# Physics 207: Lecture 20. Today s Agenda Homework for Monday

Size: px
Start display at page:

Transcription

1 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 Momentum Conservaton Inelastc collsons n one dmenson 1 System of Partcles: Center of Mass For partcles, the components of R are: mx my ( X, Y, Z ) =,, M M mz M For a contnuous sold, we have to do an ntegral. N M = = 1 m y x dm r R rdm rdm = = dm M where dm s an nfntesmal mass element. 2 Page 1

2 System of Partcles: Center of Mass We can use ntuton to fnd the locaton of the center of mass for symmetrc objects that have unform densty: It wll smply be at the geometrcal center! 3 System of Partcles: Center of Mass The center of mass for a combnaton of objects s the average center of mass locaton of the objects: m 2 R N mr = 1 = N m = 1 R 2 -R 1 m 1 R 1 R R 2 y x so f we have two objects: R m1r1 m2r2 = m m 1 m2 = R1 M 2 ( R R ) Page 2

3 Lecture 20, Act 1 Center of Mass The dsk shown below (1) clearly has ts at the center. Suppose the dsk s cut n half and the peces arranged as shown n (2): Where s the of (2) as compared to (1)? (a) hgher (b) lower (c) same X (1) (2) 5 Lecture 20, Act 1 Soluton The of each half-dsk wll be closer to the fat end than to the thn end (thnk of where t would balance). The of the compound object wll be halfway between the s of the two halves. Ths s hgher than the of the dsk X X X X (1) (2) 6 Page 3

4 System of Partcles: Center of Mass The center of mass () of an object s where we can freely pvot that object. pvot Gravty acts on the of an object (show later) If we pvot the object somewhere else, t wll orent tself so that the s drectly below the pvot. Ths fact can be used to fnd the of oddly-shaped objects. mg pvot pvot 7 System of Partcles: Center of Mass Hang the object from several pvots and see where the vertcal lnes through each pvot ntersect! pvot pvot pvot The ntersecton pont must be at the. 8 Page 4

5 Lecture 20, Act 2 Center of Mass An object wth three prongs of equal mass s balanced on a wre (equal angles between prongs). What knd of equlbrum s ths poston? a) stable b) neutral c) unstable 9 Lecture 20, Act 2 Soluton The center of mass of the object s at ts center and s ntally drectly over the wre If the object s pushed slghtly to the left or rght, ts center of mass wll not be above the wre and gravty wll make the object fall off mg mg (front vew) 10 Page 5

6 Lecture 20, Act 2 Soluton Consder also the case n whch the two lower prongs have balls of equal mass attached to them: mg mg In ths case, the center of mass of the object s below the wre When the object s pushed slghtly, gravty provdes a restorng force, creatng a stable equlbrum 11 Velocty and Acceleraton of the Center of Mass If ts partcles are movng, the of a system can also move. Suppose we know the poston r of every partcle n the system as a functon of tme. R 1 N r M m = = 1 N M = = 1 m So: V = dr N r N v dt = 1 M m d dt = 1 M m = 1 = 1 And: A = dv N v N a dt = 1 M m d dt = 1 M m = 1 = 1 The velocty and acceleraton of the s just the weghted average velocty and acceleraton of all the partcles. 12 Page 6

7 Lnear Momentum: Defnton: For a sngle partcle, the momentum p s defned as: (p s a vector snce v s a p = mv vector). So p x = mv x etc. Newton s 2nd Law: F = ma = m dv dt d = ( mv) dt F d p = dt Unts of lnear momentum are kg m/s. 13 Lnear Momentum: For a system of partcles the total momentum P s the vector sum of the ndvdual partcle momenta: N N m = 1 = 1 P = p = v But we just showed that N mv = MV = 1 V 1 = M N m = 1 v So P = MV 14 Page 7

8 Lnear Momentum: So the total momentum of a system of partcles s just the total mass tmes the velocty of the center of mass. P = MV Observe: dp M d V = = MA = ma = F, dt dt net dp We are nterested n so we need to fgure out dt F net, 15 Lnear Momentum: Suppose we have a system of three partcles as shown. Each partcle nteracts wth every other, and n addton there s an external force pushng on partcle 1. F = F, NET 1, EXT = ( F F F ) , EXT ( F F ) ( F F ) (snce the other forces cancel n pars...newton s 3rd Law) All of the nternal forces cancel!! Only the external force matters!! F 13 m 1 F 31 F 12 F 1,EXT m 3 F32 F 23 F 21 m 2 16 Page 8

