# Chapter Seven - Potential Energy and Conservation of Energy

Size: px
Start display at page:

Transcription

1 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 o a system. The am ount o potental energy n the system s determned by the conguraton o the system. Movng members o the system to derent postons or rotatng them may change the conguraton o the system and thereore ts potental energy as shown n gure (7-1). Fgure (7-1), representng external agent lts a boo slowly rom a heght y to a heght y. In mechancs, the specc orms o potental energy are: (a) Gravtatonal potental energy.(b) Elastc potental energy. Potental energy s a scalar quantty and ts SI unt s. Chapter Seven - Potental Energy and Conservaton o Energy 7 Gravtatonal Potental Energy: The gravtatonal potental energy U g s the energy that an object o mass m has by vrtue o ts poston relatve to the surace o the earth. That poston s measured by the heght y o the object relatve to an arbtrary zero level. The wor done by the external agent on the system (object and the Earth) as the object undergoes ths upward dsplacement s gven by the product o the upward led orce F and the upward dsplacement o ths orce, Δr = Δy Ug mgy ( F ). y ( mg)( y y ) mgy mgy ext App Gravtatonal potental energy s the potental energy o the object - Earth system. As an object alls ts energy s transerred rom gravtatonal energy to netc energy by the gravtatonal orce.as shown n g (7-). Fgure (7-), representng transerred rom gravtatonal energy to netc energy. 10

2 From ths result, we see that the wor done on any object by the gravtatonal orce s equal to the negatve o the change n the system s gravtatonal potental energy. 7 Elastc Potental Energy The elastc potental energy o the system can be thought o as the energy stored n the deormed sprng (one that s ether compressed or stretched rom ts equlbrum poston). The wor done by an external led orce F on a system consstng o a bloc connected t o the sprng s gven by Equaton: ext 1 x 1 x ext : The wor done by sprng and s the orce constant. In ths stuaton, the ntal and nal x coordnates o the bloc are measured rom ts equlbrum poston, x = 0. Thereore the elastc potental energy stored n a sprng o orce constant s gven by: U S 1 x Notes 1- The elastc potental energy stored n a sprng s zero whenever the sprng s unreormed (x =0) as shown n (a) n the ollowng gure. Chapter Seven - Potental Energy and Conservaton o Energy - Energy s stored n the sprng only when the sprng s ether stretched or compressed. - hen an object o mass m pushed aganst the sprng compressng t a dstance x, as shown n (b) n the ollowng, the elastc potental energy stored n the sprng s (1/)x. 4- hen the object s released rom rest, the sprng snaps bac to ts orgnal length and the stored elastc potental energy s transormed nto netc energy o the object, as shown n (c) n the ollowng gure. 5- The elast c potental energy s a maxmum when the sprng has reached ts maxmum compresson or extenson (that s, when x s a maxmum). 6- Because the elastc potental s proportonal to x, U s s always postve n deormed sprng. Fgure (7-), representng the elastc potental energy stored n a sprng. 7-4 Conservatve and Non-conservatve Forces 104

3 7 4 1 Conservatve Force: Conservatve orces have two mportant propertes: 1- A orce s sad to be conservatve the wor done by the orce actng on a partcle movng between two ponts s ndependent o the path the partcle taes between the ponts. - The wor done by a conservatve orce movng through any closed path s zero. (A clo sed path s one n whch the begnnng and end ponts are dented). Fgure (7-4), representng the wor done by a conservatve orce movng through any closed path. Chapter Seven - Potental Energy and Conservaton o Energy For example o a conservat ve orce: (1) the gravtatonal orce : The wor done by the gravtatonal orce on an object s ndependent on the path o the object s movements. It only depends on the derence o the object s ntal and nal poston n the drecton o the orce. Fgure (7-5), representng the gravtatonal orce. mgy mgy g g ) The orce that a sprng exerts on any object attached to the sprng : The wor done on the object by the sprng depends only on the ntal and nal poston o the dstorted sprng. 1 1 g x x 7 4 Non-conservatve Forces: A orce or whch the wor depends on the path s called a non-conservatve orce. For example o a non- conservatve orce s the rctonal orce. 105

