ANSWER - KEY 1. PHYSICAL WORLD AND MEASUREMENT Q Ans. A B B C A D B A A C B A A C B

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1 NSWE - KEY HYSIC WOD ND MESUEMEN Q ns C D C C Q ns D C D D, D C D So (a Caoie is he ni of hea ie, eney So diensions of eney M So (b G F d Fd G [ M ][ ] [ G ] [ M ] [ M ] So (b na oen = [ M So (c ][ ] [ M E h [ M So (a ] ] [ h][ ] [ h] [ M y pincipe of diensiona hooeniy 6 a ] [ a] [ ][ ] [ M ] [ ] = [ M ] 6 So (d y pin he diensions of each qaniy boh he sides y we e [ ] [ M] [ M ] Now copain he diensions of qaniies in boh sides we e y and y 7 So (b Fo he pincipe of hooeniy 8 So (a, y has diensions of y sbsiin diension of each qaniy in HS of opion (a we e M = [ ] M his opion ies he diension of eociy 9 So (a e S y z by sbsiin he diension of [ ] [ ] [ S] [ M ],[ ] [ ],[ ] [ M and by copain he powe of boh he sides ] /, y /, z / so So (c e C / y S k z G h S y sbsiin he foowin diensions : [ C ] ;[ G] [ M ] and [ h ] [ M ] Now copain boh sides we wi e / ; y /, z / So c So (b / G / / F [ ] [ M z h s F [ M ], [ ], [ ] z So (a k [ M N So (a So (c / / ] ] Hee % eo in = % c and % eo in = % % eo in = % eo in + (% eo in = % So (b % eo is oe % eo in adis = % 6 So (b Hee, S ( 8 and ( sec J ISION d SOUION (ae: 8

2 Epessin i in pecenae eo, we hae, S 8 % 8 % 8 and % 7% s 8 ( / s 7 7 So (a Since pecenae incease in enh = % Hence, pecenae incease in aea of sqae shee % = % 8 So (d a b c / d e So ai eo in a is ien by a a a d e d e b c d % 9 So (b e b c b c eae ae 6 sec Now Mean absoe eo So (b H I 8 sec H I H I 6% So (c ( 6% Qaniy C has ai powe So i bins ai eo in So (a ecenae eo in X a b c So (d f C C does no epesen he diension of feqency So (d [n] = Nbe of paices cossin a ni aea in ni ie = [ ] n nbe of paices pe ni oe = [ ] n [ ] [ ] = posiions [ n [ ] D = ] n So (b, d n enh G c y h z = [M ] [ ] y z [ ] [ M ] y copain he powe of M, and in boh sides we e z, y z and y z y soin aboe hee eqaions we e 6 So (a, y, z e adis of yaion [ k ] [ h] [ c] [ G] y sbsiin he diension of [ k] [ ] [ h ] [ M ],[ c] [ y ],[ G] [ M z and by copain he powe of boh sides we can e /, y /, z / So diension of adis of yaion ae [ h] [ c] [ G] 7 So (c F / F F M [ ] 8 So (d Densiy, [ M M M M M 6 6 ] ] / / ecenae eo % / 9 So (a J ISION d SOUION (ae: 8

3 KINEMICS ( ECO Q ns D C D D C Q ns D C C So (d 7 ( 6 Uni eco in he diecion of wi be So eqied eco = ˆi So (a ˆj ˆ ˆi ˆj ˆi ˆj e he coponens of akes anes, y and z ais especiey hen cos cos cos cos cos cos So (a y z, and wih iˆ ˆj kˆ ( ( ( cos So (a iˆ, I is a ni eco cos So (c iˆ ˆj ˆ 6 So (d Q ˆ QQˆ 7 So (a iˆ ˆj 7kˆ, cos ˆ j = iˆ ˆj a, kˆ 7 a sec, ˆ i ˆj 7kˆ, ˆ i ˆj ( ( 8 So (b esan of ecos and iˆ ˆj 8ˆ i 8 ˆj iˆ ˆj ˆ 9 So (a ˆ i ˆj ( ( iˆ ˆj (ˆ i ˆj kˆ(ˆ i ˆj kˆ cos cos, 9 So (a Fo 7 N boh he eco shod be paae ie ane beween he shod be zeo Fo 7 N boh he ecos shod be anipaae ie ane beween he shod be 8 Fo N boh he ecos shod be pependica o each ohe ie ane beween he shod be 9 So (d Fo fie ˆ j and iˆ = /s (ˆ i ˆ j and diecion an ( ie S W So (a ˆi cos ˆ sin j and O = /s ˆi cos ˆ sin j J ISION d SOUION (ae: 8

4 So chane in oen So (b cos ˆ, cos j e ( C so (b y sbsiin, cos So (a so (c 6 so (d 7 So (a cos F, F and F we e 6 (ien (i sin an an cos 9 cos cos (ii 8 cos (iii y soin eq (i, (ii and (iii we e 6N, N 8 So (b Q sin an 9 Qcos Q cos cos cos Q Q 9 So (a ccodin o pobe Q and Q y soin we e and Q Q Q Q Hee C Which is pependica o boh eco and C So (b F ˆi ˆj kˆ ( ( ˆi ( ( j ˆ ( ( kˆ So (a 7ˆi 6 ˆj kˆ Q a a a So (c Do podc of wo pependica eco wi be zeo So (b Manide of ni eco = ( (8 c y soin we e c So (b ˆj kˆ, ˆi ˆj kˆ ˆi C ˆj kˆ 7ˆi ˆj kˆ Hence aea = C sqni 6 So (a ccodin o ai's heoe Q sin sin sin 7 So (c sin sin sin cos cos cos (cos cos cos 8 So (b ne ( J ISION d SOUION (ae: 8

5 KINEMICS ( MOION IN -D Q ns D D C C C C C Q ns D D C D C C Q ns C C C C,,D,D D C C Q ns C D C So (b Hoizona disance coeed by he whee in haf eoion = So he dispaceen of he poin which was iniiay in conac wih a ond = So (d ( ( oadisanceaeed eae speed = oaieaken = So (d ( / (/ 6 ( 6 ( 6 d d = ( 6 d d If eociy = hen, 6 sec Hence a =, = ( ( + = ees So (b a b = d d iniia (/ new (/ b = sec, = = 8 c / sec (s b c So (b In his pobe oa disance is diided ino wo eqa pas So 6 So (c Gien d a = d d a eociy d d d d d d / d / a = a 6b b d ( b d a and acceeaion (a = When acceeaion = a 6b a a 6b b 7 So (c Gien y a b c d = dy d b c d in, iniia = b So iniia eociy = b Now, acceeaion (a in =, a iniia = c 8 So (a d d c d diffeeniain ie wih espec o disance d d So, acceeaion (a = d d d d d d d d J ISION d SOUION (ae: 86

