# Work is the change in energy of a system (neglecting heat transfer). To examine what could

Size: px
Start display at page:

Download "Work is the change in energy of a system (neglecting heat transfer). To examine what could"

Transcription

1 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 a length. Consder a constant orce F on a block. The block genercally could be dsplaced n three ways. (Ths dsplacement could be due to other d R orces or an ntal momentum.) What s the work d U done by F on the block? Work s a scalar and so the uncton must take two vectors, F and d, and produce a scalar. It turns out that the correct way to do ths s to look at parallel components o F and d. Work s dened as W F d Fy dy Fz dzor W F d. Ths s correct only or a constant orce. Lookng at the dagram above, what s the work done durng a dsplacement d R? It s smply the product Fd R. The orce s dong postve work on the block. What about or a dsplacement d L? In ths case the orce and dsplacement are antparallel and the work s FdL. The orce reduces the energy o the block,.e. t would slow the block down and reduce the knetc energy. How about or a dsplacement that s perpendcular to the orce? Here the work done s zero. The orce does not change the energy o the block as t moves along ths perpendcular path. There s an mportant specal case that has ths perpendcular relatonshp between the orce and dsplacement. Consder your crcular orbt and the arrows that you drew. The orce (or acceleraton) was radally nward whle the velocty (nstantaneous dsplacement d v t ) s tangent to the crcle and thereore perpendcular to the orce. Ths means the work done by ths orce s zero. Another case were the work s zero s when there s no dsplacement o the object. An nterestng case s that o jumpng o the loor. What s the work done by the loor on your body? Clearly the loor places a orce on your body, but yet the loor does not move so the work done by the loor s zero. Ths may seem strange and you should ask where you get your energy to jump up when the loor does no work. We wll return to ths nterestng queston n a couple o weeks. d L F The last case wth constant orces to consder s a dsplacement at an arbtrary angle to the orce. Usng the denton or the dot product W F d Fd cos. d F

2 Now the general case or non-constant orces along an arbtrary trajectory needs to be eamned. The red arrows show the orce that a partcle eperences along a trajectory shown n black. I the trajectory s broken up nto many small segments as shown n the gure below, then these segments wll be appromately straght and the work done s W F r F cos r where s the angle between the orce and the small dsplacement at locaton. O course, we want to take the lmt o ths so that r dr and the sum becomes an ntegral, a lne ntegral. Takng ths lmt, the general epresson or the work done by a non-constant orce along an arbtrary trajectory s y z. You wll learn how W F dr F d F dy F dz y z y z to evaluate ths ntegral n your class on calculus o several varables. Rght now you could evaluate ths usng numercal technques. Let s nd the work done by a varable orce rom a sprng, where F k( ) ˆ. The dsplacement s along the -as. Here s the stuaton: F dr

3 The work done by the sprng on the block as t moves rom to s gven by: 1 1 ˆˆ W k( ) dr k k k( ) k ( ). You need to be careul when dong ths calculaton to make sure that F dr has the correct sgn. Qute oten t s convenent to place the orgn o the coordnate system at the pont o equlbrum or the sprng, 1.e.. Wth ths choce the work smples to W k ( ). Note that n ths case the orce that the sprng does on the block s negatve. It s removng knetc energy rom the block. Eventually, the block wll come to rest and begn to move back towards ; durng ths part o the cycle the sprng wll do postve work on the block. I a complete cycle s consdered the sprng wll perorm zero net work and the block wll return to the orgnal pont wth eactly the same knetc energy as t had the cycle beore. Let s consder a slghtly more complcated case where the moton s not one dmensonal. A mass s launched wth an ntal velocty v ( v, v ) n a unorm gravtatonal eld and we want to calculate the work done by gravty. Frst let there be no ar resstance. Ths means that we have to calculate W F dr along the trajectory. Consderng the orce o gravty 1 F (, mg). The parabolc trajectory s gven by r( t) ( vt, vyt gt ) thereore dr( t) ( v, v gt) dt so the epresson or the work done by gravty s y t t 1 W( t) (, mg) ( v, vy gt) dt mg ( vy gt) dt mg( vyt gt ) now note that the quantty n the parenthess s just the negatve o the y component o the trajectory,.e. the negatve o the heght about the ground. So the work done by gravty s W() t mgry. Snce the orce o gravty has no component, there s no dependence on how ar laterally the partcle has moved. Only the vertcal dsplacement matters or the work done by gravty. Or more generally, only the component o the dsplacement that s (ant)parallel to the orce contrbutes to the work. Now that we know how to calculate the work done by a orce you mght ask how does ths aect the knetc energy o a body? Startng wth W F dr we consder or an nntesmal tme dt that an nntesmal amount o work s done dw dp dv orce, we can use Newton s law F net m dt dt we nd that dv dr dw m dr m dv mv dv dt dt ntegratng ths yelds, y F dr Let the orce be the net or total