9 Lnear Momentum: Only the total external force matters! dp dt = F, = F EXT NET, EXT m 3 Whch s the same as: dp F = = MA NET, EXT dt m 1 F 1,EXT m 2 Newton s 2nd law appled to systems! 17 Center of Mass Moton: Recap (really know ths!) We have the followng law for moton: F EXT dp = =MA dt Ths has several nterestng mplcatons: It tells us that the of an extended object behaves lke a smple pont mass under the nfluence of external forces: We can use t to relate F and A lke we are used to dong. It tells us that f F EXT = 0, the total momentum of the system can not change. The total momentum of a system s conserved f there are no external forces actng. 18 Page 9

10 Example: Astronauts & Rope Two astronauts at rest n outer space are connected by a lght rope. They begn to pull towards each other. Where do they meet? M = 1.5m m 19 Example: Astronauts & Rope... They start at rest, so V = 0. V remans zero because there are no external forces. So, the does not move! They wll meet at the. M = 1.5m m L x=0 x=l Fndng the : If we take the astronaut on the left to be at x = 0: M m(l m(l x cm = ( 0 ) ) ) 2 = = M m 25. m 5 L 20 Page 10

11 Lecture 20, Act 3 Center of Mass Moton A man weghs exactly as much as hs 20 foot long canoe. Intally he stands n the center of the motonless canoe, a dstance of 20 feet from shore. Next he walks toward the shore untl he gets to the end of the canoe. What s hs new dstance from the shore. (There no horzontal force on the canoe by the water). 20 ft? ft 20 ft before (a) 10 ft (b) 15 ft (c) 16.7 ft after 21 Lecture 20, Act 3 Soluton Snce the man and the canoe have the same mass, the of the man-canoe system wll be halfway between the of the man and the of the canoe. Intally the of the system s 20 ft from shore. 20 ft X X of system x 22 Page 11

12 Lecture 20, Act 3 Soluton Snce there s no force actng on the canoe n the x-drecton, the locaton of the of the system can t change! Therefore, the man ends up 5 ft to the left of the system, and the center of the canoe ends up 5 ft to the rght. He ends up movng 5 ft toward the shore (15 ft away). 15 ft 20 ft X 10 ft 5 ft X x of system 23 Center of Mass Moton: Revew We have the followng law for moton: F EXT dp = =MA dt Ths has several nterestng mplcatons: It tells us that the of an extended object behaves lke a smple pont mass under the nfluence of external forces: We can use t to relate F and A lke we are used to dong. It tells us that f F EXT = 0, the total momentum of the system does not change. The total momentum of a system s conserved f there are no external forces actng. 24 Page 12

13 Lecture 20, Act 4 Center of Mass Moton Two pucks of equal mass are beng pulled at dfferent ponts wth equal forces. Whch experences the bgger acceleraton? (a) A 1 > A 2 (b) A 1 < A 2 (c) A 1 = A 2 A 1 (1) M T A 2 F (2) M T 25 Lecture 20, Act 4 Soluton We have just shown that MA = F EXT Acceleraton depends only on external force, not on where t s appled! Expect that A 1 and A 2 wll be the same snce F 1 = F 2 = T = F / 2 The answer s (c) A 1 = A 2. So the fnal veloctes should be the same! 26 Page 13

14 Lecture 20, Act 4 Soluton The fnal velocty of the of each puck s the same! Notce, however, that the moton of the partcles n each of the pucks s dfferent (one s spnnng). V Ths one has more knetc energy (rotaton) ω V 27 Momentum Conservaton F EXT dp = dt dp = 0 F EXT = 0 dt The concept of momentum conservaton s one of the most fundamental prncples n physcs. Ths s a component (vector) equaton. We can apply t to any drecton n whch there s no external force appled. You wll see that we often have momentum conservaton even when energy s not conserved. 28 Page 14

15 Elastc vs. Inelastc Collsons A collson s sad to be elastc when knetc energy as well as momentum s conserved before and after the collson. K before = K after Carts colldng wth a sprng n between, bllard balls, etc. v A collson s sad to be nelastc when knetc energy s not conserved before and after the collson, but momentum s conserved. K before K after Car crashes, collsons where objects stck together, etc. 29 Inelastc collson n 1-D: Example 1 A block of mass M s ntally at rest on a frctonless horzontal surface. A bullet of mass m s fred at the block wth a muzzle velocty (speed) v. The bullet lodges n the block, and the block ends up wth a speed V. In terms of m, M, and V : What s the ntal speed of the bullet v? What s the ntal energy of the system? What s the fnal energy of the system? Is knetc energy conserved? x v V before after 30 Page 15

