CHPTER 1 Introductory Concpts Elmnts of Vctor nalysis Nwton s Laws Units Th basis of Nwtonian Mchanics D lmbrt s Principl 1 Scinc of Mchanics: It is concrnd with th motion of matrial bodis. odis hav diffrnt scals: Microscropic, macroscopic and astronomic scals. In mchanics - mostly macroscopic bodis ar considrd. Spd of motion - srvs as anothr important variabl - small and high (approaching spd of light). 2 1
In Nwtonian mchanics - study motion of bodis much biggr than particls at atomic scal, and moving at rlativ motions (spds) much smallr than th spd of light. Two gnral approachs: Vctorial dynamics: uss Nwton s laws to writ th quations of motion of a systm, motion is dscribd in physical coordinats and thir drivativs; nalytical dynamics: uss nrgy lik quantitis to dfin th quations of motion, uss th gnralizd coordinats to dscrib motion. 3 1.1 Vctor nalysis: Scalars, vctors, tnsors: Scalar: It is a quantity xprssibl by a singl ral numbr. Exampls includ: mass, tim, tmpratur, nrgy, tc. Vctor: It is a quantity which nds both dirction and magnitud for complt spcification. ctually (mathmatically), it must also hav crtain transformation proprtis. 4 2
Ths proprtis ar: vctor magnitud rmains unchangd undr rotation of axs. x: forc, momnt of a forc, vlocity, acclration, tc. gomtrically, vctors ar shown or dpictd as dirctd lin sgmnts of propr magnitud and dirction. 5 = (unit vctor) if w us a coordinat systm, w dfin a basis st ( iˆ, ˆ, kˆ ): w can writ = xi + y + zk Z or, w can also us th thr componnts and Y dfin X { } = {,, } x y z T 6 3
Th thr componnts x, y, z can b usd as 3-dimnsional vctor lmnts to spcify th vctor. Thn, laws of vctor-matrix algbra apply. Tnsors: scalar - an array of zro dimnsion vctor - an array of on dimnsion 7 quantitis which nd arrays of two or highr dimnsion to spcify thm compltly - calld tnsors of appropriat rank. gain - to b a tnsor, th obct must also satisfy crtain transformation proprtis of rotation and translation. Exs: Scond-ordr tnsors: strss at a point in dformabl body - strss tnsor has nin componnts (a 3x3 matrix in a rprsntation whn th basis is dfind), inrtia tnsor (again, a 3x3 matrix in usual notation) xprssing mass distribution in a rigid body. 8 4
TYPES OF VECTORS: Considr a forc F acting on a body at point P. Th forc has a lin F of action. This forc P can lad to translation of th rigid body, rotation of th rigid body about som point, as wll as dformation of th body. 9 P 1 F or P 2 F F Th sam forc is now acting at two diffrnt points P 1, P 2 of th body, i.., th lins of action ar distinct. sam translational ffct th translational ffct dpnds only on magnitud and dirction of th forc, not on its point of application or th lin of actionfr vctors 10 5
P 1 F F or Th forc has th sam lin of action in th two cass. Th points of application (P 1 and P 2 ) ar diffrnt but momnt about vry point is th sam sam rotational ffct (as wll as translational ffct): ffct of vctor F dpnds on magnitud, dirction as wll as lin of action - sliding vctors P 2 F 11 If th body is dformabl, th ffct of forc is diffrnt dpnding its point of application; whthr th forc acts at point P 1 or P 2. Thus, in such a cas, th point of application is also crucial - bound vctors. P 1 F or F P 2 dformation 12 6
