Spring 2002 Lecture #13


 Bruce Gibbs
 4 years ago
 Views:
Transcription
1 4450 Sprng 00 ecture # Dr. Jaehoon Yu. Rotatonal Energy. Computaton of oments of nerta. Parallelas Theorem 4. Torque & Angular Acceleraton 5. Work, Power, & Energy of Rotatonal otons Remember the mdterm eam on Wednesday, ar.. Wll cover Chapters 0. Today s Homework Assgnment s the Homework #5!!!
2 y r v θ m O Snce a rgd body s a collecton of masslets, the total knetc energy of the rgd object s By defnng a new quantty called, oment of nerta,, as Rotatonal Energy What do you thnk the knetc energy of a rgd object that s undergong a crcular moton s? Knetc energy of a masslet, m, movng at a tangental speed, v, s What are the dmenson and unt of oment of nerta? What do you thnk the moment of nerta s? What smlarty do you see between rotatonal and lnear knetc energes? mr K K m v mr ω ω R K mr The above epresson s smplfed as kg m [ ] easure of resstance of an object to changes n ts rotatonal moton. mr ω K R ω ass and speed n lnear knetc energy are replaced by moment of nerta and angular speed. ar. 6, Sprng 00 Dr. J. Yu, ecture #
3 Eample 0.4 n a system conssts of four small spheres as shown n the fgure, assumng the rad are neglgble and the rods connectng the partcles are massless, compute the moment of nerta and the rotatonal knetc energy when the system rotates about the yas at ω. l y O m m b b Thus, the rotatonal knetc energy s l mr ar. 6, Sprng 00 Dr. J. Yu, ecture # Why are some 0s? Snce the rotaton s about y as, the moment of nerta about y as, y, s l + l + m 0 + m 0 l Ths s because the rotaton s done about y as, and the rad of the spheres are neglgble. ( l ) ω l ω K R ω Fnd the moment of nerta and rotatonal knetc energy when the system rotates on the y plane about the zas that goes through the orgn O. m r l + l + mb + mb ( l mb ) ( ) ( ) + K R ω l + mb ω l + mb ω
4 Calculaton of oments of nerta oments of nerta for large objects can be computed, f we assume the object conssts of small volume elements wth mass, m. The moment of nerta for the large rgd object s lm r m t s sometmes easer to compute moments of nerta n terms How can we do ths? of volume of the elements rather than ther mass Usng the volume densty, ρ, replace dm The moments of ρ ; dm ρdv ρ dm n the above equaton wth dv. dv nerta becomes Eample 0.5: Fnd the moment of nerta of a unform hoop of mass and radus R about an as perpendcular to the plane of the hoop and passng through ts center. y dm The moment of nerta s m r dm R dm 0 R r dm r dv O R What do you notce from ths result? The moment of nerta for ths object s the same as that of a pont of mass at the dstance R. ar. 6, Sprng 00 Dr. J. Yu, ecture # 4
5 Eample 0.6 Calculate the moment of nerta of a unform rgd rod of length and mass about an as perpendcular to the rod and passng through ts center of mass. y d What s the moment of nerta when the rotatonal as s at one end of the rod. Wll ths be the same as the above. Why or why not? The lne densty of the rod s so the masslet s The moment of nerta s ar. 6, Sprng 00 Dr. J. Yu, ecture # r dm / / r dm 0 λ dm λd d d [( ) 0] ( ) 4 0 d / / Snce the moment of nerta s resstance to moton, t makes perfect sense for t to be harder to move when t s rotatng about the as at one end. 5
6 y Parallel As Theorem oments of nerta for hghly symmetrc object s relatvely easy f the rotatonal as s the same as the as of symmetry. However f the as of rotaton does not concde wth as of symmetry, the calculaton can stll be done n smple manner usng parallelas theorem. + C D y y y C r C D What does ths theorem tell you? (, y) C ( C, y C ) ar. 6, Sprng 00 Dr. J. Yu, ecture # oment of nerta s defned r dm ( + y ) dm () Snce and y are C + ' ; y yc + One can substtute and y n Eq. to obtan [( + ) + ( y + y ) ] ' ' dm ( + y ) dm+ ' dm+ y y' dm+ ( ' + y' )dm C C C C C Snce the and y are the dstance from C, by defnton Therefore, the parallelas theorem C y' ' dm y ' dm + C 0 0 D oment of nerta of any object about any arbtrary as are the same as the sum of moment of nerta for a rotaton about the C and that of 6 the C about the rotaton as.
