerta. Parallelas Theorem 4. Torque & Angular Acceleraton 5. Work, Power, & Energy of Rotatonal otons Remember the mdterm
More informationWeek 6, Chapter 7 Sect 15
Week 6, Chapter 7 Sect 15 Work and Knetc Energy Lecture Quz The frctonal force of the floor on a large sutcase s least when the sutcase s A.pushed by a force parallel to the floor. B.dragged by a force
More informationK = 100 J. [kg (m/s) ] K = mv = (0.15)(36.5) !!! Lethal energies. m [kg ] J s (Joule) Kinetic Energy (energy of motion) E or KE.
Knetc Energy (energy of moton) E or KE K = m v = m(v + v y + v z ) eample baseball m=0.5 kg ptche at v = 69 mph = 36.5 m/s K = mv = (0.5)(36.5) [kg (m/s) ] Unts m [kg ] J s (Joule) v = 69 mph K = 00 J
More informationPHYS 1441 Section 002 Lecture #16
PHYS 1441 Secton 00 Lecture #16 Monday, Mar. 4, 008 Potental Energy Conservatve and Nonconservatve Forces Conservaton o Mechancal Energy Power Today s homework s homework #8, due 9pm, Monday, Mar. 31!!
More informationStudy Guide For Exam Two
Study Gude For Exam Two Physcs 2210 Albretsen Updated: 08/02/2018 All Other Prevous Study Gudes Modules 0106 Module 07 Work Work done by a constant force F over a dstance s : Work done by varyng force
More informationPhysics 141. Lecture 14. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 14, Page 1
Physcs 141. Lecture 14. Frank L. H. Wolfs Department of Physcs and Astronomy, Unversty of Rochester, Lecture 14, Page 1 Physcs 141. Lecture 14. Course Informaton: Lab report # 3. Exam # 2. MultPartcle
More informationRecitation: Energy, Phys Energies. 1.2 Three stones. 1. Energy. 1. An acorn falling from an oak tree onto the sidewalk.
Rectaton: Energy, Phys 207. Energy. Energes. An acorn fallng from an oak tree onto the sdewalk. The acorn ntal has gravtatonal potental energy. As t falls, t converts ths energy to knetc. When t hts the
More informationin state i at t i, Initial State E = E i
Physcs 01, Lecture 1 Today s Topcs n More Energy and Work (chapters 7 & 8) n Conservatve Work and Potental Energy n Sprng Force and Sprng (Elastc) Potental Energy n Conservaton of Mechanc Energy n Exercse
More informationPlease initial the statement below to show that you have read it
EN40: Dynamcs and Vbratons Mdterm Examnaton Thursday March 5 009 Dvson of Engneerng rown Unversty NME: Isaac Newton General Instructons No collaboraton of any knd s permtted on ths examnaton. You may brng
More informationCHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)
CHAPTER 6 LAGRANGE S EQUATIONS (Analytcal Mechancs) 1 Ex. 1: Consder a partcle movng on a fxed horzontal surface. r P Let, be the poston and F be the total force on the partcle. The FBD s: mgk F 1 x O
More informationChapter Seven  Potential Energy and Conservation of Energy
Chapter Seven  Potental Energy and Conservaton o Energy 7 1 Potental Energy Potental energy. e wll nd that the potental energy o a system can only be assocated wth specc types o orces actng between members
More informationSCALARS 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 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 informationChapter 8 Potential Energy and Conservation of Energy Important Terms (For chapters 7 and 8)
Pro. Dr. I. Nasser Chapter8_I November 3, 07 Chapter 8 Potental Energy and Conservaton o Energy Important Terms (For chapters 7 and 8) conservatve orce: a orce whch does wor on an object whch s ndependent
More informationEMU Physics Department
Physcs 0 Lecture 8 Potental Energy and Conservaton Assst. Pro. Dr. Al ÖVGÜN EMU Physcs Department www.aovgun.com Denton o Work W q The work, W, done by a constant orce on an object s dened as the product
More informationPY2101 Classical Mechanics Dr. Síle Nic Chormaic, Room 215 D Kane Bldg
PY2101 Classcal Mechancs Dr. Síle Nc Chormac, Room 215 D Kane Bldg s.ncchormac@ucc.e Lectures stll some ssues to resolve. Slots shared between PY2101 and PY2104. Hope to have t fnalsed by tomorrow. Mondays
More informationThe equation of motion of a dynamical system is given by a set of differential equations. That is (1)
Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence
More informationLagrange Multipliers. A Somewhat Silly Example. Monday, 25 September 2013
Lagrange Multplers Monday, 5 September 013 Sometmes t s convenent to use redundant coordnates, and to effect the varaton of the acton consstent wth the constrants va the method of Lagrange undetermned
More information= 1.23 m/s 2 [W] Required: t. Solution:!t = = 17 m/s [W]! m/s [W] (two extra digits carried) = 2.1 m/s [W]
Secton 1.3: Acceleraton Tutoral 1 Practce, page 24 1. Gven: 0 m/s; 15.0 m/s [S]; t 12.5 s Requred: Analyss: a av v t v f v t a v av f v t 15.0 m/s [S] 0 m/s 12.5 s 15.0 m/s [S] 12.5 s 1.20 m/s 2 [S] Statement:
