KINEMATICS OF RIGID BODIES
|
|
|
- Gwen Melton
- 5 years ago
- Views:
Transcription
1 KINEMTICS OF RIGID ODIES
2 Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body.
3 Description of the motion of rigid bodies is importnt for to resons: 1) To generte, trnsmit or control motions by using cms, gers nd linkges of vrious types nd nlyze the displcement, velocity nd ccelertion of the motion to determine the design geometry of the mechnicl prts. Furthermore, s result of the motion generted, forces my be developed hich must be ccounted for in the design of the prts.
4 ) To determine the motion of rigid body cused by the forces pplied to it. Clcultion of the motion of rocket under the influence of its thrust nd grvittionl ttrction is n exmple of such problem.
5 Rigid ody ssumption rigid body is system of prticles for hich the distnces beteen the prticles nd the ngle beteen the lines remin unchnged. Thus, if ech prticle of such body is locted by position vector from reference xes ttched to nd rotting ith the body, there ill be no chnge in ny position vector s mesured from these xes.
6 Of course this is n ideliztion since ll solid mterils chnge shpe to some extent hen forces re pplied to them. Nevertheless, if the movements ssocited ith the chnges in shpe re very smll compred ith the movements of the body s hole, then the ssumption of rigidity is usully cceptble.
7 Plne Motion ll prts of the body move in prllel plnes. The plne motion of rigid body is divided into severl ctegories: 1. Trnsltion. Rottion 3. Generl Motion
8 1. TRNSLTION It is ny motion in hich every line in the body remins prllel to its originl position t ll times. In trnsltion, there is no rottion of ny line in the body. 1. Rectiliner Trnsltion: ll points in the body move in prllel stright lines. Rocket test sled
9 . Curviliner Trnsltion: ll points move on congruent curves. In ech of the to cses of trnsltion, the motion of the body is completely specified by the motion of ny point in the body, since ll the points hve the sme motion.
10 . Fixed xis Rottion Rottion bout fixed xis is the ngulr motion bout the xis. ll prticles in rigid body move in circulr pths bout the xis of rottion nd ll lines in the body hich re perpendiculr to the xis of rottion rotte through the sme ngle t the sme time. C C
11 3. Generl Plne Motion It is the combintion of trnsltion nd rottion.
12 Crnk (Krnk) (Rottion) Piston (Trnsltion) O Connecting rod (Generl Motion) hinge
13 Rottion The rottion of rigid body is described by its ngulr motion. The figure shos rigid body hich is rotting s it undergoes plne motion in the plne of the figure. The ngulr positions of ny to lines 1 nd ttched to the body re specified by q 1 nd q mesured from ny convenient fixed reference direction. ecuse the ngle b is invrint, the reltion q = q 1 + b upon differentition ith respect to time gives q q 1 nd q q during finite intervl, D q 1 = D q 1. ll lines on rigid body in its plne of motion hve the sme ngulr displcement, the sme ngulr velocity nd the sme ngulr ccelertion.
14 ngulr Motion Reltions The ngulr velocity nd ngulr ccelertion of rigid body in plne rottion re, respectively, the first nd second time derivtives of the ngulr position coordinte q of ny line in the plne of motion of the body. These definitions give dq q dt d dt ωdω αdθ or or d q q dt qd q qdθ
15 For rottion ith constnt ngulr ccelertion, the reltionships become q q 0 t 0 0 t 0 q q 1 t 0
16 Rottion bout Fixed xis When rigid body rottes bout fixed xis, ll points other thn those on the xis move in concentric circles bout the fixed xis. Thus, for the rigid body in the figure rotting bout fixed xis norml to the plne of the figure through O, ny point such s moves in circle of rdius r. So the velocity nd the ccelertion of point cn be ritten s v r n t r r v r v
17 These quntities my be expressed using the cross product reltionship of vector nottion, r r v k k, n t r v r r dt dr r dt d r dt d v dt d
18 PROLEMS 1. The ngulr velocity of ger is controlled ccording to = 1 3t, here in rds nd t is the time in seconds. Find the net ngulr displcement Dq from the time t = 0 to t = 3 s. lso find the totl number of revolutions N through hich the ger turns during the three seconds. SOLUTION dq dq dt dt q 0 dq t dt, q 1t t 13 Dq 9 rd rd
19 SOLUTION Does the ger stop beteen t = 0 nd t = 3 seconds? 1 3t 0 1 3t t s ( it stops t t s) q q dq dq rd t dt q 1t t t dt q 1 1t t rd 3 16 rd 1revolution rd N revolutions 3 rd N 3.66 revolutions
20 PROLEMS. The belt-driven pulley nd ttched disk re rotting ith incresing ngulr velocity. t certin instnt the speed v of the belt is 1.5 ms, nd the totl ccelertion of point is 75 ms. For this instnt determine () the ngulr ccelertion of the pulley nd disk, (b) the totl ccelertion of point, nd (c) the ccelertion of point C on the belt.