6 = d d 9 So (b ( o cos 9 eae acceeaion /s owad noh-wes (s cea fo he fie So (b Gien ha d, C, a, C When paice is away fo he oiin ( ( sec d a d d d a fo sec a a / s So (a When disance ie aph is a saih ine wih consan sope han oion is nifo So (c In fis insan yo wi appy o an /s an and say, i is won becase foa an is aid when ane is eased wih ie ais Hee ane is aken fo dispaceen ais So ane fo o o o ie ais 9 6 o Now an 6 So (c eae eociy = / 9 oa dispacee n oa / ie = = /s So (c In fis haf of oion he acceeaion is nifo & eociy aday deceases, so sope wi be neaie b fo ne haf acceeaion is posiie So sope wi be posiie hs aph 'C' is coec No inoin ai esisance eans pwad oion i hae acceeaion (a + and he downwad oion wi hae ( a So (a We know ha he eociy of body is ien by he sope of dispaceen ie aph So i is cea ha iniiay sope of he aph is posiie and afe soe ie i becoes zeo (coespondin o he peak of he aph and hen i wi be neaie 6 So (b Disance coeed in oa 7 seconds = ea of apezi CD ( 6 = Disance coeed in as second = aea of iane CDQ= So eqied facion 7 So (a ea of ecane CO = = 8 ea of ecane CDEF = ( = ea of ecane FGHI = = Dispaceen = s of aea wih hei sin = 8 + ( + = 8 Disance = s of aea wih o sin = = 6 8 So (d In he posiie eion he eociy deceases ineay (din ise and in neaie eion eociy incease ineay (din fa and he diecion is opposie o each ohe din ise and fa, hence fa is shown in he neaie eion 9 So (a When ba is dopped fo heih d is eociy wi be zeo s ba coes downwad h deceases and inceases js befoe he ebond fo he eah h = and = ai and js afe ebond eociy edces o haf and diecion becoes opposie s soon as he heih inceases is eociy deceases and becoes zeo a d h his inepeaion is ceay shown by aph (a J ISION d SOUION (ae: 87

7 eociy eociy So (d aice can no possess wo eociies a a sine insan so aph (d is no possibe So (c a ceain insan sope of is eae han (, so acceeaion in is eae han So (a Iniiay when ba fas fo a heih is eociy is zeo and oes on inceasin when i coes down Js afe ebond fo he eah is eociy deceases in anide and is diecion es eesed his pocess is epeaed ni ba coes o a es his inepeaion is we epained in aph (a So (a cceeaion aon O cceeaion aon O So (d cceeaion aon C as as s s / s /s s s (s a = consan a So (a a s y sin n S n s in h second = a (, Disance aeed by body Disance aeed by body in d second is = a ( ccodin o pobe : a ( = a ( a 9a a a 9 6 So (a ie ie e iniia ( eociy of paice = fo fis sec of oion s a s ee, so by sin a( a (i fo fis 8 sec of oion s8 ee 8 a(8 8a (ii y soin (i and (ii 7 6 / s a / s Now disance aeed by paice in oa sec s a y sbsiin he ae of and a we wi e s 8 So he disance in as sec = s s So (c Since he body sas fo es heefoe S S a( S a a( a S S = S S S a a 7 a a( S S S = a a hs Ceay S S S 8 So (a s S n ( n, S S 7 9 So (a If ba is hown wih eociy, hen ie of fih eociy afe sec : disance in as '' sec : ( ( h h So (c e boh bas ee a poin afe ie = So, J ISION d SOUION (ae: 88

8 he disance aeed by ba ( h (i he disance aeed by ba ( h (ii y addin (i and (ii h h = (Gien h h h / 8sec and h, h 8 So (b e a poin iniia eociy of body is eqa o zeo Fo pah : h (i Fo pah C : ( (ii Soin (i and (ii So (c h Fo fis case of doppin h Fo second case of downwad howin h ( (i Fo hid case of pwad howin h ( (ii on soin hese wo eqaions : So (b e he inea be hen fo qesion C = Fo fis dop ( (i h h h Fo second dop (ii y soin (i and (ii and hence eqied heih h 7 So (c s he baoon is oin p we wi ake iniia eociy of fain body / s, h 8, / s y appyin h ; 8 ( 8 6 So (b 76 sec Fo eica downwad oion we wi conside iniia eociy = y appyin h, ( ( h h 6 So (a d d 9 7 So (c d = a d = ( b d S b b b d d b 6 b 8 So (a If and ae he ie aken by paice o coe fis and second haf disance especiey / (i 6 and 7 So, 9 So (c 7 (ii oa ie 6 So, aeae speed / sec J ISION d SOUION (ae: 89

9 d b b d b d d, K We e b K d b ain b K d, K b 6 d d So (a,b,d 6 d d 6 Ineain boh sides, o (6 e K oe (6 K d 6 d (i, oe 6 K Sbsiin he ae of K in eqaion (i o (6 o e 6 oe 6 e 6 e e 6( e ( e ina / s (When d d cceeaion a e 6e d d Iniia acceeaion = 6 / s So (a,d he body sas fo es a and hen aain coes o es a I eans iniiay acceeaion is posiie and hen neaie So we can concde ha can no eains posiie fo a in he inea ie s chane sin din he oion So (b he aea nde acceeaion ie aph ies chane in eociy s acceeaion is zeo a he end of sec ie ea of O a / s So (d e he ca acceeae a ae fo ie hen ai eociy aained, Now, he ca deceeaes a a ae fo ie and finay coes o es hen, ( ( So (c If a sone is dopped fo heih h hen h (i If a sone is hown pwad wih eociy hen h (ii If a sone is hown downwad wih eociy hen h (iii Fo (i (ii and (iii we e (i ( Diidin (i and ( we e o ( ( y soin So (c Since diecion of is opposie o he diecion of and h so fo eqaion of oion h 6 So (c h 8h h h / eociy afe aein disance h ( Fo second ee disance 8 8 J ISION d SOUION (ae: 9

10 akin +e sin ( / / and so on ( / 7 So (d Inea of ba how = sec If we wan ha ini hee (oe han wo ba eain in ai hen ie of fih of fis ba s be eae han sec sec sec 96 / s fo =96 Fis ba wi js sike he ond (in sky Second ba wi be a hihes poin (in sky hid ba wi be a poin of pojecion o a ond (no in sky 8 So (a he disance coeed by he ba din he as seconds of is pwad oion = Disance coeed by i in fis seconds of is downwad oion Fo h h [s = fo i downwad oion] 9 So (c J ISION d SOUION (ae: 9

11 KINEMICS ( C MOION IN -D Q ns C D C C D C C Q ns C C C C D C D D D Q ns C D C C C C C D C C Q ns D D D C C C Q ns D C D D C So (c y copain he coefficien of in ien eqaion wih sandad eqaion y an an 6 cos So (b y copain he coefficien of in ien eqaion wih sandad eqaion y an cos cos Sbsiin = 6 o we e / sec So (d Sandad eqaion of pojecie oion y an Gien eqaion : y 6 o y 6 6 / y copain aboe eqaions So (c O 6 =8 E y ( a F So is disance fo O is ien by d y ( (6 d So (c Dispaceen aon X- ais : a 6 ( Dispaceen aon Y- ais : 8 y y ay 8 ( 6 oa disance fo he oiin 6 So (a y (8 (6 8 6 Fo he foa of insananeos eociy sin ( ( sin 7 So (b 7 / s o y d F ody oes hoizonay wih consan iniia eociy /s po seconds and in pependica diecion i oes nde he effec of consan foce wih zeo iniia eociy po seconds Hoizona eociy a poin ' O' cos Hoizona eociy a poin ' ' sin In pojecie oion hoizona coponen of eociy eains consan hoho he oion sin O cos 9 o sin cos co sin 9 o J ISION d SOUION (ae: 9