More informationPHYS 705: Classical Mechanics. Newtonian Mechanics
1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationCenter of Mass and Linear Momentum
PH 2212A Fall 2014 Center of Mass and Lnear Momentum Lectures 1415 Chapter 9 (Hallday/Resnck/Walker, Fundamentals of Physcs 9 th edton) 1 Chapter 9 Center of Mass and Lnear Momentum In ths chapter we
More informationPhysics 2A Chapters 6  Work & Energy Fall 2017
Physcs A Chapters 6  Work & Energy Fall 017 These notes are eght pages. A quck summary: The workenergy theorem s a combnaton o Chap and Chap 4 equatons. Work s dened as the product o the orce actng on
More informationLinear Momentum. Center of Mass.
Lecture 6 Chapter 9 Physcs I 03.3.04 Lnear omentum. Center of ass. Course webste: http://faculty.uml.edu/ndry_danylov/teachng/physcsi Lecture Capture: http://echo360.uml.edu/danylov03/physcssprng.html
More informationEnergy and Energy Transfer
Energy and Energy Transer Chapter 7 Scalar Product (Dot) Work Done by a Constant Force F s constant over the dsplacement r 1 Denton o the scalar (dot) product o vectors Scalar product o unt vectors = 1
More informationLecture 22: Potential Energy
Lecture : Potental Energy We have already studed the workenergy theorem, whch relates the total work done on an object to the change n knetc energy: Wtot = KE For a conservatve orce, the work done by
More informationWeek 9 Chapter 10 Section 15
Week 9 Chapter 10 Secton 15 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 informationPhysics 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 informationwhere v means the change in velocity, and t is the
1 PHYS:100 LECTURE 4 MECHANICS (3) Ths lecture covers the eneral case of moton wth constant acceleraton and free fall (whch s one of the more mportant examples of moton wth constant acceleraton) n a more
More informationAngular Momentum and Fixed Axis Rotation. 8.01t Nov 10, 2004
Angular Momentum and Fxed Axs Rotaton 8.01t Nov 10, 2004 Dynamcs: Translatonal and Rotatonal Moton Translatonal Dynamcs Total Force Torque Angular Momentum about Dynamcs of Rotaton F ext Momentum of a
More informationPotential Energy and Conservation
PH 13A Fall 009 Potential Energy and Conservation of Energy Lecture 113 Chapter 8 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition) Chapter 8 Potential Energy and Conservation of Energy
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 informationPhysics 207 Lecture 6
Physcs 207 Lecture 6 Agenda: Physcs 207, Lecture 6, Sept. 25 Chapter 4 Frames of reference Chapter 5 ewton s Law Mass Inerta s (contact and noncontact) Frcton (a external force that opposes moton) Free
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 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 informationPhysics 111: Mechanics Lecture 11
Physcs 111: Mechancs Lecture 11 Bn Chen NJIT Physcs Department Textbook Chapter 10: Dynamcs of Rotatonal Moton q 10.1 Torque q 10. Torque and Angular Acceleraton for a Rgd Body q 10.3 RgdBody Rotaton
More informationElectric Potential Energy & Potential. Electric Potential Energy. Potential Energy. Potential Energy. Example: Charge launcher
Electrc & Electrc Gravtatonal Increases as you move farther from Earth mgh Sprng Increases as you ncrease sprng extenson/comp resson Δ Increases or decreases as you move farther from the charge U ncreases
More informationˆ (0.10 m) E ( N m /C ) 36 ˆj ( j C m)
7.. = = 3 = 4 = 5. The electrc feld s constant everywhere between the plates. Ths s ndcated by the electrc feld vectors, whch are all the same length and n the same drecton. 7.5. Model: The dstances to
More informationAdiabatic Sorption of AmmoniaWater System and Depicting in ptx Diagram
Adabatc Sorpton of AmmonaWater System and Depctng n ptx Dagram J. POSPISIL, Z. SKALA Faculty of Mechancal Engneerng Brno Unversty of Technology Techncka 2, Brno 61669 CZECH REPUBLIC Abstract:  Absorpton
More information10/23/2003 PHY Lecture 14R 1
Announcements. Remember  Tuesday, Oct. 8 th, 9:30 AM Second exam (coverng Chapters 94 of HRW) Brng the followng: a) equaton sheet b) Calculator c) Pencl d) Clear head e) Note: If you have kept up wth
More informationPeriod & Frequency. Work and Energy. Methods of Energy Transfer: Energy. WorkKE 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 informationCHAPTER 10 ROTATIONAL MOTION
CHAPTER 0 ROTATONAL MOTON 0. ANGULAR VELOCTY Consder argd body rotates about a fxed axs through pont O n xy plane as shown. Any partcle at pont P n ths rgd body rotates n a crcle of radus r about O. The
More informationSo 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 informationA particle in a state of uniform motion remain in that state of motion unless acted upon by external force.