4 The wor done by rcton along that path 1 s gven by: 1 The wor done by the rcton orce along path 14 s gven by: d d d d 14 d Notes Fgure (7-6), representng the wor done aganst the orce o netc rcton depends on the path taen as the boo s moved rom to. 7 5 Relatonshp between Conservatve Forces and Potental Energy e can dene a potental energy uncton U, such that the wor done by a conservatve orce equals the decrease n the potental energy o the system. The wor done by a conservatve orce F as a partcle moves along the x-axs s: Chapter Seven - Potental x Energy and Conservaton o Energy c Fx dx Uؤ x here F x s the component o F n the drecton o the dsplacement. That s, the wor done by a conservatve orce equals the negatve o the change n the potental energy assocated wth that orce. The above equaton can also express as: In general, U = U(x, y, z) c ÄU U F dr ÄU U U U F dx 1- U s negatve when F x and dx are n the same drecton, as when an object s lowered n a gravtatonal eld or when a sprng pushes an object toward equlbrum. - I a conservatve orce does postve wor on a system, potental energy s lost. x x x - I a conservatve orce does negatve wor, potental energy s ganed. 4- I the conservatve orce s nown as a uncton o poston, we can use the equaton: 106

5 U x x x F dx U 5- The above equaton s used to calculate the change n potental energy o a system as an object wthn the system moves rom x to x. x 7 6 How to nd F x the potental energy o the system s nown I the pont o lcaton o the orce undergoes an nntesmal dsplacement dx, we can express the nntesmal change n the potental energy o the system du as: du F dx Thereore, the conservatve orce s related to the potental energy uncton through the relatonshp: du F x dx That s, any conservatve orce actng on an object wthn a system equals the negatve dervatve o the potental energy o the system wth respect to x. In three dmensons, the expresson s: U U U F î ĵ ˆ U x y z Snce U=U(x, y, z) and U x Chapter Seven - Potental F x Energy ; F y and ; F z Conservaton o Energy Let's very ths expresson correctly gves the gravtatonal orce and the elastc orce when usng the gravtatonal potental energy and the elastc potental energy: x U y (1) The gravtatonal potental energy uncton s U g = mgy From the equaton: (x) F y du(y) dy U z It ollows that: (x) F y du(y) dy d dy mgy mg () The elastc potental energy o the deormed sprng s U s = (1/)x From the equaton: (x) F x du(x) dx It ollows that: du(x) Fx (x) dx 7 7 Conservaton o Mechancal Energy d dx 1 x x 107

6 The total mechancal energy o a system s dened as: the sum o the netc energy K and the potental energy U. The prncple o mechancal energy conservaton states that: the total mechancal energy o a system remans constant n any solated system o objects that nteracts only through conservatve orces. An solated system: s a system that there s no net wor s done on the system by external orces. You can wrte the conservaton o energy statement n many derent mathematcal orms. Here are some o them: E E E ÄE ÄK ÄU K K U U K E The above equaton s vald only when no energy s added to or removed rom the system. Furthermore, there must be no non-conservatve orces dong wor wthn the system. 0 U I more than one conservatve orce acts on an object wthn a system, a potental energy uncton s assocated wth each orce. In such a case, we can ly the prncple o mechancal energy conservaton or the system as: K U K U Chapter Seven - Potental Energy and Conservaton o Energy here the number o terms n the sums equals the numbers o conservaton orces present. For example, an object connected to a sprng oscllates vertcally, two conservatve orces act on the object; the sprng orce and the gravtatonal orce. So that we can express the above equaton as: K U E K U g, U s, I some o the orces actng on objects wthn the system are not conservatve, then the mechancal energy o the system does not reman constant. Let us examne two types o non-conservatve orces, (a) An led orce; and (b) The orce o netc rcton. 0 K K (a) or Done by an Appled Force I s the wor done by an led orce on an object and c s the wor done by the conservatve orce on the object, then the net wor done on the object s related to the change n ts netc energy accordng to the wor-netc energy theorem and s gven by: ÄK net net c 0 U U Kؤ c g, U s, 0 Potental energy s related to conservatve orces only, so we can use the expresson: c Uؤ From substtuton: 108

