Week 9 Chapter 10 Section 15


 Mildred McGee
 3 years ago
 Views:
Transcription
1 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, but the rgd object model s very useful n many stuatons where the deformaton s neglgble Ths smplfcaton allows analyss of the moton of an extended object Angular Poston Axs of rotaton s the center of the dsc Choose a fxed reference lne Pont P s at a fxed dstance r from the orgn A small element of the dsc can be modeled as a partcle at P Angular Poston, Pont P wll rotate about the orgn n a crcle of radus r Every partcle on the dsc undergoes crcular moton about the orgn, O Polar coordnates are convenent to use to represent the poston of P (or any other pont) P s located at (r, ) where r s the dstance from the orgn to P and s the measured counterclockwse from the reference lne Angular Poston, 3 As the partcle moves, the only coordnate that changes s As the partcle moves through, t moves though an arc length s. The arc length and r are related: s = r Radan Ths can also be expressed as s r s a pure number, but commonly s gven the artfcal unt, radan One radan s the angle subtended by an arc length equal to the radus of the arc Whenever usng rotatonal equatons, you must use angles expressed n radans 1
2 Conversons Comparng degrees and radans 360 1rad 57.3 Convertng from degrees to radans 180 rad degrees Angular Dsplacement The angular dsplacement s defned as the angle the object rotates through durng some tme nterval f Ths s the angle that the reference lne of length r sweeps out Average Angular Speed The average angular speed, ω avg, of a rotatng rgd object s the rato of the angular dsplacement to the tme nterval avg f t t t f Instantaneous Angular Speed The nstantaneous angular speed s defned as the lmt of the average speed as the tme nterval approaches zero lm t 0 d t dt Angular Speed, fnal Unts of angular speed are radans/sec rad/s or s 1 snce radans have no dmensons Angular speed wll be postve f θ s ncreasng (counterclockwse) Angular speed wll be negatve f θ s decreasng (clockwse) Average Angular Acceleraton The average angular acceleraton,, of an object s defned as the rato of the change n the angular speed to the tme t takes for the object to undergo the change: f avg t t t f
3 Instantaneous Angular Acceleraton The nstantaneous angular acceleraton s defned as the lmt of the average angular acceleraton as the tme goes to 0 lm d t 0 t dt Angular Acceleraton, fnal Unts of angular acceleraton are rad/s² or s  snce radans have no dmensons Angular acceleraton wll be postve f an object rotatng counterclockwse s speedng up Angular acceleraton wll also be postve f an object rotatng clockwse s slowng down Angular Moton, General Notes When a rgd object rotates about a fxed axs n a gven tme nterval, every porton on the object rotates through the same angle n a gven tme nterval and has the same angular speed and the same angular acceleraton So all characterze the moton of the entre rgd object as well as the ndvdual partcles n the object Drectons, detals Strctly speakng, the speed and acceleraton ( are the magntudes of the velocty and acceleraton vectors The drectons are actually gven by the rghthand rule Rotatonal Knematcs Under constant angular acceleraton, we can descrbe the moton of the rgd object usng a set of knematc equatons These are smlar to the knematc equatons for lnear moton The rotatonal equatons have the same mathematcal form as the lnear equatons The new model s a rgd object under constant angular acceleraton Analogous to the partcle under constant acceleraton model Rotatonal Knematc Equatons t f 1 f t t f f 1 f f t all wth consant 3
4 Comparson Between Rotatonal and Lnear Equatons Relatonshp Between Angular and Lnear Quanttes Dsplacements s r Speeds v r Acceleratons a r Every pont on the rotatng object has the same angular moton Every pont on the rotatng object does not have the same lnear moton Speed Comparson The lnear velocty s always tangent to the crcular path Called the tangental velocty The magntude s defned by the tangental speed Acceleraton Comparson The tangental acceleraton s the dervatve of the tangental velocty dv d at r r dt dt ds d v r r dt dt Clcker Queston Alex and Bran are rdng on a merrygoround. Alex rdes on a horse at the outer rm of the crcular platform, twce as far from the center of the crcular platform as Bran, who rdes on an nner horse. When the merrygoround s rotatng at a constant angular speed, what s Alex s tangental speed? A. twce Bran s B. the same as Bran s C. half of Bran s D. four tmes of Bran s E. mpossble to determne Speed and Acceleraton Note All ponts on the rgd object wll have the same angular speed, but not the same tangental speed All ponts on the rgd object wll have the same angular acceleraton, but not the same tangental acceleraton The tangental quanttes depend on r, and r s not the same for all ponts on the object 4