The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,
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 informationChapter 11 Angular Momentum
Chapter 11 Angular Momentum Analyss Model: Nonsolated System (Angular Momentum) Angular Momentum of a Rotatng Rgd Object Analyss Model: Isolated System (Angular Momentum) Angular Momentum of a Partcle
More informationConservation of Angular Momentum = "Spin"
Page 1 of 6 Conservaton of Angular Momentum = "Spn" We can assgn a drecton to the angular velocty: drecton of = drecton of axs + rght hand rule (wth rght hand, curl fngers n drecton of rotaton, thumb ponts
More informationGAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME PHYSICAL SCIENCES GRADE 12 SESSION 1 (LEARNER NOTES)
PHYSICAL SCIENCES GRADE 1 SESSION 1 (LEARNER NOTES) TOPIC 1: MECHANICS PROJECTILE MOTION Learner Note: Always draw a dagram of the stuaton and enter all the numercal alues onto your dagram. Remember to
More informationGravitational Acceleration: A case of constant acceleration (approx. 2 hr.) (6/7/11)
Gravtatonal Acceleraton: A case of constant acceleraton (approx. hr.) (6/7/11) Introducton The gravtatonal force s one of the fundamental forces of nature. Under the nfluence of ths force all objects havng
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationChapter 9. The Dot Product (Scalar Product) The Dot Product use (Scalar Product) The Dot Product (Scalar Product) The Cross Product.
The Dot Product (Scalar Product) Chapter 9 Statcs and Torque The dot product of two vectors can be constructed by takng the component of one vector n the drecton of the other and multplyng t tmes the magntude
More information6. Stochastic processes (2)
Contents Markov processes Brthdeath processes Lect6.ppt S38.45  Introducton to Teletraffc Theory Sprng 5 Markov process Consder a contnuoustme and dscretestate stochastc process X(t) wth state space
More information6. Stochastic processes (2)
6. Stochastc processes () Lect6.ppt S38.45  Introducton to Teletraffc Theory Sprng 5 6. Stochastc processes () Contents Markov processes Brthdeath processes 6. Stochastc processes () Markov process
More informationOpen Systems: Chemical Potential and Partial Molar Quantities Chemical Potential
Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,
More information12. The HamiltonJacobi Equation Michael Fowler
1. The HamltonJacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
More informationOne 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 informationEN40: Dynamics and Vibrations. Homework 4: Work, Energy and Linear Momentum Due Friday March 1 st
EN40: Dynamcs and bratons Homework 4: Work, Energy and Lnear Momentum Due Frday March 1 st School of Engneerng Brown Unversty 1. The fgure (from ths publcaton) shows the energy per unt area requred to
More informationSupplemental Instruction sessions next week
Homework #4 Wrtten homework due now Onlne homework due on Tue Mar 3 by 8 am Exam 1 Answer keys and scores wll be posted by end of the week Supplemental Instructon sessons next week Wednesday 8:45 10:00
More informationHow Differential Equations Arise. Newton s Second Law of Motion
page 1 CHAPTER 1 FrstOrder Dfferental Equatons Among all of the mathematcal dscplnes the theory of dfferental equatons s the most mportant. It furnshes the explanaton of all those elementary manfestatons
More informationKinematics in 2Dimensions. Projectile Motion
Knematcs n Dmensons Projectle Moton A medeval trebuchet b Kolderer, c1507 http://members.net.net.au/~rmne/ht/ht0.html#5 Readng Assgnment: Chapter 4, Sectons 6 Introducton: In medeval das, people had
More informationcoordinates. Then, the position vectors are described by
Revewng, what we have dscussed so far: Generalzed coordnates Any number of varables (say, n) suffcent to specfy the confguraton of the system at each nstant to tme (need not be the mnmum number). In general,