7 Eؤ Uؤ Kؤ The rght sde o the above equaton represents the change n the mechancal energy o the system. e conclude that, an object s part o a system, then an led orce can transer energy nto or out o the system. (b) Force o Knetc Frcton Knetc rcton s an example o a non-conservatve orce. hen an object moves through a dstance d on a horzontal surace that s not rctonless, the only orce that does wor s the orce o netc rcton. Ths orce causes a decrease n the netc energy o the object, so that: ÄK d rcton I an object moves on an nclne that s not rctonless, a change n the gravtatonal potental energy o the object-earth system also occurs, and d s the amount by whch the mechancal energy o the system changes because o the orce o the netc rcton. In such cases: ÄE ÄK ÄU d here: E ÄE E Example 7.1: An Esmo returnng rom a successul shng trp pulls a sled loaded wth salmon. The total mass o the sled and salmon s 50.0 g, and the Esmo exerts a orce o Chapter Seven - Potental Energy and Conservaton o Energy N on the sled by pullng on the rope. (a) How much wor does he do on the sled the rope s horzontal to the ground (θ = 0 n Fgure) and he pulls the sled 5.00 m? (b) How much wor does he do on the sled θ = 0 and he pulls the sled the same dstance? (Treat the sled as a pont partcle, so detals Such as the pont o attachment o the rope mae no derence.) Soluton: (a) Fnd the wor done when the orce s horzontal. FÄx ( N)(5.00) (b) Fnd the wor done when the orce s exerted at a 0 angle. ( F cos )Äx ( N)(cos0)(5.00) Example 7.: Suppose that n Example 7.1 the coecent o netc rcton between the loaded 50.0g sled and snow s (a) The Esmo agan pulls the sled 5.00 m, exertng a orce o N at an angle o 0. Fnd the wor done on the sled by rcton, and the net wor. (b) Repeat the calculaton the led orce s exerted at an angle o 0.0 wth the horzontal Soluton: (a) Fnd the wor done by rcton on the sled and the net wor, the led orce s horzontal. F y n mg 0 n mg rc x nx mgx ( 0.)(50)(9.8)(5)

8 110 Sum the rctonal wor wth the wor done by the led orce rom Example 7.1 to get the net wor (The normal and gravty orces are perpendcular to the dsplacement, so they don t contrbute): 610 ( ) net rc n g (b) Recalculate the rctonal wor and net wor the led orce s exerted at a 0.0 angle. F n mg F sn 0 n mg F sn y rc x nx ( mg F sn ) x (0.)( sn 0)(5) 4.10 net rc n g 5.10 ( 4.10 ) Example 7.: A 60.0-g ser s at the top o a slope, as shown n Fgure. At the ntal pont A, she s 10.0 m vertcally above pont B.(a) Settng the zero level or gravtatonal potental energy at B, nd the gravtatonal potental energy o ths system when the ser s at A and then at B. Fnally, nd the change n potental energy o the ser Earth system as the ser goes rom pont A to pont B. (b) Repeat ths problem wth the zero level at pont A. (c) Repeat agan, wth the zero level.00 m hgher than pont B. Soluton: (a) Let y = 0 at B. Calculate the potental energy at A, at B, and the change n potental energy. Chapter Seven - Potental Energy and Conservaton o Energy PE PE mgy PE (b) Repeat the problem y = 0 at A, the new reerence pont, so that PE = 0 at A. PE mgy ( 10) PE PE (c) Repeat the problem, y = 0 two meters above B. PE PE mgy mgy ( ) PE PE

9 111 H. [1] A ball o mass m s dropped rom a heght h above the ground. (a) Neglectng ar resstance, determne the speed o the ball when t s at a heght y above the ground. (b) Determne the speed o the ball at y at the nstant o relea se t already has an ntal speed v at the ntal alttude h. [] A 4g partcle moves rom the orgn to poston C whch has coordnates x=5m and y=5m as shown n the gure. One orce on t s the orce o gravty n the negatve y drecton. Calcu late the wor done by gravty as the partcle moves rom O to C along (a) OAC; (b) OC. Is the orce conservatve? Explan. Chapter Seven - Potental Energy and Conservaton o Energy [] A pendulum conssts o a sphere o mass m attached to a lght cord o length l, as shown n the ollowng gure. The sphere s released rom rest when the cord maes an angle θ 0 wth the vertcal, and the pvot at P s rctonless. (a) Fnd the speed o the sphere when t s at the lowest pont B. (b) hat s the tenson T B n the cord at B?

