NUPOC STUDY GUIDE ANSWE KEY N cruiing Commnd
SOLUTIONS TO MATHEMATICS POBLEMS. Susiuing gin soluions sos,i is corrc nsr.. On - ln, coos o foci Fc, nd F -c, nd dno consn disnc s. Tn, using disnc formul nd squring o sids is cn rrin s ic is quion for rol... Trfor, cnr is,. 5. d P F P F d ʹ,, n c l c c c, Fc, F-c, P, 6 m in susiuing d /,,, 7 squr coml 7
6. i i 9 i i i i 9 i 5 i 7. Using Crmr's ul: D 5 5 6 X 6 6 8, Y 5 6 6 7, Z 5 6 6 Tn, 8, 7, z 8. Logrims of s numrs r usd o simlif comlicd numricl comuions inoling roducs, quoins, nd ors of rl numrs. Bs is commonl mlod s i is ll suid o dciml form. T nurl logrim of s, ic gorns mn nurl nomn is inrs funcion of nurl onnil funcion. Tus, ln if nd onl if. 9. r in squr f circumfrnc in f πr r r r πr. A circl i cnr, C, nd rdius r > consiss of ll oins in ln r r unis from C. A oin P, is on circl if nd onl if dc,p r, or disnc formul, [ ] d C,k, P, k k ic is quion of circl of rdius r nd cnr,k. r P, C,k r r r. l l A l l susiuing in for nd in rms of l from rious quions, l l l l/ l/ l l l/ l l l,,.
. A B C B C A A B C A A B B B A A A B A B A ± B ± B AC A B AC A B AC A C A. A sr ncloss mimum olum i minimum surfc r. Tis is rifil using rori formuls for rious gomric ss.. Gring s oins sos is o rol. Is quion is -. -, 8,, - - - -8 - -6 5.. For ringl of s,, nd ig : m A A A d d. An incrmnl scor of circl cn roimd s ringl of s dθ nd ig. T r of ringl is / or / dθ. Ingring is or rng <θ<π ilds: A π A A π dθ π dθ
c. Using olr coordins, diffrnil olum of con of s rdius,, nd ig, is: d. Find olum in firs qudrn using ril ingrl nd mulil four. T ingrion coms: 6... r r rdr r rdrd r dz rdrd V r π π π θ θ π π V z z z dz z z dz z z dddz dddz V z z z z z < < < < < < A d A 6 A d A.5.5.5.5.5.5.5.5
dv 7. To find r, rk ingrl u ino r rgions: A A d [ 6 ] 9 d dd [ 6 ] 6 d d d 8-6 - - 6 - -8 8. T diffrnil olum cn roimd s clindr i diffrnil olum dvπ H or dvπ d: V π V π π d π d.5 6 8 -.5-9... A π π π A π rdrdθ r dθ θ π u u dθ sin d du du d cos cos d cos sin c - - - d sin d - - - cos. / d / c
cos sin c. d.. Using susiuion, f. g. c d d d / c d d d / d sin c 8 cos sin d sin sin du u sin d cos d d du u d cos c scu c cosu u cos sinu cosu sinu cosu scu nudu c c d du u d d d du u d
. i..... π / π / π / π r d d c r sin θdφdθdr π r sin θdθ dr π r cos θ π d c d cos sin d cos π / sin dr cos cos sin d d c c c π rdr sin cos cos c. d d 5 d. d d. sin ʹ cos cos ʹ sin nʹ d d cos sc sin cos cos cos sin sin cos cos sin cos