16 Example 1... Consder the bullet & block as a system. After the bullet s shot, there are no external forces actng on the system n the x-drecton. Momentum s conserved n the x drecton! P x, = P x, f mv = (Mm)V v = M m m V v V x ntal fnal 31 v = M m m V Example 1... Now consder the knetc energy of the system before and after: Before: E mv m M m 2 M m B = 1 2 = 1 1 V = ( M m) V 2 2 m 2 m After: EA = 1 ( M 2 mv ) So E A m = M m E B Knetc energy s NOT conserved! (frcton stopped the bullet) However, momentum was conserved, and ths was useful. 32 Page 16

17 Inelastc Collson n 1-D: Example 2 M m V v = 0 ce (no frcton) M m v =? 33 Example 2... Use conservaton of momentum to fnd v after the collson. Before the collson: After the collson: P = MV m( 0 ) Pf = ( M m) v Conservaton of momentum: P = P f MV = ( M m) v M v = V vector equaton (M m ) 34 Page 17

18 v = M m m V Example 2... Now consder the K.E. of the system before and after: Before: E MV M M m 2 M m BUS = 1 2 = 1 1 v = ( M m) v 2 2 M 2 M After: EA = 1 ( M m) v So E A M = M m E B Knetc energy s NOT conserved n an nelastc collson! 35 Lecture 20, Act 5 Momentum Conservaton Two balls of equal mass are thrown horzontally wth the same ntal velocty. They ht dentcal statonary boxes restng on a frctonless horzontal surface. The ball httng box 1 bounces back, whle the ball httng box 2 gets stuck. Whch box ends up movng faster? (a) Box 1 (b) Box 2 (c) same Page 18

19 Lecture 20, Act 5 Momentum Conservaton Snce the total external force n the x-drecton s zero, momentum s conserved along the x-axs. In both cases the ntal momentum s the same (mv of ball). In case 1 the ball has negatve momentum after the collson, hence the box must have more postve momentum f the total s to be conserved. The speed of the box n case 1 s bggest! V 1 V x 37 Lecture 20, Act 5 Momentum Conservaton mv nt = MV 1 -mv fn V 1 = (mv nt mv fn ) / M mv nt = (Mm)V 2 V 2 = mv nt / (Mm) V 1 numerator s bgger and ts denomnator s smaller than that of V 2. V 1 > V 2 x V 1 V Page 19

20 Exploson (nelastc un-collson) Before the exploson: M After the exploson: v 1 v 2 m 1 m 2 39 Exploson... No external forces, so P s conserved. Intally: P = 0 Fnally: P = m 1 v 1 m 2 v 2 = 0 m 1 v 1 = - m 2 v 2 M v 1 v 2 m 1 m 2 40 Page 20

### 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 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 211: Lecture 14. Today s Agenda

Physics 211: Lecture 14 Today s Agenda Systems of Particles Center of mass Linear Momentum Example problems Momentum Conservation Inelastic collisions in one dimension Ballistic pendulum Physics 211: Lecture

### 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

### Page 1. Physics 131: Lecture 14. Today s Agenda. Things that stay the same. Impulse and Momentum Non-constant forces

Physcs 131: Lecture 14 Today s Agenda Imulse and Momentum Non-constant forces Imulse-momentum momentum thm Conservaton of Lnear momentum Eternal/Internal forces Eamles Physcs 201: Lecture 1, Pg 1 Physcs

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Linear Momentum. Center of Mass.

Lecture 16 Chapter 9 Physcs I 11.06.2013 Lnear oentu. Center of ass. Course webste: http://faculty.ul.edu/ndry_danylov/teachng/physcsi Lecture Capture: http://echo360.ul.edu/danylov2013/physcs1fall.htl

### 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

### 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

### 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

### 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

### Physics 207: Lecture 27. Announcements

Physcs 07: ecture 7 Announcements ake-up labs are ths week Fnal hwk assgned ths week, fnal quz next week Revew sesson on Thursday ay 9, :30 4:00pm, Here Today s Agenda Statcs recap Beam & Strngs» What

### 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

### Rotational Dynamics. Physics 1425 Lecture 19. Michael Fowler, UVa

Rotatonal Dynamcs Physcs 1425 Lecture 19 Mchael Fowler, UVa Rotatonal Dynamcs Newton s Frst Law: a rotatng body wll contnue to rotate at constant angular velocty as long as there s no torque actng on t.

