Chapter 3: Vectors and Coordinate Systems


 Bertram Cross
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1 Coodinate Systems Chapte 3: Vectos and Coodinate Systems Used to descibe the position of a point in space Coodinate system consists of a fied efeence point called the oigin specific aes with scales and labels instuctions on how to label a point elative to the oigin and the aes Catesian Coodinate System Pola Coodinate System Also called ectangula coodinate system  and y aes intesect at the oigin Points ae labeled (,y) Oigin and efeence line ae noted Point is distance fom the oigin in the diection of angle θ, ccw fom efeence line Points ae labeled (,θ) 3 4
2 Pola to Catesian Coodinates Based on foming a ight tiangle fom and θ cos θ y sin θ Catesian to Pola Coodinates is the hypotenuse and θ an angle tanθ y + y θ must be ccw fom positive ais fo these equations to be valid 5 6 Vectos and Scalas Vecto Notation A scala quantity is completely specified by a single value with an appopiate unit and has no diection. A vecto quantity is completely descibed by a numbe and appopiate units plus a diection. When handwitten, use an aow: When pinted, will be in bold pint: A When dealing with just the magnitude of a vecto in pint, an italic lette will be used: A o A The magnitude of the vecto has physical units The magnitude of a vecto is always a positive numbe A
3 Adding Vectos Gaphically Subtacting Vectos Continue dawing the vectos tiptotail The esultant is dawn fom the oigin of A to the end of the last vecto Measue the length of R and its angle Use the scale facto to convet length to actual magnitude Special case of vecto addition If A B, then use A+(B) Continue with standad vecto addition pocedue 9 10 Multiplying o Dividing a Vecto by a Scala The esult of the multiplication o division is a vecto The magnitude of the vecto is multiplied o divided by the scala If the scala is positive, the diection of the esult is the same as of the oiginal vecto If the scala is negative, the diection of the esult is opposite that of the oiginal vecto A component is a pat It is useful to use ectangula components Components of a Vecto These ae the pojections of the vecto along the  and yaes 1
4 Vecto Component Teminology A and A y ae the component vectos of A They ae vectos and follow all the ules fo vectos A and A y ae scalas, and will be efeed to as the components of A Components of a Vecto The component of a vecto is the pojection along the ais A Acosθ The ycomponent of a vecto is the pojection along the yais A Asinθ y Components of a Vecto The pevious equations ae valid only if θ is measued with espect to the ais The components ae the legs of the ight tiangle whose hypotenuse is A 1 Ay A A + Ay and θ tan A May still have to find θ with espect to the positive ais Unit Vectos A unit vecto is a dimensionless vecto with a magnitude of eactly 1. Unit vectos ae used to specify a diection and have no othe physical significance
5 Unit Vectos, cont. Unit Vectos in Vecto Notation The symbols î, ĵ, andkˆ epesent unit vectos They fom a set of mutually pependicula vectos The complete vecto can be epessed as ĵ A A ˆi + A ˆj + A kˆ y z î Adding Vectos Using Unit Vectos Using R A + B Then ( A ˆ ˆ ) ( ˆ ˆ Ay B By ) R i + j + i + j ( A ) ˆ B ( Ay By ) R + i + + ˆj R R + Ry and so R A + B and R y A y + B y Tig Function Waning The component equations (A A cos θ and A y A sin θ) apply only when the angle is measued with espect to the ais (pefeably ccw fom the positive ais). The esultant angle (tan θ A y / A ) gives the angle with espect to the ais. R R + R θ tan 1 y R R y
6 Adding Vectos with Unit Vectos Adding Vectos Using Unit Vectos Thee Diections Using R A + B R A ˆi + A ˆj + A kˆ + B ˆi + B ˆj + B kˆ ( y z ) ( y z ) ( A B ) ( A B ) ( A B ) R + ˆi + + ˆj + + kˆ y y z z R R + Ry + Rz R A + B, R y A y + B y and R z A z + B z R R + R + R θ tan 1 y z R etc. R Which figue shows A + A + A? 1 3 Chapte 3. Questions
7 Which figue shows A + A + A? 1 3 Which figue shows A B? Which figue shows A B? What ae the  and ycomponents C and C y of vecto C? A. C 1, Cy 1 B. C 3, C y 1 C. C, Cy 1 D. C 4, C y E. C 3, C y 1