21 C ? c)? b)? ) s m r s m s m r s m r s rd R s m s m R s rd r v s m s m v C C C n t t t n SOLUTION
22 PROLEMS 3. The design chrcteristics of ger-reduction unit re under revie. Ger is rotting clockise (c) ith speed of 300 revmin hen torque is pplied to ger t time t= s to give ger counterclockise (cc) ccelertion hich vries ith time for durtion of 4 seconds s shon. Determine the speed N of ger hen t=6 s.
23 SOLUTION s rd rev N s t min 300 The velocities of gers nd re sme t the contct point. min ) 6 ( rev N s rd b b s t s t t s rd t t dt t d dt d t s rd b b v v t
24 bsolute Motion In this pproch, e mke use of the geometric reltions hich define the configurtion of the body involved nd then proceed to tke the time derivtives of the defining geometric reltions to obtin velocities nd ccelertions.
25 PROLEM 1) heel of rdius r rolls on flt surfce ithout slipping. Determine the ngulr motion of the heel in terms of the liner motion of its center O. lso determine the ccelertion of point on the rim of the heel s the point comes into contct ith the surfce on hich the heel rolls.
26 PROLEM ) Motion of the equilterl tringulr plte C in its plne is controlled by the hydrulic cylinder D. If the piston rod in the cylinder is moving uprd t the constnt rte of 0.3 ms during n intervl of its motion, clculte for the instnt hen q=30 o the velocity nd ccelertion of the center of the roller in the horizontl guide nd the ngulr velocity nd ngulr ccelertion of edge C.
27 PROLEM 3) Derive n expression for the uprd velocity v of the cr hoist in terms of q. The piston rod of the hydrulic cylinder is extending t the rte. s
28 PROLEM 4) Clculte the ngulr velocity of the slender br s fuction of the distnce x nd the constnt ngulr velocity o of the drum.
29 Reltive Motion chnge. The second pproch to rigid body kinemtics uses the principles of reltive motion. In kinemtics of prticles for motion reltive to trnslting xes, e pplied the reltive velocity eqution v v v to the motions of to prticles nd. We no choose to points on the sme rigid body for our to prticles. The consequence of this choice is tht the motion of one point s seen by n observer trnslting ith the other point must be circulr since the rdil distnce to the observed point from the reference point does not
30 The figure shos rigid body moving in the plne of the figure from position to during time Dt. This movement my be visulized s occurring in to prts. First, the body trnsltes to the prllel position ith the displcement. Second, the body rottes bout through the ngle Dq, from the nonrotting reference xes x -y ttched to the reference point, giving rise to the displcement D of ith respect to. r Dr D r
31 With s the reference point, the totl displcement of is Where D r D r Dr Dr hs the mgnitude rdq s Dq pproches zero. Dividing the time intervl Dt nd pssing to the limit, e obtin the reltive velocity eqution v v v The distnce r beteen nd remins constnt.