7 Eample 0.8 Calculate the moment of nerta of a unform rgd rod of length and mass about an as that goes through one end of the rod, usng parallelas theorem. y C d The moment of nerta about the C The lne densty of the rod s so the masslet s C r dm / / λ dm λd d 4 d / / Usng the parallel as theorem C + D The result s the same as usng the defnton of moment of nerta. Parallelas theorem s useful to compute moment of nerta of a rotaton of a rgd object wth complcated shape about an arbtrary as ar. 6, Sprng 00 Dr. J. Yu, ecture # 7
8 P Torque Torque s the tendency of a force to rotate an object about some as. Torque, t, s a vector quantty. d r d oment arm F F φ ne of Acton agntude of torque s defned as the product of the force eerted on the object to rotate t and the moment arm. ar. 6, Sprng 00 Dr. J. Yu, ecture # Consder an object pvotng about the pont P by the force F beng eerted at a dstance r. The lne that etends out of the tal of the force vector s called the lne of acton. The perpendcular dstance from the pvotng pont P to the lne of acton s called oment arm. When there are more than one force beng eerted on certan ponts of the object, one can sum up the torque generated by each force vectorally. The conventon for sgn of the torque s postve f rotaton s n counterclockwse and negatve f clockwse. τ rf snφ Fd τ τ + τ Fd F d 8
9 Eample 0.9 A one pece cylnder s shaped as n the fgure wth core secton protrudng from the larger drum. The cylnder s free to rotate around the central as shown n the pcture. A rope wrapped around the drum whose radus s R eerts force F to the rght on the cylnder, and another force eerts F on the core whose radus s R downward on the cylnder. A) What s the net torque actng on the cylnder about the rotaton as? R F The torque due to F τ R F and due to F τ R F R So the total torque actng on the system by the forces s τ τ + τ R F + RF F Suppose F 5.0 N, R.0 m, F 5.0 N, and R 0.50 m. What s the net torque about the rotaton as and whch way does the cylnder rotate from the rest? Usng the above result τ R F + R F N m The cylnder rotates n counterclockwse. ar. 6, Sprng 00 Dr. J. Yu, ecture # 9
10 What does ths mean? r Torque & Angular Acceleraton F t F r m et s consder a pont object wth mass m rotatng n a crcle. What forces do you see n ths moton? The tangental force F t and radal force F r The tangental force F t s F ma The torque due to tangental force F t s What do you see from the above relatonshp? What law do you see from ths relatonshp? ar. 6, Sprng 00 Dr. J. Yu, ecture # τ t t mrα Ft r mar t τ α mr α Torque actng on a partcle s proportonal to the angular acceleraton. How about a rgd object? The eternal tangental force df t s df t The torque due to tangental force F dm t s τ α dm α r r The total torque s O What s the contrbuton due to radal force and why? Analogs to Newton s nd law of moton n rotaton. dft dmat dmrα dτ df r ( r dm)α Contrbuton from radal force s 0, because ts lne of acton passes through the pvotng pont, makng the moment arm 0. 0 t
11 Eample 0.0 A unform rod of length and mass s attached at one end to a frctonless pvot and s free to rotate about the pvot n the vertcal plane. The rod s released from rest n the horzontal poston what s the ntal angular acceleraton of the rod and the ntal lnear acceleraton of ts rght end? / g Snce the moment of nerta of the rod when t rotates about one end We obtan α g g The only force generatng torque s the gravtatonal force g g τ Fd F g α ar. 6, Sprng 00 Dr. J. Yu, ecture # r dm d 0 λ 0 Usng the relatonshp between tangental and angular acceleraton a t α g What does ths mean? The tp of the rod falls faster than an object undergong a free fall. 0