Equality of vctors: For fr vctors and, = if and only if and hav th sam magnitud and th sam dirction. Unit vctors: If is a vctor with magnitud, / is a vctor along with unit lngth = / or =. 13 = 1 unit vctor in th dirction of. ddition of vctors: Considr two vctors and. Thir addition is a vctor C givn by C = +. lso C= + (addition is commutativ). Th rsult is also a vctor. 14 7
C = + Graphically, on can us th paralllogram rul of vctor addition. For mor than two vctors, on can add C squntially - polygon of vctors. Considr Th addition of vctors,, and D. C = +, E = ( + ) + D = C + D D or E E = ( + D) + C = + ( + D). O 15 COMPONENTS OF VECTOR: Considr th vctor addition for and : graphically: O C = + W can intrprt and to b componnts of th vctor C. Clarly, th componnts of ar non-uniqu. s an F E C xampl, E and F ar also componnts. O W can mak it mor systmatic. 16 C C 8
Lt 1, 2, 3 - thr linarly indpndnt unit vctors (not ncssarily orthogonal) and lt b a vctor. W can writ = 1 + 2 + 3 whr i, i = 1, 2, 3 ar componnts of along th dirctions spcifid by unit vctors, i = 1, 2, 3. i Thn : = 1 1 + 22 + 33 i, 1 2 3 i = 1, 2, 3 ar uniqu scalar componnts 3 1 2 17 Lt = 1 1 + 2 2 + 33 b anothr vctor, with componnts xprssd in sam unit vctors. W can thn writ th sum as C= + = ( 1 + 1 ) 1 + ( 2 + 2 ) 2 + ( 3 + 3) 3 Th componnts of th vctor C ar thn C 1 = 1 + 1, C 2 = 2 + 2 and C 3 = 3 + 3. 18 9
Th mor familiar cas of unit vctors is th Cartsian coordinat systm - (x, y, z) Lt i,, k - unit vctors along x, y and z dirctions. Thn =xi+y + 2k whr x, y, z componnts of Z along axs. z k X x i ar y 19 Y SCLR PRODUCT: Dfinition: (DOT) For two vctors and, th dot product is dfins as = cosθ Dot product is Commutativ, i.., = If 3 3 = = θ cosθ i i, i i, i= 1 i= 1 thn = + + 1 1 2 2 3 3 providd th unit vctors ar an orthogonal st, i.., = = =. 1 2 2 3 3 1 0 20 10
VECTOR PRODUCT: Lt, b two vctors that mak an angl θ with rspct to ach othr. Thn, th vctor or cross product is dfind as a vctor C with magnitud C = = sinθ. Lt k b th unit vctor normal to th k θ plan formd by vctors and. It is fixd by th right hand scrw rul. Thn = sinθ k 21 Som proprtis of cross product ar : =- Considr unit vctors for th Cartsian coordinat systm (x,y,z), (i,, k): Thn i = k = k i = 1 k i right-hand rul i i = = k k = 0 i = k ; k = i k i = 22 11
Now, considr cross-product again. Whn vctors and ar xprssd in componnt form: =x i +y +zk = x i + y + z k, Th cross product is valuatd by th opration i k = x y z x y z ( y + ( x z ) i + ( y z ) k y y x z x ) x z 23 SCLR TRIPLE PRODUCT: Considr thr vctors,, and C. Th scalar tripl product is givn by R= C = C ( ) ( ) C Not that th rsult is th sam scalar quantity. It can b intrprtd as th volum of th paralllpipd having th vctors, and C as th dgs. Th sign can b +v or -v. 24 12
VECTOR TRIPLE PRODUCT: Considr vctors,, and C. Thn, vctor tripl product is dfind as a vctor D, givn by D= ( C) Not that ( C) ( ) C On can show that D= ( C)=( C)-( )C DERIVTIVE OF VECTOR: Suppos that a vctor is a function of a scalar u, i.., =( u). W can thn considr chang in vctor associatd with chang in th scalar u. 25 Lt =( u) and ( u + u) ( u) + Thn d ( u) + -( u) = du lim u u 0 or d du = lim u 0 u This is th drivativ of with rspct to u. Ex: Th position vctor r(t) for a particl moving dpnds on tim. W dfin th dr r vlocity to b v( t) = dt lim t t 0 26 13
