ENGI 1313 Mechanics I Lectue 05: Catesian Vects Shawn Kenny, Ph.D., P.Eng. ssistant Pfess Faculty f Engineeing and pplied Science Memial Univesity f Newfundland spkenny@eng.mun.ca
Chapte Objectives t eview cncepts fm linea algeba t sum fces, detemine fce esultants and eslve fce cmpnents f D vects using Paallelgam Law t expess fce and psitin in Catesian vect fm t intduce the cncept f dt pduct 007 S. Kenny, Ph.D., P.Eng.
Lectue 05 Objectives t futhe examine Catesian vect ntatin and extend t epesentatin f a 3D vect t sum 3D cncuent fce systems 3 007 S. Kenny, Ph.D., P.Eng.
Recall D Catesian Vect Cplana Fce Vect Summatin Unit vect dimensinless F F R R F 1 F + F 3 ( F F + F ) i + ( F + F F ) j 1x x 3 x ) 1y y 3 y ) Unit Vect; ^ i F X F X ^ Unit Vect; j F y F y Fce Vects Cmpnent Vects Resultant Fce Vect 4 007 S. Kenny, Ph.D., P.Eng.
Extend t 3D Why Use Vects Simplifies mathematical peatins Rectangula Cdinate System Right-hand ule 5 007 S. Kenny, Ph.D., P.Eng.
Catesian Vect in 3D Thee Unit Vects Cmpnent magnitude x, y, z scala Cmpnent sense +, - Catesian quadant Cmpnent diectin i, j, k unit vects 6 007 S. Kenny, Ph.D., P.Eng.
7 007 S. Kenny, Ph.D., P.Eng. Summatin Catesian Vects in 3D Use D Pinciples Vect Vect B ( ) ( ) ( )kˆ B ĵ B î B B z z y y x x + + + + + + kˆ ĵ î z y x + + kˆ B ĵ B î B B z y x + + ( ) ( ) ( )kˆ B ĵ B î B B z z y y x x + +
Catesian Vect Magnitude Successive pplicatin f D Pinciple Pythagean theem Find cmpnents x and y + x y Cmbine with z cmpnent + z x + y + z Magnitude f vect 8 007 S. Kenny, Ph.D., P.Eng.
Catesian Vect Diectin Cdinate Diectin ngle Vect tail with cdinate axis α, β, and γ Range fm 0 t 180 Visualizatin aid using ight ectangula pism 9 007 S. Kenny, Ph.D., P.Eng.
Catesian Vect Diectin (cnt.) Cdinate Diectin ngle, α Measued +x-axis t tail f vect csα x 10 007 S. Kenny, Ph.D., P.Eng.
Catesian Vect Diectin (cnt.) Cdinate Diectin ngle, β Measued +y-axis t tail f vect csα cs β x y 11 007 S. Kenny, Ph.D., P.Eng.
Catesian Vect Diectin (cnt.) Cdinate Diectin ngle, γ Measued +z-axis t tail f vect csα cs β cs γ x y z 1 007 S. Kenny, Ph.D., P.Eng.
13 007 S. Kenny, Ph.D., P.Eng. Catesian Vect Diectin (cnt.) Diectin Csines Cdinate diectin angles (α, β, & γ) detemined by cs -1 cs x α cs y β cs z γ cs x 1 α cs y 1 β cs z 1 γ
Catesian Vect Diectin (cnt.) Expess as Catesian Vect, x î + y ĵ + z kˆ Recall Unit Vect Fm Unit Vect, u^ û x y z î + ĵ + kˆ + + x y ^ Unit Vect; u z u 14 007 S. Kenny, Ph.D., P.Eng.
Catesian Vect Diectin (cnt.) Unit Vect, ^u x y û î + ĵ + Relate t Diectin Csines csα x Theefe û cs β z y kˆ csα î + cs β ĵ + cs γ csγ kˆ z u 15 007 S. Kenny, Ph.D., P.Eng.
Catesian Vect Diectin (cnt.) Find Unit Vect mplitude, u ^ Recall geneal case f vect and magnitude x î + y ĵ + z kˆ + + Whee the unit vect and magnitude is x y z û csα î + cs β ĵ + cs γ kˆ Theefe cs û cs α + cs β + cs γ 1 α + cs β + cs γ 1 16 007 S. Kenny, Ph.D., P.Eng.
Catesian Vect Oientatin (cnt.) Typical Catesian Vect Pblems Magnitude and cdinate angles Example.8 Magnitude and pjectin angles Example.10 Only need t knw angles 17 007 S. Kenny, Ph.D., P.Eng.
Cmpehensin Quiz 5-01 If yu nly knw u (unit vect) yu can detemine the f uniquely. ) magnitude () B) angles (α, β, and γ) C) cmpnents ( x, y, & z ) D) ll f the abve nswe B Unit vect (u ) defines diectin defines magnitude 18 007 S. Kenny, Ph.D., P.Eng.
Example Pblem 5-01 Fces F and G ae applied t a hk. Fce F makes 60 angle with the X-Y plane. Fce G has a magnitude f 80 lb with α 111 and β 69.3. G 80lb α 111 β 69.3 Find the esultant fce in Catesian vect fm 19 007 S. Kenny, Ph.D., P.Eng.