32 The mgnitude of the reltive velocity is thus seen to be hich, ith becomes r v Using to represent the vector, e my rite the reltive velocity s the vector dt d r t r t r v t t q q D D D D D D 0 0 lim lim q r v r r Therefore, the reltive velocity eqution becomes r v v
33 Here, is the ngulr velocity vector norml to the plne of the motion in the sense determined by the right hnd rule. It should be noted tht the direction of the reltive velocity ill lys be perpendiculr to the line joining the points nd. Interprettion of the Reltive Velocity Eqution We cn better understnd the reltive velocity eqution by visulizing the trnsltion nd rottion components seprtely. Trnsltion Fixed xid rottion
34 In the figure, point is chosen s the reference point nd the velocity of is the vector sum of the trnsltionl portion the rottionl portion q v =r, here v r, plus, hich hs the mgnitude, the bsolute ngulr velocity of. The reltive liner velocity is lys perpendiculr to the line joining the to points nd. v
35 Reltive ccelertion Eqution of reltive velocity is v v v y differentiting the eqution ith respect to time, e obtin the reltive ccelertion eqution, hich is or v v v This eqution sttes tht the ccelertion of point equls the vector sum of the ccelertion of point nd the ccelertion hich ppers to hve to nonrotting observer moving ith.
36 If points nd re locted on the sme rigid body, the distnce r beteen them remins constnt. ecuse the reltive motion is circulr, the reltive ccelertion term ill hve both norml component directed from tord due to the chnge of direction of v nd tngentil component perpendiculr to due to the chnge in mgnitude of. Thus, e my rite, v n t Where the mgnitudes of the reltive ccelertion components re v r v r n t r
37 In vector nottion the ccelertion components re r r t n The reltive ccelertion eqution, thus, becomes r r
38 The figure shos the ccelertion of to be composed of to prts: the ccelertion of nd the ccelertion of ith respect to.
39 Solution of the Reltive ccelertion Eqution s in the cse of the reltive velocity eqution, the reltive ccelertion eqution my be crried out by sclr or vector lgebr or by grphicl construction. ecuse the norml ccelertion components depend on velocities, it is generlly necessry to solve for the velocities before the ccelertion clcultions cn be mde.
40 PROLEMS 1. The center O of the disk hs the velocity nd ccelertion shon. If the disk rolls ithout slipping on the horizontl surfce, determine the velocity of nd the ccelertion of for the instnt represented.
41 PROLEMS. If the velocity of point is 3 ms to the right nd is constnt for n intervl including the position shon, determine the tngentil ccelertion of point long its pth nd the ngulr ccelertion of the br.
42 PROLEMS 3. The flexible bnd F ttched to the sector t E is given constnt velocity of 4 ms s shon. For the instnt hen D is perpendiculr to O, determine the ngulr ccelertion of D.
43 PROLEMS 4. mechnism for pushing smll boxes from n ssembly line onto conveyor belt is shon ith rm OD nd crnk C in their verticl positions. For the configurtion shon, crnk C hs constnt clockise ngulr velocity of rds. Determine the ccelertion of E.
44 PROLEMS 5. t given instnt, the ger hs the ngulr motion shon. Determine the ccelertion of points nd on the link nd the link s ngulr ccelrtion t this instnt.
45 PROLEMS 6. The center O of the disk rolling ithout slipping on the horizontl surfce hs the velocity nd ccelertion shon. Rdius of the disk is 4.5 cm. Clculte the velocity nd ccelertion of point. v o =45 cms o =90 cms 37 o O 4 cm 4.5 cm y 6 cm 1 y x 4 x x= cm 10 cm
KINEMATICS OF RIGID BODIES
KINEMTICS OF RIGI OIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. escription
THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES
THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES 1. TRANSLATION Figure shows rigid body trnslting in three-dimensionl spce. Any two points in the body, such s A nd B, will move long prllel stright lines if
INTRODUCTION. The three general approaches to the solution of kinetics problems are:
INTRODUCTION According to Newton s lw, prticle will ccelerte when it is subjected to unblnced forces. Kinetics is the study of the reltions between unblnced forces nd the resulting chnges in motion. The
Dynamics and control of mechanical systems. Content
Dynmics nd control of mechnicl systems Dte Dy 1 (01/08) Dy (03/08) Dy 3 (05/08) Dy 4 (07/08) Dy 5 (09/08) Dy 6 (11/08) Content Review of the bsics of mechnics. Kinemtics of rigid bodies plne motion of
Problems (Motion Relative to Rotating Axes)
1. The disk rolls without slipping on the roblems (Motion Reltie to Rotting xes) horizontl surfce, nd t the instnt represented, the center O hs the elocity nd ccelertion shown in the figure. For this instnt,
2/20/ :21 AM. Chapter 11. Kinematics of Particles. Mohammad Suliman Abuhaiba,Ph.D., P.E.