12 Work, Power, and Energy n Rotaton O dθr ds F φ et s consder a moton of a rgd body wth a sngle eternal force F eerted on the pont P, movng the object by ds. The work done by the force F as the object rotates through nfntesmal dstance dsrdθ n a tme s dw F d s ( F snφ ) rdθ What s Fsnφ? The tangental component of force F. What s the work done by radal component Fcosφ? Snce the magntude of torque s rfsnφ, The rate of work, or power becomes The rotatonal work done by an eternal force equals the change n rotatonal energy. The work put n by the eternal force then W ar. 6, Sprng 00 Dr. J. Yu, ecture # Zero, because t s perpendcular to the dsplacement. dw τdθ P dw dw τ τdθ α τdθ θι θ ι τω dω ωdω τdθ ω ω ι f How was the power defned n lnear moton? ωdω dω dθ ω dθ ω f
13 Smlarty Between near and Rotatonal otons All physcal quanttes n lnear and rotatonal motons show strkng smlarty. Smlar Quantty ass ength of moton Speed Acceleraton Force Work Power omentum Knetc Energy ass Dstance Force Work Knetc near v a ar. 6, Sprng 00 Dr. J. Yu, ecture # W P dr dv F f Fd ma oment of nerta Angle Torque Work Rotatonal Rotatonal r dm ω α d θ d ω τ F v P τω p m v mv (Radan) α K K R ω θ W ω θ θ f τdθ
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.!
More informationCHAPTER 10 ROTATIONAL MOTION
CHAPTER 0 ROTATONAL MOTON 0. ANGULAR VELOCTY Consder argd body rotates about a fxed axs through pont O n xy plane as shown. Any partcle at pont P n ths rgd body rotates n a crcle of radus r about O. The
More informationWeek 9 Chapter 10 Section 15
Week 9 Chapter 10 Secton 15 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,
More informationPart C Dynamics and Statics of Rigid Body. Chapter 5 Rotation of a Rigid Body About a Fixed Axis
Part C Dynamcs and Statcs of Rgd Body Chapter 5 Rotaton of a Rgd Body About a Fxed Axs 5.. Rotatonal Varables 5.. Rotaton wth Constant Angular Acceleraton 5.3. Knetc Energy of Rotaton, Rotatonal Inerta
More informationSo far: simple (planar) geometries
Physcs 06 ecture 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap. to 3 Rotatonal quanttes as vectors Cross product Torque epressed as a vector Angular momentum defned Angular momentum as a vector
More informationStudy Guide For Exam Two
Study Gude For Exam Two Physcs 2210 Albretsen Updated: 08/02/2018 All Other Prevous Study Gudes Modules 0106 Module 07 Work Work done by a constant force F over a dstance s : Work done by varyng force
More informationRotational 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.
More informationPhysics 111: Mechanics Lecture 11
Physcs 111: Mechancs Lecture 11 Bn Chen NJIT Physcs Department Textbook Chapter 10: Dynamcs of Rotatonal Moton q 10.1 Torque q 10. Torque and Angular Acceleraton for a Rgd Body q 10.3 RgdBody Rotaton
More informationPHYS 1443 Section 002
PHYS 443 Secton 00 Lecture #6 Wednesday, Nov. 5, 008 Dr. Jae Yu Collsons Elastc and Inelastc Collsons Two Dmensonal Collsons Center o ass Fundamentals o Rotatonal otons Wednesday, Nov. 5, 008 PHYS PHYS
More informationImportant 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 MakeUp Sessons n ARC 03: Sat Dec. 6, 0 AM noon, and Sun Dec. 7, 8 PM 0 PM. Post
More informationChapter 11 Angular Momentum
Chapter 11 Angular Momentum Analyss Model: Nonsolated System (Angular Momentum) Angular Momentum of a Rotatng Rgd Object Analyss Model: Isolated System (Angular Momentum) Angular Momentum of a Partcle
More informationChapter 11 Torque and Angular Momentum
Chapter Torque and Angular Momentum I. Torque II. Angular momentum  Defnton III. Newton s second law n angular form IV. Angular momentum  System of partcles  Rgd body  Conservaton I. Torque  Vector