co ʹ d d sin csc cos sin d sc ʹ d cos cos sin cos cos sc n d csc ʹ d sin sin cos sin sin csc co sin sin cos cos sin sin cos sin cos sin f. g.. i. d ln d d d ln sin cos sin d d d d 5 sin cos cos 5 d d sin cos cos sin cos cos cos sin sin sin cos sin cos sin cos sin sin cos cos. An ingrl is summion rocss, ing sum of roducs of lng of n inrl nd lu of funcion oin in inrl. B Fundmnl Torm of Clculus, if f nd F r coninuous on, nd F' f for, n fd r n F F * f Δ * is som oin n nd nd Δ
An indfini ingrl s no ounds, il dfini ingrl is oundd coninuous funcions. An ingrl in D sc rrsns r undr gr of funcion. In D sc, ingrl rrsns olum. Ingrls cn lso usd o find moun of ork ssocid i forc cing on moing oc.. A diffrnil is n incrmnl lu of n indndn ril, d, or funcion, d'd. In gnrl, drii is usd o rrsn r of cng. A drii is limi of diffrnc-quoin: f fc fʹ c lim c c is drii of f c. T s drii rrsns slo of funcion oin. T nd drii rrsns curur of funcion oin.. D D f Δ f lim Δ Δ Δ lim Δ Δ Δ Δ lim Δ Δ lim Δ Δ 5. sin lim : Sinc f / g is indr min, us L'Hôil's ul, fʹ f lim L imlis lim L c c g ʹ g lim sin lim cos 6. Ingrion Prs : Cin ul : Quoin ul : d d fg d f d g ud u du [ fʹ ] g g ʹ gf ʹ fg ʹ [ g ] 7.. f fʹ fʹ ʹ, locl mimum - - - -
π/ π π/ π. f sin fʹ cos f ʹ ʹ sin π/, locl mimum π/ locl minimum π, inflcion oin. - c. π f fʹ fʹ ʹ no rm 6 - - - d. f 7 fʹ 6 7 fʹ ʹ 6 7/6, -9/ locl minimum - 6 - -. f fʹ f ʹ ʹ 6, locl mimum /, -/7 locl minimum /, -/7 inflcion oin - - - -8 - f. f fʹ fʹ ʹ 5.5 - - ±,/ locl mimum, locl minimum ±.67,.77 nd ±.5,. r inflcion oins.
,z 8. ʹ ʹ ʹ no rm no inf lcion o in s - 9. For rol, i ricl is, r,k nd focus,k quion in sndrd form is, ʹ ʹ ʹ k k Erm r found r ', nd rol ill conc urd if '' > or conc donrd if '' <. Tus for ', if > n,k is minimum, if < n,k is mimum.. A sr z cn rrsnd in form, ru, cos cosu î cos sinu ĵ sin kˆ r u π π π T surfc ingrl is rrsnd As da r n S r r u r r u u r r u cos cosu î [ cos cos u cos sin u cos sin ] [ cos cos u sin u cos cos ] / [ cos cos cos ] cos dud cos sinu ĵ cos sin kˆ T limis of ingrion on dnd on ll of ll o surfc. For ml, if lf sr s osd, As π / π π π π / cos dud cos d sin π / π / / ic is lf surfc r of sr.
.. ʹ ʹ 6ʹ 9 5 Tr soluion, λ, n crcrisic quion is λ 6λ λ λ 6 i rl roos λ, 6 Tn gnrl soluion for omognous quion '' 6' is: C 6 C Sinc gin quion is nonomognous, gnrl soluion consiss of gnrl soluion of omognous quion lus riculr soluion of nonomognous quion. Sinc rig nd sid is consn coic of riculr soluion is C C. Susiuing is soluion ino diffrnil quion gis 6C C 9 5 Tus, gnrl soluion o diffrnil quion is C 6 C. c. dn N d dn d N dn d N lnn N d d d d d d C C Using iniil condiion, n C-/
W. L f s Drii nd Drii. T soluion cn rcd roug ril nd rror ring diffrn ss nd diffrniing Ar o find mimum. cngl: L W 8 A L W A 8 WW da 8 W n W, L A m, rcngl 8 f Smi-circl: π 8 A π A A π 8 π m,smicircl f Tus, smi-circl inscris lrgs r.