//15 11:1 M Chpter 11 Kinemtics of Prticles 1 //15 11:1 M Introduction Mechnics Mechnics = science which describes nd predicts the conditions of rest or motion of bodies under the ction of forces It is
DYNAMICS. Kinematics of Rigid Bodies VECTOR MECHANICS FOR ENGINEERS: Tenth Edition CHAPTER
Tenth E CHTER 15 VECTOR MECHNICS FOR ENGINEERS: YNMICS Ferdinnd. eer E. Russell Johnston, Jr. hillip J. Cornwell Lecture Notes: rin. Self Cliforni olytechnic Stte Uniersity Kinemtics of Rigid odies 013
13.4 Work done by Constant Forces
13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push
Partial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
KINETICS OF RIGID BODIES PROBLEMS
KINETICS OF RIID ODIES PROLEMS PROLEMS 1. The 6 kg frme C nd the 4 kg uniform slender br of length l slide with negligible friction long the fied horizontl br under the ction of the 80 N force. Clculte
Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.
1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples
Version 001 HW#6 - Circular & Rotational Motion arts (00223) 1
Version 001 HW#6 - Circulr & ottionl Motion rts (00223) 1 This print-out should hve 14 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. Circling
ME 141. Lecture 10: Kinetics of particles: Newton s 2 nd Law
ME 141 Engineering Mechnics Lecture 10: Kinetics of prticles: Newton s nd Lw Ahmd Shhedi Shkil Lecturer, Dept. of Mechnicl Engg, BUET E-mil: sshkil@me.buet.c.bd, shkil6791@gmil.com Website: techer.buet.c.bd/sshkil
2/2/ :36 AM. Chapter 11. Kinematics of Particles. Mohammad Suliman Abuhaiba,Ph.D., P.E.
//16 1:36 AM Chpter 11 Kinemtics of Prticles 1 //16 1:36 AM First Em Wednesdy 4//16 3 //16 1:36 AM Introduction Mechnics Mechnics = science which describes nd predicts the conditions of rest or motion
1/31/ :33 PM. Chapter 11. Kinematics of Particles. Mohammad Suliman Abuhaiba,Ph.D., P.E.
1/31/18 1:33 PM Chpter 11 Kinemtics of Prticles 1 1/31/18 1:33 PM First Em Sturdy 1//18 3 1/31/18 1:33 PM Introduction Mechnics Mechnics = science which describes nd predicts conditions of rest or motion
E S dition event Vector Mechanics for Engineers: Dynamics h Due, next Wednesday, 07/19/2006! 1-30
Vector Mechnics for Engineers: Dynmics nnouncement Reminders Wednesdy s clss will strt t 1:00PM. Summry of the chpter 11 ws posted on website nd ws sent you by emil. For the students, who needs hrdcopy,
PROBLEM deceleration of the cable attached at B is 2.5 m/s, while that + ] ( )( ) = 2.5 2α. a = rad/s. a 3.25 m/s. = 3.
![PROBLEM deceleration of the cable attached at B is 2.5 m/s, while that + ] ( )( ) = 2.5 2α. a = rad/s. a 3.25 m/s. = 3.](/thumbs/71/66176966.jpg "PROBLEM deceleration of the cable attached at B is 2.5 m/s, while that + ] ( )( ) = 2.5 2α. a = rad/s. a 3.25 m/s. = 3.")