8 What ae the  and ycomponents C and C y of vecto C? Angle φ that specifies the diection of is given by C A. C 1, Cy 1 B. C 3, C y 1 C. C, Cy 1 D. C 4, C y E. C 3, C y 1 A. tan 1 (C y /C ) B. tan 1 (C / C y ) C. tan 1 (C y / C ) D. tan 1 (C /C y ) E. tan 1 ( C / C y ) Angle φ that specifies the diection of is given by C Back to the concepts of motion: Chapte 1 A. tan 1 (C y /C ) B. tan 1 (C / C y ) C. tan 1 (C y / C ) D. tan 1 (C /C y ) E. tan 1 ( C / C y )
9 Chapte 1. Concepts of Motion The univese we live in is one of change and motion. Although we all have intuition about motion, based on ou epeiences, some of the impotant aspects of motion tun out to be athe subtle. Chapte Goal: To intoduce the fundamental concepts of motion. Displacement  vecto Velocity  vecto Acceleation vecto Diffeent types of motion Diffeent types of motion Tanslational Motion Cicula Motion Pojectile Motion Rotational Motion
10 1 sec sec 3 sec 4 sec 1 sec sec 3 sec 4 sec How can we chaacteize the motion? 1 sec sec 3 sec 4 sec 1 sec sec 3 sec 4 sec What is the diffeence between these motions? How can we chaacteize these motions? The fist step: PARTICLE MODEL MOTION DIAGRAM We conside object as a single point without size o shape, disegad intenal motion of the object. How can we chaacteize the motion? 1 sec sec 3 sec 4 sec Diffeent oigins diffeent coodinates 1 Physical meaning displacement  1sec sec 3sec 4sec Oigin (0) The second step: POSITION OF THE OBJECT (POINT) COORDIANTE SYSTEM  DISPLACEMENT We intoduce coodinate system: fo motion along a line  only (which means that y0); fo a motion in a plane and y. Oigin (0) Oigin (0) ( 10) Copyight 008 Peason Education, Inc., publishing as 30 Peason AddisonWesley.
11 Displacement y a A b d B A  initial position of the object If O is an oigin then vecto chaacteizes initial position of the object B  final position of the O (oigin) object Vecto b chaacteizes the final position of the object Vecto d is a displacement (final position minus initial position does not depend on coodinate system) d b a Standad notation fo displacement is final initial a Displacement y initial O (oigin) final 1,initial 1, final Displacement does not depend on coodinate system final initial 1, final 1, initial Displacement Displacement is a vecto, it does not depend on coodinate system A B How can we chaacteize the motion? The fist step: PARTICLE MODEL MOTION DIAGRAM, The second step: POSITION OF THE OBJECT (POINT) DISPLACEMENT The thid step: (AVERAGE) VELOCITY 1sec sec 3sec 4sec Aveage velocity is a vecto: displacement v avg time t A B C Oigin 10 (0) vavg 1sec vavg 0 sec 30 Copyight 008 Peason Education, Inc., publishing sec as Peason AddisonWesley. Fo a motion along the line diection of velocity is along the line and the magnitude v avg t
12 AVERAGE VELOCITY v avg displacement time t The magnitude of velocity (vecto) is called speed Eample: We know initial position of the object (in some coodinate system) 1 We know the aveage velocity v of the object duing time Then: What is the final position of the object? displacement 1 v time t t + v t 1 How can we chaacteize the motion? The fist step: PARTICLE MODEL MOTION DIAGRAM The second step: POSITION OF THE OBJECT (POINT) DISPLACEMENT The thid step: (AVERAGE) VELOCITY The foth step: (AVERAGE) ACCELERATION Oigin (0) a v1, avg 0 sec avg v, avg 1sec 30 sec The change in position is chaacteized by aveage velocity, The change in velocity is chaacteized by aveage acceleation v a Copyight 008 Peason Education, Inc., publishing avg as Peason AddisonWesley. t v (30 0), avg v 1, avg sec 10 t 1sec sec y How can we chaacteize the motion? The fist step: PARTICLE MODEL MOTION DIAGRAM The second step: POSITION OF THE OBJECT (POINT) DISPLACEMENT The thid step: (AVERAGE) VELOCITY v v avg t aavg The foth step: (AVERAGE) ACCELERATION t 1 v1 t v 1 3 v t v 1 t v v v v 1 a1 a t t v3 t 3 v 3 Acceleation Because velocity is a vecto, it can change in two possible ways. 1.The magnitude can change, indicating a change in speed, o. The diection can change, indicating that the object has changed diection.
