C2 Vectors C3 Interactions transfer momentum. General Physics GP7-Vectors (Ch 4) 1

Transcription

1 C2 Vectos C3 Inteactions tansfe momentum Geneal Phsics GP7-Vectos (Ch 4) 1

2 Solutions to HW When ou homewok is gaded and etuned, solutions will be available. Download PobViewe Passwod fo C1 dail HW poblems 1234 Coections fo C1 dail homewok ae due on Thusda, 9/11 Geneal Phsics GP7-Vectos (Ch 4) 2

3 Vectos and Scalas scala quantit completel specified b a single value with an appopiate unit and has no diection. vecto quantit completel descibed b a numbe and appopiate units plus a diection. Geneal Phsics GP7-Vectos (Ch 4) 3

4 Vectos and Scalas scala quantit completel specified b a single value with an appopiate unit and has no diection. Use an italic lette: vecto quantit completel descibed b a numbe and appopiate units plus a diection. Use an aow: Geneal Phsics GP7-Vectos (Ch 4) 4

5 Vecto Eample paticle tavels fom to along the path shown b the dotted ed line This is the distance taveled and is a scala The displacement is the solid line fom to The displacement is independent of the path taken between the two points Displacement is a vecto Geneal Phsics GP7-Vectos (Ch 4) 5

6 Equalit of Two Vectos Two vectos ae equal if the have the same magnitude and the same diection = if = and the point along paallel lines ll of the vectos shown in the diagam at ight ae equal Geneal Phsics GP7-Vectos (Ch 4) 6

7 dding Vectos Gaphicall Daw the vectos tipto-tail The esultant is dawn fom the oigin of to the end of the last vecto Measue the length of and its angle Use the scale facto to convet length to actual magnitude R Geneal Phsics GP7-Vectos (Ch 4) 7

8 dding Multiple Vectos tip-to-tail fo all vectos The esultant, R, is still dawn fom the oigin of the fist vecto to the end of the last vecto Geneal Phsics GP7-Vectos (Ch 4) 8

9 dding Vectos, Rules When two vectos ae added, the sum is independent of the ode of the addition. This is the commutative law of addition = Geneal Phsics GP7-Vectos (Ch 4) 9

10 dding Vectos, Rules cont. When adding thee o moe vectos, thei sum is independent of the wa in which the individual vectos ae gouped This is called the ssociative Popet of ddition ( ) C = ( C) Geneal Phsics GP7-Vectos (Ch 4) 10

11 dding Vectos, Rules final When adding vectos, all of the vectos must have the same units ll of the vectos must be of the same tpe of quantit Fo eample, ou cannot add a displacement to a velocit Geneal Phsics GP7-Vectos (Ch 4) 11

12 Negative of a Vecto The negative of a vecto is defined as the vecto that, when added to the oiginal vecto, gives a esultant of zeo Repesented as (! ) = 0 The negative of the vecto will have the same magnitude, but point in the opposite diection Geneal Phsics GP7-Vectos (Ch 4) 12

13 Subtacting Vectos Special case of vecto addition! If, then use (! ) Continue with standad vecto addition pocedue Geneal Phsics GP7-Vectos (Ch 4) 13

14 Vecto epesentation vecto has both magnitude (numbe value) and diection. p = p p p z z Show magnitude o value Show diection Geneal Phsics GP7-Vectos (Ch 4) 14

15 Geneal Phsics GP7-Vectos (Ch 4) 15 Column Vecto How can we descibe this vecto using a using column vecto? z p p p p z = tpical vecto notation column vecto notation!!! " # $ $ $ % & = z p p p p

16 Components of a Vecto component is a pat We will use ectangula components These ae the pojections of the vecto along the - and -aes and ae the component vectos of The ae vectos and follow all the ules fo vectos and ae scalas, and will be efeed to as the components of Geneal Phsics GP7-Vectos (Ch 4) 16

17 Magnitude and Diection vecto ma also be epesented b it s magnitude and diection. Magnitude p = mag ( ) p = p p p The magnitude of the vecto has phsical units The magnitude of a vecto is alwas a positive numbe z Geneal Phsics GP7-Vectos (Ch 4) 17

18 Multipling o Dividing a Vecto b a Scala The esult of the multiplication o division is a vecto The magnitude of the vecto is multiplied o divided b 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 Geneal Phsics GP7-Vectos (Ch 4) 18

19 Think-Pai-Shae boat is cossing a ive that flows fom Noth to South at a ate of 3 m/s. The boat stats at the east end of the ive and heads diectl west with a speed of 4 m/s. (a) What is the boats total velocit? (b) If the ive is 1000 m in the E-W diection, how long does it take the boat to coss? Geneal Phsics GP7-Vectos (Ch 4) 19