PROLEM 15.105 A 5-m steel bem is lowered by mens of two cbles unwinding t the sme speed from overhed crnes. As the bem pproches the ground, the crne opertors pply brkes to slow the unwinding motion. At
a < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
Conducting Ellipsoid and Circular Disk
1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,
Lecture 5. Today: Motion in many dimensions: Circular motion. Uniform Circular Motion
Lecture 5 Physics 2A Olg Dudko UCSD Physics Tody: Motion in mny dimensions: Circulr motion. Newton s Lws of Motion. Lws tht nswer why questions bout motion. Forces. Inerti. Momentum. Uniform Circulr Motion
The Properties of Stars
10/11/010 The Properties of Strs sses Using Newton s Lw of Grvity to Determine the ss of Celestil ody ny two prticles in the universe ttrct ech other with force tht is directly proportionl to the product
- 5 - TEST 2. This test is on the final sections of this session's syllabus and. should be attempted by all students.
- 5 - TEST 2 This test is on the finl sections of this session's syllbus nd should be ttempted by ll students. Anything written here will not be mrked. - 6 - QUESTION 1 [Mrks 22] A thin non-conducting
Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
Plane curvilinear motion is the motion of a particle along a curved path which lies in a single plane.
Plne curiliner motion is the motion of prticle long cured pth which lies in single plne. Before the description of plne curiliner motion in n specific set of coordintes, we will use ector nlsis to describe
AP Physics 1. Slide 1 / 71. Slide 2 / 71. Slide 3 / 71. Circular Motion. Topics of Uniform Circular Motion (UCM)
Slide 1 / 71 Slide 2 / 71 P Physics 1 irculr Motion 2015-12-02 www.njctl.org Topics of Uniform irculr Motion (UM) Slide 3 / 71 Kinemtics of UM lick on the topic to go to tht section Period, Frequency,
CAPACITORS AND DIELECTRICS
Importnt Definitions nd Units Cpcitnce: CAPACITORS AND DIELECTRICS The property of system of electricl conductors nd insultors which enbles it to store electric chrge when potentil difference exists between
BME 207 Introduction to Biomechanics Spring 2018
April 6, 28 UNIVERSITY O RHODE ISAND Deprtment of Electricl, Computer nd Biomedicl Engineering BME 27 Introduction to Biomechnics Spring 28 Homework 8 Prolem 14.6 in the textook. In ddition to prts -e,
[ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves
![[ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves](/thumbs/74/70299306.jpg "[ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves")
Gols: 1. To find the re etween two curves Section 6.1 Are of Regions etween two Curves I. Are of Region Between Two Curves A. Grphicl Represention = _ B. Integrl Represention [ ( ) ( )] f x g x dx = C.
#6A&B Magnetic Field Mapping
#6A& Mgnetic Field Mpping Gol y performing this lb experiment, you will: 1. use mgnetic field mesurement technique bsed on Frdy s Lw (see the previous experiment),. study the mgnetic fields generted by
2A1A Vector Algebra and Calculus I
Vector Algebr nd Clculus I (23) 2AA 2AA Vector Algebr nd Clculus I Bugs/queries to sjrob@robots.ox.c.uk Michelms 23. The tetrhedron in the figure hs vertices A, B, C, D t positions, b, c, d, respectively.
PhysicsAndMathsTutor.com
1. A uniform circulr disc hs mss m, centre O nd rdius. It is free to rotte bout fixed smooth horizontl xis L which lies in the sme plne s the disc nd which is tngentil to the disc t the point A. The disc
2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the
APPLICATIONS OF THE DEFINITE INTEGRAL
APPLICATIONS OF THE DEFINITE INTEGRAL. Volume: Slicing, disks nd wshers.. Volumes by Slicing. Suppose solid object hs boundries extending from x =, to x = b, nd tht its cross-section in plne pssing through
PROBLEM SOLUTION
PROLEM 15.11 The 18-in.-rdius flywheel is rigidly ttched to 1.5-in.-rdius shft tht cn roll long prllel rils. Knowing tht t the instnt shown the center of the shft hs velocity of 1. in./s nd n ccelertion
Plane curvilinear motion is the motion of a particle along a curved path which lies in a single plane.