x: x Considr a particl P moving along a curvd path. Its position dpnds on distanc from som landmark, O, i.. rop rop ( s) whr s is th distanc along th curv. W shall considr chaptr. z O O y r OP (s) drop ( s) ds s P r s P latr in th nxt 27 Som usful proprtis and ruls of diffrntiation ar: d ( + ) = d + d du du du d ( ) ( g( u) ( u)) = g( u) d + dg u ( u) du du du d ( ) ( ) ( ( u) ( u)) = ( u) d u + d u ( u) du du du d ( ) ( ) ( ( u) ( u)) = ( u) d u + d u ( u) du du du Finally, if thn = + + = 1 1 2 2 3 3 i i, d di di = ( ) i + i ( ) du du du 28 14
Som mor usful proprtis: Concpt of a Dyad and Dyadic: Considr two vctors a and b Dyad: It consists of a pair of vctors ab for two vctors a and b. a - calld an antcdnt, b - calld a consqunt. Dyadic: It is a sum of dyads. Suppos that th vctors a and b ar xprssd in trms of a st of unit basis vctors, so that 1, 2, 3 a = a1 1 + a2 2 + a3 3 and b = b1 1 + b2 2 + b3 Thn, a b = ( a1 1 + a2 2 + a3 3)( b1 1 + b2 2 + b3 3) 29 3 and, a b = 3 3 i= 1 = 1 a b i i= 1 = 1 In th Txt of Grnwood, th unit vctors 1 2 3 ar mostly limitd to th Cartsian basis ( i,, k) Conugat Dyadic for th dyadic is obtaind by intrchanging th ordr of vctors a and b and is T dnotd by. Thus, T b a = 3 3 i= 1 = 1 b a i T dyadic is symmtric if =, that is i = i i i = = 3 3 3 3 i= 1 = 1 i i,, i i 30 15
n xampl of a symmtric dyadic is th inrtia dyadic: I 3 3 i= 1 = 1 I i i = I + I + I xx yx zx i i + I + I k i + I + I k + I k k k T dyadic is skw-symmtric if =, that is, it is ngativ of its conugat. Not that symmtry proprty of a dyadic is indpndnt of th unit vctor basis or its orthogonality. i xy yy zy i + I xz i k yz zz 31 Som oprational proprtis: 1. Th sum of two dyadics is a dyadic obtaind by adding th corrsponding lmnts in th sam basis: C = + if and onlyif Ci = i + i 2. Th dot product of a dyadic with a vctor is a vctor. Considr vctors a and b, and th drivd dyadic = a: b. Th dot product with th vctor c is th vctor d givn by d c = ( ab) c = a( b c) which is a vctor in th dirction of vctor a. Not that c = c ( ab) = ( c b) c d 32 16
In gnral, pr-multiplying a vctor by a dyadic post-multiplying th vctor by th sam dyadic. For a symmtric dyadic, th ordr dos not mattr. Considr th product of Inrtia dyadic with th angular vlocity vctor: I ω = ( I i i + I i + I i k + I i + I + I k + I k i + I k + I k k) ω = I ( ω i + ω + ω k) Now, not that xx xy xz yx yy yz zx zy zz x y k I i i ( ω i + ω + ω k) xx x y k = I i( ω i i + ω i + ω i k) = I ω i tc. xx x y k xx x 33 Thus, I ω = ( I ω + I ω + I ω ) i xx x xy y xz z + ( I ω + I ω + I ω ) yx x yy y yz z + ( I ω + I ω + I ω ) k = ω I zx x zy y zz z Sinc dot product of a dyadic and a vctor is a nw vctor, dyadic is rally an oprator acting on a vctor. intrsting symmtric dyadic is th unit dyadic: U i i + + k k For any vctor a, U a i ( i a) + ( a) + k( k a) = a U = a So, it lavs vry vctor unchangd. 34 17
Finally, considr th cross product of a dyadic with a vctor: 3 3 c c. If =, thn 3 3 3 3 i= 1 = 1 c = ( c ) i i i i i= 1 = 1 i= 1 = 1 anothr dyadic i i S Sction 7.5 for Grnwood. 35 1.2 Nwton s Laws: Thr ar thr laws 1. Evry body continus in its stat of rst, or of uniform motion in a straight lin, unlss complld to chang that stat by forcs acting upon it. 2. Th tim rat of chang of linar momntum of a body is proportional to th forc acting upon it and occurs in th dirction in which th forc acts. 36 18
