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 300º-60º

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. Rght-hnded coordnte sstem Left-hnded 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 one-hlf 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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