Third In-Class Exam Soluions Mah 6, Profssor David Lvrmor Tusday, 5 April 07 [0] Th vrical displacmn of an unforcd mass on a spring is givn by h 5 3 cos 3 sin a [] Is his sysm undampd, undr dampd, criically dampd, or ovr dampd? Giv your rasoning! b [5] Exprss h in h ampliud-phas form h A 3 cos δ wih A > 0 and 0 δ < π Labl h ampliud and phas Th phas may b xprssd in rms of an invrs rig funcion c [3] Giv h naural frquncy and naural priod of his spring-mass sysm Soluion a Th sysm is undr dampd bcaus h vrical displacmn h ariss from a characrisic polynomial wih h conjuga pair of roos 3 ± i Alrnaiv Soluion a Th sysm is undr dampd bcaus h displacmn h is a dcaying oscillaion, which is vidn from h dcaying xponnial 3 muliplying h oscillaory rigonomric funcions cos and sin Rmark Boh h 3 and h cos and sin mus play a rol in your rasoning for full crdi! Soluion b By comparing A 3 cos δ A 3 cosδ cos + A 3 sinδ sin, wih h 5 3 cos 3 sin, w s ha A cosδ 5, A sinδ This shows ha A, δ ar h polar coordinas of h poin in h plan whos Carsian coordinas ar 5, Clarly A is givn by A 5 + 5 + 69 3 Bcaus 5, lis in h hird quadran, h phas δ mus saisfy π < δ < 3 π W can xprss δ svral ways A picur shows ha if w us π as a rfrnc hn cosδ π 5, sinδ π, anδ π, 3 3 5 whrby w can xprss h phas by any on of h formulas δ π + cos 5 3, δ π + sin 3, δ π + an 5 Th sam picur shows ha if w us 3 π as a rfrnc hn cos 3 π δ, sin 3 π δ 5, an 3 π δ 5, 3 3 whrby w can xprss h phas by any on of h formulas, δ 3 π sin 5 3 δ 3π cos 3 Only on xprssion for δ is rquird, δ 3 π an 5 Rmark I is incorrc o giv h phas by on of h formulas δ cos 5 3, δ sin 3, δ an 5,
bcaus, by our convnions for h rang of h invrs rigonomric funcions, cos 5 3 lis in π, π, sin 3 lis in π, 0, and an 5 lis in 0, π Soluion c Bcaus h undrlying characrisic polynomial has h conjuga pair of roos 3 ± i, i mus b pz z + 3 + z + 6z + 9 + 6 z + 6z + 5 Thrfor h vrical displacmn h saisfis h diffrnial quaion ḧ + 6ḣ + 5h 0 W can rad off ha h naural frquncy is ω o 5 5 radians pr sc, whrby h naural priod T o is givn by T o π ω o π 5 [6] Whn a 0 gram mass is hung vrically from a spring, a rs i srchs h spring 50 cm Graviaional acclraion is g 980 cm/sc Th mdium impars a damping forc of 60 dyns dyn gram cm/sc whn h spd of h mass is cm/sc A 0 h mass is displacd 3 cm blow is rs posiion and is rlasd wih a upward vlociy of cm/sc Assum ha h spring forc is proporional o displacmn, ha h damping is proporional o vloiciy, and ha hr ar no ohr forcs Formula an iniial-valu problm ha govrns h moion of h mass for > 0 DO NOT solv his iniial-valu problm, jus wri i down! Soluion L h b h displacmn in cnimrs of h mass from is rs posiion a im in sconds, wih upward displacmns bing posiiv Th govrning iniial-valu problm hn has h form sc mḧ + γḣ + kh 0, h0 3, ḣ0, whr m is h mass, γ is h damping cofficin, and k is h spring consan W ar givn ha m 0 grams W obain k by balancing h forc applid by h spring whn i is schd 50 cm wih h wigh of h mass mg 0 980 dyns This givs k 50 0 980, or 0 980 k 980 dyns/cm 50 W obain γ by balancing h damping forc whn h spd of h mass is cm/sc wih 60 dyns This givs γ 60, or γ 60 dyns sc/cm Thrfor h govrning iniial-valu problm is 0ḧ + 60 ḣ + 980h 0, h0 3, ḣ0 Rmark Had w chosn h convnion of downward displacmns bing posiiv hn h govrning iniial-valu problm is 0ḧ + 60 ḣ + 980h 0, h0 3, ḣ0