. V L W LW W L Vʹ LW W L S V' o find mimum or minimum olum LW W L W L LW Sol using qudric formul W L ± W L 8LW To find mimum, find n V'' <, V ʹ ʹ W L Eluing using, mimum iss for W L W L 8LW 5. T moun of im fl snds in ir is sm s im i ks for o runnrs o rc c or. Tus, d 5 mils r 5 m d m r mils fl fl 6. Llc Trnsform: Ls Fourir Sris: Tlor Sris: f f n n s n n fd n f c n c n! 7. L'Hoil's ul is usd n luing limi of funcion of form q f/g roducs n indrmin form / n f nd g r coninuous nd diffrnil. 8. Of 6 ossil cominions, 6 of m roduc rsul of 7. Tus, 6 7 6 6 9. dn λn d dn λd N λ N C, C N N N λ Tn, in rs N N N N λ ln λ λ ln λ For oulion N N N ln ln N ln ln ln ln.7 rs
f A. L f g', n on uni sis A for conninc r undr funcion cur is d g g A Also, rcngulr r is f g' A Tis is sisfid g Tus, f g'. T olg of or sourc quls olg dro cross rsisor nd ccior. Aloug V,, nd C r consn, crg on ccior, Q, nd currn, I, r funcions of im, nd IdQ/d. Q V I C dq Q V d C dq d Q CV C Ingring, dq Q CV C lnq CV C d K A, Q, n K ln-cv ln Q CV C k onnil of c sid, Q Q CV CV dq I d V C C C. An ml of o quions in o unknon funcions nd r gin ʹ ʹ Tis cn rin in Mri form s, ʹ A ʹ ʹ A ʹ T ignlus r found from ril soluion,! λ! n, A Iλ soling for ignlus, λ, nd igncors, nonriil soluion of diffrnil quions coms,!!,, λ λ n n λn!. n n nn! n 55
. Using mod of comosi rs, l M ol mss, cnr of mss coordin, nd ρ dnsi. Tn, M m ρ [ ] ρ [] ρ[ ] [ ] m 5. Diffrnil quions cn clssifid ccording o r criri: Ordr: Tis is igs ordr drii of unknon funcion,. ', '', ''' r s, nd, rd ordr Linri: A diffrnil quion is linr in unknon funcion nd is driis if i conins rms of form, ', '', ''' A nonlinr quion m conin rms suc s, '', ''' / Homogni: If diffrnil quion, i is omognous. Emls: nd ordr, linr, omognous quion ʹ ʹ nd ordr, nonlinr ʹ ʹ ʹ 6. A gnrl soluion, of nonomognous quion is of form r is gnrl soluion, nd is riculr soluion. To find gnrl soluion, sol omognous quion. Assum, ʹ ʹ ʹ C λ Cλ λ Cλ λ susiuing ino originl omognous quion gis λ C λ λ 5λC 5λ 6 λ, λ 6C λ Tus, gnrl soluion is C C
To oin riculr soluion, ssum, ʹ ʹ ʹ B susiuing ino quion nd diiding - gis, B Tus, riculr soluion is Finll, B B B 5B 6B C C 7. Ls s s fd d Using ingrion rs u, du d d s s d s s s s s s d s s s 8. Assum C λ n, λ λ λ λ i roos, Tus, gnrl soluion is C C To oin riculr soluion ssum, A sin B cos ʹ A cos B sin ʹ ʹ A sin B cos susiuing in gis, A sin B sin A cos B sin A sin B cos sin A B A sin B A B cos sin
from cofficins in fron of sin nd cos funcions, A B A B n, A/ nd B-/5 nd riculr soluion coms sin cos 5 Finll, gnrl soluion is C C sin cos 5 9. d d k d k ln C ln k C k d k C 5. Gnrl Soluion Priculr Soluion ʹ K ʹ K d ln K Kd K K K ʹ K Assum ʹ K K /K, ssumion olds 5. Assum soluion i undrmind cofficins. Diffrni ril soluion for s mn driis is, nd susiu ino originl quion. Vlus for cofficin cn drmind iniil condiions. T soluion ould of form AcosBsin or onnils, λ r λ is roo of crcrisic quion. 5. ʹ d ln d C C C C Using iniil condiion C or C