4 1 1 1 So the work done by the net orce W dw mv dv mv v mv mv KE on an object s equal to the change n the knetc energy o the object. Ths s an mportant result and named the Work-Energy Theorem. Lastly, t s nterestng to know the tme rate o change o the work done by a orce. Usng dw dr dw F dr and dvdng by dt you can see that F F v. Ths quantty s call the dt dt dw power, P F v and tells the rate at whch energy o a system s beng changed by a dt orce. The unts s joules/sec whch s called a watt. Let s end ths dscusson by eamnng a problem usng all that we have learned. Consder the same problem above, projectle n a unorm gravtatonal eld, but now let s nclude ar drag. We can no longer compute the trajectory analytcally, but numercally t s no problem. The orces on the mass are gravty and ar drag. The total orce s gven by 1 F ˆ net Fgrav Fdrag mg ACv v and dr vdt. By takng the dot product o these epressons t s straghtorward to nd the nntesmal work done by gravty and ar drag at any tme. Then a numercal ntegraton (sum) can be perormed. Ths s how the man loop looks or the mplementaton n Python:

5 You should be able to understand each lne o ths man loop. In the last block o code, ponted to by the red arrow, the rst two lnes nd the delta work done by ar and gravty durng the last tme ncrement. The last two lnes sum up the delta work done to nd the total work done by ar and gravty. Here s a pcture o the object n lght. The red arrow s the orce and the cyan s proportonal to dr. The dot produce o these vectors gves the nntesmal work done durng that tme step. Notce that the orce vector s not vertcal. Ths s due to the ncluson o ar drag. The entre trajectory looks lke ths. Notce that the net orce s much reduced. The object s approachng termnal velocty. Now let s look at the power and work beng done by the orces. The power s equal to dw dr F so t s a relecton o the dt dt nstantaneous work beng done. Below s the graph o the power. The red curve s the rate o work that gravty s dong and the black curve s the power or the ar drag. There are a couple o thngs to notce. The work beng done by

6 gravty changes sgn at 1 sec. Whereas the work beng done by ar drag s always negatve. What happens at 1 sec to change the sgn o the gravtatonal power? Ths s when the partcle has reached mamum heght and the y component o the velocty changes rom postve to negatve. Gravty does negatve works as the partcle ncreases heght,.e. t s slowng down the partcle, but ater the ape gravty does postve work and tres to accelerate the partcle downward. Now at the same tme the drag orce s always dong negatve work and reducng the velocty. You mght correctly epect that at termnal velocty the negatve power due to drag s equal to the postve power rom gravty. The object s close to reachng termnal velocty near the end o the trajectory. Notce that the two powers are appromately equal, but wth derent sgns. I the power s ntegrated then you get the total work done by the orce. Below s a graph o ths work. Agan red s gravty and black s ar drag. The cyan lne s the knetc energy o the partcle and the orange lne shows the ntal total energy o the system. The work energy theorem says that work done by the sum o the orces s equal to the change n knetc energy. In other words, the sum o the red and black lnes should equal the cyan mnus orange. Ths s true or all tmes. There are a couple other thngs to notce. Frst the total work done by gravty at the end o the trajectory s zero. The partcle has gone up and back down by the same change n heght, and gravty beng a conservatve orce only depends on the startng and endng heght so the work done must be zero. Ths s qute derent rom the work done by ar drag. Here the work s always an ncreasng negatve value. The drag s always removng mechancal energy rom the system. Lastly, notce that the knetc energy has just about reached a steady value. Ths s agan a relecton o the partcle nearng termnal velocty so the knetc energy does not change.