10 [4] An electrcally charged partcle s held at rest at the pont x = 0, whle a second partcle wth equal charge s ree to move along the postve x-axs. The potental energy o the system s U(x) = c / x, where c s a postve constant that depends on the magntude o the charges. Derve an expresson or the x-component o orce actng on the movable charge, as a uncton o ts poston. [5] A ser starts rom rest at the top o a rctonless nclne o heght 0m. At the bottom o the nclne, she encounters a horzontal surace where the coecent o netc rcton betwe en the ss and the snow s 0.1. (a ) How ar does travel on the horzontal surace beore comng to rest? (b) Fnd the horzontal dstance the ser travels beore comng to rest the nclne also has a coecent o netc rcton equal to 0.1. [6] A sngle conservatve orce acts on a 5g partcle. The equaton F x = (x + 4) N, where x s n m, descrbes ths orce. As the partcle moves along the x-axs rom x=1m to x = 5m, calculate Chapter Seven - Potental Energy and Conservaton o Energy (a) The wor done by ths orce; (b) The change n the potental energy o the system; (c) The netc energy o the partcle at x = 5m ts speed at 1m s m/s. [7 Two blocs are connected by a lght strng that passes over a rctonless pulley. The bloc o mass m 1 les on a horzontal surace and s connect to a sprng o orce constant. The system s released rom rest when the sprng s unstretched. I the hangng bloc o mass m alls a dstance h beore comng to rest. Calculate the coecent o netc rcton between the bloc o mass m 1 and the surace. [8] A 5g bloc s set nto moton up an nclned plane wth an ntal speed o 8m/s. The bloc comes to rest ater travelng m along the plane, whch s nclned an angle o 0 º to the horzontal. For ths moton determne: (a) The change n the blocۥs netc energy. (b) The change n the potental energy. (c) The rctonal orce exerted on t (assumed to be constant). 11

11 [9] The coecent o rcton between the g bloc and surace n the ollowng gure s 0.4. The system starts rom rest. hat s the speed o the 5g ball when t has allen 1.5m, usng energy methods? [10] Suppose the ntal netc and potental energes o a system are 75 and 50 respectvely, and that the nal netc and potental energes o the same system are 00 a nd - 5 respectvely. How much wor was done on the system by non-conservatve orces? [11] A dver o mass m drops rom a board 10m above the water surace, as n the gure. Fnd hs speed 5m above the water surace. Neglect ar resstance Chapter Seven - Potental Energy and Conservaton o Energy [1] A 0.5g bloc rests on a horzontal, rctonless surace as n the gure; t s pressed aganst a lght sprng havng a sprng constant o = 800N/m, wth an ntal compresson o cm. To what heght h does the bloc rse when movng up the nclne? [1] An 8.0 g mountan clmber s n the nal stage o the ascent o Pe s Pea, whch 401m above sea level. 11

12 114 (a) hat s the change n gravtatonal potental energy as the clmber gans the last 100m o alttude? Use U=0 at sea level. (b) Do the same calculaton wth U=0 at the top o the pea. [14] A g mass starts rom rest and sldes a dstance d down a rctonless 0 º nclne. hle sldng, t comes nto contact wth an unstressed sprng o neglgble mass, as shown n the ollowng gur e. The mass sldes an addtonal 0.m as t brought momentarly to rest by compresson o the sprng (=400N/m). Fnd the ntal separaton d between the mass and the sprng. Chapter Seven - Potental Energy and Conservaton o Energy [15] A g bloc s attached to a sprng o a orce constant 500N/m on a horzontal table. The bloc s pulled 5cm to the rght o equlbrum and released rom rest. Fnd the speed o the bloc as t passes through equlbrum (a) The horzontal surace s rctonless and (b) The coecent o rcton between bloc and surace s 0.50.