More informationUnit 5: Quadratic Equations & Functions
Date Perod Unt 5: Quadratc Equatons & Functons DAY TOPIC 1 Modelng Data wth Quadratc Functons Factorng Quadratc Epressons 3 Solvng Quadratc Equatons 4 Comple Numbers Smplfcaton, Addton/Subtracton & Multplcaton
More informationChapter 11 Torque and Angular Momentum
Chapter Torque and Angular Momentum I. Torque II. Angular momentum  Defnton III. Newton s second law n angular form IV. Angular momentum  System of partcles  Rgd body  Conservaton I. Torque  Vector
More informationMechanics Physics 151
Mechancs Physcs 151 Lecture 3 Lagrange s Equatons (Goldsten Chapter 1) Hamlton s Prncple (Chapter 2) What We Dd Last Tme! Dscussed multpartcle systems! Internal and external forces! Laws of acton and
More informationPES 1120 Spring 2014, Spendier Lecture 6/Page 1
PES 110 Sprng 014, Spender Lecture 6/Page 1 Lecture today: Chapter 1) Electrc feld due to charge dstrbutons > charged rod > charged rng We ntroduced the electrc feld, E. I defned t as an nvsble aura
More informationChapter 2: Electric Energy and Capacitance
Chapter : Electrc Energy and Capactance Potental One goal of physcs s to dentfy basc forces n our world, such as the electrc force as studed n the prevous lectures. Expermentally, we dscovered that the
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
More informationChapter 11: Angular Momentum
Chapter 11: ngular Momentum Statc Equlbrum In Chap. 4 we studed the equlbrum of pontobjects (mass m) wth the applcaton of Newton s aws F 0 F x y, 0 Therefore, no lnear (translatonal) acceleraton, a0 For
More information11. Dynamics in Rotating Frames of Reference
Unversty of Rhode Island DgtalCommons@URI Classcal Dynamcs Physcs Course Materals 2015 11. Dynamcs n Rotatng Frames of Reference Gerhard Müller Unversty of Rhode Island, gmuller@ur.edu Creatve Commons
More informationSymmetric Lie Groups and Conservation Laws in Physics
Symmetrc Le Groups and Conservaton Laws n Physcs Audrey Kvam May 1, 1 Abstract Ths paper eamnes how conservaton laws n physcs can be found from analyzng the symmetrc Le groups of certan physcal systems.
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 informationCorrelation and Regression. Correlation 9.1. Correlation. Chapter 9
Chapter 9 Correlaton and Regresson 9. Correlaton Correlaton A correlaton s a relatonshp between two varables. The data can be represented b the ordered pars (, ) where s the ndependent (or eplanator) varable,
More informationCelestial Mechanics. Basic Orbits. Why circles? Tycho Brahe. PHY celestialmechanics  J. Hedberg
PHY 454  celestalmechancs  J. Hedberg  207 Celestal Mechancs. Basc Orbts. Why crcles? 2. Tycho Brahe 3. Kepler 4. 3 laws of orbtng bodes 2. Newtonan Mechancs 3. Newton's Laws. Law of Gravtaton 2. The
More informationENGN 40 Dynamics and Vibrations Homework # 7 Due: Friday, April 15
NGN 40 ynamcs and Vbratons Homework # 7 ue: Frday, Aprl 15 1. Consder a concal hostng drum used n the mnng ndustry to host a mass up/down. A cable of dameter d has the mass connected at one end and s wound/unwound
More informationPHYSICS 231 Lecture 18: equilibrium & revision
PHYSICS 231 Lecture 18: equlbrum & revson Remco Zegers Walkn hour: Thursday 11:3013:30 am Helproom 1 gravtaton Only f an object s near the surface of earth one can use: F gravty =mg wth g=9.81 m/s 2
More informationMath1110 (Spring 2009) Prelim 3  Solutions
Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3  Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers.
More informationTemperature. Chapter Heat Engine
Chapter 3 Temperature In prevous chapters of these notes we ntroduced the Prncple of Maxmum ntropy as a technque for estmatng probablty dstrbutons consstent wth constrants. In Chapter 9 we dscussed the
More information