Chapter 3  Vectors. Arithmetic operations involving vectors. A) Addition and subtraction.  Graphical method  Analytical method Vector components


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1 Chpter 3  Vectors I. Defnton II. Arthmetc opertons nvolvng vectors A) Addton nd sutrcton  Grphcl method  Anltcl method Vector components B) Multplcton Revew of ngle reference sstem 90º<θ <80º 90º 0º<θ <90º 80º θ θ 0º Orgn of ngle reference sstem 80º<θ 3 <70º 70º θ 3 θ 4 70º<θ 4 <360º Angle orgn Θ 4 300º60º
2 I. Defnton Vector quntt: quntt wth mgntude nd drecton. It cn e represented vector. Emples: dsplcement, veloct, ccelerton. Dsplcement does not descre the oect s pth. Sme dsplcement Sclr quntt: quntt wth mgntude, no drecton. Emples: temperture, pressure II. Arthmetc opertons nvolvng vectors Vector ddton:  Geometrcl method Rules: s commuttve lw) 3.) s ) c c) ssoctve lw) 3.)
3 Vector sutrcton: d ) 3.3) Vector component: proecton of the vector on n s. cosθ snθ 3.4) Sclr components of tnθ 3.5) Vector mgntude Vector drecton Unt vector: Vector wth mgntude. No dmensons, no unts. ˆ, ˆ, ˆ unt vectors n postve drecton of,, es ˆ ˆ 3.6) Vector component Vector ddton:  Anltcl method: ddng vectors components. r )ˆ ) ˆ 3.7) 3
4 4 Vectors & Phscs: The reltonshps mong vectors do not depend on the locton of the orgn of the coordnte sstem or on the orentton of the es.  The lws of phscs re ndependent of the choce of coordnte sstem. θ θ ' 3.8) ' ' Multplng vectors:  Vector sclr:  Vector vector: Sclr product sclr quntt s f 3.9) cos dot product) ) 90 0 cos 0 ) 0 cos Rule: ) cos90 cos0 Multplng vectors:  Vector vector Vector product vector sn ˆ ) ˆ ) )ˆ c c cross product) Mgntude Angle etween two vectors: cosϕ
5 0 sn 0 0 ) sn 90 ) Vector product Drecton rght hnd rule Rule: ) 3.) c perpendculr to plne contnng, ) Plce nd tl to tl wthout lterng ther orenttons. ) c wll e long lne perpendculr to the plne tht contns nd where the meet. 3) Sweep nto through the smllest ngle etween them. Rghthnded coordnte sstem Lefthnded coordnte sstem 5
6 sn ) ) ) ˆ ˆ 4: If B s dded to C 3 4, the result s vector n the postve drecton of the s, wth mgntude equl to tht of C. Wht s the mgntude of B? Method Method Isosceles trngle B C B 3ˆ 4 ˆ) D D ˆ θ C D C tnθ 3/ 4) θ 36.9 B 3ˆ 4 ˆ) 5 ˆ B 3ˆ ˆ D B 9 3. θ B / θ sn B Dsn 3. D B/ B 50: A fre nt goes through three dsplcements long level ground: d for 0.4m SW, d 0.5m E, d 3 0.6m t 60º North of Est. Let the postve drecton e Est nd the postve drecton e North. ) Wht re the nd components of d, d nd d 3? ) Wht re the nd the components, the mgntude nd the drecton of the nt s net dsplcement? c) If the nt s to return drectl to the strtng pont, how fr nd n wht drecton should t move? ) ) d4 d d 0.8ˆ 0.8 ˆ) 0.5ˆ 0.ˆ 0.8 ˆ) m N d 0.4cos45 0.8m D d4 d3 0.ˆ 0.8 ˆ) 0.3ˆ 0.5 ˆ) 0.5ˆ 0.4 ˆ) m D d m E 0.4sn D m d 0.5m d 45º d d 4 d 3 d 0 d3 0.6cos m d3 0.6sn m 0.4 θ tn North of Est c) Return vector negtve of net dsplcement, D0.57m, drected 5º South of West 6
7 53: d 5 ˆ 6ˆ 4ˆ d ˆ ˆ 3ˆ d 4ˆ 3 ˆ ˆ 3 ) r d d d3? ) Angle etween r nd? c) Component of d long d? d) Component of d perpendculr to d nd n plne of d, d? ) r d d d 4ˆ 5 ˆ 6ˆ) ˆ ˆ 3ˆ) 4ˆ 3 ˆ ˆ) 9ˆ 6 ˆ 7ˆ 3 r ˆ 7 ) r cosθ 7 θ cos 3.88 r m d perp d d c) d d dd cosθ cosθ d d θ d // d d d// d cosθ d 3.m d d 3.74 d m d d d) d d// dperp d perp m d m 30: If d ˆ 4ˆ 3ˆ d 5ˆ ˆ ˆ d d) d 4d)? d d) contned n d, d) plne d 4d) 4 d d) 4 perpendculr to d, d) plne perpendculr to cos Tp: Thn efore clculte!!! 54: Vectors A nd B le n n plne. A hs mgntude 8.00 nd ngle 30º; B hs components B 7.7, B Wht re the ngles etween the negtve drecton of the s nd ) the drecton of A, ) the drecton of AB, c) the drecton of AB3)? ˆ A 30º ) Angle etween nd A B ) Angle, A B) C ngle ˆ, ˆ ecuse C perpendculr plne A, B) ) 90 c) Drecton A B 3ˆ) D E B 3ˆ 7.7ˆ 9. ˆ 3ˆ ˆ D A E ˆ ˆ ˆ 5.4 ˆ 94.6ˆ 3 D ˆ D ˆ 8.39ˆ 5.4 ˆ 94.6ˆ) 5.4 ˆ 5.4 cos D θ θ 99 D
8 39: A wheel wth rdus of 45 cm rolls wthout sleepng long horontl floor. At tme t the dot P pnted on the rm of the wheel s t the pont of contct etween the wheel nd the floor. At lter tme t, the wheel hs rolled through onehlf of revoluton. Wht re ) the mgntude nd ) the ngle reltve to the floor) of the dsplcement P durng ths ntervl? Vertcl dsplcement: R 0. 9m Horontl dsplcement: πr). 4m r.4m )ˆ 0.9m) ˆ r m R tnθ θ 3.5 πr d 6: Vector hs mgntude of 5.0 m nd s drected Est. Vector hs mgntude of 4.0 m nd s drected 35º West of North. Wht re ) the mgntude nd drecton of )?. ) Wht re the mgntude nd drecton of )?. c) Drw vector dgrm for ech comnton. W   N 5º S E 5ˆ ˆ 4sn 35 4cos35 ˆ.9ˆ 3.8 ˆ ).7ˆ 3.8 ˆ ) m 3.8 tnθ θ ) 7.9ˆ 3.8 ˆ m 3.8 tnθ θ or ) North of West 8
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