### 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

### 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

### 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

### 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

### 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 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

### 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!!

### 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

### 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

### 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

### 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

### 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

### 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

### 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.

### 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

### 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

### 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

### 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 207, Lecture 13, Oct. 15. Energy

Physcs 07 Lecture 3 Physcs 07, Lecture 3, Oct. 5 Goals: Chapter 0 Understand the relatonshp between moton and energy Dene Potental Energy n a Hooke s Law sprng Deelop and explot conseraton o energy prncple

### 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

### 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,

### 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

### Page 1. Clicker Question 9: Physics 131: Lecture 15. Today s Agenda. Clicker Question 9: Energy. Energy is Conserved.

Physcs 3: Lecture 5 Today s Agenda Intro to Conseraton o Energy Intro to some derent knds o energy Knetc Potental Denton o Mechancal Energy Conseraton o Mechancal Energy Conserate orces Examples Pendulum

### 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

### 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

### Work 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

### 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

### 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

### 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

### 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

### AP Physics Enosburg Falls High School Mr. Bushey. Week 6: Work, Energy, Power

AP Physcs Enosburg Falls Hgh School Mr. Bushey ee 6: or, Energy, Power Homewor! Read Gancol Chapter 6.1 6.10 AND/OR Read Saxon Lessons 1, 16, 9, 48! Read Topc Summary Handout! Answer Gancol p.174 Problems

### 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

### 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/

### 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

### Physics 101 Lecture 9 Linear Momentum and Collisions

Physcs 0 Lecture 9 Lnear Momentum and Collsons Dr. Al ÖVGÜN EMU Physcs Department www.aogun.com Lnear Momentum and Collsons q q q q q q q Conseraton o Energy Momentum Impulse Conseraton o Momentum -D Collsons

### 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

### Linear Momentum and Collisions

Lnear Momentum and Collsons Chater 9 Lnear Momentum [kg m/s] x y mv x mv y Newton s nd Law n terms o momentum: Imulse I - [kg m/s] I t t Fdt I = area under curve bounded by t axs Imulse-Momentum Theorem

### Physics 131: Lecture 16. Today s Agenda

Physcs 131: Lecture 16 Today s Agenda Intro to Conseraton o Energy Intro to some derent knds o energy Knetc Potental Denton t o Mechancal Energy Conseraton o Mechancal Energy Conserate orces Examples Pendulum

### 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

### EMU Physics Department.

Physcs 0 Lecture 9 Lnear Momentum and Collsons Assst. Pro. Dr. Al ÖVGÜN EMU Physcs Department www.aogun.com Lnear Momentum q Conseraton o Energy q Momentum q Impulse q Conseraton o Momentum q -D Collsons

### 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 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 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

### 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

### 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 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

### Physics 106 Lecture 6 Conservation of Angular Momentum SJ 7 th Ed.: Chap 11.4

Physcs 6 ecture 6 Conservaton o Angular Momentum SJ 7 th Ed.: Chap.4 Comparson: dentons o sngle partcle torque and angular momentum Angular momentum o a system o partcles o a rgd body rotatng about a xed

### 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

### Physics 105: Mechanics Lecture 13

Physcs 05: Mechancs Lecture 3 Wenda Cao NJIT Physcs Department Momentum and Momentum Conseraton Momentum Impulse Conseraton o Momentum Collsons Lnear Momentum A new undamental quantty, lke orce, energy

### EN40: Dynamics and Vibrations. Homework 7: Rigid Body Kinematics

N40: ynamcs and Vbratons Homewor 7: Rgd Body Knematcs School of ngneerng Brown Unversty 1. In the fgure below, bar AB rotates counterclocwse at 4 rad/s. What are the angular veloctes of bars BC and C?.

### Conservation of Energy

Conservaton o nergy The total energy o a system can change only by amounts o energy that are transerred nto or out o the system W mec th nt Ths s one o the great conservaton laws n nature! Other conservaton

### v c motion is neither created nor destroyed, but transferred via interactions. Fri. Wed (.18,.19) Introducing Potential Energy RE 6.

r. 6.5-.7 (.) Rest Mass,ork by Changng orces Columba Rep 3pm, here RE 6.b (last day to drop) ed. 6.8-.9(.8,.9) Introducng Potental Energy RE 6.c Tues. H6: Ch 6 Pr s 58,59, 99(a-c), 05(a-c) moton s nether