PHYSICS. In -dircion, comonn of gri forc lncs norml forc, nd in -dircion, forc of fricion lncs comonn of gri forc us for lock srs sliding. F N Mgcos θ F N F F g Mgsin θ µ N g F fr susiuing in for norml forc, F Mgsin θ µ Mgcos θ θ n µ n.8 θ! 8.66. consrion of momnum: m m m ʹ mʹ MV MVʹ mʹ consrion of nrg: m ʹ ʹ m m m MV MV m In n lsic collision, us, ʹ ʹ V V ʹ ʹ ʹ ʹ m m m m m m m M ʹ V M m M m Vʹ V M m Assuming comll inlsic collision MV M m ʹ V M ʹ M m. T sring-mss ssm ill undrgo siml rmonic moion F s k d M k d d k d M k l ϖ M Iniil condiions r ʹ Tn, im, cos ϖ sin ϖ sin ϖ cos ϖ T gnrl soluion is Finll, cos ϖ sin ϖ cos ϖ
. T griionl forc nd lcrosic forc o dnd uon inrs of disnc squrd, m m r q q F k r F G gr lc Tus, if disnc is dould, Forc n o msss or o crgd ricls ill ¼ originl. 5. d V! d V g f 555 f. s 5.87 s d 5 f / s5.87 s d 9.5 f 6. Non's scond l, cn sd in rms of momnum, M, s!! d d! F M d d dm d For rock si, nding ful r dm/d, loci of dm is rli loci i rsc o M rl u rusrs rock If scsi is in our sc sufficinl fr from n lrg msss, n rnl forc du o gri is ngligil. T rm rl dm/d, rfrrd o s rus nd is forc rd on scsi lld gs. Tus, d dm M rl d d d dm u M For scsi o com o rs from iniil loci V, V d u lnu dm M lnm u V M ln ln u M u V M u M V u M M M M V M M Sinc finl mss, M, ill lss n iniil mss, M, loci ill ngi, indicing rrs rus. To find finl mss rquirs knoldg of im i ks o so scsi from iniil loci nd knoldg of ful consumion r, dm/d.
7. Tis rolm cn rokn don ino numr of smllr rolms nd sold using consrion of nrg rincils, ssuming no losss. For cr of mss, M, sring ig,, loci oom of dclin, V, cn found conring onil nrg of cr ino kinic nrg KE PE V MV Mg g As cr gos round loo, if forc of gri orcoms cnrifugl forc, cr ill fll off rck. T minimum loci of cr ill occur igs oin on loo, sinc som of kinic nrg of cr s n conrd ck ino onil nrg. Tus, V M r V F,min Mg gr r V is minimum loci llod o k cr from flling off rck. From consrion of nrg ΔKE ΔPE MV V Mg MV V g r V MV g g r Mgr Mgr To sol for, qu V,min o V o g g r g r rg 5 r rg Wi no losss, loci fr loo ill V sinc ll onil nrg of loo s n conrd ck ino iniil kinic nrg. Tus, from sring-mss quions dislcmn,, cn found ssuming ll kinic nrg of cr is conrd ino onil nrg of sring PE s k MV k MV
8. Soling for im,, i ks unil rocil is ground gis Tn disnc rocil rls cn gin s Tis is mimum n sinθ, or θ 5 dgrs. 9. Surfc nsion ill rn r from silling ou. T surfc is unrurd nd ill rmin unrurd s long s r is r. T r is ld ogr srong cosi forcs n r molculs, ic is r srong i nsil srng of 6 N/m. Hor, sligs disurnc nd r ill gin o flo undr influnc of gri.. T clindrs qul our rdius, nd ollo clindr s innr rdius i. Tus, For solid clindr: For ollo clindr: s > rfor, solid clindr ill rc oom firs. g,, sin,, cos θ θ g sin g sin θ θ θ θ θ θ θ θ sin g cos sin g g sin cos cos, r Mg, I M KEs iniil PE, KE KE cm roionl rnslion l ϖ ϖ Δ Δ Δ g g Mg M M M I s s s s cm i i i i cm g g Mg M M M M I