5 Centrpetal Acceleraton An object travelng n a crcle, even though t moves wth a constant speed, wll have an acceleraton Therefore, each pont on a rotatng rgd object wll experence a centrpetal acceleraton v ac r r Resultant Acceleraton The tangental component of the acceleraton s due to changng speed The centrpetal component of the acceleraton s due to changng drecton Total acceleraton can be found from these components a a a r r r 4 4 t r Rotatonal Moton Example For a compact dsc player to read a CD, the angular speed must vary to keep the tangental speed constant (v t = r) At the nner sectons, the angular speed s faster than at the outer sectons Rotatonal Knetc Energy An object rotatng about some axs wth an angular speed, ω, has rotatonal knetc energy even though t may not have any translatonal knetc energy Each partcle has a knetc energy of K = ½ m v Snce the tangental velocty depends on the dstance, r, from the axs of rotaton, we can substtute v = r Rotatonal Knetc Energy, cont The total rotatonal knetc energy of the rgd object s the sum of the energes of all ts partcles 1 KR K mr 1 1 KR mr I Where I s called the moment of nerta Rotatonal Knetc Energy, fnal There s an analogy between the knetc energes assocated wth lnear moton (K = ½ mv ) and the knetc energy assocated wth rotatonal moton (K R = ½ I ) Rotatonal knetc energy s not a new type of energy, the form s dfferent because t s appled to a rotatng object The unts of rotatonal knetc energy are Joules (J) 5
6 Moment of Inerta The defnton of moment of nerta s I r m The dmensons of moment of nerta are ML and ts SI unts are kg. m We can calculate the moment of nerta of an object more easly by assumng t s dvded nto many small volume elements, each of mass m Moment of Inerta, cont We can rewrte the expresson for I n terms of m I r m r dm lm m 0 Wth the small volume segment assumpton, I r dv If s constant, the ntegral can be evaluated wth known geometry, otherwse ts varaton wth poston must be known Notes on Varous Denstes Volumetrc Mass Densty mass per unt volume: = m / V Surface Mass Densty mass per unt thckness of a sheet of unform thckness, t : t Lnear Mass Densty mass per unt length of a rod of unform crosssectonal area: = m / L = Moment of Inerta of a Unform Rgd Rod The shaded area has a mass dm = dx Then the moment of nerta s L / M Iy r dm x dx L / L 1 I ML 1 Moment of Inerta of a Unform Sold Cylnder Moments of Inerta of Varous Rgd Objects Dvde the cylnder nto concentrc shells wth radus r, thckness dr and length L dm = dv = Lr dr Then for I Iz r dm r Lr dr 1 Iz MR 6
7 Clcker Queston A secton of hollow ppe and a sold cylnder have the same radus, mass and length. They both rotate about ther long central axes wth the same angular speed. Whch object has hgher rotatonal knetc energy? A. Sold cylnder B. The hollow ppe C. Same D. Informaton s not suffcent to determne ParallelAxs Theorem In the prevous examples, the axs of rotaton concded wth the axs of symmetry of the object For an arbtrary axs, the parallelaxs theorem often smplfes calculatons The theorem states I = I CM + MD I s about any axs parallel to the axs through the center of mass of the object I CM s about the axs through the center of mass D s the dstance from the center of mass axs to the arbtrary axs ParallelAxs Theorem Example The axs of rotaton goes through O The axs through the center of mass s shown The moment of nerta about the axs through O would be I O = I CM + MD Moment of Inerta for a Rod Rotatng Around One End The moment of nerta of the rod about ts center s 1 ICM ML 1 D s ½ L Therefore, I ICM MD 1 1 I ML M ML 1 3 L 7
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 1s tme nterval. The velocty of the partcle
More informationSpring 2002 Lecture #13
4450 Sprng 00 ecture # Dr. Jaehoon Yu. Rotatonal Energy. Computaton of oments of nerta. Parallelas Theorem 4. Torque & Angular Acceleraton 5. Work, Power, & Energy of Rotatonal otons Remember the mdterm
More 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 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 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 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 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 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 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 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 informationPHYS 1443 Section 003 Lecture #17
PHYS 144 Secton 00 ecture #17 Wednesda, Oct. 9, 00 1. Rollng oton of a Rgd od. Torque. oment of Inerta 4. Rotatonal Knetc Energ 5. Torque and Vector Products Remember the nd term eam (ch 6 11), onda, Nov.!
