Phys 221. Chapter 3. Vectors A. Dzyubenko Brooks/Cole

Transcription

1 Phs 221 Chapter 3 Vectors adzubenko@csub.edu Dzubenko 2014 rooks/cole 1

2 Coordinate Sstems Used to describe the position of a point in space Coordinate sstem consists of a fied reference point called the origin specific aes with scales and labels instructions on how to label a point relative to the origin and the aes 2

3 Tpes of Coordinate Sstems Cartesian Plane polar 3

4 Cartesian Coordinate Sstem also called rectangular coordinate sstem - and -aes points are labeled (,) 4

5 Plane Polar Coordinate Sstem point is distance r from the origin in the direction of angle θ, counterclockwise from the positive ais points are labeled (r,θ) 5

6 Trigonometr Review sin θ cosθ opposite side hpotenuse adjacent side hpotenuse tan θ opposite adjacent side side 6

7 Conversion between Coordinate Sstems From the plane polar coordinates to Cartesian coordinates r cosθ r sinθ From Cartesian coordinates to the plane polar coordinates tanθ r 2 2 7

8 Eample 3.1 The Cartesian coordinates are (,) (-3.50, -2.50) m, Find the polar coordinates Solution: r 2 2 ( 3.50 m) ( 2.50 m) m 2.50 m tanθ m θ 216 (signs give quadrant) 8

9 Scalar and Vector Quantities Scalar quantities are completel described b magnitude onl Vector quantities have both magnitude (size) and direction Represented b an arrow, the length of the arrow is proportional to the magnitude of vector b Head of the arrow represents the direction 9

10 Vector Notation When handwritten,!use an arrow!: When printed, will be in bold print: When dealing with just the magnitude of a vector in print, an italic letter will be used: r 10

11 Quick Quiz Which of the following are vector quantities and which are scalar quantities? (a) our age (b) acceleration (c) velocit (d) speed (e) mass 11

12 Equalit of Two Vectors onl if and if and point in the same direction along parallel lines 12

13 dding Vectors: Graphical Methods tip-to-tail method Tip-to-tail method: the resultant vector R is the vector drawn from the tail of to the tip of Tail-to-tail (parallelogram) method: the resultant vector R is the vector drawn from where the tails join, outwards to the opposite corner of the parallelogram tail-to-tail method 13

14 dding Vectors, cont dd more than two vectors: RCD R is the vector drawn from the tail of the first vector to the tip of the last vector 14

15 Sum of vectors is independent of the order of the addition Commutative law of addition: ssociative law of addition: (C) () C 15

16 Negative of a Vector Vector - is negative to vector if (-) 0 that means vectors and - have the same magnitude but point in opposite directions 16

17 Subtracting Vectors Define the operation - as vector - added to vector (-) 17

18 Quick Quiz If vector is added to vector, under what condition does the resultant vector have magnitude? (a) and are parallel and in the same direction (b) and are parallel and in opposite directions (c) and are perpendicular 18

19 Multipling a Vector b a Scalar m?? m > 0 m < 0 has the same direction as and m has the opposite direction to and m 19

20 Components of a Vector The projections of vector along coordinate aes are called the components of the vector, are the components of the vector : cos sin θ θ The signs of the components and depend on the angle θ 20

21 Components of a Vector, cont n vector can be completel described b its components θ tan The components of a vector can be epressed in an convenient coordinate sstem 21

22 Quick Quiz Choose the correct response to make the sentence true: Component of a vector is (a) alwas, (b) never, or (c) sometimes larger than the magnitude of the vector 22

23 Unit Vectors ˆ, i ˆ, j and kˆ Dimensionless vectors having a magnitude of eactl 1 ˆi ˆj kˆ 1 Use to specif a given direction Has hat on the smbol Smbols ˆi, ˆj, and kˆ represent unit vectors pointing in the positive, and z directions 23

24 Unit Vectors, cont. ˆ, i ˆ, j and kˆ form a set of mutuall perpendicular vectors in a right-handed coordinate sstem z 24

25 Unit-Vector Notation Vector lies in the plane: ˆi ˆj ˆi ˆj Consider a point (,). It can be specified b the position vector r r ˆi ˆj 25

26 Vector ddition: Using Components dd vector (, ) to vector (, ) The resultant vector R R or R ( ( ˆi ˆ) j ( )ˆi ( ˆi ˆ) j )ˆj The components of the resultant vector R R R R ˆi R 26 ˆj

27 27 The magnitude of R: use Pthagorean theorem ( ) ( ) R R R R R tanθ Using Components Using Components

28 28 Three Three-Dimensional Vectors Dimensional Vectors The angle θ that R makes with e.g., the ais: ( ) ( ) ( ) z z z R R R R k j i ˆ ˆ ˆ z R cosθ R k j i ˆ ˆ ˆ z (,, z ) (,, z ) The sum of and is k j i R ˆ ) ( )ˆ ( )ˆ ( z z The magnitude of vector R is

29 E 3.5 Taking a Hike hiker begins a trip b first walking 25.0 km southeast from her car. She stops and sets up her tent for the night. On the second da, she walks 40.0 km in a direction 60.0 north of east, Determine the components of the hiker s resultant displacement 29

30 E 3.5 Solution Use vector components: cos( 45.0 ) (25.0 km)(0.707) 17.7 km sin( 45.0 ) (25.0 km)( 0.707) 17.7 km cos60.0 (40.0 km)(0.500) 20.0 km sin60.0 (40.0 km)(0.866) 34.6 km Find the resultant: r R (37.7 ˆi 16.9 ˆj) km 30