### 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

### 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

### 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

### 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

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

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

### ENGN 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

### 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

### 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

### 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

### 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,

### 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

### Phys102 General Physics II

Electrc Potental/Energy Phys0 General Physcs II Electrc Potental Topcs Electrc potental energy and electrc potental Equpotental Surace Calculaton o potental rom eld Potental rom a pont charge Potental

### 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

### in 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

### RETURN ONLY THE SCANTRON SHEET!

Andrzej Czajkowsk PHY/ exam Page out o Prncples o Physcs I PHY PHY Instructor: Dr. Andrzej Czajkowsk Fnal Exam December Closed book exam pages questons o equal value 5 correct answers pass the test! Duraton:

### 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

### 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 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

### How does the momentum before an elastic and an inelastic collision compare to the momentum after the collision?

Experent 9 Conseraton o Lnear Moentu - Collsons In ths experent you wll be ntroduced to the denton o lnear oentu. You wll learn the derence between an elastc and an nelastc collson. You wll explore how

### Remark: Positive work is done on an object when the point of application of the force moves in the direction of the force.

Unt 5 Work and Energy 5. Work and knetc energy 5. Work - energy theore 5.3 Potenta energy 5.4 Tota energy 5.5 Energy dagra o a ass-sprng syste 5.6 A genera study o the potenta energy curve 5. Work and

### 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

### (T > w) F R = T - w. Going up. T - w = ma

ANSES Suspended Acceleratng-Objects A resultant orce causes a syste to accelerate. he drecton o the acceleraton s n the drecton o the resultant orce. As llustrated belo, hen suspended objects accelerate,

### CHAPTER 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

### Physics 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 non-contact) Frcton (a external force that opposes moton) Free

### Chapter 9 Linear Momentum and Collisions

Chapter 9 Lnear Momentum and Collsons m = 3. kg r = ( ˆ ˆ j ) P9., r r (a) p m ( ˆ ˆj ) 3. 4. m s = = 9.. kg m s Thus, p x = 9. kg m s and p y =. kg m s (b) p px p y p y θ = tan = tan (.33) = 37 px = +

### Classical Mechanics Virtual Work & d Alembert s Principle

Classcal Mechancs Vrtual Work & d Alembert s Prncple Dpan Kumar Ghosh UM-DAE Centre for Excellence n Basc Scences Kalna, Mumba 400098 August 15, 2016 1 Constrants Moton of a system of partcles s often

### 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

### Chapter Eight. Review and Summary. Two methods in solid mechanics ---- vectorial methods and energy methods or variational methods

Chapter Eght Energy Method 8. Introducton 8. Stran energy expressons 8.3 Prncpal of statonary potental energy; several degrees of freedom ------ Castglano s frst theorem ---- Examples 8.4 Prncpal of statonary

### 9/19/2013. PHY 113 C General Physics I 11 AM-12:15 PM MWF Olin 101

PHY 3 C General Physcs I AM-:5 PM MF Oln 0 Plan or Lecture 8: Chapter 8 -- Conservaton o energy. Potental and knetc energy or conservatve orces. Energy and non-conservatve orces 3. Power PHY 3 C Fall 03--

### Please 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

### 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

### 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

### Linear 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

### Physics 106a, Caltech 11 October, Lecture 4: Constraints, Virtual Work, etc. Constraints

Physcs 106a, Caltech 11 October, 2018 Lecture 4: Constrants, Vrtual Work, etc. Many, f not all, dynamcal problems we want to solve are constraned: not all of the possble 3 coordnates for M partcles (or

### 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

### 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

### 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

### 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

### Physics 111 Final Exam, Fall 2013, Version A

Physcs 111 Fnal Exam, Fall 013, Verson A Name (Prnt): 4 Dgt ID: Secton: Honors Code Pledge: For ethcal and farness reasons all students are pledged to comply wth the provsons of the NJIT Academc Honor

### 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

### GAUTENG 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

### a) No books or notes are permitted. b) You may use a calculator.

PHYS 050 Sprng 06 Name: Test 3 Aprl 7, 06 INSTRUCTIONS: a) No books or notes are permtted. b) You may use a calculator. c) You must solve all problems begnnng wth the equatons on the Inormaton Sheet provded

### 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)]

### 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