3 [6] Rcas h ordinary diffrnial quaion v sinvv + v 3 v + cosv as a firs-ordr sysm of ordinary diffrnial quaions Soluion Bcaus h quaion is fourh ordr, h firs-ordr sysm mus hav dimnsion a las four Th simpls such firs-ordr sysm is x x x v d x d x 3 x 3 x, whr x x 3 v v x sinx x + x 3 x 3 + cosx x v [] Considr h vcor-valud funcions x, x a [] Compu h Wronskian W [x, x ] b [] Find A such ha x, x is a fundamnal s of soluions o h sysm x Ax whrvr W [x, x ] 0 c [] Giv a gnral soluion o h sysm ha you found in par b d [] Find h naural fundamnal marix associad wih h iniial im 0 for h sysm ha you found in par b Soluion a Th Wronskian is W [x, x ] d + Soluion b L Ψ Bcaus Ψ AΨ, w hav A Ψ Ψ 0 + 0 + + Soluion c A gnral soluion is x c x + c x c + c Soluion d By using h fundamnal marix Ψ from par b w find ha h naural fundamnal marix associad wih h iniial im 0 is Φ ΨΨ0 0 + 0 3
5 [8] Find a gnral soluion of h sysm d x x d y 3 3 y Soluion Th characrisic polynomial of A is 3 3 pz z raz + da z + z 5 z 3z + 5 Th ignvalus of A ar h roos of his polynomial, which ar 3 and 5 Ths can b xprssd as ± Thn [ A coshi + sinh ] A I 0 [cosh + sinh ] 0 3 cosh + sinh sinh 3 sinh cosh sinh Chck ha A I has rac zro! Thrfor a gnral soluion of h sysm is cosh + x A c c sinh sinh 3 sinh + c cosh sinh 6 [8] Find a gnral soluion of h sysm d x 0 x d y y Soluion Th characrisic polynomial of A 0 is pz z raz + da z z + z Th ignvalus of A ar h roos of his polynomial, which is only Thn A [I + A I] [ ] 0 + 0, + Chck ha A I has rac zro! Thrfor a gnral soluion of h sysm is x A c c + c + 7 [0] Solv h iniial-valu problm d x x d y y, x0 3 y0 3
Soluion Th characrisic polynomial of A is 5 pz z raz + da z z + 5 z + Th ignvalus of A ar h roos of his polynomial, which ar + i and i Thn [ A cosi + sin ] A I ] 0 0 [cos + sin cos sin sin cos 0 0 Chck ha A I has rac zro! Thrfor a gnral soluion of h iniial-valu problm is cos x A x I sin 3 3 cos + sin cos 3 3 sin 6 sin + 3 cos 8 [6] Two inrconncd anks ar filld wih brin sal war A 0 h firs ank conains 5 lirs and h scond conains 30 lirs Brin wih a sal concnraion of 5 grams pr lir flows ino h firs ank a 6 lirs pr hour Wll-sirrd brin flows from h firs ank ino h scond a 8 lirs pr hour, from h scond ino h firs a 7 lirs pr hour, from h firs ino a drain a lir pr hour, and from h scond ino a drain a 3 lirs pr hour A 0 hr ar 7 grams of sal in h firs ank and 8 grams in h scond Giv an iniial-valu problm ha govrns h amoun of sal in ach ank as a funcion of im Soluion L V and V b h volums li of brin in h firs and scond ank a im hours L S and S b h mass gr of sal in h firs and scond ank a im hours Bcaus mixurs ar assumd o b wll-sirrd, h sal concnraion of h brin in h anks a im ar C S /V and C S /V rspcivly In paricular, hs ar h concnraions of h brin ha flows ou of hs anks W hav h following picur 5 gr/li 6 li/hr C gr/li li/hr V li S gr V 0 5 li S 0 7 gr C gr/li 8 li/hr C gr/li 7 li/hr V li S gr V 0 30 li S 0 8 gr C gr/li 3 li/hr W ar askd o wri down an iniial-valu problm ha govrns S and S Th ras work ou so hr will b V 5 + lirs of brin in h firs ank and V 30 lirs in h scond Thn S and S ar govrnd by h