### Spring 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

### Chapter 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.

### Week3, 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 1-s tme nterval. The velocty of the partcle

### PHYS 1441 Section 002 Lecture #16

PHYS 1441 Secton 00 Lecture #16 Monday, Mar. 4, 008 Potental Energy Conservatve and Non-conservatve Forces Conservaton o Mechancal Energy Power Today s homework s homework #8, due 9pm, Monday, Mar. 31!!

### PHYS 1443 Section 004 Lecture #12 Thursday, Oct. 2, 2014

PHYS 1443 Secton 004 Lecture #1 Thursday, Oct., 014 Work-Knetc Energy Theorem Work under rcton Potental Energy and the Conservatve Force Gravtatonal Potental Energy Elastc Potental Energy Conservaton o

### A 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

### Chapter 8. Potential Energy and Conservation of Energy

Chapter 8 Potental Energy and Conservaton of Energy In ths chapter we wll ntroduce the followng concepts: Potental Energy Conservatve and non-conservatve forces Mechancal Energy Conservaton of Mechancal

### 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

### CHAPTER 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 non-conservatve orces Physcs 1 Knetc energy: Potental energy: Energy assocated

### Physics 2A Chapters 6 - Work & Energy Fall 2017

Physcs A Chapters 6 - Work & Energy Fall 017 These notes are eght pages. A quck summary: The work-energy theorem s a combnaton o Chap and Chap 4 equatons. Work s dened as the product o the orce actng on

### Chapter 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

### PHYS 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

### Period & 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

### Conservation 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,

### Chapter 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

### 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

### Name: PHYS 110 Dr. McGovern Spring 2018 Exam 1. Multiple Choice: Circle the answer that best evaluates the statement or completes the statement.

Name: PHYS 110 Dr. McGoern Sprng 018 Exam 1 Multple Choce: Crcle the answer that best ealuates the statement or completes the statement. #1 - I the acceleraton o an object s negate, the object must be

### Chapter 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

### Chapter 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

### Lecture 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

### Energy 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

### Physics 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

### You 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

### Lecture 22: Potential Energy

Lecture : Potental Energy We have already studed the work-energy 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

### EMU 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

### 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

### Force = F Piston area = A

CHAPTER III Ths chapter s an mportant transton between the propertes o pure substances and the most mportant chapter whch s: the rst law o thermodynamcs In ths chapter, we wll ntroduce the notons o heat,

### PHYS 1101 Practice problem set 12, Chapter 32: 21, 22, 24, 57, 61, 83 Chapter 33: 7, 12, 32, 38, 44, 49, 76

PHYS 1101 Practce problem set 1, Chapter 3: 1,, 4, 57, 61, 83 Chapter 33: 7, 1, 3, 38, 44, 49, 76 3.1. Vsualze: Please reer to Fgure Ex3.1. Solve: Because B s n the same drecton as the ntegraton path s

### TIME OF COMPLETION NAME SOLUTION DEPARTMENT OF NATURAL SCIENCES. PHYS 2211, Exam 2 Section 1 Version 1 October 18, 2013 Total Weight: 100 points

TIME OF COMPLETION NAME SOLUTION DEPARTMENT OF NATURAL SCIENCES PHYS, Exam Secton Verson October 8, 03 Total Weght: 00 ponts. Check your examnaton or completeness pror to startng. There are a total o nne

### Chapter 3 Differentiation and Integration

MEE07 Computer Modelng Technques n Engneerng Chapter Derentaton and Integraton Reerence: An Introducton to Numercal Computatons, nd edton, S. yakowtz and F. zdarovsky, Mawell/Macmllan, 990. Derentaton

### 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

### Week 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,

### Physics 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

### Physics 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

### Chapter 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

### Chapter 2. Pythagorean Theorem. Right Hand Rule. Position. Distance Formula

Chapter Moton n One Dmenson Cartesan Coordnate System The most common coordnate system or representng postons n space s one based on three perpendcular spatal axes generally desgnated x, y, and z. Any

### Problem 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

### Chapter Twelve. Integration. We now turn our attention to the idea of an integral in dimensions higher than one. Consider a real-valued function f : D

Chapter Twelve Integraton 12.1 Introducton We now turn our attenton to the dea of an ntegral n dmensons hgher than one. Consder a real-valued functon f : R, where the doman s a nce closed subset of Eucldean