More informationτ 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
More information10/9/2003 PHY Lecture 11 1
Announcements 1. Physc Colloquum today The Physcs and Analyss of Nonnvasve Optcal Imagng. Today s lecture Bref revew of momentum & collsons Example HW problems Introducton to rotatons Defnton of angular
More informationWeek 11: Chapter 11. The Vector Product. The Vector Product Defined. The Vector Product and Torque. More About the Vector Product
The Vector Product Week 11: Chapter 11 Angular Momentum There are nstances where the product of two vectors s another vector Earler we saw where the product of two vectors was a scalar Ths was called the
More informationChapter 11: Angular Momentum
Chapter 11: ngular Momentum Statc Equlbrum In Chap. 4 we studed the equlbrum of pontobjects (mass m) wth the applcaton of Newton s aws F 0 F x y, 0 Therefore, no lnear (translatonal) acceleraton, a0 For
More informationPhysics 207: Lecture 20. Today s Agenda Homework for Monday
Physcs 207: Lecture 20 Today s Agenda Homework for Monday Recap: Systems of Partcles Center of mass Velocty and acceleraton of the center of mass Dynamcs of the center of mass Lnear Momentum Example problems
More informationAngular Momentum and Fixed Axis Rotation. 8.01t Nov 10, 2004
Angular Momentum and Fxed Axs Rotaton 8.01t Nov 10, 2004 Dynamcs: Translatonal and Rotatonal Moton Translatonal Dynamcs Total Force Torque Angular Momentum about Dynamcs of Rotaton F ext Momentum of a
More informationMoments 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
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1s tme nterval. The velocty of the partcle
More informationChapter 10 Rotational motion
Prof. Dr. I. Nasser Chapter0_I November 6, 07 Important Terms Chapter 0 Rotatonal moton Angular Dsplacement s, r n radans where s s the length of arc and r s the radus. Angular Velocty The rate at whch
More information10/23/2003 PHY Lecture 14R 1
Announcements. Remember  Tuesday, Oct. 8 th, 9:30 AM Second exam (coverng Chapters 94 of HRW) Brng the followng: a) equaton sheet b) Calculator c) Pencl d) Clear head e) Note: If you have kept up wth
More informationChapter 9. The Dot Product (Scalar Product) The Dot Product use (Scalar Product) The Dot Product (Scalar Product) The Cross Product.
The Dot Product (Scalar Product) Chapter 9 Statcs and Torque The dot product of two vectors can be constructed by takng the component of one vector n the drecton of the other and multplyng t tmes the magntude
More informationLinear Momentum. Center of Mass.
Lecture 6 Chapter 9 Physcs I 03.3.04 Lnear omentum. Center of ass. Course webste: http://faculty.uml.edu/ndry_danylov/teachng/physcsi Lecture Capture: http://echo360.uml.edu/danylov03/physcssprng.html
More informationPhysics 181. Particle Systems
Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system
More informationPHYSICS 231 Review problems for midterm 2
PHYSICS 31 Revew problems for mdterm Topc 5: Energy and Work and Power Topc 6: Momentum and Collsons Topc 7: Oscllatons (sprng and pendulum) Topc 8: Rotatonal Moton The nd exam wll be Wednesday October
More informationtotal 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 pulseoentu theore. In words, the theore states that the change n lnear oentu
More informationMEASUREMENT OF MOMENT OF INERTIA
1. measurement MESUREMENT OF MOMENT OF INERTI The am of ths measurement s to determne the moment of nerta of the rotor of an electrc motor. 1. General relatons Rotatng moton and moment of nerta Let us
More informationFirst Law: A body at rest remains at rest, a body in motion continues to move at constant velocity, unless acted upon by an external force.