Example Pblem 5-01 (cnt.) Reslve Fce F cmpnents F z 100 lb sin60 86.60 lb F 100 lb cs60 50.00 lb F z G 80lb α 111 β 69.3 F x 50 lb cs 45 35.36 lb F y 50 lb sin45 35.36 lb F y F x Catesian Vect Ntatin F { 35.36 î 35.36 ĵ + 86.60 kˆ }lb F 0 007 S. Kenny, Ph.D., P.Eng.
Example Pblem 5-01 (cnt.) Detemine γ f Vect G cs α + cs β + cs γ 1 G 80lb α 111 β 69.3 cs 111 + cs 69.3 + cs γ 1 γ csγ 1 ( 0.3584) + ( 0.3535) ± 0. 8641 γ 30. 149.78 1 007 S. Kenny, Ph.D., P.Eng.
Example Pblem 5-01 (cnt.) Cdinate Diectin ngles α111 z α 111 -x G 80lb G csα x G G csα G x x y Unit cicle G x G x 80 cs111 80 cs G x 8.67lb lb ( 180 111 ) lb 007 S. Kenny, Ph.D., P.Eng.
Example Pblem 5-01 (cnt.) Cdinate Diectin ngles β and γ z -x G 80lb γ 30. β 69.3 y x 3 007 S. Kenny, Ph.D., P.Eng.
Example Pblem 5-01 (cnt.) Catesian Vect G -x z G 80lb α 111 γ 30. x β 69.3 y { G 80 cs111 î + 80 cs69.3 ĵ + 80 cs 30. kˆ }lb G { 8.67 î + 8.8 ĵ + 69.13 kˆ }lb 4 007 S. Kenny, Ph.D., P.Eng.
Example Pblem 5-01 (cnt.) Cmbine Fce Vects R F + G F { 35.36 î 35.36 ĵ + 86.60 kˆ }lb G { 8.67 î + 8.8 ĵ + 69.13 kˆ }lb G 80lb α 111 β 69.3 Resultant Vect R + R { 6.69 î 7.08 ĵ + 156 kˆ }lb {( 35.36 8.67) î + ( 35.36 + 8.8) ĵ + ( 86.60 69.13) kˆ }lb 5 007 S. Kenny, Ph.D., P.Eng.
Gup Pblem 5-01 Pblem -57 (Hibbele, 007) Detemine the magnitude and cdinate diectin angles f F and F 1. Sketch each fce n an x, y, z efeence. F { 1 60 î 50 ĵ + 40 kˆ }N F { 40 î 85 ĵ + 30 kˆ }N 6 007 S. Kenny, Ph.D., P.Eng.
Gup Pblem 5-01 (cnt.) Fce F 1 F { 1 60 î 50 ĵ + 40 kˆ }N F ( 60N) + ( 50N) + ( 40N) 87.7 N 1 F1x 60N csα1 α1 46. 9 F 87.75N 1 F1y 50N cs β 1 β1 15 F 87.75N F1z 40N cs γ1 γ1 6. 9 F 87.75N 1 1 7 007 S. Kenny, Ph.D., P.Eng.
Gup Pblem 5-01 (cnt.) Fce F F { 40 î 85 ĵ + 30 kˆ }N F ( 40N) + ( 85N) + ( 30N) 98.6 N F x 40N csα α 114 F 98.6N F y 85N cs β β 150 F 98.6N Fz 30N cs γ γ 7. 3 F 98.6N 8 007 S. Kenny, Ph.D., P.Eng.
Gup Pblem 5-0 Pblem -59 (Hibbele, 007) Detemine the magnitude and cdinate angles f F acting n the stake. 9 007 S. Kenny, Ph.D., P.Eng.
Gup Pblem 5-0 Detemine F and cmpnents 5 F 40N 50N 4 3 F z 40N 30N 4 F x 40N cs70 13.7N F x F F y F z F y 40N sin70 37.6N 30 007 S. Kenny, Ph.D., P.Eng.
Gup Pblem 5-0 Detemine Cdinate Diectin ngles Fx 13.68N csα α 74. 1 F 50N Fy 37.58N cs β β 41. 3 F 50N Fz 30N cs γ γ 53. 1 F 50N 31 007 S. Kenny, Ph.D., P.Eng.
Classificatin f Textbk Pblems Hibbele (007) Pblem Set Cncept Degee f Difficulty Estimated Time -57 Catesian: Magnitude & diectin Medium 10-15min -58 t -60 Catesian: Fces Easy 5-10min -61 t -69 Catesian; Fce cmpnents & esultant Easy 5-10min -70 t -71 Catesian; Fce cmpnents & esultant Had 0-5min -7 t -78 Catesian; Fce cmpnents & esultant Medium 15-0min 3 007 S. Kenny, Ph.D., P.Eng.
Refeences Hibbele (007) http://wps.penhall.cm/esm_hibbele_eng mech_1 en.wikipedia.g 33 007 S. Kenny, Ph.D., P.Eng.