ENGI 1313 Mechanics I

Transcription

1 ENGI 1313 Mechanics I Lectue 04: oce Vectos and System of Coplana oces Shawn Kenny, Ph.D., P.Eng. Assistant Pofesso aculty of Engineeing and Applied Science Memoial Univesity of Newfoundland spkenny@eng.mun.ca

2 Tutoial Questions SI Units and Use Section 1.4 Page 9 Use a single pefix Magnitude between 0.1 and 1000 Pa N m MPa 3 ( + 3 ( 10 N 10 N 6 5( 10 m 100 kn 10 5 mm 3 m 5 10 mm mm + 3 ( 6 N 11 N GN m m m ( 6 MN ( 10 N ( 10 N ( 10 N 6 m ( 10 mm mm 3 mm 10 m ( ( GPa 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04 N mm

3 Tutoial Questions SI Units and Use Section 1.4 Page 9 Use a single pefix Magnitude between 0.1 and ( 50 kn( 60nm [ 50( 10 N] [ 60( 10 m] 3+ ( 9 ( ( 3 6 ( N m 3( 10 ( 10 N m 3 ( N m 3 mn m S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

4 Tutoial Questions SI Units and Use Section 1.4 Page 9 Do not use compound pefixes μkg ( 10 ( 10 g ( 10 g ( 10 g mg S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

5 Chapte Objectives to eview concepts fom linea algeba to sum foces, detemine foce esultants and esolve foce components fo D vectos using Paallelogam Law to expess foce and position in Catesian vecto fom to intoduce the concept of dot poduct S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

6 Lectue 04 Objectives to sum foce vectos, detemine foce esultants, and esolve foce components fo D vectos using Scala o Catesian Vecto Notation to demonstate by example S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

7 Why an Altenate Appoach? Application of Paallelogam Law Cumbesome with a lage numbe of coplana foces due to successive application Recall Lectue 0 (Slide S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

8 Paallelogam Law (Lectue 0 Multiple oce Vectos R ( S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

9 What is the Altenate Appoach? Resolve oce Components Algebaic Summation Rx Ry Rx x Ry y oce Vectos Component Vectos Recall Lectue 0 (Slide S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

10 What is the Altenate Appoach? Resolve oce Components Algebaic Summation om Resultant oce oce Vectos Component Vectos Resultant oce Vecto S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

11 Coplana oce Vecto Summation How to esolve a system of foces into ectangula components and detemine the esultant foce? Two Notations Used (1 Scala Notation Moe familia appoach ( Catesian Vecto Notation Useful in applications of linea algeba Advantageous ove scala notation fo 3D S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

12 Catesian Coodinate System Chaacteistics Rectangula coodinate system Unique spatial position Vecto algeba Analytical geomety Odinate Abscissa S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

13 Rectangula oce Components Axes Must be Othogonal Axes Oientation Does not Matte x + y x + y S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

14 Resolve oce Components Known: oce Vecto and Oientation Angle θ x y x + y x y cosθ sinθ S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

15 15 ENGI 1313 Statics I Lectue S. Kenny, Ph.D., P.Eng. Resolve oce Components Known: oce Vecto and Slope x y + h x x x L L h y y y L L L x L y L h

16 Detemine Resultant oce Known: oce Components Resultant oce Magnitude Pythagoean theoem x + Resultant oce Diection y y Tigonomety 1 y θ tan x y θ x x S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

17 Notation Summation Coplana oces Scala Notation R X + Y Catesian Vecto Notation ^ ^ X i + Y j Common 3. Diection: Othogonal X & Y axes 3. Diection: Unit vectos Unit Vecto; ^ i X Unit Vecto; ^ j Y 1. Magnitude: X & Y. Sense: + & - -X -i R X +Y +j Y +X +i X Y -Y -j S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

18 Unit Vecto Lectue 3 Scala Magnitude and sense (+,- Vecto Magnitude, sense (+,- and diection Unit Vecto Vecto Magnitude 4 units Sense Positive Diection X-axis û A A 4 units A A A + x S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

19 Coplana oce Vecto Summation Step 1: Define System of oces Rectangula coodinate system oce vectos 1, and 3 oce Vectos S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

20 Coplana oce Vecto Summation Step : Resolve Component oces n nx + cosθ ny nx nx n sinθ ny ny n oce Vectos Component Vectos S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

21 Coplana oce Vecto Summation Step 3: Sum System oce Components Obtain esultant foce vecto components Rx Rx N n 1 x Ry Ry N n 1 y oce Vectos Component Vectos Resultant oce Vecto S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

22 Coplana oce Vecto Summation Step 3: Sum System oce Components Scala notation + + N n 1 + x y Rx Ry 1x 1y + x y 3 x 3 y oce Vectos Component Vectos Resultant oce Vecto 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

23 Coplana oce Vecto Summation Step 4: Detemine Resultant oce Vecto Magnitude, sense and diection + x y θ tan 1 y x oce Vectos Component Vectos Resultant oce Vecto S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

24 Coplana oce Vecto Summation Step 3: Sum System oce Components Catesian vecto notation R R R j x i + j ( i + j + ( i + ( 1x 1y ( + i + ( + j 1x x 3 x x 1y y Unit Vecto; ^ i X y 3 y 3 y 3 oce Vectos Component Vectos Resultant oce Vecto X S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