13 Acceleation is the change of velocity (speed can be the same) v 1 v v v a t 1 v 1 1sec sec 3sec 4sec v Velocity is the same zeo acceleation v aavg 0 t 1sec sec 3sec 4sec Velocity is inceasing acceleation has the same diection as velocity v v 1 sec sec v 3 sec v v 3 4 sec v 3 a 3 t a v v 3 Velocity is deceasing acceleation has the opposite diection v v a t 3 a v v 3 What is the diffeence between these motions? 1sec sec 3sec 4sec 1sec sec 3sec 4sec 1 sec a 0 sec 3 sec 4 sec v v a v a
14 SI units Units of velocity: Basic Units: m Time seconds (s) vavg t s Length metes (m) Units of acceleation: Mass kilogam v m / s m (kg) aavg t s s EXAMPLE 1.7 Intepeting a position gaph EXAMPLE 1.7 Intepeting a position gaph Geneal PoblemSolving Stategy
15 Chapte 1. Summay Slides Geneal Stategy Geneal Stategy
16 Impotant Concepts Impotant Concepts Impotant Concepts Applications
17 Applications Chapte 1. Questions Which ca is going faste, A o B? Assume thee ae equal intevals of time between the fames of both movies. Which ca is going faste, A o B? Assume thee ae equal intevals of time between the fames of both movies. B is going faste.
18 Thee motion diagams ae shown. Which is a dust paticle settling to the floo at constant speed, which is a ball dopped fom the oof of a building, and which is a descending ocket slowing to make a soft landing on Mas? A. (a) is ball, (b) is dust, (c) is ocket B. (a) is ball, (b) is ocket, (c) is dust C. (a) is ocket, (b) is dust, (c) is ball D. (a) is ocket, (b) is ball, (c) is dust E. (a) is dust, (b) is ball, (c) is ocket Thee motion diagams ae shown. Which is a dust paticle settling to the floo at constant speed, which is a ball dopped fom the oof of a building, and which is a descending ocket slowing to make a soft landing on Mas? A. (a) is ball, (b) is dust, (c) is ocket B. (a) is ball, (b) is ocket, (c) is dust C. (a) is ocket, (b) is dust, (c) is ball D. (a) is ocket, (b) is ball, (c) is dust E. (a) is dust, (b) is ball, (c) is ocket A paticle moves fom position 1 to position duing the inteval t. Which vecto shows the paticle s aveage velocity? A paticle moves fom position 1 to position duing the inteval t. Which vecto shows the paticle s aveage velocity?
19 A paticle undegoes acceleation a while moving fom point 1 to point. Which of the choices shows the velocity vecto v as the object moves away fom point? A paticle undegoes acceleation a while moving fom point 1 to point. Which of the choices shows the velocity vecto v as the object moves away fom point? Rank in ode, fom the most to the least, the numbe of significant figues in the following numbes. Fo eample, if b has moe than c, c has the same numbe as a, and a has moe than d, you could give you answe as b > c a > d. a. 800 b c d A. a b d > c B. b d > c > a C. d > c > b a D. d > c > a > b E. b > a c d Rank in ode, fom the most to the least, the numbe of significant figues in the following numbes. Fo eample, if b has moe than c, c has the same numbe as a, and a has moe than d, you could give you answe as b > c a > d. a. 800 b c d A. a b d > c B. b d > c > a C. d > c > b a D. d > c > a > b E. b > a c d
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