Secton 1. Dynamcs (Newton s Laws of Moton) Two approaches: 1) Gven all the forces actng on a body, predct the subsequent (changes n) moton. 2) Gven the (changes n) moton of a body, nfer what forces act
More informationWeek 6, Chapter 7 Sect 15
Week 6, Chapter 7 Sect 15 Work and Knetc Energy Lecture Quz The frctonal force of the floor on a large sutcase s least when the sutcase s A.pushed by a force parallel to the floor. B.dragged by a force
More informationSpinrotation coupling of the angularly accelerated rigid body
Spnrotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 Emal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s
More informationENGN 40 Dynamics and Vibrations Homework # 7 Due: Friday, April 15
NGN 40 ynamcs and Vbratons Homework # 7 ue: Frday, Aprl 15 1. Consder a concal hostng drum used n the mnng ndustry to host a mass up/down. A cable of dameter d has the mass connected at one end and s wound/unwound
More informationSCHOOL OF COMPUTING, ENGINEERING AND MATHEMATICS SEMESTER 2 EXAMINATIONS 2011/2012 DYNAMICS ME247 DR. N.D.D. MICHÉ
s SCHOOL OF COMPUTING, ENGINEERING ND MTHEMTICS SEMESTER EXMINTIONS 011/01 DYNMICS ME47 DR. N.D.D. MICHÉ Tme allowed: THREE hours nswer: ny FOUR from SIX questons Each queston carres 5 marks Ths s a CLOSEDBOOK
More informationA Tale of Friction Basic Rollercoaster Physics. Fahrenheit Rollercoaster, Hershey, PA max height = 121 ft max speed = 58 mph
A Tale o Frcton Basc Rollercoaster Physcs Fahrenhet Rollercoaster, Hershey, PA max heght = 11 t max speed = 58 mph PLAY PLAY PLAY PLAY Rotatonal Movement Knematcs Smlar to how lnear velocty s dened, angular
More informationChapter 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 nonconservatve forces Mechancal Energy Conservaton of Mechancal
More informationPhysics 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
More informationWork is the change in energy of a system (neglecting heat transfer). To examine what could
Work Work s the change n energy o a system (neglectng heat transer). To eamne what could cause work, let s look at the dmensons o energy: L ML E M L F L so T T dmensonally energy s equal to a orce tmes
More informationDynamics 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 =
More informationPES 1120 Spring 2014, Spendier Lecture 6/Page 1
PES 110 Sprng 014, Spender Lecture 6/Page 1 Lecture today: Chapter 1) Electrc feld due to charge dstrbutons > charged rod > charged rng We ntroduced the electrc feld, E. I defned t as an nvsble aura
More informationPhysics 2A Chapters 6  Work & Energy Fall 2017
Physcs A Chapters 6  Work & Energy Fall 017 These notes are eght pages. A quck summary: The workenergy theorem s a combnaton o Chap and Chap 4 equatons. Work s dened as the product o the orce actng on
More informationPhysics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding.
Physcs 53 Rotatonal Moton 3 Sr, I have found you an argument, but I am not oblged to fnd you an understandng. Samuel Johnson Angular momentum Wth respect to rotatonal moton of a body, moment of nerta plays
More informationChapter 3. r r. Position, Velocity, and Acceleration Revisited
Chapter 3 Poston, Velocty, and Acceleraton Revsted The poston vector of a partcle s a vector drawn from the orgn to the locaton of the partcle. In two dmensons: r = x ˆ+ yj ˆ (1) The dsplacement vector
More informationPHYS 1441 Section 001 Lecture #15 Wednesday, July 8, 2015
PHYS 1441 Secton 001 Lecture #15 Wednesday, July 8, 2015 Concept of the Center of Mass Center of Mass & Center of Gravty Fundamentals of the Rotatonal Moton Rotatonal Knematcs Equatons of Rotatonal Knematcs
More informationEN40: 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?.