### Center of Mass and Linear Momentum

PH 221-2A Fall 2014 Center of Mass and Lnear Momentum Lectures 14-15 Chapter 9 (Hallday/Resnck/Walker, Fundamentals of Physcs 9 th edton) 1 Chapter 9 Center of Mass and Lnear Momentum In ths chapter we

### 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

### 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

### 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

### 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 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

### 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

### 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

### 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.

### 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

### Dynamics of Rotational Motion

Dynamcs of Rotatonal Moton Torque: the rotatonal analogue of force Torque = force x moment arm = Fl moment arm = perpendcular dstance through whch the force acts a.k.a. leer arm l F l F l F l F = Fl =

### 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

### 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

### PHYSICS 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

### 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

### 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

### 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

### 10/9/2003 PHY Lecture 11 1

Announcements 1. Physc Colloquum today --The Physcs and Analyss of Non-nvasve Optcal Imagng. Today s lecture Bref revew of momentum & collsons Example HW problems Introducton to rotatons Defnton of angular

### 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

### 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

### 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)]

### 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

### 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

### 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

### 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

### 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

### Important Dates: Post Test: Dec during recitations. If you have taken the post test, don t come to recitation!

Important Dates: Post Test: Dec. 8 0 durng rectatons. If you have taken the post test, don t come to rectaton! Post Test Make-Up Sessons n ARC 03: Sat Dec. 6, 0 AM noon, and Sun Dec. 7, 8 PM 0 PM. Post

### Chapter 12 Equilibrium & Elasticity

Chapter 12 Equlbrum & Elastcty If there s a net force, an object wll experence a lnear acceleraton. (perod, end of story!) If there s a net torque, an object wll experence an angular acceleraton. (perod,

### 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,

### Ground Rules. PC1221 Fundamentals of Physics I. Linear Momentum, cont. Linear Momentum. Lectures 17 and 18. Linear Momentum and Collisions

PC Fundamentals of Physcs I Lectures 7 and 8 Lnear omentum and Collsons Dr Tay Seng Chuan Ground Rules Swtch off your handphone and pager Swtch off your laptop computer and keep t No talkng whle lecture

### 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 231 Lecture 18: equilibrium & revision

PHYSICS 231 Lecture 18: equlbrum & revson Remco Zegers Walk-n hour: Thursday 11:30-13:30 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

### 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

### 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

### 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

### Rotational and Translational Comparison. Conservation of Angular Momentum. Angular Momentum for a System of Particles

Conservaton o Angular Momentum 8.0 WD Rotatonal and Translatonal Comparson Quantty Momentum Ang Momentum Force Torque Knetc Energy Work Power Rotaton L cm = I cm ω = dl / cm cm K = (/ ) rot P rot θ W =

### Physics 3A: Linear Momentum. Physics 3A: Linear Momentum. Physics 3A: Linear Momentum. Physics 3A: Linear Momentum

Recall that there was ore to oton than just spee A ore coplete escrpton of oton s the concept of lnear oentu: p v (8.) Beng a prouct of a scalar () an a vector (v), oentu s a vector: p v p y v y p z v

### 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

### Physics 231. Topic 8: Rotational Motion. Alex Brown October MSU Physics 231 Fall

Physcs 231 Topc 8: Rotatonal Moton Alex Brown October 21-26 2015 MSU Physcs 231 Fall 2015 1 MSU Physcs 231 Fall 2015 2 MSU Physcs 231 Fall 2015 3 Key Concepts: Rotatonal Moton Rotatonal Kneatcs Equatons

### 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 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

### Physics 240: Worksheet 30 Name:

(1) One mole of an deal monatomc gas doubles ts temperature and doubles ts volume. What s the change n entropy of the gas? () 1 kg of ce at 0 0 C melts to become water at 0 0 C. What s the change n entropy

### 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

### p p +... = p j + p Conservation Laws in Physics q Physical states, process, and state quantities: Physics 201, Lecture 14 Today s Topics

Physcs 0, Lecture 4 Conseraton Laws n Physcs q Physcal states, process, and state quanttes: Today s Topcs Partcle Syste n state Process Partcle Syste n state q Lnear Moentu And Collsons (Chapter 9.-9.4)