25 Compehension Quiz 4-01 Resolve along x and y axes in Catesian vecto notation. { } N A 80 cos 30 i - 80 sin 30 j B 80 sin 30 i + 80 cos 30 j C 80 sin 30 i - 80 cos 30 j D 80 cos 30 i + 80 sin 30 j C y ĵ -80 cos 30 y 30 { o o 80 sin30 i 80 cos 30 j }N 80 N x 80 sin30 x S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

26 Compehension Quiz 4-0 Detemine the magnitude of the esultant foce when 1 { 10 î + 0 ĵ } N { 0 î + 0 ĵ } N j R 1 1 i A 30 N B 40 N C 50 N D 60 N E 70 N {( i + ( j } N { 30 i + 40 j }N ( 30 N + ( 40 N 50 N C 50 N S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

27 Example Poblem 4-01 ind the magnitude and angle of the esultant foce acting on the backet. Solution Plan Step 1: Define system of foces Step : Resolve component foces Step 3: Sum system foce components Step 4: Detemine esultant foce vecto, magnitude and diection S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

28 Example Poblem 4-01 (cont. Step : Resolve Components Catesian vecto fom, 1 1 {( 15 sin 40 i + ( 15 cos 40 j }kn 1x 15kN sin 40 1y 15kN cos 40 1x 1y S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

29 Example Poblem 4-01 (cont. Step : Resolve Components Catesian vecto fom, x -6kN (1/ i j kn y 6kN (5/13 y x S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

30 Example Poblem 4-01 (cont. Step : Resolve Components Catesian vecto fom, 3 3x 36kN cos 30 3 {( 36 cos 30 i ( 36 sin 30 j }kn 3y 36kN sin 30 3x 3y S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

31 Example Poblem 4-01 (cont. Step : Resolve Components Catesian vecto fom i + 13 {( 15 sin 40 i ( 15 cos 40 j } kn j kn {( 36 cos 30 i ( 36 sin 30 j }kn 1x Theefoe 1 3 { 9.64 i j } { 4 i + 10 j } kn { i 18 j }kn kn y x 1y 3x 3y S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

32 Example Poblem 4-01 (cont. Step 3: Sum Collinea oces Collinea Catesian vecto fom 1 3 R { 9.64 i j } { 4 i + 10 j } kn { i 18 j }kn kn {( i + ( j }kn x 1y y 3y 1x 3x S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

33 Example Poblem 4-01 (cont. Step 3: Sum Collinea oces Resultant components Catesian vecto fom 1 3 { 9.64 i j } { 4 i + 10 j } kn { i 18 j }kn R { R 16.8 i j }kn kn Ry {( i + ( j }kn Rx S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

34 Example Poblem 4-01 (cont. Step 4: Detemine Resultant oce Vecto ( 16.8 kn + ( 3.49 N 17. kn θ tan kn 16.8 kn 11.7 o ( ccw x axis R θ 11.7 Ry Rx S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

35 Goup Poblem 4-01 ind the magnitude and angle of the esultant foce acting on the backet. Solution Plan Step 1: Define system of foces Step : Resolve component foces Step 3: Sum system foce components Step 4: Detemine esultant foce vecto, magnitude and diection S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

36 Goup Poblem 4-01 (cont. Step : Resolve oce Components 1x v i 850 j 5 5 { 680 i 510 j }N N 1y S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

37 Goup Poblem 4-01 (cont. Step : Resolve oce Components v i 850 j N 5 5 { 680 i 510 j }N v { 31.5 i j }N o o { ( 65 sin 30 i ( 65 cos 30 j } N x y S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

38 Goup Poblem 4-01 (cont. Step : Resolve oce Components 3x 3y v i 850 j N 5 5 { 680 i 510 j }N v o o ( 65 sin 30 i ( 65 cos 30 j { 31.5 i j }N v 3 { i j }N { } o o { ( 750 cos 45 i + ( 750 sin 45 j } N N S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

39 Goup Poblem 4-01 (cont. Step 3: Sum Collinea oces v i 850 j N 5 5 { 680 i 510 j }N v o o ( 65 sin 30 i ( 65 cos 30 j { 31.5 i j }N { } o o { ( 750 cos 45 i + ( 750 sin 45 j } v 3 { i j }N v R v { R 16.8 i 51 j }N N N {( i + ( j }N Ri Rj R S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

40 Goup Poblem 4-01 (cont. Step 4: Detemine Resultant oce Vecto v { R 16.8 i 51 j }N ( 16.8 N + ( 51N 546 N R θ tan 1 51N 16.8 N 7.6 o (local 53 o (ccw x axis S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

41 Classification of Textbook Poblems Hibbele (007 Poblem Set Concept Degee of Difficulty Estimated Time -31 to -3 Vecto Addition Paallelogam Law Medium 10-15min -33 to -38 Vecto Addition Paallelogam Law Easy 5-10min -39 to -41 Resultant oce Easy 5-10min -4 to -55 Resultant & Components Medium 10-15min -56 Resultant & Components Had 0min S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04

42 Refeences Hibbele (007 mech_1 en.wikipedia.og S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04