More informationPhysics 207: Lecture 27. Announcements
Physcs 07: ecture 7 Announcements akeup 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
More informationˆ (0.10 m) E ( N m /C ) 36 ˆj ( j C m)
7.. = = 3 = 4 = 5. The electrc feld s constant everywhere between the plates. Ths s ndcated by the electrc feld vectors, whch are all the same length and n the same drecton. 7.5. Model: The dstances to
More informationThe classical spinrotation coupling
LOUAI H. ELZEIN 2018 All Rghts Reserved The classcal spnrotaton couplng Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 louaelzen@gmal.com Abstract Ths paper s prepared to show that a rgd
More information10/24/2013. PHY 113 C General Physics I 11 AM 12:15 PM TR Olin 101. Plan for Lecture 17: Review of Chapters 913, 1516
0/4/03 PHY 3 C General Physcs I AM :5 PM T Oln 0 Plan or Lecture 7: evew o Chapters 93, 56. Comment on exam and advce or preparaton. evew 3. Example problems 0/4/03 PHY 3 C Fall 03  Lecture 7 0/4/03
More informationPhysics 207 Lecture 6
Physcs 207 Lecture 6 Agenda: Physcs 207, Lecture 6, Sept. 25 Chapter 4 Frames of reference Chapter 5 ewton s Law Mass Inerta s (contact and noncontact) Frcton (a external force that opposes moton) Free
More informationPHYS 1443 Section 004 Lecture #12 Thursday, Oct. 2, 2014
PHYS 1443 Secton 004 Lecture #1 Thursday, Oct., 014 WorkKnetc Energy Theorem Work under rcton Potental Energy and the Conservatve Force Gravtatonal Potental Energy Elastc Potental Energy Conservaton o
More informationWeek 8: Chapter 9. Linear Momentum. Newton Law and Momentum. Linear Momentum, cont. Conservation of Linear Momentum. Conservation of Momentum, 2
Lnear omentum Week 8: Chapter 9 Lnear omentum and Collsons The lnear momentum of a partcle, or an object that can be modeled as a partcle, of mass m movng wth a velocty v s defned to be the product of
More informationConservation of Angular Momentum = "Spin"
Page 1 of 6 Conservaton of Angular Momentum = "Spn" We can assgn a drecton to the angular velocty: drecton of = drecton of axs + rght hand rule (wth rght hand, curl fngers n drecton of rotaton, thumb ponts
More informationPHYS 1441 Section 002 Lecture #16
PHYS 1441 Secton 00 Lecture #16 Monday, Mar. 4, 008 Potental Energy Conservatve and Nonconservatve Forces Conservaton o Mechancal Energy Power Today s homework s homework #8, due 9pm, Monday, Mar. 31!!
More informationPHYS 705: Classical Mechanics. Newtonian Mechanics
1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]
More informationChapter 20 Rigid Body: Translation and Rotational Motion Kinematics for Fixed Axis Rotation
Chapter 20 Rgd Body: Translaton and Rotatonal Moton Knematcs for Fxed Axs Rotaton 20.1 Introducton... 1 20.2 Constraned Moton: Translaton and Rotaton... 1 20.2.1 Rollng wthout slppng... 5 Example 20.1
More informationFour Bar Linkages in Two Dimensions. A link has fixed length and is joined to other links and also possibly to a fixed point.
Four bar lnkages 1 Four Bar Lnkages n Two Dmensons lnk has fed length and s oned to other lnks and also possbly to a fed pont. The relatve velocty of end B wth regard to s gven by V B = ω r y v B B = +y
More informationCelestial Mechanics. Basic Orbits. Why circles? Tycho Brahe. PHY celestialmechanics  J. Hedberg
PHY 454  celestalmechancs  J. Hedberg  207 Celestal Mechancs. Basc Orbts. Why crcles? 2. Tycho Brahe 3. Kepler 4. 3 laws of orbtng bodes 2. Newtonan Mechancs 3. Newton's Laws. Law of Gravtaton 2. The
More informationPhysics 141. Lecture 14. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 14, Page 1
Physcs 141. Lecture 14. Frank L. H. Wolfs Department of Physcs and Astronomy, Unversty of Rochester, Lecture 14, Page 1 Physcs 141. Lecture 14. Course Informaton: Lab report # 3. Exam # 2. MultPartcle
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationPhysics 231. Topic 8: Rotational Motion. Alex Brown October MSU Physics 231 Fall
Physcs 231 Topc 8: Rotatonal Moton Alex Brown October 2126 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
More informationMechanics 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
More informationRigid body simulation
Rgd bod smulaton Rgd bod smulaton Once we consder an object wth spacal etent, partcle sstem smulaton s no longer suffcent Problems Problems Unconstraned sstem rotatonal moton torques and angular momentum
More informationPhysics 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
More informationPHYS 1443 Section 002 Lecture #20
PHYS 1443 Secton 002 Lecture #20 Dr. Jae Condtons for Equlbru & Mechancal Equlbru How to Solve Equlbru Probles? A ew Exaples of Mechancal Equlbru Elastc Propertes of Solds Densty and Specfc Gravty lud
More informationPhysics 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
More informationChapter 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,
More informationProblem While being compressed, A) What is the work done on it by gravity? B) What is the work done on it by the spring force?