Plne curiliner motion is the motion of prticle long cured pth which lies in single plne. Before the description of plne curiliner motion in n specific set of coordintes, we will use ector nlsis to describe
Review of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
Plate Theory. Section 11: PLATE BENDING ELEMENTS
Section : PLATE BENDING ELEMENTS Plte Theor A plte is structurl element hose mid surfce lies in flt plne. The dimension in the direction norml to the plne is referred to s the thickness of the plte. A
Plate Theory. Section 13: PLATE BENDING ELEMENTS
Section : PLATE BENDING ELEENTS Wshkeic College of Engineering Plte Theor A plte is structurl element hose mid surfce lies in flt plne. The dimension in the direction norml to the plne is referred to s
Forces from Strings Under Tension A string under tension medites force: the mgnitude of the force from section of string is the tension T nd the direc
Physics 170 Summry of Results from Lecture Kinemticl Vribles The position vector ~r(t) cn be resolved into its Crtesin components: ~r(t) =x(t)^i + y(t)^j + z(t)^k. Rtes of Chnge Velocity ~v(t) = d~r(t)=
Physics 207 Lecture 7
Phsics 07 Lecture 7 Agend: Phsics 07, Lecture 7, Sept. 6 hpter 6: Motion in (nd 3) dimensions, Dnmics II Recll instntneous velocit nd ccelertion hpter 6 (Dnmics II) Motion in two (or three dimensions)
DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS
3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive
Problem Solving 7: Faraday s Law Solution
MASSACHUSETTS NSTTUTE OF TECHNOLOGY Deprtment of Physics: 8.02 Prolem Solving 7: Frdy s Lw Solution Ojectives 1. To explore prticulr sitution tht cn led to chnging mgnetic flux through the open surfce
Week 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
7.6 The Use of Definite Integrals in Physics and Engineering
Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems
Chapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
Math 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
Simulation of Eclipsing Binary Star Systems. Abstract
Simultion of Eclipsing Binry Str Systems Boris Yim 1, Kenny Chn 1, Rphel Hui 1 Wh Yn College Kowloon Diocesn Boys School Abstrct This report briefly introduces the informtion on eclipsing binry str systems.
Sample Problems for the Final of Math 121, Fall, 2005
Smple Problems for the Finl of Mth, Fll, 5 The following is collection of vrious types of smple problems covering sections.8,.,.5, nd.8 6.5 of the text which constitute only prt of the common Mth Finl.
MEE 214 (Dynamics) Tuesday Dr. Soratos Tantideeravit (สรทศ ต นต ธ รว ทย )
MEE 14 (Dynmics) Tuesdy 8.30-11.0 Dr. Sortos Tntideerit (สรทศ ต นต ธ รว ทย ) sortos@oep.go.th Lecture Notes, Course updtes, Extr problems, etc No Homework Finl Exm (Dte & Time TBD) 1/03/58 MEE14 Dynmics
Distributed Forces: Centroids and Centers of Gravity
Distriuted Forces: Centroids nd Centers of Grvit Introduction Center of Grvit of D Bod Centroids nd First Moments of Ares nd Lines Centroids of Common Shpes of Ares Centroids of Common Shpes of Lines Composite
Electrical Drive 4 th Class
University Of Technology Electricl nd Electronics Deprtment Dr Nofl ohmmed Ther Al Kyt A drive consist of three min prts : prime mover; energy trnsmitting device nd ctul pprtus (lod), hich perform the
Physics 207 Lecture 5
Phsics 07 Lecture 5 Agend Phsics 07, Lecture 5, Sept. 0 Chpter 4 Kinemtics in or 3 dimensions Independence of, nd/or z components Circulr motion Cured pths nd projectile motion Frmes of reference dil nd
Job No. Sheet 1 of 8 Rev B. Made by IR Date Aug Checked by FH/NB Date Oct Revised by MEB Date April 2006
Job o. Sheet 1 of 8 Rev B 10, Route de Limours -78471 St Rémy Lès Chevreuse Cedex rnce Tel : 33 (0)1 30 85 5 00 x : 33 (0)1 30 5 75 38 CLCULTO SHEET Stinless Steel Vloristion Project Design Exmple 5 Welded
KEY. Physics 106 Common Exam 1, Spring, 2004
Physics 106 Common Exm 1, Spring, 2004 Signture Nme (Print): A 4 Digit ID: Section: Instructions: Questions 1 through 10 re multiple-choice questions worth 5 points ech. Answer ech of them on the Scntron
A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.
