Couse Oulne. MATLAB uoal. Moon of syses ha can be dealzed as pacles Descpon of oon, coodnae syses; Newon s laws; Calculang foces equed o nduce pescbed oon; Deng and solng equaons of oon 3. Conseaon laws fo syses of pacles Wo, powe and enegy; Lnea pulse and oenu Angula oenu 4. Vbaons Exa opcs Chaacescs of baons; baon of fee DO syses Vbaon of daped DO syses oced Vbaons 5. Moon of syses ha can be dealzed as gd bodes Descpon of oaonal oon neacs; geas, pulleys and he ollng wheel Ineal popees of gd bodes; oenu and enegy Dynacs of gd bodes
Pacle Dynacs concep checls Undesand he concep of an neal fae Be able o dealze an engneeng desgn as a se of pacles, and now when hs dealzaon wll ge accuae esuls Descbe he oon of a syse of pacles (eg coponens n a fxed coodnae syse; coponens n a pola coodnae syse, ec) Be able o dffeenae poson ecos (wh pope use of he chan ule!) o deene elocy and acceleaon; and be able o negae acceleaon o elocy o deene poson eco. Be able o descbe oon n noal-angenal and pola coodnaes (eg be able o we down eco coponens of elocy and acceleaon n es of speed, adus of cuaue of pah, o coodnaes n he cylndcal-pola syse). Be able o cone beween Caesan o noal-angenal o pola coodnae descpons of oon Be able o daw a coec fee body daga showng foces acng on syse dealzed as pacles Be able o we down Newon s laws of oon n ecangula, noal-angenal, and pola coodnae syses Be able o oban an addonal oen balance equaon fo a gd body ong whou oaon o oang abou a fxed axs a consan ae. Be able o use Newon s laws of oon o sole fo unnown acceleaons o foces n a syse of pacles Use Newon s laws of oon o dee dffeenal equaons goenng he oon of a syse of pacles Be able o e-we second ode dffeenal equaons as a pa of fs-ode dffeenal equaons n a fo ha MATLAB can sole
Ineal fae non acceleang, non oang efeence fae Pacle pon ass a soe poson n space Poson Veco Velocy Veco ( ) = x( ) + y( ) + z( ) () = () + () + () x y z d dx dy dz = ( x+ y+ z) = + + d d d d dx dy dz x() = y() = z() = d d d Decon of elocy eco s paallel o pah Magnude of elocy eco s dsance aeled / e Acceleaon Veco O () (+d) d d dy () () () () x d a = a z x + ay + az = ( x+ y+ z) = + + d d d d Also Pacle Kneacs d d () x y d () () z x y z d x d y d z a = = a = = a = = d d d d d d d dy () x d a () () z x = x ay = y az = z dx dy dz () d pah of pacle
Pacle Kneacs Sagh lne oon wh consan acceleaon = X + V + a = ( V + a) = a a Te/elocy/poson dependen acceleaon use calculus = X + () d = V + a() d d g() a = = f ( ) d = g() d d f () V x () dx g() = = f ( x) d = () d d f () X d = ax ( ) d d dx d = ax ( ) = ax ( ) dx d dx () x () V d = a( x) dx
Pacle Kneacs Ccula Moon a cons speed θ = ω s= Rθ V = ωr ( cosθ snθ ) R( sn cos ) = R + = ω θ+ θ = V V a= ω R(cosθ+ sn θ) = ω Rn= n R Geneal ccula oon ( cosθ snθ ) R( sn cos ) = R + = ω θ+ θ = V a= Rα( snθ+ cos θ) Rω (cosθ+ sn θ) dv V = αr + ω Rn= + n d R ω = dθ / d α = dω / d = d θ / d s = Rθ V = ds / d = Rω snθ cosθ R n θ=ω Rsn θ Rcosθ snθ cosθ R n θ Rsn θ Rcosθ
Pacle Kneacs Moon along an abay pah = V dv V a= + n d R R n s Radus of cuaue R = d x d y + ds ds = xs ( ) + y(s) Pola Coodnaes e θ e d dθ = e + e d d θ d dθ d θ d dθ a= e + + e d d d d d θ θ
Usng Newon s laws Calculang foces equed o cause pescbed oon Idealze syse ee body daga Kneacs (descbe oon usually goal s o fnd foula fo acceleaon) =a fo each pacle. M c = (fo seadly o non-oang gd bodes o faes only) Sole fo unnown foces o acceleaons (us le sacs)
Usng Newon s laws o dee equaons of oon. Idealze syse. Inoduce aables o descbe oon (ofen x,y coods, bu we wll see ohe exaples) 3. We down, dffeenae o ge a 4. Daw BD 5. = a 6. If necessay, elnae eacon foces 7. Resul wll be dffeenal equaons fo coods defned n (), e.g. d x dx + λ + x = Y snω d d 8. Idenfy nal condons, and sole ODE