Problem 0750 A 0.25 kg block s dropped on a relaed sprng that has a sprng constant o k 250.0 N/m (2.5 N/cm). The block becomes attached to the sprng and compresses t 0.12 m beore momentarl stoppng. Whle
More informationSYSTEMS OF PARTICLES AND ROTATIONAL MOTION
CHAPTER SEVEN SYSTES OF PARTICLES AND ROTATIONAL OTION 7.1 Introducton 7.2 Centre of mass 7.3 oton of centre of mass 7.4 Lnear momentum of a system of partcles 7.5 Vector product of two vectors 7.6 Angular
More informationPhysics 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
More informationELASTIC 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
More informationChapter 07: Kinetic Energy and Work
Chapter 07: Knetc Energy and Work Conservaton o Energy s one o Nature s undamental laws that s not volated. Energy can take on derent orms n a gven system. Ths chapter we wll dscuss work and knetc energy.
More information. You need to do this for each force. Let s suppose that there are N forces, with components ( N) ( N) ( N) = i j k
EN3: Introducton to Engneerng and Statcs Dvson of Engneerng Brown Unversty 3. Resultant of systems of forces Machnes and structures are usually subected to lots of forces. When we analyze force systems
More informationGravitational Acceleration: A case of constant acceleration (approx. 2 hr.) (6/7/11)
Gravtatonal Acceleraton: A case of constant acceleraton (approx. hr.) (6/7/11) Introducton The gravtatonal force s one of the fundamental forces of nature. Under the nfluence of ths force all objects havng
More informationSlide. King Saud University College of Science Physics & Astronomy Dept. PHYS 103 (GENERAL PHYSICS) CHAPTER 5: MOTION IN 1D (PART 2) LECTURE NO.
Slde Kng Saud Unersty College of Scence Physcs & Astronomy Dept. PHYS 103 (GENERAL PHYSICS) CHAPTER 5: MOTION IN 1D (PART ) LECTURE NO. 6 THIS PRESENTATION HAS BEEN PREPARED BY: DR. NASSR S. ALZAYED Lecture
More informationPhysics 207 Lecture 13. Lecture 13
Physcs 07 Lecture 3 Goals: Lecture 3 Chapter 0 Understand the relatonshp between moton and energy Defne Potental Energy n a Hooke s Law sprng Develop and explot conservaton of energy prncple n problem
More informationχ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body
Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown
More informationEN40: Dynamics and Vibrations. Homework 4: Work, Energy and Linear Momentum Due Friday March 1 st
EN40: Dynamcs and bratons Homework 4: Work, Energy and Lnear Momentum Due Frday March 1 st School of Engneerng Brown Unversty 1. The fgure (from ths publcaton) shows the energy per unt area requred to
More informationPHYS 1441 Section 002 Lecture #15
PHYS 1441 Secton 00 Lecture #15 Monday, March 18, 013 Work wth rcton Potental Energy Gravtatonal Potental Energy Elastc Potental Energy Mechancal Energy Conservaton Announcements Mdterm comprehensve exam
More informationModeling 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
More informationA particle in a state of uniform motion remain in that state of motion unless acted upon by external force.