### PHYS 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

### 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

### PHYS 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)]

### Gravitational 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

### Chapter 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

### CHAPTER 10 ROTATIONAL MOTION

CHAPTER 0 ROTATONAL MOTON 0. ANGULAR VELOCTY Consder argd body rotates about a fxed axs through pont O n x-y plane as shown. Any partcle at pont P n ths rgd body rotates n a crcle of radus r about O. The

### 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

### EN40: 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

### Celestial Mechanics. Basic Orbits. Why circles? Tycho Brahe. PHY celestial-mechanics - J. Hedberg

PHY 454 - celestal-mechancs - 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

### Physics 40 HW #4 Chapter 4 Key NEATNESS COUNTS! Solve but do not turn in the following problems from Chapter 4 Knight

Physcs 40 HW #4 Chapter 4 Key NEATNESS COUNTS! Solve but do not turn n the ollowng problems rom Chapter 4 Knght Conceptual Questons: 8, 0, ; 4.8. Anta s approachng ball and movng away rom where ball was

### Modeling motion with VPython Every program that models the motion of physical objects has two main parts:

1 Modelng moton wth VPython Eery program that models the moton o physcal objects has two man parts: 1. Beore the loop: The rst part o the program tells the computer to: a. Create numercal alues or constants

### Prof. Dr. I. Nasser T /16/2017

Pro. Dr. I. Nasser T-171 10/16/017 Chapter Part 1 Moton n one dmenson Sectons -,, 3, 4, 5 - Moton n 1 dmenson We le n a 3-dmensonal world, so why bother analyzng 1-dmensonal stuatons? Bascally, because

### Week 6, Chapter 7 Sect 1-5

Week 6, Chapter 7 Sect 1-5 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

### Physics 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. Mult-Partcle

### Physics 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

### Moments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.

Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these

### Chapter 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

### 10/24/2013. 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

### Week 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

### Study 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

### = 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:

### General Tips on How to Do Well in Physics Exams. 1. Establish a good habit in keeping track of your steps. For example, when you use the equation

General Tps on How to Do Well n Physcs Exams 1. Establsh a good habt n keepng track o your steps. For example when you use the equaton 1 1 1 + = d d to solve or d o you should rst rewrte t as 1 1 1 = d

### PHYSICS 203-NYA-05 MECHANICS

PHYSICS 03-NYA-05 MECHANICS PROF. S.D. MANOLI PHYSICS & CHEMISTRY CHAMPLAIN - ST. LAWRENCE 790 NÉRÉE-TREMBLAY QUÉBEC, QC GV 4K TELEPHONE: 48.656.69 EXT. 449 EMAIL: smanol@slc.qc.ca WEBPAGE: http:/web.slc.qc.ca/smanol/

### How Differential Equations Arise. Newton s Second Law of Motion

page 1 CHAPTER 1 Frst-Order 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

### One Dimension Again. Chapter Fourteen

hapter Fourteen One Dmenson Agan 4 Scalar Lne Integrals Now we agan consder the dea of the ntegral n one dmenson When we were ntroduced to the ntegral back n elementary school, we consdered only functons

### Conservation 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

### Part 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

### Spin-rotation coupling of the angularly accelerated rigid body

Spn-rotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 E-mal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s

### Angular 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

### Chapter 5. Answers to Even Numbered Problems m kj. 6. (a) 900 J (b) (a) 31.9 J (b) 0 (c) 0 (d) 31.9 J. 10.

Answers to Even Numbered Problems Chapter 5. 3.6 m 4..6 J 6. (a) 9 J (b).383 8. (a) 3.9 J (b) (c) (d) 3.9 J. 6 m s. (a) 68 J (b) 84 J (c) 5 J (d) 48 J (e) 5.64 m s 4. 9. J 6. (a). J (b) 5. m s (c) 6.3

### 12. The Hamilton-Jacobi Equation Michael Fowler

1. The Hamlton-Jacob 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

### Chapter 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

### 11. 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

### First 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

### ˆ (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

### Supplemental 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

### A 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,

### Single Variable Optimization

8/4/07 Course Instructor Dr. Raymond C. Rump Oce: A 337 Phone: (95) 747 6958 E Mal: rcrump@utep.edu Topc 8b Sngle Varable Optmzaton EE 4386/530 Computatonal Methods n EE Outlne Mathematcal Prelmnares Sngle

### Physics 2A Chapter 9 HW Solutions

Phscs A Chapter 9 HW Solutons Chapter 9 Conceptual Queston:, 4, 8, 13 Problems: 3, 8, 1, 15, 3, 40, 51, 6 Q9.. Reason: We can nd the change n momentum o the objects b computng the mpulse on them and usng

