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
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 10 0. 10 0 0 5m m m 6 6 6 ( 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
Tutoial Questions SI Units and Use Section 1.4 Page 9 Use a single pefix Magnitude between 0.1 and 1000 3 9 ( 50 kn( 60nm [ 50( 10 N] [ 60( 10 m] 3+ ( 9 ( 50 60 10 3 10 ( 3 6 ( N m 3( 10 ( 10 N m 3 ( N m 3 mn m 3 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
Tutoial Questions SI Units and Use Section 1.4 Page 9 Do not use compound pefixes μkg 6 3 6+ 3 3 ( 10 ( 10 g ( 10 g ( 10 g mg 4 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
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 5 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
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 6 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
Why an Altenate Appoach? Application of Paallelogam Law Cumbesome with a lage numbe of coplana foces due to successive application Recall Lectue 0 (Slide 13 7 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
Paallelogam Law (Lectue 0 Multiple oce Vectos + 1 1 R + 1 + + 3 ( 1 + + 3 1 3 8 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
What is the Altenate Appoach? Resolve oce Components Algebaic Summation Rx Ry Rx x Ry y oce Vectos Component Vectos Recall Lectue 0 (Slide 9 9 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
What is the Altenate Appoach? Resolve oce Components Algebaic Summation om Resultant oce oce Vectos Component Vectos Resultant oce Vecto 10 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
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 11 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
Catesian Coodinate System Chaacteistics Rectangula coodinate system Unique spatial position Vecto algeba Analytical geomety Odinate Abscissa 1 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
Rectangula oce Components Axes Must be Othogonal Axes Oientation Does not Matte x + y x + y 13 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
Resolve oce Components Known: oce Vecto and Oientation Angle θ x y x + y x y cosθ sinθ 14 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
15 ENGI 1313 Statics I Lectue 04 007 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
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 16 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
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 17 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
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 18 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
Coplana oce Vecto Summation Step 1: Define System of oces Rectangula coodinate system oce vectos 1, and 3 oce Vectos 19 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
Coplana oce Vecto Summation Step : Resolve Component oces n nx + cosθ ny nx nx n sinθ ny ny n oce Vectos Component Vectos 0 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
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 1 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
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
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 3 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
Coplana oce Vecto Summation Step 3: Sum System oce Components Catesian vecto notation R R R j + 1 3 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 4 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
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 5 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
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 {( 10 + 0 i + ( 0 + 0 j } N { 30 i + 40 j }N ( 30 N + ( 40 N 50 N C 50 N 6 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
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 7 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
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 8 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
Example Poblem 4-01 (cont. Step : Resolve Components Catesian vecto fom, x -6kN (1/13 1 6 13 i + 6 5 13 j kn y 6kN (5/13 y x 9 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
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 30 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
Example Poblem 4-01 (cont. Step : Resolve Components Catesian vecto fom 1 3 + 1 6 i + 13 {( 15 sin 40 i ( 15 cos 40 j } 5 6 13 kn j kn {( 36 cos 30 i ( 36 sin 30 j }kn 1x Theefoe 1 3 { 9.64 i + 11.491 j } { 4 i + 10 j } kn { 31.18 i 18 j }kn kn y x 1y 3x 3y 31 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
Example Poblem 4-01 (cont. Step 3: Sum Collinea oces Collinea Catesian vecto fom 1 3 R { 9.64 i + 11.491 j } { 4 i + 10 j } kn { 31.18 i 18 j }kn kn {( 9.64 4 + 31.18 i + ( 11.49 + 10 18 j }kn x 1y y 3y 1x 3x 3 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
Example Poblem 4-01 (cont. Step 3: Sum Collinea oces Resultant components Catesian vecto fom 1 3 { 9.64 i + 11.491 j } { 4 i + 10 j } kn { 31.18 i 18 j }kn R { R 16.8 i + 3.49 j }kn kn Ry {( 9.64 4 + 31.18 i + ( 11.49 + 10 18 j }kn Rx 33 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
Example Poblem 4-01 (cont. Step 4: Detemine Resultant oce Vecto ( 16.8 kn + ( 3.49 N 17. kn θ tan 1 3.49 kn 16.8 kn 11.7 o ( ccw x axis R θ 11.7 Ry Rx 34 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
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 35 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
Goup Poblem 4-01 (cont. Step : Resolve oce Components 1x v 4 3 1 850 i 850 j 5 5 { 680 i 510 j }N N 1y 36 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
Goup Poblem 4-01 (cont. Step : Resolve oce Components v 4 3 1 850 i 850 j N 5 5 { 680 i 510 j }N v { 31.5 i 541.3 j }N o o { ( 65 sin 30 i ( 65 cos 30 j } N x y 37 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
Goup Poblem 4-01 (cont. Step : Resolve oce Components 3x 3y v 4 3 1 850 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 541.3 j }N v 3 { 530.3 i + 530.3 j }N { } o o { ( 750 cos 45 i + ( 750 sin 45 j } N N 38 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
Goup Poblem 4-01 (cont. Step 3: Sum Collinea oces v 4 3 1 850 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 541.3 j }N { } o o { ( 750 cos 45 i + ( 750 sin 45 j } v 3 { 530.3 i + 530.3 j }N v R v { R 16.8 i 51 j }N N N {( 680 31.5 530.3 i + ( 510 541.3 + 530.3 j }N Ri Rj R 39 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
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 40 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
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 41 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04
Refeences Hibbele (007 http://wps.penhall.com/esm_hibbele_eng mech_1 en.wikipedia.og 4 007 S. Kenny, Ph.D., P.Eng. ENGI 1313 Statics I Lectue 04