### 10/23/2003 PHY Lecture 14R 1

Announcements. Remember -- Tuesday, Oct. 8 th, 9:30 AM Second exam (coverng Chapters 9-4 of HRW) Brng the followng: a) equaton sheet b) Calculator c) Pencl d) Clear head e) Note: If you have kept up wth

### 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

### Page 1. SPH4U: Lecture 7. New Topic: Friction. Today s Agenda. Surface Friction... Surface Friction...

SPH4U: Lecture 7 Today s Agenda rcton What s t? Systeatc catagores of forces How do we characterze t? Model of frcton Statc & Knetc frcton (knetc = dynac n soe languages) Soe probles nvolvng frcton ew

### 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

### Physic 231 Lecture 14

Physc 3 Lecture 4 Man ponts o last lecture: Ipulses: orces that last only a short te Moentu p Ipulse-Moentu theore F t p ( ) Ipulse-Moentu theore ptot, p, p, p, p, ptot, Moentu and external orces F p ext

### 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

### 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

### 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

### 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,

### PES 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

### Conservation Laws (Collisions) Phys101 Lab - 04

Conservaton Laws (Collsons) Phys101 Lab - 04 1.Objectves The objectves o ths experment are to expermentally test the valdty o the laws o conservaton o momentum and knetc energy n elastc collsons. 2. Theory

### 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 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

### 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

### 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 231. Topic 8: Rotational Motion. Alex Brown October MSU Physics 231 Fall

Physcs 231 Topc 8: Rotatonal Moton Alex Brown October 21-26 2015 MSU Physcs 231 Fall 2015 1 MSU Physcs 231 Fall 2015 2 MSU Physcs 231 Fall 2015 3 Key Concepts: Rotatonal Moton Rotatonal Kneatcs Equatons

### 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

### The 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

### 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

### PY2101 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

### Modeling of Dynamic Systems

Modelng of Dynamc Systems Ref: Control System Engneerng Norman Nse : Chapters & 3 Chapter objectves : Revew the Laplace transform Learn how to fnd a mathematcal model, called a transfer functon Learn how

### total If no external forces act, the total linear momentum of the system is conserved. This occurs in collisions and explosions.

Lesson 0: Collsons, Rotatonal netc Energy, Torque, Center o Graty (Sectons 7.8 Last te we used ewton s second law to deelop the pulse-oentu theore. In words, the theore states that the change n lnear oentu

### NEWTON S LAWS. These laws only apply when viewed from an inertial coordinate system (unaccelerated system).

EWTO S LAWS Consder two partcles. 1 1. If 1 0 then 0 wth p 1 m1v. 1 1 2. 1.. 3. 11 These laws only apply when vewed from an nertal coordnate system (unaccelerated system). consder a collecton of partcles

### From Newton s 2 nd Law: v v. The time rate of change of the linear momentum of a particle is equal to the net force acting on the particle.

From Newton s 2 nd Law: F ma d dm ( ) m dt dt F d dt The tme rate of change of the lnear momentum of a artcle s equal to the net force actng on the artcle. Conseraton of Momentum +x The toy rocket n dee

### 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

### Physics 114 Exam 3 Spring Name:

Physcs 114 Exam 3 Sprng 015 Name: For gradng purposes (do not wrte here): Queston 1. 1... 3. 3. Problem 4. Answer each of the followng questons. Ponts for each queston are ndcated n red. Unless otherwse

### Angular momentum. Instructor: Dr. Hoi Lam TAM ( 譚海嵐 )

Angular momentum Instructor: Dr. Ho Lam TAM ( 譚海嵐 ) Physcs Enhancement Programme or Gted Students The Hong Kong Academy or Gted Educaton and Department o Physcs, HKBU Department o Physcs Hong Kong Baptst

### Newton s Laws of Motion

Chapter 4 Newton s Laws of Moton 4.1 Forces and Interactons Fundamental forces. There are four types of fundamental forces: electromagnetc, weak, strong and gravtatonal. The frst two had been successfully

### Physics 114 Exam 2 Fall 2014 Solutions. Name:

Physcs 114 Exam Fall 014 Name: For gradng purposes (do not wrte here): Queston 1. 1... 3. 3. Problem Answer each of the followng questons. Ponts for each queston are ndcated n red. Unless otherwse ndcated,

### 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

### 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 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

### 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