### Section 8.3 Polar Form of Complex Numbers

80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the

### K = 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

### ONE-DIMENSIONAL COLLISIONS

Purpose Theory ONE-DIMENSIONAL COLLISIONS a. To very the law o conservaton o lnear momentum n one-dmensonal collsons. b. To study conservaton o energy and lnear momentum n both elastc and nelastc onedmensonal

### PHYS 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.!

### Complex Variables. Chapter 18 Integration in the Complex Plane. March 12, 2013 Lecturer: Shih-Yuan Chen

omplex Varables hapter 8 Integraton n the omplex Plane March, Lecturer: Shh-Yuan hen Except where otherwse noted, content s lcensed under a BY-N-SA. TW Lcense. ontents ontour ntegrals auchy-goursat theorem

### Chapter 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

### Physics for Scientists & Engineers 2

Equpotental Surfaces and Lnes Physcs for Scentsts & Engneers 2 Sprng Semester 2005 Lecture 9 January 25, 2005 Physcs for Scentsts&Engneers 2 1 When an electrc feld s present, the electrc potental has a

### Shuai Dong. Isaac Newton. Gottfried Leibniz

Computatonal pyscs Sua Dong Isaac Newton Gottred Lebnz Numercal calculus poston dervatve ntegral v velocty dervatve ntegral a acceleraton Numercal calculus Numercal derentaton Numercal ntegraton Roots

### ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look

### Physics 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 Rgd-Body Rotaton

### 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

### Physics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1

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

### OPTIMISATION. Introduction Single Variable Unconstrained Optimisation Multivariable Unconstrained Optimisation Linear Programming

OPTIMIATION Introducton ngle Varable Unconstraned Optmsaton Multvarable Unconstraned Optmsaton Lnear Programmng Chapter Optmsaton /. Introducton In an engneerng analss, sometmes etremtes, ether mnmum or

### Momentum. Momentum. Impulse. Momentum and Collisions

Momentum Momentum and Collsons From Newton s laws: orce must be present to change an object s elocty (speed and/or drecton) Wsh to consder eects o collsons and correspondng change n elocty Gol ball ntally

### MTH 263 Practice Test #1 Spring 1999

Pat Ross MTH 6 Practce Test # Sprng 999 Name. Fnd the area of the regon bounded by the graph r =acos (θ). Observe: Ths s a crcle of radus a, for r =acos (θ) r =a ³ x r r =ax x + y =ax x ax + y =0 x ax

### where 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

### 36.1 Why is it important to be able to find roots to systems of equations? Up to this point, we have discussed how to find the solution to

ChE Lecture Notes - D. Keer, 5/9/98 Lecture 6,7,8 - Rootndng n systems o equatons (A) Theory (B) Problems (C) MATLAB Applcatons Tet: Supplementary notes rom Instructor 6. Why s t mportant to be able to

### τ rf = Iα I point = mr 2 L35 F 11/14/14 a*er lecture 1

A mass s attached to a long, massless rod. The mass s close to one end of the rod. Is t easer to balance the rod on end wth the mass near the top or near the bottom? Hnt: Small α means sluggsh behavor

### Electricity and Magnetism - Physics 121 Lecture 10 - Sources of Magnetic Fields (Currents) Y&F Chapter 28, Sec. 1-7

Electrcty and Magnetsm - Physcs 11 Lecture 10 - Sources of Magnetc Felds (Currents) Y&F Chapter 8, Sec. 1-7 Magnetc felds are due to currents The Bot-Savart Law Calculatng feld at the centers of current

### CHAPTER 9 LINEAR MOMENTUM, IMPULSE AND COLLISIONS

CHAPTER 9 LINEAR MOMENTUM, IMPULSE AND COLLISIONS 103 Phy 1 9.1 Lnear Momentum The prncple o energy conervaton can be ued to olve problem that are harder to olve jut ung Newton law. It ued to decrbe moton

### Mechanics Cycle 3 Chapter 9++ Chapter 9++

Chapter 9++ More on Knetc Energy and Potental Energy BACK TO THE FUTURE I++ More Predctons wth Energy Conservaton Revst: Knetc energy for rotaton Potental energy M total g y CM for a body n constant gravty