20 Components of a Vecto, 2 The -component of a vecto is the pojection along the -ais The -component of a vecto is the pojection along the -ais Then, = Geneal Phsics GP7-Vectos (Ch 4) 20

21 Components of a Vecto, 3 The pevious equations ae valid onl if θ is measued with espect to the -ais The components ae the legs of the ight tiangle whose hpotenuse is Geneal Phsics GP7-Vectos (Ch 4) 21

22 Components of a Vecto, final The components can be positive o negative and will have the same units as the oiginal vecto The signs of the components will depend on the angle Geneal Phsics GP7-Vectos (Ch 4) 22

23 Think-Pai-Shae displacement vecto in the - plane is 15 m long and diected at an angle of 30 degees above the positive ais. Detemine (a) the -component (b) the -component Geneal Phsics GP7-Vectos (Ch 4) 23

24 Think-Pai-Shae displacement vecto in the - plane is 10 m long and diected at an angle of 190 degees counteclockwise fom the positive ais. Detemine (a) the -component (b) the -component Geneal Phsics GP7-Vectos (Ch 4) 24

25 Think-Pai-Shae fl lands on one wall of a oom. The lowe left-hand cone of the wall is selected as the oigin of the twodimensional Catesian coodinate sstem. If the fl is located at the point having coodinates (2.00, 1.00) m, (a) how fa is it fom the cone of the oom? (b) What is its location in pola coodinates? Geneal Phsics GP7-Vectos (Ch 4) 25

26 Think-Pai-Shae peson walks noth of east fo 3.10 km. How fa would she have to walk due noth and due east to aive at the same location? Geneal Phsics GP7-Vectos (Ch 4) 26

27 Think-Pai-Shae Jane leaves he house and walks 5.0 blocks east and then poceeds noth until she is 7.8 blocks fom home at an angle of 50 degees Noth of east. How man blocks noth did she tavel? Geneal Phsics GP7-Vectos (Ch 4) 27

28 Eample The Catesian coodinates of a point in the plane ae (,) = (-3.50, -2.50) m, as shown in the figue. Find the pola coodinates of this point. Solution: and, Geneal Phsics GP7-Vectos (Ch 4) 28

29 Unit Vectos unit vecto is a dimensionless vecto with a magnitude of eactl 1. Unit vectos ae used to specif a diection (,, and z) and have no othe phsical significance The smbols fo,, z ae, and z ŷ ẑ The fom a set of mutuall pependicula vectos Geneal Phsics GP7-Vectos (Ch 4) 29

30 Geneal Phsics GP7-Vectos (Ch 4) 30 Unit Vectos in Vecto Notation The complete vecto can be epessed as = = z z = ŷ

31 Geneal Phsics GP7-Vectos (Ch 4) 31 dding Vectos Using Unit Vectos Using Then and so R = and R = ( ) ( ) ( ) ( ) R R R R R R = = = = R = " # $ % & ' " # $ % & ' = " # $ % & '

32 dding Vectos with Unit Vectos Geneal Phsics GP7-Vectos (Ch 4) 32

33 Geneal Phsics GP7-Vectos (Ch 4) 33 dding Vectos Using Unit Vectos Thee Diections Using R =, R = and R z = z z ( ) ( ) ( ) ( ) ( )!!! " # $ $ $ % & =!!! " # $ $ $ % & =!!! " # $ $ $ % & = = = = = z z z z z z z z z R z R R R R z R z z R R

34 Think-Pai-Shae (a) In unit-vecto notation, what is the sum of a b, whee a = (4.0 m) i (3.0 m) ŷj b = (-13.0 m) i (7.0 m) ŷj What ae the (b) magnitude and (c) diection of a b (elative to )? Geneal Phsics GP7-Vectos (Ch 4) 34

35 Think-Pai-Shae Repeat the poblem using column vecto notation. Geneal Phsics GP7-Vectos (Ch 4) 35

36 Coodinate Sstems Used to descibe the position of a point in space Coodinate sstem 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 Geneal Phsics GP7-Vectos (Ch 4) 36

37 Catesian Coodinate Sstem lso called ectangula coodinate sstem - and - aes intesect at the oigin Points ae labeled (,) Geneal Phsics GP7-Vectos (Ch 4) 37

38 Pola Coodinate Sstem Oigin and efeence line ae noted Point is distance fom the oigin in the diection of angle θ, counte-clockwise fom efeence line Points ae labeled (, θ) Geneal Phsics GP7-Vectos (Ch 4) 38

39 Pola to Catesian Coodinates ased on foming a ight tiangle fom and θ = cos θ = sin θ SOH-CH-TO H O Geneal Phsics GP7-Vectos (Ch 4) 39

40 Catesian to Pola Coodinates is the hpotenuse and θ an angle θ must be counteclockwise fom positive ais fo these equations to be valid Geneal Phsics GP7-Vectos (Ch 4) 40