More 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 information= 1.23 m/s 2 [W] Required: t. Solution:!t = = 17 m/s [W]! m/s [W] (two extra digits carried) = 2.1 m/s [W]
Secton 1.3: Acceleraton Tutoral 1 Practce, page 24 1. Gven: 0 m/s; 15.0 m/s [S]; t 12.5 s Requred: Analyss: a av v t v f v t a v av f v t 15.0 m/s [S] 0 m/s 12.5 s 15.0 m/s [S] 12.5 s 1.20 m/s 2 [S] Statement:
More 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 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 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 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 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 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 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 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 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 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 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 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 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 informationMotion in One Dimension
Moton n One Dmenson Speed ds tan ce traeled Aerage Speed tme of trael Mr. Wolf dres hs car on a long trp to a physcs store. Gen the dstance and tme data for hs trp, plot a graph of hs dstance ersus tme.
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 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 informationkq r 2 2kQ 2kQ (A) (B) (C) (D)
PHYS 1202W MULTIPL CHOIC QUSTIONS QUIZ #1 Answer the followng multple choce questons on the bubble sheet. Choose the best answer, 5 pts each. MC1 An uncharged metal sphere wll (A) be repelled by a charged
More informationMath1110 (Spring 2009) Prelim 3  Solutions
Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3  Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers.
More 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 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 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 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 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 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 2A Chapter 3 HW Solutions
Phscs A Chapter 3 HW Solutons Chapter 3 Conceptual Queston: 4, 6, 8, Problems: 5,, 8, 7, 3, 44, 46, 69, 70, 73 Q3.4. Reason: (a) C = A+ B onl A and B are n the same drecton. Sze does not matter. (b) C
More informationPhysics 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 informationAP Physics 1 & 2 Summer Assignment
AP Physcs 1 & 2 Summer Assgnment AP Physcs 1 requres an exceptonal profcency n algebra, trgonometry, and geometry. It was desgned by a select group of college professors and hgh school scence teachers
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 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 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 informationcoordinates. Then, the position vectors are described by
Revewng, what we have dscussed so far: Generalzed coordnates Any number of varables (say, n) suffcent to specfy the confguraton of the system at each nstant to tme (need not be the mnmum number). In general,
More informationChapter 10. Rotation of a Rigid Object about a Fixed Axis
Chapter 10 Rotation of a Rigid Object about a Fixed Axis Angular Position Axis of rotation is the center of the disc Choose a fixed reference line. Point P is at a fixed distance r from the origin. A small
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationOne Dimensional Axial Deformations
One Dmensonal al Deformatons In ths secton, a specfc smple geometr s consdered, that of a long and thn straght component loaded n such a wa that t deforms n the aal drecton onl. The as s taken as the
More informationLesson 5: Kinematics and Dynamics of Particles
Lesson 5: Knematcs and Dynamcs of Partcles hs set of notes descrbes the basc methodology for formulatng the knematc and knetc equatons for multbody dynamcs. In order to concentrate on the methodology and
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 informationPhysics 4B. Question and 3 tie (clockwise), then 2 and 5 tie (zero), then 4 and 6 tie (counterclockwise) B i. ( T / s) = 1.74 V.