A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
Simple Harmonic Motion I Sem
Simple Hrmonic Motion I Sem Sllus: Differentil eqution of liner SHM. Energ of prticle, potentil energ nd kinetic energ (derivtion), Composition of two rectngulr SHM s hving sme periods, Lissjous figures.
3. Vectors. Vectors: quantities which indicate both magnitude and direction. Examples: displacemement, velocity, acceleration
Rutgers University Deprtment of Physics & Astronomy 01:750:271 Honors Physics I Lecture 3 Pge 1 of 57 3. Vectors Vectors: quntities which indicte both mgnitude nd direction. Exmples: displcemement, velocity,
Mathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
Magnetic forces on a moving charge. EE Lecture 26. Lorentz Force Law and forces on currents. Laws of magnetostatics
Mgnetic forces on moving chrge o fr we ve studied electric forces between chrges t rest, nd the currents tht cn result in conducting medium 1. Mgnetic forces on chrge 2. Lws of mgnetosttics 3. Mgnetic
Student Session Topic: Particle Motion
Student Session Topic: Prticle Motion Prticle motion nd similr problems re on the AP Clculus exms lmost every yer. The prticle my be prticle, person, cr, etc. The position, velocity or ccelertion my be
Math 0230 Calculus 2 Lectures
Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two
Centre of Mass, Moments, Torque
Centre of ss, oments, Torque Centre of ss If you support body t its center of mss (in uniform grvittionl field) it blnces perfectly. Tht s the definition of the center of mss of the body. If the body consists
Year 12 Mathematics Extension 2 HSC Trial Examination 2014
Yer Mthemtics Etension HSC Tril Emintion 04 Generl Instructions. Reding time 5 minutes Working time hours Write using blck or blue pen. Blck pen is preferred. Bord-pproved clcultors my be used A tble of
IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB
` K UKATP ALLY CE NTRE IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB 7-8 FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd - 5 7 Ph: -66 Regd
( ) as a fraction. Determine location of the highest
AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if
l 2 p2 n 4n 2, the total surface area of the
Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n n-sided regulr polygon of perimeter p n with vertices on C. Form cone
6.5 Plate Problems in Rectangular Coordinates
6.5 lte rolems in Rectngulr Coordintes In this section numer of importnt plte prolems ill e emined ug Crte coordintes. 6.5. Uniform ressure producing Bending in One irection Consider first the cse of plte
( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
The Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
Time : 3 hours 03 - Mathematics - March 2007 Marks : 100 Pg - 1 S E CT I O N - A
Time : hours 0 - Mthemtics - Mrch 007 Mrks : 100 Pg - 1 Instructions : 1. Answer ll questions.. Write your nswers ccording to the instructions given below with the questions.. Begin ech section on new
5.2 Volumes: Disks and Washers
4 pplictions of definite integrls 5. Volumes: Disks nd Wshers In the previous section, we computed volumes of solids for which we could determine the re of cross-section or slice. In this section, we restrict
We divide the interval [a, b] into subintervals of equal length x = b a n
![We divide the interval [a, b] into subintervals of equal length x = b a n](/thumbs/80/80533932.jpg "We divide the interval [a, b] into subintervals of equal length x = b a n")
Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:
HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)
HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time llowed Two hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions
Kirchhoff and Mindlin Plates
Kirchhoff nd Mindlin Pltes A plte significntly longer in two directions compred with the third, nd it crries lod perpendiculr to tht plne. The theory for pltes cn be regrded s n extension of bem theory,
10 Vector Integral Calculus
Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve
Dynamics: Newton s Laws of Motion
Lecture 7 Chpter 4 Physics I 09.25.2013 Dynmics: Newton s Lws of Motion Solving Problems using Newton s lws Course website: http://fculty.uml.edu/andriy_dnylov/teching/physicsi Lecture Cpture: http://echo360.uml.edu/dnylov2013/physics1fll.html
Thomas Whitham Sixth Form
Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos
Motion of Electrons in Electric and Magnetic Fields & Measurement of the Charge to Mass Ratio of Electrons
n eperiment of the Electron topic Motion of Electrons in Electric nd Mgnetic Fields & Mesurement of the Chrge to Mss Rtio of Electrons Instructor: 梁生 Office: 7-318 Emil: shling@bjtu.edu.cn Purposes 1.
