SCALAR A physical quantity that is completely characterized by a real number (or by its numerical value) is called a scalar. In other words, a scalar
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2 SCALAR A phscal quantt that s completel charactered b a real number (or b ts numercal value) s called a scalar. In other words, a scalar possesses onl a magntude. Mass, denst, volume, temperature, tme, energ, area, speed and length are eamples to scalar quanttes.
3 ECTOR Several quanttes that occur n mechancs requre a descrpton n terms of ther drecton as well as the numercal value of ther magntude. Such quanttes behave as vectors. Therefore, vectors possess both magntude and drecton; and the obe the parallelogram law of addton. Force, moment, dsplacement, veloct, acceleraton, mpulse and momentum are vector quanttes.
4 Tpes of ectors Phscal quanttes that are vectors fall nto one of the three classfcatons as free, sldng or fed. A free vector s one whose acton s not confned to or assocated wth a unque lne n space. For eample f a bod s n translatonal moton, veloct of an pont n the bod ma be taen as a vector and ths vector wll descrbe equall well the veloct of ever pont n the bod. Hence, we ma represent the veloct of such a bod b a free vector. In statcs, couple moment s a free vector.
5 A sldng vector s one for whch a unque lne n space must be mantaned along whch the quantt acts. hen we deal wth the eternal acton of a force on a rgd bod, the force ma be appled at an pont along ts lne of acton wthout changng ts effect on the bod as a whole and hence, consdered as a sldng vector.
6 A fed vector s one for whch a unque pont of applcaton s specfed and therefore the vector occupes a partcular poston n space. The acton of a force on a deformable bod must be specfed b a fed vector.
7 Prncple of Transmssblt The eternal effect of a force on a rgd bod wll reman unchanged f the force s moved to act on ts lne of acton. In other words, a force ma be appled at an pont on ts gven lne of acton wthout alterng the resultant eternal effects on the rgd bod on whch t acts.
8 Equalt and Equvalence of ectors Two vectors are equal f the have the same dmensons, magntudes and drectons. Two vectors are equvalent n a certan capact f each produces the ver same effect n ths capact.
9 PROPERTIES OF ECTORS Addton of ectors s done accordng to the parallelogram law of vector addton. ( ) ( ) M M or
10 Subtracton of ectors s done accordng to the parallelogram law. ( ) Z Z Multplcaton of a Scalar and a ector a a a ( b ) ab ( a b) a b a( ) a a
11 nt ector A unt vector s a free vector havng a magntude of 1 (one) as n ( n or e) It descrbes drecton. The most convenent wa to descrbe a vector n a certan drecton s to multpl ts magntude wth ts unt vector. n n 1 and have the same unt, hence the unt vector s dmensonless. Therefore, ma be epressed n terms of both ts magntude and drecton separatel. (a scalar) epresses the magntude and n (a dmensonless vector) epresses the drectonal sense of.
12 ector Components and Resultant ector Let the sum of and be. Here, and are named as the components and s named as the resultant. Sne theorem sn β snα sn γ Cosne theorem cosγ
13 The relatonshp between a force and ts vector components must not be confused wth the relatonshp between a force and ts perpendcular (orthogonal) proectons onto the same aes. For eample, the perpendcular proectons of force F onto aes a and b are F a and F b, whch are parallel to the vector components of F 1 and F 2. a b F 1 a F //a //b F a F a F 2 b F b b Components: F 1 and F 2 Proectons: F a and F b
14 It s seen that the components of a vector are not necessarl equal to the proectons of the vector onto the same aes. The components and proectons of F are equal onl when the aes a and b are perpendcular. F 1 a F //a //b F a F a b a F 2 b F b b Components: F 1 and F 2 Proectons: F a and F b
15 Cartesan Coordnates Cartesan Coordnate Sstem s composed of 90 (orthogonal) aes. It conssts of and aes n two dmensonal (planar) case,, and aes n three dmensonal (spatal) case. - aes are generall taen wthn the plane of the paper, ther postve drectons can be selected arbtrarl; the postve drecton of the aes must be determned n accordance wth the rght hand rule.
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17 ector Components n Two Dmensonal (Planar) Cartesan Coordnates unt vector along the as,, unt vector along the as, ( ) ( ) tan θ 2 2 θ
18 ector Components n Three Dmensonal (Spatal) Cartesan Coordnates ( ) ( ) ( ) unt vector along the as,, unt vector along the as,, unt vector along the as,, ( ) 2 2 2
19 Poston ector: It s the vector that descrbes the locaton of one pont wth respect to another pont. In two dmensonal case B ( B, B ) r B/A A ( A, A ) r B/A ( ) ( ) B A B A
20 In three dmensonal case B ( B, B, B ) r B/A ( ) ( ) ( ) B r B/A A ( A, A, A ) A B A B A
21 Dot (Scalar) Product A scalar quantt s obtaned from the dot product of two vectors. cos cos a s rrelevant multplcaton of order a α α,, cos,, cos α In terms of unt vectors n Cartesan Coordnates;
22 Normal and Parallel Components of a ector wth respect to a Lne λ n // θ Magntude of parallel component: // cosθ n n cosθ cosθ 1 Parallel component:, // // n ( n) n Normal (Orthogonal) component: //
23 Cross (ector) Product: The multplcaton of two vectors n cross product results n a vector. Ths multplcaton vector s normal to the plane contanng the other two vectors. Its drecton s determned b the rght hand rule. Its magntude equals the area of the parallelogram that the vectors span. The order of multplcaton s mportant. θ a, snθ ( ) ( a ) ( a ) ( Y ) Y snθ θ
24 ,,,, sn,, sn In terms of unt vectors n Cartesan Coordnates;
25 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] [ ] [ ]
26 Med Trple Product: It s used when tang the moment of a force about a lne. ( ) ( ) ( ) or
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