Physcs 4 Solutons to Chapter 3 HW Chapter 3: Questons:, 4, 1 Problems:, 15, 19, 7, 33, 41, 45, 54, 65 Queston 31 and 3 te (clockwse), then and 5 te (zero), then 4 and 6 te (counterclockwse) Queston 34
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 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 informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
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 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 informationSCALARS AND VECTORS All physical quantities in engineering mechanics are measured using either scalars or vectors.
SCALARS AND ECTORS All phscal uanttes n engneerng mechancs are measured usng ether scalars or vectors. Scalar. A scalar s an postve or negatve phscal uantt that can be completel specfed b ts magntude.
More informationPhysics 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 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 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 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 informationPY2101 Classical Mechanics Dr. Síle Nic Chormaic, Room 215 D Kane Bldg
PY2101 Classcal Mechancs Dr. Síle Nc Chormac, Room 215 D Kane Bldg s.ncchormac@ucc.e Lectures stll some ssues to resolve. Slots shared between PY2101 and PY2104. Hope to have t fnalsed by tomorrow. Mondays
More 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 informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationIterative General Dynamic Model for SerialLink Manipulators
EEL6667: Knematcs, Dynamcs and Control of Robot Manpulators 1. Introducton Iteratve General Dynamc Model for SeralLnk Manpulators In ths set of notes, we are gong to develop a method for computng a general
More informationTHERMAL DISTRIBUTION IN THE HCL SPECTRUM OBJECTIVE
ame: THERMAL DISTRIBUTIO I THE HCL SPECTRUM OBJECTIVE To nvestgate a system s thermal dstrbuton n dscrete states; specfcally, determne HCl gas temperature from the relatve occupatons of ts rotatonal states.
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationProblem Points Score Total 100
Physcs 450 Solutons of Sample Exam I Problem Ponts Score 1 8 15 3 17 4 0 5 0 Total 100 All wor must be shown n order to receve full credt. Wor must be legble and comprehensble wth answers clearly ndcated.
More informationLecture 20: Noether s Theorem
Lecture 20: Noether s Theorem In our revew of Newtonan Mechancs, we were remnded that some quanttes (energy, lnear momentum, and angular momentum) are conserved That s, they are constant f no external
More informationAn Algorithm to Solve the Inverse Kinematics Problem of a Robotic Manipulator Based on Rotation Vectors
An Algorthm to Solve the Inverse Knematcs Problem of a Robotc Manpulator Based on Rotaton Vectors Mohamad Z. Alaz*, Mazn Z. Othman**, and Baker B. AlBahr* *ALNahran Unversty, Computer Eng. Dep., Baghdad,
More informationChapter 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
More informationIndeterminate pinjointed frames (trusses)
Indetermnate pnjonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all
More informationMEV442 Introduction to Robotics Module 2. Dr. Santhakumar Mohan Assistant Professor Mechanical Engineering National Institute of Technology Calicut
MEV442 Introducton to Robotcs Module 2 Dr. Santhakumar Mohan Assstant Professor Mechancal Engneerng Natonal Insttute of Technology Calcut Jacobans: Veloctes and statc forces Introducton Notaton for tmevaryng
More informationACTM State Calculus Competition Saturday April 30, 2011
ACTM State Calculus Competton Saturday Aprl 30, 2011 ACTM State Calculus Competton Sprng 2011 Page 1 Instructons: For questons 1 through 25, mark the best answer choce on the answer sheet provde Afterward
More informationPHY2049 Exam 2 solutions Fall 2016 Solution:
PHY2049 Exam 2 solutons Fall 2016 General strategy: Fnd two resstors, one par at a tme, that are connected ether n SERIES or n PARALLEL; replace these two resstors wth one of an equvalent resstance. Now
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 informationπ e ax2 dx = x 2 e ax2 dx or x 3 e ax2 dx = 1 x 4 e ax2 dx = 3 π 8a 5/2 (a) We are considering the Maxwell velocity distribution function: 2πτ/m
Homework Solutons Problem In solvng ths problem, we wll need to calculate some moments of the Gaussan dstrbuton. The bruteforce method s to ntegrate by parts but there s a nce trck. The followng ntegrals
More information