. For TV i lcric fild srng, E, rndiculr o iniil loci. F m qe m qe m qe m / T lcron follos of rol. Wr i is scrn cn cngd lring E nd.. sin θ sin θ g6 9.6 sin θ cos θ 9.6 cos θ sin θ nθ.6 θ.66! T rocil rcs mimum ig im sconds 9.6 f sin θ 9.6 sin.66 g. s L: A od coninus in is s of rs or of uniform sd in srig lin unlss i is comlld o cng s forcs cing on i. Tis l is lso knon s l of inri. nd L: T cclrion of n oc is dircl roorionl o n forc cing on i nd is inrsl roorionl o is mss. T dircion of cclrion is in dircion of lid n forc. Fm. rd L: Wnr on oc rs forc on scond oc, scond rs n qul nd oosi forc on firs. Mor commonl sd s, "For r cion, r is n qul nd oosi rcion.". Assuming n inlsic collision m g m / s g f m 9.9 m / s m m f f T cng in kinic nrg ill qul ris in onil nrg m m f g 5 m f m m g
5. sin θ g, sin θ g sin θ g g cos θ sin θ cos θ g sinθ T disnc rock sld rls dnds on iniil loci nd ngl of rm. 6. g g g,, 7. mn us,mn mn z,us us,mn z us If mn > us n ill cc us. Tis ill occur if srion disnc z is lss n crin disnc gin,mn z < > z us 8. Wn lock is rlsd, onil nrg is cngd ino kinic nrg, nd som of nrg ill rnsfrrd o sionr lock. M M g g For oll lsic collision M M M ʹ M ʹ M M M ʹ M ʹ M M ʹ g M M M M
9. Du o ls of consrion of nrg nd momnum, firs ll rnsfrs is momnum nd nrg o innr srs, c on rnsfrring is nrg o n sr. Bcus no sr is locd n o nd ll, i rins momnum nd nrg nd sings ourd.. Using consrion of momnum, ssuming n inlsic collision M f M M M g m / s 9.9 m / s M M g f Tis rolm cnno sold using consrion of nrg sinc collision is inlsic nd som of iniil kinic nrg s n rnsformd ino or s suc s rml. Tis cn sn comring kinic nrg for nd fr collision. kg m / s M 5 J. kg9.9 m / s M M f 55 J Tus, ou 55 J of nrg s conrd o or forms. Sinc scific of ood is 7 J/kg C o, if ll rsidul nrg s rnsformd ino rml nrg, mrur of ood lock ould n risd o C.. Momnum is roduc of mss of od nd is loci, m. Non's Scond L ss, Forc mss cclrion! d F m m d d m, d F d d d m d. g,, f / s To find im mimum ig, diffrni nd s '. ʹ g g m / s. m / s.6 s.6..6 55. f. T onil nrg of mss s n lmos ll conrd ino kinic nrg us for i sriks ground. Tus, mg m g
... Assumions md r, no rnl, non-consri forcs c on ssm. T sring is iglss nd in-nsil. c. T diffrnc is loss of ir fricion, ngligil ffc. d. For n lsic collision, consrion of momnum nd kinic nrg,. If collision is non-lsic 5. Work is roduc of forc disnc. T moun of ork rquird o mo lock unis is 6. Work is dfind o roduc of mgniud of dislcmn ims comonn of forc rlll o dislcmn. Enrg is ili o do ork. Por is r ic nrg is rnsformd. g m mg m m ʹ ʹ m m m m m m m ʹ m m m g d Fd d F W