12 8 So (b e in sec body eaches po poin and afe one oe sec po poin oa ie of ascen fo a body is ien sec ie sin sin (i [Usin = + as] / s So (c Since hoizona coponen of eociy eain aways consan heefoe ony eica coponen of eociy chanes Iniiay eica coponen becoes zeo So chane in eociy So (b sin Finay i sin O cos o cos sin = ( So (b s we know fo copeenay anes [s = o ] Hoizona coponen of eociy eains aways consan cos cos (ii Fo eica pwad oion beween poin O and sin o sin Usin o sin s sin / s Sbsiin his ae in eqaion (ii o (iii cos cos Fo eqaion (i and (iii and 6 9 So (d poin N ane of pojecion of he body wi be e eociy of pojecion a his poin is If he body js anaes o coss he we hen sin ane Diaee of we s / s we hae o cacae he eociy ( of he body a poin M Fo oion aon he incined pane (fo M o N Fina eociy ( = /s, acceeaion (a = sin = sin o, disance of incined pane (s = ( M N o So (b e and ae he hoizona disances aeed by paice and especiey in ie cos (i and cos 6 o (ii So (b o cos cos 6 / o When body pojeced wih iniia eociy by akin ane wih he hoizona hen afe ie, (a poin i s diecion is pependica o sin Manide of eociy a poin is ien by co (fo sape pobe no 9 Fo eica oion : Iniia eociy (a poin O Fina eociy (a poin ie of fih (fo poin O o = ppyin fis eqaion of oion co cos sin sin co cos sin sin 9 o cos O cos cos (9 sin cos co cos cosec J ISION d SOUION (ae: 9

13 So (c Foa fo cacaion of ie o each he body on he ond fo he owe of heih h (If i is hown eicay p wih eociy is ien by h So we can esoe he ien eociy in eica diecion and can appy he aboe foa Iniia eica coponen of eociy sin sin / s 98 = 6 sec 6 So (c 98 7 ( sin sin and and (If and ae consan In he ien condiion o ake ane dobe, eociy s be inceased po aoaicay ie of fih wi becoes 7 So (c ies ha of peios ae So ies Fo fis paices ane of pojecion fo he hoizona is So sin Fo second paice ane of pojecion fo he eica is i ean fo he hoizona is ( 9 sin(9 cos So aio of ie of fih 8 So (c sin an Faciona decease in ie of fih 9 So (a sin o 7 o ecenae decease = 9% Sipy we hae o cacae he ane of pojecie So (b sin ecase ane So (c ( 866 ee (eociy of pojecion sin( sin (ne of pojec ion a = (when 6 / s So (d ane hoizona coponen of eociy Gaph shows ai ane, so fooba possesses ai hoizona eociy in his case So (c Fo ai hoizona ane Fo H co = H [s = o, fo ai ane] Speed of he paice wi be ini a he hihes poin of paaboa So he co-odinae of he hihes poin wi be (/, / So (d We know Whee y hoizona coponen of iniia eociy, eica coponen of iniia eociy So (a We know H co H H co co ; cos s H ien ane o sin cos y sin ; J ISION d SOUION (ae: 9

14 6 So (d a (when H a ee (when 9 7 So (d When wo sones ae pojeced wih sae eociy hen fo copeenay anes and (9 o aio of ai heihs : H H 8 So (b an an H H Y ie of fih fo he ba hown by ankaj ie of fih fo he ba hown by Sdhi sin(9 pobe o cos ccodin o cos cos Heih of he ba hown by ankaj Heih of he ba hown by Sdhi H sin (9 o 9 o H cos So, H So (b sin ( ojecie possess ini kineic eney a he hihes poin of he ajecoy ie a a hoizona disance / So (b Kineic eney a he hihes poin o K cos K / So (c K' K cos oenia eney a he hihes poin is ien by E sin Fo fis ba 9 ( E Fo second ba (9 6 fo he hoizona ( E sin 8 E ( E So (d II : KE a hihes poin cos cos I cos 6 So (c o K' K cos o H H 9 So (a / cos / = [s cos ] a [when ] So, fo he eaion So (b H co H co H Y X H H So (a sin sin sin sin sin sec 6 o o sin he pah of he ba appeas paaboic o a obsee nea he ae becase i is a es o a io he pah appeas saih ine becase he hoizona eociy of aeopane and he ba ae eqa, so he eaie hoizona dispaceen is zeo J ISION d SOUION (ae: 9

15 7 So (c h h h h nh b h So (d = 7 k/h nb y sin eqaion of ajecoy y ( nb fo ien condiion nh 8 So (c n h b ie of descen s h Fo fis paice : h h Fo second paice : h h y So he aio of eociies wi be : 9 So (c Foa fo his condiion is ien by So (c h h h b h n b whee h = heih of each sep, b = widh of sep, = hoizona eociy of pojecion, n = nbe of sep h y c / sec / sec So (a sec and hoizona disance S 7 S Mai ane p he incined pane ( a p ( sin Mai ane down he incined pane ( a down and accodin o pobe : ( sin ( sin ( sin y soin = o So (c Mai ane on hoizona pane 6k (ien Mai ane on a incined pane a ( sin in = o a So (c ( sin o ad Minehand and 6 in Eah ad ad h 6 in Minehand : Eah 6 k J ISION d SOUION (ae: 96

16 So (b So (a C a M na eociy of abo na eociy of abo C a C a : ] 6 So (b C 8 o Mass pefos nifo cica oion on he abe e n is he feqency of eoion hen cenifa foce n Fo eqiibi his foce wi be eqa o weih M n M M n So (d a So (d / s sin 8 sin 7 So (b ˆi o o sin9 Jˆ 6 kˆ 6 / s ( 6 i (8 Jˆ ( 8 kˆ 8 ˆi Jˆ kˆ 8 So (d Speed ain by body fain hoh a disance h is eqa o h [s 9 So (b ension o h ien] [ension and ass ae consan] ecenae chane in ie peiod (pecenae chane in enh [If % chane is ey sa] ( % % a So (b, / s, (ien an 6 9 So (b an / s 9 k, 98 / s an J ISION d SOUION (ae: 97 ( 6 k / h / s, k, 98 / s (ien ne of bankin an So (b h / 98 an o an,,, / s (ien h 6 So (c Fo he foa N s < N N h N 86 / s

17 7 So (a k 7 / s,, k (ien h eacion a owes poin ( 8 So (a N KN Noa eacion a he hihes poin of he pah Fo ai, ae of he adis of cae ( shod be ini and i is ini in fis condiion 9 So (c 6 (ways owes poin 6 So (c Hihes poin e he sin beaks a poin ension cos eakin senh cos (i If he bob is eeased fo es (fo poin hen eociy acqied by i a poin h cos (ii [s h= cos ] y sbsiin his ae in eqaion (i cos ( cos o cos cos cos 6 So (d When ca os down he incined pane fo heih H, hen eociy acqied by i a he owes poin (i H and fo oopin of oop, eociy a he owes poin shod be (ii Fo eq n (i and (ii C h = cos cos / H H (iii Fo he fie H h H h Sbsiin he ae of in eqaion (iii we e H h H H h 6 So (a y he conseaion of eney oenia eney a poin = Kineic eney a poin and ension ( 6 So (c e he paice eae he sphee a heih h fo he boo We know fo ien condiion and h [s = ] 6 So (a 6 6 / s 6 So (d 66 So (b : : : : : : c cos s inceases deceases So 67 So (d 68 So (b cos 9 89 Newon 7 ad / sec ( 98 cos 6 h J ISION d SOUION (ae: 98