3. To vry action thr is an qual and opposit raction. Laws of motion for a particl: Lt F = applid forc, m = mass of th particl, = vlocity at an instant, p mv = linar momntum of th particl. d d Thn, F=k (p)=k (mv) dt dt =kma (for a body with constant mass) v 37 Hr, k > 0 constant; it is chosn such that k = 1 dpnding on th choic of units. d d F= (p)= (mv) dt dt =ma (for constant mass systm) Sinc, F and a ar vctors, w can xprss thm in th appropriat coordinat systm. x: in a Cartsian coordinat systm (x,y,z): or F=F i + F +F k=m(a i + a +a k) x y z x y z F =ma, F =ma, F =ma x x y y z z 38 19
Imp: Nwton s laws ar applicabl in a spcial rfrnc fram - calld th inrtial rfrnc fram. Not that in practic, any fixd rfrnc fram or rigid body will suffic. REDING SSIGNMENT #1: Th discussion in txt. 1.3 UNITS: On first introducs dimnsions associatd with ach quantity. 39 Not that th quantitis rlatd by Nwton s Laws ar: F - forc, M - mass, L - lngth, T - tim. Sinc thr is on rlation among th four quantitis (scond law), thr of th units ar indpndnt, and th fourth fixd by th rquirmnt of principl of dimnsional homognity. bsolut systm - mass, lngth, and tim ar fundamntal quantitis, whr as forc is considrd a drivd quantity. 40 20
In th absolut systm, th units of various quantitis ar: Mass - kilogram (kg); tim - scond (sc.); lngth - mtr (m) Now, considr th dimnsional rlation: F = ML/T 2 = kg.m/s 2 This unit is calld a Nwton: it is th forc ndd to giv an obct of mass 1 kg an acclration of 1 m/s 2. Gravitational systm - In this cas, lngth, tim, and forc ar assumd fundamntal, whr as mass is drivd. 1 slug = lb.s 2 /ft 41 REDING SSIGNMENT #2: 1.4. Th asics of Nwtonian Mchanics. 42 21
1.5 D LEMERT S PRINCIPLE: Considr Nwton s scond law: F=ma If w writ (F-ma)=0, on can imagin (-ma) to rprsnt anothr forc, th so calld inrtia forc. Thn, w ust hav summation of forcs = 0, that is, an quivalnt statics problm. W will s that this principl has profound significanc whn considring drivation of Lagrang s quations. 43 Ex 1.1: (Txt) O a Considr a masslss rigid rod suspndd from point O in a l θ box which is acclrating to th right at a constant m acclration a. Find: Th tnsion in th Fr body diagram: cabl and th angl θ, whn rod has rachd a stady position rlativ to th box. ma θ mg T 44 g 22
Ex 1.1 O a Fr body diagram: l θ g ma θ T y m mg x D lmbrt s principl: F -ma =0: Tsin-ma=0; x y x F -ma =0: y 2 2 T=m (a +g ), Tcos-mg=0; tan=a/g 45 Ex 1.2: (to clarly point out th diffrnc in componnts and orthogonal proctions) Considr th vctor with = 5, at an angl of 60 with th horizontal. θ=60 +v horizontal i Suppos that w want to xprss it in trms of unit vctors 1, 2 whr = ( i ), = 1135 (=(- i +)/ 2). 1 2 46 23
Th vctor sum can b rprsntd as = + 1 2 or Now =1 1+22 2 =5(cos60 i +sin60 )= + 1 1 2 2 =1 i 1+ 2(- i +)/ 2 (5 i +5 3 )/2= i + (- i +)/ 2 1 1 2 1 2 2 2 135 or i :5/ 2 = / 2 ; : = 5 6 / 2 1 60 1 = [5(1 + 3) + 5 6 ]/ 2 1 2 47 In th abov, componnt of componnt of along along 1 2 is 5(1 + 3) / 2; is 5 6 / 2. 2 1 What about orthogonal proctions of? 2 2 2 135 1 60 1 1 48 24