Moon of a poecle n eahs gay X Y Z d d = + + = Vx Vy Vz = + + X V ( ) ( ) = X + Vx + Y + Vy + Z + Vz g ( V ) ( V ) ( V g) = + + a= g x y z
Reaangng dffeenal equaons fo MATLAB Exaple d y d dy + ζωn + ωny = d Inoduce = dy / d Then d y d = ζωn ωny Ths has fo dw d y = f(, w) w =
Conseaon Conseaon laws fo Laws pacles: concep Concep checls Checls Know he defnons of powe (o ae of wo) of a foce, and wo done by a foce Know he defnon of nec enegy of a pacle Undesand powe-wo-nec enegy elaons fo a pacle Be able o use wo/powe/nec enegy o sole pobles nolng pacle oon Be able o dsngush beween conseae and non-conseae foces Be able o calculae he poenal enegy of a conseae foce Be able o calculae he foce assocaed wh a poenal enegy funcon Know he wo-enegy elaon fo a syse of pacles; (enegy conseaon fo a closed syse) Use enegy conseaon o analyze oon of conseae syses of pacles Know he defnon of he lnea pulse of a foce Know he defnon of lnea oenu of a pacle Undesand he pulse-oenu (and foce-oenu) elaons fo a pacle Undesand pulse-oenu elaons fo a syse of pacles (oenu conseaon fo a closed syse) Be able o use pulse-oenu o analyze oon of pacles and syses of pacles Know he defnon of esuon coeffcen fo a collson Pedc changes n elocy of wo colldng pacles n D and 3D usng oenu and he esuon foula Know he defnon of angula pulse of a foce Know he defnon of angula oenu of a pacle Undesand he angula pulse-oenu elaon Be able o use angula oenu o sole cenal foce pobles/pac pobles
Wo-Enegy elaons fo a sngle pacle Rae of wo done by a foce (powe deeloped by foce) P = O P Toal wo done by a foce W d W = = d O () P Knec enegy T= = + + ( ) x y z Powe-nec enegy elaon Wo-nec enegy elaon dt P = d W = d = T T O P
Poenal Enegy Poenal enegy of a conseae foce (pa) U( ) = d+ consan = gad( U ) Type of foce Gay acng on a pacle nea eahs suface Gaaonal foce exeed on ass by ass M a he ogn Poenal enegy U = gy GM U = y O P oce exeed by a spng wh sffness and unseched lengh L oce acng beween wo chaged pacles U = ( L ) QQ +Q +Q U = 4πε oce exeed by one olecule of a noble gas (e.g. He, A, ec) on anohe (Lennad Jones poenal). a s he equlbu spacng beween olecules, and E s he enegy of he bond. 6 a a U = E
Enegy Relaon fo a Conseae Syse Inenal oces: (foces exeed by one pa of he syse on anohe) Exenal oces: (any ohe foces) R Syse s conseae f all nenal foces ae conseae foces (o consan foces) 4 R R 3 R 3 R R 3 R 3 3 3 Enegy elaon fo a conseae syse 4 3 3 = = Toal KE T Toal PE U Exenal Powe P () Exenal wo W P() d = Toal KE T Toal PE U 4 3 3 ( ) W = T + U T + U Specal case zeo enal wo: + = + T U T U KE+PE = consan
Ipulse-Moenu fo a sngle pacle Defnons Lnea Ipulse of a foce Lnea oenu of a pacle I= () d p= O () Ipulse-Moenu elaons d = p d I= p p = p=p O () = p=p
Ipulse-Moenu Ipulse-oenu fo fo a syse a syse of pacles of pacles R oce exeed on pacle by pacle Exenal foce on pacle Velocy of pacle 4 R R 3 R 3 R R 3 R 3 3 3 Toal Exenal oce Toal Exenal Ipulse 4 3 () I = () d 4 3 Ipulse-oenu fo he syse: d = p d I = p p = 3 Toal oenu p = 3 Toal oenu p Specal case zeo enal pulse: p = p (Lnea oenu conseed)
Collsons x A A x A A * x B x B B B Moenu Resuon foula + = + A B A B A x B x A x B x ( ) = e B A B A = ( + e) + ( ) B B A B A A = + ( + e) + B ( ) A A B B A A B A A A A B n B B B Moenu Resuon foula + = + B A B A B A B A B A B A B A ( ) ( ) ( e) ( ) = + n n + + ( ) ( ) B B A B A = + e n n B A ( ) ( ) A A B B A = + + e n n B A
Angula Ipulse-Moenu Equaons fo a Pacle pacle Angula Ipulse Angula Moenu Ipulse-Moenu elaons A= () () d h= p= d = h d x O y () () z A= h h Specal Case A= h = h Angula oenu conseed Useful fo cenal foce pobles (when foces on a pacle always ac hough a sngle pon, eg planeay gay)