6 iniial-valu problm ds d 5 6 + S 30 7 S 5 + 8 S 5 +, S 0 7, ds d S 5 + 8 S 30 7 S 30 3, S 0 8 You could lav h answr in h abov form Howvr, i can b simplifid o ds d 30 + 7 30 S 5 + S, S 0 7, ds d 8 5 + S 5 5 S, S 0 8 Noic ha h inrval of dfiniion for his iniial-valu problm is 5, 5 9 [] Considr h following MATLAB commands >> syms s Y; f [ ˆ3 + havisid *8 ˆ3 ]; >> diffqn sym DDy 6*Dy + 3*y f; >> qnrans laplacdiffqn,, s; >> algqn subsqnrans, { laplacy,,s,,s, y0, Dy0 }, {Y,, 5}; >> yrans simplifysolvalgqn, Y; >> y ilaplacyrans, s, a [] Giv h iniial-valu problm for y ha is bing solvd b [8] Find h Laplac ransform Y s of h soluion y You may rfr o h abl on h las pag DO NOT ak h invrs Laplac ransform o find y, jus solv for Y s! Soluion a Th iniial-valu problm for y ha is bing solvd is y 6y + 3y f, y0, y 0 5, whr h forcing f can b xprssd ihr as { 3 for 0 <, f 8 for, or in rms of h uni sp funcion as f 3 + u 8 3 Soluion b Th Laplac ransform of h iniial-valu problm is Bcaus L[y ]s 6L[y ]s + 3L[y]s L[f]s L[y]s Y s, L[y ]s sy s y0 sy s +, L[y ]s s Y s sy0 y 0 s Y s + s 5, h Laplac ransform of h iniial-valu problm bcoms s Y s + s 5 6 sy s + + 3Y s L[f]s
7 This simplifis o whrby s 6s + 3Y s + s 7 L[f]s, Y s To compu L[f]s, w wri f as s + 7 + L[f]s s 6s + 3 f 3 + u 8 3 3 + u j, whr by sing j 8 3 w s ha j 8 + 3 8 3 + 6 + + 8 3 6 Rfrring o h abl on h las pag, im wih a 0 and n 3, wih a 0 and n, and wih a 0 and n shows ha L[ 3 ]s 6 s, L[ ]s s 3, L[]s s, whrby im 6 wih c and j 3 6 shows ha L [ u j ] s s L[j]s s L [ 3 + 6 + ] s 6 s s + s + 3 s Thrfor L[f]s L [ 3 + u j ] s 6 6 s s s + s + 3 s Upon placing his rsul ino h xprssion for Y s found arlir, w obain Y s s + 7 + 6 6 s 6s + 3 s s s + s + 3 s 0 [6] Compu h Grn funcion g for h diffrnial opraor D + 3 whr D d d Soluion Th opraor D + 3 has characrisic polynomial ps s + 3 Thrfor is Grn funcion g is givn by [ ] [ ] g L L ps s + 3 Rfrring o h abl on h las pag, im wih a and n givs [ ] g L s + 3
8 [8] Compu h Laplac ransform of f u from is dfiniion Hr u is h uni sp funcion Soluion Th dfiniion of Laplac ransform givs L[f]s lim T 0 s u d lim T s+ d Whn s his limi divrgs o + bcaus in ha cas w hav for vry T > s+ d which clarly divrgs o + as T Whn s > w hav for vry T > s+ d s+ s + whrby T d T, s+t s + + s+ s +, [ L[f]s lim s+t T s + ] + s+ s+ s + s + for s > [8] Find h invrs Laplac ransform L [Y s] of h funcion Y s 3s 3s + 3 s 3s You may rfr o h abl on h las pag Soluion Rfrring o h abl on h las pag, im 6 wih c 3 implis ha L [ 3s Js ] u 3j 3, whr j L [Js] W apply his formula o Js 3s + 3 s 3s Bcaus h dnominaor facors as s s+, w hav h parial fracion idniy 3s + 3 s 3s 3s + 3 s s + 5 s + s + Rfrring o h abl on h las pag, im wih a and n 0, and wih a and n 0 implis ha [ ] [ ] L, L s s + Ths formulas also can b obaind from im wih a and b 0, and wih a and b 0
9 Th abov formulas and h linariy of h invrs Laplac ransform yild [ ] 3s + 3 j L [Js] L s 3s [ 5 L s + 5L [ s s + ] ] L [ s + ] 5 Thrfor L [ Y s ] L [ 3s Js] u 3j 3 u 3 5 3 3 A Shor Tabl of Laplac Transforms L[ n a n! ]s s a n+ for s > a L[ a s a cosb]s s a + b for s > a L[ a sinb]s b s a + b for s > a L[ n j]s n J n s whr Js L[j]s L[ a j]s Js a whr Js L[j]s L[u cj c]s cs Js whr Js L[j]s and u is h uni sp funcion