18 69 So (c Mini eociy a owes poin o copee eica oop So ini kineic eney ( ( 7 and ae wo ships ( k apa which ae oin wih eociies and ( k/h especiey in diecions noh and wes he eociy of eaie o = ( he eaie eociy is shown in fi(b by C (C = ( + (C o (C = [( ( ] = 66 and an = / = = º akin hoizona coponens eq ( ies sin = = k/h o = = k/h sin Hence coec answe is ( 7 So he siaion is shown in fi e he boaan sas fo he poin on one side of ie and wans o each he poin on he opposie bank eacy in fon of In ode o each he poin, he has o diec his boa owads poin C ie in diecion C which akes an ane wih Hee he eociy of wae is k/h and eociy of boa wih espec o wae is 8 k/h Fo fi sin = /8 = / ie = º so he boaan shod ow his boa a an ane º wih o a an ane º + 9º = º wih he fow of wae Fo fi (E = (D (DE, If be he eociy of boaan in diecion, hen = (8 ( = 8 o N C =k/h C = 8 = 69 k/h ie aken in cossin he ie C W S E =k/h k 9º ( eaie D eociy k/h D 8k/h E hs he ship shod oe aon a diecion C eaie o he ship whee C is º wih I is obios fo he fie ha he disance of coses appoach is D Now D = sin º = (/ = 77 k and D = cos º = (/ = 77 k ie o each D = h Hence coec answe is ( 7 So When he an is a es w he ond, he ain coes o hi a an ane º wih he eica his is he diecion of he eociy of aindops wih espec o he ond dis an ce = speed in his diecion k 69 k / h 6 = in = in 69 Hence coec answe is ( = Hee, = eociy of ain wih espec o he ond, = eociy of he an wih espec o he ond nd, = eociy of he ain wih espec o he an, we hae, =, +, ( J ISION d SOUION (ae: 99

19 NEWON S WS OF MOION Q ns D C C D C D C D Q ns C C C D C So (d ecase he hoizona coponen of eociy ae sae fo boh ca and ba so hey coe eqa hoizona disances in ien ie inea So (a Gien hee foces ae in eqiibi ie ne foce wi be zeo I eans he paice wi oe wih sae eociy So (b e wo foces ae F and F ( F F ccodin o pobe: F F 8 (i ne beween F and esan ( is 9 F sin an 9 F F cos F F F cos F cos (ii` F = F 6 f F ( C W( EC ( C [fo he fie EC= cos 6] F( sin6 W( cos 6 ( cos 6 F W s W 8W 8 F 7b 6 So (c De o boyan foce on he aini bock he eadin of spin baance wi be ess han k b i incease he eadin of baance 7 So (b Fo he fee body diaa of an and cae syse: f 6 o F E C D W Wa and F F F F cos F F F F cos (iii by soin (i, (ii and (iii we e So (b F N and F N e foces ae F and F and ane beween he is and esan akes an ane wih he foce F F sin an an 9 F Fcos F Fcos cos / o So (c Since he syse is in eqiibi heefoe F and F y F and W Now by akin he oen of foces abo poin 8 So (a Fo eica eqiibi ( M ( M esan of wo ecos and, which ae wokin a an ane, can be ien by (M + cos [s F and F ] J ISION d SOUION (ae:

20 F F 9 9 So (d F F F F F cos 7 F F cos 9 cos 7 7 cos o cos 8 8 Moen acqied by he paice is neicay eqa o he aea encosed beween he F- ce and ie is Fo he ien diaa aea in a ppe haf is posiie and in owe haf is neaie (and eqa o he ppe haf So ne aea is zeo Hence he oen acqied by he paice wi be zeo So (c eaion beween inea oen (, an ( and kineic eney (E E [as E is consan] So (d Moen of body fo ien opions ae : (a (b (c (d k E h / sec k 6 6 So fo opion (d oen is ai So (b Up hs foce d F d So (c N eadin of weihin scae ( a 8 ( So (d N ( a ( / sec k 8k [becase he if is oin pwad wih a =] So (b N N o k weih of a an in saionayif weih of a an in downwad oin if ( a / sec / sec a 6 So (c a o a Fo eadin oion of a if ( a fo downwad oion ( a fo pwad oion Since he weih of he body decease fo a whie and hen coes back o oiina ae i eans he if was oin pwad and sops sddeny Noe : Geneay we se ( a fo pwad oion ( a fo downwad oion hee a= acceeaion, b fo he ien pobe a= eadaion 7 So (c e k, k and F N (ien 6 Foce on he ihe ass = 8 So (b F 6 N When he foce is appied on ass conac foce f When he foce is appied on ass conac foce f aio of conac foces 9 So (c f f y copain he aboe pobe wih enea epession So (a F Fo FD of ass k 8 Newon a ' (i Fo FD of ass k a ' (ii Fo oa syse pwad foce F a 6 8 N = 78 N J ISION d SOUION (ae:

21 by sbsiin he ae of in eqaion (i and (ii and soin we e So (b a So (d ' 7 N / s 7 sin cceeaion sin = So (b k sin sin a sin sin 7 / s a k a So (c Fo fis case a (i Fo second case a fo fee body diaa of a a [s = ] a (ii a / Fo (i and (ii / a J ISION d SOUION (ae:

22 FICION Q ns C D D D C Q ns C C, C So (c wo ficiona foce wi wok on bock f F cos 6 F f f G a G( = + ( = + 9 = N (his is he eqied ini foce So (d When he if oes down wad wih acceeaion 'a' hen effecie acceeaion de o aiy ' = a ' [s he if fas feey, so a = ] So foce of ficion ' So (d Fo ppe haf by he eqaion of oion as ( sin / sin [s, s /, a sin] Fo owe haf (sin cos / [s, s /, a (sin cos ] sin (sin cos [s fina eociy of ppe haf wi be eqa o he iniia eociy of owe haf] f f G sin cos an So (a Ficiona foce Gond f F oh Sooh F cos 6 ( W F sin6 F cos 6 F sin 6 So (d F N iiin ficion = = 6 6 cceeaion = = 6 So (b 6 K F 6 = 6 s ppied foce = 6 Mass of he Kineic body = ficion Kineic eney acqied by body = oa wok done on he body Wok done aains ficion = F S S = = = J 7 So (a wi bein o side on if sedo foce is oe han iiin ficion F F' F s M F M 8 So (a F 76 N iiin ficion beween he bock and he sface F M ( 8 S S N b he appied foce is N so he owe bock wi no oe ie hee is no psedo foce on ppe bock Hence hee wi be no foce of ficion beween and W+F sin 6 J ISION d SOUION (ae:

23 9 So (a Fo he aboe epession, fo he eqiibi cos and F sin Sbsiin hese ae in co So (b F we e an o y copain he ien condiion wih enea epession M M M M M M So (a Fo he epession So (a ' ' [s = ] = % of he enh of he chain an an n So (a / s (a y dawin he fee body diaa of he bock fo (c ciica condiion F Q sin ( Q cos Q sin Q cos 6 (b iiin ficion F s ( Since downwad foce is ess han iiin ficion heefoe bock is a es so he saic foce N of ficion wi wok on i F s = downwad foce = Weih N 7 (c Mai foce by sface when ficion woks F f Mini foce ( F when hee is no ficion + Q cos +Q sin N 8 (a Hence anin fo o We e, M F M f F cos 6 ( W F sin6 Sbsiin & W we e F N 9 (b When wo bocks pefos sipe haonic oion oehe hen a he eee posiion ( a apide = K esoin foce F K a a hee wi be no eaie oion beween and Q if psedo foce on bock is ess han o js eqa o iiin ficion beween and Q K ie iiin ficion Mai ficion (c Noa eacion K sin iiin ficion beween body and sface is ien by, F (a iiin ficion beween bock and sab s o N appied foce on bock is N So he bock wi sip oe a sab Now kineic ficion woks beween bock and sab F 98 9 N k k his kineic ficion heps o oe he sab 9 cceeaion of sab (a iiin ficion F cos f F sin 6 F 6 W + sin F F cos / s F 7 cos N (appoiaey cos J ISION d SOUION (ae:

24 when he bock is yin on he incined pane hen coponen of weih down he pane 98 sin 98 N sin I eans he body is saionay, so saic ficion wi wok on i Saic ficion = ppied foce = 98 N (a,c In cycin, he ea whee oes by he foce conicaed o i by pedain whie fon whee oes by isef So, whie pedain a bicyce, he foce eeed by ea whee on ond akes foce of ficion ac on i in he fowad diecion (ike wakin Fon whee oin by isef epeience foce of ficion in backwad diecion (ike oin of a ba [Howee, if pedain is sopped boh whees oe by hesees and so epeience foce of ficion in backwad diecion] J ISION d SOUION (ae:

25 WOK, ENEGY OWE & COISION Q ns C D D D D C D D C D Q ns C D D C D D C Q 6 7 ns C C So (c Wok done F (i ˆ(ˆ j i ˆ j 7 J So (d Wok done by cenipea foce in cica oion is aways eqa o zeo So (b Wok done F d So (d Cd C C Wok done = Fs = a a s acceeai on ( a ien So (a a Wok done = Coeed aea on foce dispaceen aph = + + = e 6 So (d Iniia kineic eney of he body ( J Fina kineic eney = Iniia eney wok done aains esisie foce (ea beween aph and dispaceen ais 7 So (b J s aiaiona fied is conseaie in nae So wok done in oin a paice fo o does no depends pon he pah foowed by he body I aways eains sae 8 So (d 6 7 and 8 6 Wok done = Incease in kineic eney [ ] [6 7] 96J 9 So (c e = ass of he boy, M = ass of he an, = eociy of he boy and = eociy of he an Iniia kineic eney of an M s J ISION d SOUION (ae: 6 M ien M When he an speeds p by /s, M( So (d ( (ii M (i Fo (i and (ii we e speed of he an / s s he foces ae wokin a ih ane o each ohe heefoe ne foce on he body F N Kineic eney of he body = wok done = F s F a F ( J So (b Kineic eney E F E

26 ecenae incease in kineic eney = (% incease in oen [If chane is ey sa] So (d = (% = % E E E E E E E of E So (c E % If he kineic eney wod becoe haf, hen oenia eney = (Iniia kineic eney h 98 h [9] [9] h So (d du F du F d d U ( k a d k a U We e k so we e U neaie a fo k a U a = and Fo he ien fncion we can see ha F = a = ie sope of U- aph is zeo a = So (b du d F ( d d F 6 So (c s sope of pobe aph is posiie and consan po disance a hen i becoes zeo heefoe fo F du d we can say ha po disance a foce wi be consan (neaie and sddeny i becoes zeo 7 So (b F du d d d (8 du Fo he eqiibi condiion F d / So (d s / pa of he chain is hanin fo he ede of he abe So by sbsiin n = in sandad epession M W n 9 So (d M M ( 8 M W J n ( So (b n ( 87 /s (appo So (a y he conseaion of eney oa eney a poin = oa eney a poin h h / s So (b oenia eney of bock a sain poin = Kineic eney a poin = Wok done aains ficion in aein a disance s fo poin h = s h s 7 ie bock coe o es a he idpoin beween and Q So (c Foce d d h = d d d ( [ ] d d d owe = F = h = J ISION d SOUION (ae: 7

27 So (d wok done ie h ie o obain wice wae fo he sae pipe in he sae ie, he powe of oo has o be inceased o ies So (a wok done h ( owe MW ie he syse is 8% efficien owe op = 8% = 8 MW 6 So (b d F F F d = d d Now = F F F ae consans hen 7 So (a F If foce and ass Sbsiin =, 6 ( / s ie he ihe paice wi oe in oiina diecion wih he speed of /s 8 So (d Iniia eociies of bas ae + and especiey and we know ha fo ien condiion eociies e inechaned afe coision So he eociies of wo bas afe coision ae and especiey 9 So (c Fina eociy of he ae s iniiay ae is a es so by sbsiin we e So (b M M M oa kineic eney of he ba wi ansfe o he bob of sipe pend e i ises o heih h by he aw of conseaion of eney h h So (b Ony his condiion saisfies he aw of conseaion of inea oen So (c Kineic eney eained by pojecie K K 8 9 K E = E 6E 8 So (a Facion of kineic eney eained by pojecie K K Mass of neon ( = and Mass of ao ( = K K So (b o Kineic eney ansfeed o saionay ae (cabon nces K K K K K So (c 6 So (a 69 8 (6Me 7Me 69 e / / e / / 8 69 y conseaion of oen, Moen of he be ( = oen of he coposie bock ( + M eociy of coposie bock M Kineic eney ( 7 So (a M ( M M M y he conseaion of oen ( M and if he syse oes po heih h hen h ( M h M h h M J ISION d SOUION (ae: 8

28 MOION OF SYSEM OF ICES & IGID ODY Q ns C C D C D C C Q ns D C D C C Q ns D D D So (c e cabon ao is a he oiin and he oyen ao is paced a -ais, 6, iˆ ˆand j ˆ i 6iˆ 6 î 8 ˆj ie 6 Å fo cabon ao y co-odinaes of cene of ass (, and co-odinaion of he cone [,] Fo he foa of disance beween wo poins ( y, and, disance = ( y ( y ( y ( = c y (, a D ( = 9 k C (a, a k C O CM (, 8k k (a, So (b e cone of sqae CD is a he oiin and he ass 8 k is paced a his cone (ien in pobe Diaona of sqae d a 8c a c 8k, k, k, k e,,, ae he posiion ecos of especie asses ˆ i ˆj, aiˆ ˆj, aiˆ aˆj ˆ i aj ˆ Fo he foa of cene of ass i ˆj, So (c 7,, ( k and ˆi ˆj ˆ, k (i j 7, (ˆi ˆj ˆ osiion eco of cene ass k 7(ˆ i ˆj kˆ (ˆ i ˆj 7kˆ (ˆ i ˆj kˆ 7 ( ˆ i 8 ˆj kˆ iˆ 7 ass So (d 8 ˆj 7 kˆ 8,, 7 7 So coodinaes of cene of Nbe of eoion = ea beween he aph and ie ais = ea of apezi = ( = eoion J ISION d SOUION (ae: 9

29 So (c inea dispaceen (S = adis ( na dispaceen ( S (if consan Disance aeed by ass Disance aeed by ass adis of pey concened adis of pey concened ( ( y wih ass ( wih ass ( y 6 So (a Moen of Ineia of disc I = [s M =densiy] I I y ( M = y y I y ( 6 I I 6I y 7 So (b whee hickness, [If = consan] [Gien y, Moen of ineia of disc abo a diaee = (ien M I y ] M I Now oen of ineia of disc abo an ais pependica o is pane and passin hoh a poin on is i = M (I 6I 8 So (d MI of syse abo YY ' I I I I whee I = oen of ineia of in abo diaee, I = I = MI of ineia of in abo a anen in a pane 9 So (a I C D C D 7 e I Z is he oen of ineia of sqae pae abo he ais which is passin hoh he cene and pependica o he pane I Z I I I I [y he heoe of ' ' pependica ais] I Z CD C' D' [s, ' ' I I ' ' I CD IC' D' and CD, C' D' ae syeic ais] Hence So (a I CD I Moen of ineia of he syse abo z-ais can be find o by cacain he oen of ineia of indiida od abo z-ais I M I becase z-ais is he ede of od and and I becase od in yin on z-ais I So (c M syse I I I M M he oen of ineia of syse abo side of iane I I a So (b I I C a Moen of ineia of syse abo an aes which is pependica o pane of ods and passin hoh he coon cene of ods I z pependica aes heoe I z I I y a M M M ain fo 6 M 6 I I I [s M I z C a a I I ] J ISION d SOUION (ae:

30 So (b If and ae he especie disances of paices and fo he cene of ass hen ( ( 97Å and 7Å Moen of ineia of he syse abo cene of ass I I ( a (97Å a (7Å Sbsiin a = 67 7k and Å = I So (b 7 6 k M I of sphee abo is diaee I O ' M Now MI of sphee abo an ais pependica o he pane of sqae and passin hoh is cene wi be I O I O' M M M [by he heoe of paae ais] Moen of ineia of syse (ie fo sphee M M = I O M So (c Moen of ineia of sphee abo i diaee M I [s I = 6 So (b O M = ] F ( ˆ i ˆj kˆ N D and ( i kˆ / ee O C oqe F iˆ iˆ ˆj 6kˆ ˆj and ( ( ( 7 So (d F kˆ 6 = N- e he ass of he od is M Weih (W = M Iniiay fo he eqiibi F M / F F M When one an wihdaws, he oqe on he od I M na acceeaion and inea acceeaion M M [s I = M / ] a Now if he new noa foce a is M M F' Ma F' M Ma M M 8 So (a W M M O F ' hen Iniia ana oen of he syse abo poin O = inea oen ependica disance of inea oen fo he ais of oaion M (i Fina ana oen of he syse abo poin O I I ( I I M M (ii ppyin he aw of conseaion of ana oen M M F / / F M M J ISION d SOUION (ae:

31 9 So (b he ana oen ( of he syse is conseed ie = I = consan When he ooise waks aon a chod, i fis oes cose o he cene and hen away fo he cene Hence, MI fis deceases and hen inceases s a es, wi fis incease and hen decease so he chane in wi be non-inea fncion of ie So (c y he aw of conseaion of ana oen I I I I na eociy of syse I I I I oaiona kineic eney I So (d I I I I I I I ( I I ( I I Since hee ae no eena foces heefoe he ana oen of he syse eains consan Iniiay when he beads ae a he cene of he od ana oen M (i When he beads each he ends of he od hen ana oen and (ii M M ' (ii Eqain (i So (c M ' ' o MM 6 Iniia ana oen of he syse = na oen of be befoe coision M M (i e he od oaes wih ana eociy Fina ana oen of he syse M M y eqaion (i and (ii / So (a (ii M M M o na oen of he cyinde befoe coision So (a I M ( ( K K k disc d k s k in k s ccodin o pobe d So (b s E I E E % = J-s fo disc fo in d Kdisc Kin = ] E 9E E 8% of E 6 So (a owe 6 7 So (b d [s ( ( i ˆj kˆ(ˆ i ˆj kˆ = W eociy a he boo ( h K h h 8 So (c ad/sec,, sec So ana eadaion ad / sec J ISION d SOUION (ae:

32 Now ana speed afe sec = ad/sec Wok done by oqe in sec = oss in kineic eney = I ((( ( = J 9 So (a sin sin cceeaion (a sin K So (b ie of descen oen of ineia k sphee sphee k 7 k,, disc disc in k in So (b ccodin o aw of conseaion of eney h So (d We know he anenia acceeaion I M [s fo cyinde] fe ie, inea eociy of ass, So ana eociy of he cyinde So (d ( I / M M M ( M h I a I a So (d ie of descen shpee disc = So (a k h sin [s, sin and ien] h k k k 7 / sin h k sphee disc sin / s 7 k On eeasin fo es he bock fas hoh heih in sec a( S a [s ] a / s Sbsiin he ae of a in he foa a I and by soin we e I ( I k J ISION d SOUION (ae:

33 6 GIION Q ns C C C C D Q ns D C, D D D D Q ns C C C C C C C D C Q ns D C D C C C So (a Gaiaio na consan podc of he asses F (Disance beween he asses So (a Gaiaiona foce G GM ( M GM F ( df Fo ai ae of foce d d GM ( d / So (a d d ( e paice ies a oiin, paice and C on y and -ais especiey F C G ˆi 667 ˆ 9 i 67 ( Siiay F 67 Ne foce on paice 9 ˆj N ˆi N 9 F F C F 67 (ˆi ˆj N So (c If wo paices of ass ae paced disance apa hen foce of aacion G F (e Now accodin o pobe paice of ass is paced a he cene ( of sqae hen i wi epeience fo foces G F foce a poin de o paice F Siiay G G F F, F F C and G F D F Hence he ne foce on Fne F F F C F D F F ne G G ( a / a [ haf of he diaona of he sqae] G a So (a cceeaion de o aiy o [s e 6 So (c e We know 6 e and e e e G e (ien] e 6 GM GM GM ( D / D 8 If ass of he pane M and diaee of he pane D 7 So (b hen GM D cceeaion de o aiy oon oon eah M M oon eah eah oon 6 eah 8 GM 8 e J ISION d SOUION (ae:

34 8 So (b We know 9 So (c GM GM cceeaion de o aiy pane eah M M pane eah 7 GM eah pane If a sone is hown wih eociy fo he sface of he pane hen ai heih H H pane eah H So (a pane eah pane 9 h So (c H H 9 eah 9 = ee Weih of he body a heih, W W h W 9 W W N So (d 9 9 If is he poin whee ne aiaiona foce is zeo hen F F y soin d D, G G ( d F Fo he ien pobe eah, oon and 8 d d F d 8 So So (b D D D 9D 8 9 ecenae chane in when he body is aised o heih h, h % % ecenae chane in when he body is aken ino deph d, d h % % % [s So (b ecenae decease in weih h % % d d d So (a cceeaion de o aiy a deph d, 6 6 So (a d h ] d / s Effecie acceeaion de o aiy de o oaion of eah cos cos 6 7 So (a o 8 ad sec o [s and 6 ] ad sec When eah sops sddeny, cenifa foce on he an becoes zeo so is effecie weih inceases 8 So (d 9 So (c Eah d Moon J ISION d SOUION (ae:

35 oin of zeo inensiy eah M ass of he d, Mass of he oon disance beween eah & oon d inensiy fo he Eah M M M 8 M and 8 6 oin of zeo So disance fo he oon 6 6 So (a, b We know ha aiaiona foce Inensiy [s GM I ] F if F foce Inensiy when So (d when and and aiaiona [s I G F if and F ] Inensiy a he oiin I I I I I GM GM GM GM GM 8 GM 6 6 O GM [s s of G GM k k k k 8 a ] G G [s M k ien] So (b s So (d d I, if I hen = consan d E d So (b oenia inceases by K K d J /k a So (b O Ne poenia a oiin G 8 G G 6 So (d J / k eey whee so i wi be G G G Wok done = Chane in poenia eney U U 7 So (b GM GM GM GM GM 6 When body sas fain owad eah s sface is poenia eney deceases so kineic eney inceases Incease in kineic eney = Decease in poenia eney Fina kineic eney Iniia kineic eney = Iniia poenia eney Fina poenia eney Fina kineic eney GM GM Fina kineic eney GM GM h h GM GM GM GM 8 So (b Wok done / 8 GM 6 h h/, If h hen wok done J ISION d SOUION (ae: 6