The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,
More informationNEWTON 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
More informationRotational 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 =
More informationPhysics 231. Topic 8: Rotational Motion. Alex Brown October MSU Physics 231 Fall
Physcs 231 Topc 8: Rotatonal Moton Alex Brown October 2126 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
More informationPhysics 2A Chapter 3 HW Solutions
Phscs A Chapter 3 HW Solutons Chapter 3 Conceptual Queston: 4, 6, 8, Problems: 5,, 8, 7, 3, 44, 46, 69, 70, 73 Q3.4. Reason: (a) C = A+ B onl A and B are n the same drecton. Sze does not matter. (b) C
More informationLecture Torsion Properties for Line Segments and Computational Scheme for Piecewise Straight Section Calculations
Lecture  003 Torson Propertes for Lne Segments and Computatonal Scheme for Pecewse Straght Secton Calculatons ths conssts of four parts (and how we wll treat each) A  dervaton of geometrc algorthms for
More information11. Dynamics in Rotating Frames of Reference
Unversty of Rhode Island DgtalCommons@URI Classcal Dynamcs Physcs Course Materals 2015 11. Dynamcs n Rotatng Frames of Reference Gerhard Müller Unversty of Rhode Island, gmuller@ur.edu Creatve Commons
More informationSpring Force and Power
Lecture 13 Chapter 9 Sprng Force and Power Yeah, energy s better than orces. What s net? Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi IN THIS CHAPTER, you wll learn how to solve problems
More informationChapter 3 and Chapter 4
Chapter 3 and Chapter 4 Chapter 3 Energy 3. Introducton:Work Work W s energy transerred to or rom an object by means o a orce actng on the object. Energy transerred to the object s postve work, and energy
More informationI certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.
ME 270 Fall 2012 Fnal Exam Please revew the followng statement: I certfy that I have not gven unauthorzed ad nor have I receved ad n the completon of ths exam. Sgnature: INSTRUCTIONS Begn each problem
More information1 Hz = one cycle per second
Rotatonal Moton Mchael Fowler, UVa Physcs, 14E Sprng 009 Mar 5 Prelmnares: Unts for Angular Velocty The tachometer on your car dashboard tells you your car engne s angular speed n rpm, revolutons per mnute,
More informationElectricity and Magnetism  Physics 121 Lecture 10  Sources of Magnetic Fields (Currents) Y&F Chapter 28, Sec. 17
Electrcty and Magnetsm  Physcs 11 Lecture 10  Sources of Magnetc Felds (Currents) Y&F Chapter 8, Sec. 17 Magnetc felds are due to currents The BotSavart Law Calculatng feld at the centers of current
More informationPhysics for Scientists and Engineers. Chapter 9 Impulse and Momentum
Physcs or Scentsts and Engneers Chapter 9 Impulse and Momentum Sprng, 008 Ho Jung Pak Lnear Momentum Lnear momentum o an object o mass m movng wth a velocty v s dened to be p mv Momentum and lnear momentum
More informationSUMMARY Phys 2113 (General Physics I) Compiled by Prof. Erickson. v = r t. v = lim t 0. p = mv. a = v. a = lim
SUMMARY Phys 2113 (General Physcs I) Compled by Prof. Erckson Poston Vector (m): r = xˆx + yŷ + zẑ Average Velocty (m/s): v = r Instantaneous Velocty (m/s): v = lm 0 r = ṙ Lnear Momentum (kg m/s): p =
More informationCENTROID (AĞIRLIK MERKEZİ )
CENTOD (ĞLK MEKEZİ ) centrod s a geometrcal concept arsng from parallel forces. Tus, onl parallel forces possess a centrod. Centrod s tougt of as te pont were te wole wegt of a pscal od or sstem of partcles
More informationPhysics 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,
More informationFor a 1weight experiment do Part 1. For a 2weight experiment do Part 1 and Part 2
Page of 6 THE GYROSCOPE The setup s not connected to a computer. You cannot get measured values drectly from the computer or enter them nto the lab PC. Make notes durng the sesson to use them later for
More informationPhysics 607 Exam 1. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2
Physcs 607 Exam 1 Please be wellorganzed, and show all sgnfcant steps clearly n all problems. You are graded on your wor, so please do not just wrte down answers wth no explanaton! Do all your wor on
More informationPHYS 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
More informationCONDUCTORS AND INSULATORS
CONDUCTORS AND INSULATORS We defne a conductor as a materal n whch charges are free to move over macroscopc dstances.e., they can leave ther nucle and move around the materal. An nsulator s anythng else.
More informationCHAPTER 8 Potential Energy and Conservation of Energy
CHAPTER 8 Potental Energy and Conservaton o Energy One orm o energy can be converted nto another orm o energy. Conservatve and nonconservatve orces Physcs 1 Knetc energy: Potental energy: Energy assocated
More information