SCALARS AND ECTORS All phscal uanttes n engneerng mechancs are measured usng ether scalars or vectors. Scalar. A scalar s an postve or negatve phscal uantt that can be completel specfed b ts magntude. Eamples of scalar uanttes nclude length, mass, tme, denst, volume, temperature, energ, area, speed.
ECTOR A vector s an phscal uantt that reures both a magntude and a drecton for ts complete descrpton. Eamples of vectors encountered n statcs are force, poston, and moment. A vector s shown graphcall b an arrow. The length of the arrow represents the magntude of the vector, and the angle between the vector and a fed as defnes the drecton of ts lne of acton. The head or tp of the arrow ndcates the sense of drecton of the vector
TYPES O ECTORS Phscal uanttes 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 unue lne n space. or 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 euall 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.
A sldng vector s one for whch a unue lne n space must be mantaned along whch the uantt acts. When 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.
A fed vector s one for whch a unue 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.
Prncple of Transmssblt (Taşınablrl İles) 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.
Eualt and Euvalence of ectors Two vectors are eual f the have the same dmensons, magntudes and drectons. Two vectors are euvalent n a certan capact f each produces the ver same effect n ths capact.
Addton of ectors s done accordng to the parallelogram prncple of vector addton. To llustrate, the two component vectors ԦA and B are added to form a resultant vector R. Parallelogram law Trangle law A B R A B R B A R A R B
Subtracton of ectors s done accordng to the parallelogram law. R A B A B A B R B Multplcaton of a Scalar and a ector a a a b ab a b a b a a a
ector Addton of orces Epermental evdence has shown that a force s a vector uantt snce t has a specfed magntude, drecton, and sense and t adds accordng to the parallelogram law. Two common problems n statcs nvolve ether fndng the resultant force, nowng ts components, or resolvng a nown force nto two components. ndng a Resultant orce. ndng the Components of a orce.
Addton of Several orces If more than two forces are to be added, successve applcatons of the parallelogram law can be carred out n order to obtan the resultant force.
ector Components and Resultant ector Let the sum of ԦA and B be R. Here, ԦA and B are named as the components and R s named as the resultant. A B A R R B Sne theorem Cosne theorem R A sn A B B sn R sn AB (Magntude of the resultant force can be determned usng the law of nes, and ts drecton s determned from the law of snes.) Note that B A R Cosne theorem R A B AB
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. or eample, the perpendcular proectons of force onto aes a and b are and, whch are parallel to the vector components of and. 1 1 a b a a //a //b a b a b b b Components: 1 and Proectons: a and b
It s seen that the components of a vector are not necessarl eual to the proectons of the vector onto the same aes. The components and proectons of are eual onl when the aes a and b are perpendcular. 1 a //a //b a a b a b b b Components: 1 and Proectons: a and b
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.
CARTESIAN COORDINATES 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.
Cartesan nt ectors In three dmensons, the set of Cartesan unt vectors, Ԧ, Ԧ,, s used to desgnate the drectons of the,, aes, respectvel. ector Components n Two Dmensonal (Planar) Cartesan Coordnates tan
ector Components n Three Dmensonal (Spatal) Cartesan Coordnates unt vector along the as,, unt vector along the as,, unt vector along the as,,
Poston ector: It s the vector that descrbes the locaton of one pont wth respect to another pont. In two dmensonal case In three dmensonal case B ( B, B, B ) r B/A A ( A, A ) B B ( B, B ) r B/A A B A r B/A B A r B/A A ( A, A, A ) B A B A
* When the drecton angles of a force vector are gven; The angles, the lne of acton of a force maes wth the, and aes are named as drecton angles. The nes of these angles are called drecton nes; the specf the lne of acton of a vector wth respect to coordnate aes. In ths case, drecton angles are, and. Drecton nes are, and. = l = m = n
1 1 1 n m l n m l n n
1 1 1 1 1 1 AB AB n * When coordnates of two ponts along the lne of acton of a force are gven;
* When two angles descrbng the lne of acton of a force are gven; rst resolve nto horontal and vertcal components. sn Then resolve the horontal component nto - and -components. sn sn
Addton of Cartesan ectors In two dmensonal case In three dmensonal case
Dot (Scalar) Product A scalar uantt s obtaned from the dot product of two vectors. a rrelevant s multplcaton of order a,,,, 0 0 0 90 1 1 1 0 In terms of unt vectors n Cartesan Coordnates;
Normal and Parallel Components of a ector wth respect to a Lne n // Magntude of parallel component: Parallel component: // n n, // 1 // n n n Normal (Orthogonal) component: //
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 euals the area of the parallelogram that the vectors span. The order of multplcaton s mportant. W a W, sn a a Y Y W sn W
,,,, sn,, sn 1 90 0 0 0 0 In terms of unt vectors n Cartesan Coordnates;
+ +
Med Trple Product: It s used when tang the moment of a force about a lne. W W W W or W W W W W W W W