PHYSICS ASSIGNMENT-9
MPS/PHY-XII-11/A9 PHYSICS ASSIGNMENT-9 *********************************************************************************************************** 1. A wire kept long the north-south direction is llowed
Ch AP Problems
Ch. 7.-7. AP Prolems. Willy nd his friends decided to rce ech other one fternoon. Willy volunteered to rce first. His position is descried y the function f(t). Joe, his friend from school, rced ginst him,
Lecture XVII. Vector functions, vector and scalar fields Definition 1 A vector-valued function is a map associating vectors to real numbers, that is
Lecture XVII Abstrct We introduce the concepts of vector functions, sclr nd vector fields nd stress their relevnce in pplied sciences. We study curves in three-dimensionl Eucliden spce nd introduce the
Sample Exam 5 - Skip Problems 1-3
Smple Exm 5 - Skip Problems 1-3 Physics 121 Common Exm 2: Fll 2010 Nme (Print): 4 igit I: Section: Honors Code Pledge: As n NJIT student I, pledge to comply with the provisions of the NJIT Acdemic Honor
AB Calculus Review Sheet
AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is
Plates on elastic foundation
Pltes on elstic foundtion Circulr elstic plte, xil-symmetric lod, Winkler soil (fter Timoshenko & Woinowsky-Krieger (1959) - Chpter 8) Prepred by Enzo Mrtinelli Drft version ( April 016) Introduction Winkler
A. Limits - L Hopital s Rule. x c. x c. f x. g x. x c 0 6 = 1 6. D. -1 E. nonexistent. ln ( x 1 ) 1 x 2 1. ( x 2 1) 2. 2x x 1.
A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( ) lim where lim f or lim f limg. c g = c limg( ) = c = c = c How to find it: Try nd find limits by
Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
MASKING OF FERROMAGNETIC ELLIPTICAL SHELL IN TRANSVERSE MAGNETIC FIELD
POZNAN UNVE RSTY OF TE HNOLOGY AADE M JOURNALS No 7 Electricl Engineering Kzimierz JAKUUK* Mirosł WOŁOSZYN* Peł ZMNY* MASKNG OF FERROMAGNET ELLPTAL SHELL N TRANSVERSE MAGNET FELD A ferromgnetic oject,
Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.
Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)
ragsdale (zdr82) HW2 ditmire (58335) 1
rgsdle (zdr82) HW2 ditmire (58335) This print-out should hve 22 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. 00 0.0 points A chrge of 8. µc
Study Guide Final Exam. Part A: Kinetic Theory, First Law of Thermodynamics, Heat Engines
Msschusetts Institute of Technology Deprtment of Physics 8.0T Fll 004 Study Guide Finl Exm The finl exm will consist of two sections. Section : multiple choice concept questions. There my be few concept
CONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round
Not for reproduction
AREA OF A SURFACE OF REVOLUTION cut h FIGURE FIGURE πr r r l h FIGURE A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundry of solid of revolution of the type
Physics Jonathan Dowling. Lecture 9 FIRST MIDTERM REVIEW
Physics 10 Jonthn Dowling Physics 10 ecture 9 FIRST MIDTERM REVIEW A few concepts: electric force, field nd potentil Electric force: Wht is the force on chrge produced by other chrges? Wht is the force
FINALTERM EXAMINATION 9 (Session - ) Clculus & Anlyticl Geometry-I Question No: ( Mrs: ) - Plese choose one f ( x) x According to Power-Rule of differentition, if d [ x n ] n x n n x n n x + ( n ) x n+
Main topics for the Second Midterm
Min topics for the Second Midterm The Midterm will cover Sections 5.4-5.9, Sections 6.1-6.3, nd Sections 7.1-7.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the
Problem Set 4: Mostly Magnetic
University of Albm Deprtment of Physics nd Astronomy PH 102 / LeClir Summer 2012 nstructions: Problem Set 4: Mostly Mgnetic 1. Answer ll questions below. Show your work for full credit. 2. All problems
Topics Covered AP Calculus AB
Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.