36 9 So (b GM Wok done = U fina Uiniia U U GM [s k ] So (d oenia eney of hee paices syse G U G G Gien and c 667 ( U Joe So (c ( When a boy jps fo a ond ee p o heih h hen is eociy of jpin h (i and fo he ien condiion his wi becoe eqa o escape eociy escape = GM Eqain (i and (ii So (a h Gd / G d (ii h 8 Gd Escape eociy does no depend pon he ane of pojecion So (a e G 8 G e if = consan Since he pane hain dobe adis in copaison o eah heefoe he escape eociy becoes wice ie So (c k/ s Fo he aw of conseaion of eney Diffeence in poenia eney beween ond and ai heih = Kineic eney a he poin of pojecion h ( k e h/ J ISION d SOUION (ae: 7 [s e ] k e k ( k y soin heih fo he sface of eah h k So heih fo he cene of eah k So (c Escape eociy consan e k h k e and if densiy eains So if he adis edces by % hen escape eociy aso edces by % 6 So (c Kineic eney ien o ocke a he sface of eah = Chane in poenia eney of he ocke in eachin fo ond o hihes poin h h/ h h h h h h 7 So (c oenia eney of he body a a disance sface of eah e U h/ [s h e (ien] e e e e fo he So ini eney eqied o escape he body wi be e 8 So (a ccodin o Kepe s aw 9 So (c d d / eah / (/ / pane ( yea yeas and = ] [s na oen d d

37 So (a When he pane passes neae o sn hen i oes fas and ice-esa Hence he ie aken in aein D is ess han ha fo CD So (c Kepe s hid aw Nepne Nepne San San So (a / / ccodin o conseaion of ana oen consan in a in So (d a in in 6 a 6 / s a 8 ie peiod does no depend pon he ass of saeie, i ony depends pon he obia adis ccodin o Kepe's aw So (c / / ccodin o aw of conseaion of ana oen So (b d d d d Obia eociy of saeie GM 6 GM k / sec k / sec (ppo 9 So (b So (b 8 ( h GM ( [s h (ien] (9 7 [s 9 (ien] So (d / (9 ie peiod depends ony pon he obia adis So (c / / / Fo he epession h So (d GM h / na oen 6 [s GM GM ] 6 So (b Obia eociy [If deceases hen inceases] ecenae chane in G (ecenae chane in (% % obia eociy inceases by % 7 So (b If F 8 So (a hen n Obia eociy ; hee n n So (b So (c oh he paices oes diaeicay opposie posiion aon he cica pah of adis and he aiaiona foce poides eqied cenipea foce J ISION d SOUION (ae: 8 G ( G

38 6 So (b Conseaion of ana oen I M consan [If M eains sae] / n n h [s h n ] 7 So (c 8 So (c H H M CM M Eah ody H s he asses of he body and he eah ae copaabe, hey wi oe owads hei cene of ass, which eains saionay H Hence he body of ass oe hoh disance If body fas fo heih h hen ie of descen h and ie o each he eah sface h H / oon eah 6 eah oon oon 6 H J ISION d SOUION (ae: 9

39 7 OEIES OF UK ME Q ns C C C C C D C Q ns C D D C D C D,,C D Q ns D C C C D D C Q ns, C C C D C C D Q ns D C C D C D C Q ns D C D C C So (c s he wie is nifo so he weih of wie beow poin is W oa foce a poin and aea of cosssecion = S Sess a poin So (c W W W Foce ea W S When he enh of wie becoes dobe, is aea of coss secion wi becoe haf becase oe of wie is consan ( So he insananeos sess = Foce ea So (c M M / s ensie sess = N ( / cos, F F N W F Noa foce ea F Noa foce = F cos N N and hee So, ensie sess So (b Shea sess F cos F cos / cos anenia foce F sin θ F sin θ cos θ ea ( / cos θ F sin So (a ensie sess F cos cos a ie cos 6 So (b I wi be ai when o Chane in enh Sain Oiina enh 7 So (c Sain Chane in enh %of / = Oiina enh 8 So (c Fo he aph Y 9 So (a an C an an C Y Y Ybbe Yass YSee, s F Y 9 ( Y 9 N / F Y, % 6 N J ISION d SOUION (ae:

40 So (a F Y F F ie he inceen in enh wi be sae So (d F Y F F Y Y So (a F Y 8 [s F and Y ae consans] So (b F Y ( e he oiina enh of easic sin is and is foce consan is k When onidina ension N is appied on i a k (i and when onidina ension N is appied on i b k (ii y soin (i and (ii we e a b k b a and Now when onidina ension 9N is appied on easic sin hen is enh = 9 k So (a a b 9( b a Yon's ods consan F Y b a (s Y, and F ae Fo he aph i is cea ha fo sae oad eonaion is ini fo aph OD s eonaion ( is ini heefoe aea of cosssecion ( is ai So hickes wie is epesened by OD So (c F F [s d / ] Y d Y If he eonaion in boh wies (of sae enh ae sae nde he sae weih hen d d C Y Y C d Y consan Y 7 dc d 9 Y 6 So (c F Y 7 So (a C F F Y Y fo a fied oe hea sess F 8 So (a Y 7 N / Sain = F Y 9 So (d Sess = ension ea of coss-secion So (a ( 6 N consan If he syse oes wih acceeaion a and is he ension in he sin W hen by copain his condiion fo sandad case J ISION d SOUION (ae:

41 In he ien pobe ( and ension a a Sess and Sain So (d Sess Yon' s ods ay F and fo a ien sechin foce Y Y e hee wies hae yon's ods Y, Y and Y and hei coss seciona aeas ae, and : : Y : Y : Y especiey : : : : 6:: Y Y Y So (c Inceen in he enh M 6 Y = Dispaceen of poin = So (b 6 e he eqiaen yon's ods of ien cobinaion is Y and he aea of coss secion is k k k Fo paae cobinaion eq Y Y Y Y Y Y So (d, Foce consan of wie of spin k k Y Y Y k (ien F Y and foce consan Eqiaen foce consan fo ien cobinaion k eq,,, k k Y k k eq ky ky So k eq ( ky ky he aph beween appied foce and eension wi be saih ine becase in easic ane appied foce eension, b he aph beween eension and soed easic eney wi be paaboic in nae s 6 So (d k U U o eakin foce aea of coss-secion F F d d F F N 7 So (a,b,c When foce Sess in wie F (i if ( d F is appied a he owe end hen F and sess in wie = (e hen sess in wie and sess in wie = ie sess in wie > sess in wie so he wi beak befoe (ii if, (e hen Sess in wie = ( and Sess in wie ie sess in wie = sess in wie I eans eihe o ay beak (iii If hen sess in wi be oe han ie wi beak befoe 8 So (b diabaic easiciy E E isohea easiciy ie peiod of cobinaion J ISION d SOUION (ae:

42 9 So : (d Y Sess Sain Sess = Y Sain So (a K / So (b [s 8 96 N / N / hd 98 / d d e o o e e o ] d d d d / So (a K Chane in oe de o ise in epeae oeic sain bk ods sess sain So (d So (b d d d d ( = [s hee is no chane in he oe of he wie] So (b aea sain onidina sain onidina sain aea sain 6 So (c onidina sain o % aea sain % % 8% oeic sain = onidina sain aea sain 7 So (b esse a boo of he ake = ay haf he deph of a ake ccodin o ien condiion h and pesse h h ( h h 6 8 So (b h ccodin o oye's aw, pesse and oe ae inesey popoiona o each ohe ie ( h w h w [s 7c of H 7 6 ( c c 9 So (c h ] If he ise of ee in he ih ib be c he fa of ee of ecy in ef ib be c becase he aea of coss secion of ih ib ies as ha of ef ib ee of wae in ef ib is (6 + c Now eqain pesse a ineface of ecy and wae (a ' ' ( ' ' ( 6 6 (8 = % oe wi chane by % y soin we e = 6 c So (b J ISION d SOUION (ae:

43 Diffeence of pesse beween sea ee and he op of hi (i ( h h H ( 7 H and pesse diffeence de o h ee of ai = h ai (ii y eqain (i and (ii we e h ai ( 7 H h H ai Heih of he hi = k So (a Weih of cyinde = phs de o boh iqids D d d D D d So (c d D d M oe of ice, oe of wae M M Chane in oe M So (d e M ass of body in ac M ppaen weih of he body in ai = ppaen weih of sandad weihs in ai ca weih phs de o dispaced ai = ca M weih phs de o dispaced ai M W M d M d d d d M d M d d Uphs on body in iqid Uphs on body in wae a W ai W iqid b W ai Wwae So (d ossof weih in iqid ossof weih in wae s he bock oes p wih he fa of coin, deceases, siiay h wi aso decease becase when he coin is in wae, i dispaces wae eqa o is own oe ony So (c Densiy of aoy Specific aiy of aoy Densiy of wae Mass of aoy oe of aoy densiy of wae p w / w / w 6 So (b s s densiy of sbsance s specific aiy of sbsance densiy of wae e specific aiies of concee and saw ds ae and especiey ccodin o pincipe of foaaion weih of whoe sphee = phs on he sphee ( ( ( ( Mass of concee Mass of saw ds 7 So (a Facion of oe iesed in he iqidin i depends pon he densiies of he bock and iqid ie So hee wi be no chane in i if syse oes pwad o downwad wih consan eociy o soe acceeaion 8 So (b, c e he densiy of ea is and densiy of iqid If is he oe of sape hen accodin o pobe (i is J ISION d SOUION (ae:

44 8 ( (ii ( (iii y soin (i, (ii and (iii we e 7 and 9 So (c If wo diffeen bodies and ae foain in he sae iqid hen So (c ( f ( f in in / / If he iqid is incopessibe hen ass of iqid enein hoh ef end, shod be eqa o ass of iqid coin o fo he ih end M So (d / s s coss-secion aeas of boh he bes and C ae sae and be is hoizona Hence accodin eqaion of coniniy o enoi's heoe sae in boh he bes and C So (b and heefoe accodin C C o ie heih of iqid is h / s So (a When coss secion of dc deceases he eociy of wae inceases and in accodance wih enoi's heoe he pesse deceases a ha pace So (c ie aken fo he ee o fa fo H o H H ' H ' ccodin o pobe- he ie aken fo he ee o fa fo h o h h h and siiay ie aken fo he ee o fa fo h 6 So (b h / s h o zeo 7 So (b Fo enoi's heoe, Hee, d h h dh d d[ ] Now, d dh d, and hd hd d o So (b h enoi's heoe fo ni ass of iqid consan s he iqid sas fowin, i pesse eney deceases eociy of eff when he hoe is a deph h, ae of fow of wae fo sqae hoe y ae of fow of wae fo cica hoe 8 So (c (y and accodin o pobe y (y h Q a = Q a = Q Q e = he aea of coss secion of he hoe, = Iniia eociy of eff, d = Densiy of wae, Iniia oe of wae fowin o pe second = Iniia ass of wae fowin o pe second = d ae of chane of oen = d Iniia downwad foce on he o fowin wae = d So eqa aon of eacion acs pwads on he cyinde J ISION d SOUION (ae:

45 Iniia pwad eacion = d [s Iniia decease in weih d ( h 98 9 So (a h ] -w dh he heih of wae in he ank becoes ai when he oe of wae fowin ino he ank pe second becoes eqa o he oe fowin o pe second oe of wae fowin o pe second = h and oe of wae fowin in pe second 7 c / sec h 7 h 7 98 h 7 9 h c 96 6 So (d (, oise decapoise (MKS ni, d = /s and F = N F 6 So (d d d d d F When we oe fo he cene o he cicfeence eociy of iqid oes on zeo 6 So (b F 6 6 So (c he deceasin and finay becoes If wo dops of sae adis coaesce hen adis of new dop is ien by / If dop of adis is fain in iscos edi hen acqie a ciica eociy and / / / / ( ( / s 6 So (c eociy of ba when i sikes he wae sface h (i eina eociy of ba inside he wae (ii 9 i Eqain (i and (ii we e h ( 9 h 8 6 So (b ae of fow of iqid whee iqid esisance 8 Fo anohe be iqid esisance 8 8 ' 6 6 Fo he seies cobinaion New 66 So (d ' Fo So (c Fo paae cobinaion 8 68 So (b eff ( 6 69 So (b s 6 / / n / 6 / o 6% J ISION d SOUION (ae: 6

46 Iniia sface eney ( Fina sface eney ( / = / 7 So (a Incease in sface eney ( n / ( (6(8 / 6 = J = J 7 So (b 6 6c 6, c s he soap fi has wo fee sfaces W W ( W N/ 7 So (b s fi hae wo fee sfacesw W 7 J 7 So (d s oe of he bbbe / / / / / Wok done in bowin a soap bbbe W 8 / W W W / / / / ( ( W W 7 So (b F Eney eeased = 7 So (c esse inside a bbbe when i is in a iqid o 7 a 76 So (a 7 Ecess pesse inside a soap bbbe : 77 So (b Ecess pesse inside a bbbe js beow he sface wae and ecess pesse inside a dop 78 So (b cos h d [as and ae consans] W H 79 So (b W H h h W H o cos 6 cos he heih po which wae wi ise c 8 So (a cos h d W H cosw cos [h=eica heih, = ane wih eica] h [If ohe qaniies eains consan] h h 8 So (d oon eah = 6 h h eah oon 6 oon H [s eah= 6oon] d d H W of c cos 6 J ISION d SOUION (ae: 7

47 h In a if oin downwad wih acceeaion (a, he effecie acceeaion deceases So inceases 8 So (b h cos d, [ =, =, 6N / k /, = 98 /s ] 6cos h c 98 8 So (c s 8 So (b h h o h 66 h h When he be is paced eicay in wae, wae ises hoh heih h ien by Upwad foce cos cos h d Wok done by his foce in aisin wae con hoh heih h is ien by W ( cos h ( h cos hd ( hcos cos h d Howee, he incease in poenia eney he aised wae con ass of he aised con of wae, d E p of h whee is he hd 8 So (d Weih of iqid = pwad foce de o sface ension Cicfeence = 86 So (c Unde isohea condiion sface eney eain consan So (b adis of cae of coon sface of dobe bbbe 88 So (c Fo = c a ien epeae, he aio asses of ai 89 So (a h and cos = : cos cos h cos cos 6 cos o o / / So, E Fhe, h h d ( hd W E p h d he pa ( W E is sed in doin wok aains iscos foces and ficiona foces beween wae and ass sface and appeas as hea So hea eeased = W E J p h d J J ISION d SOUION (ae: 8

48 8 HE & HEMODYNMICS Q ns C D C C C D Q ns D C C D C D C C D D Q ns C D C C, D C C Q ns D C C C C Q ns D D C C D C c Q ns C C D D C C D c Q ns D D D C D D Q ns D C D C D D C D C C Q ns C C C C C D D D So (a epeae adien o C / c So (a ccodin o pobe, ae of hea oss in boh ods ae dq eqa ie d K K K K dq d [s = ( and ien] So (c dq / d K ae of fow of hea pe ni aea So (a s he epeae of oo eain consan heefoe he ae of hea eneaion fo he heae shod be eqa o he ae of fow of hea hoh a ass window K J ( ( [whee = epeae of oside] So (d = o C ae of fow of hea wi be eqa in boh es and shi K es K K 6 So (b es K shi es shi shi es K K es shi shi = hea condciiy epeae adien epeae adien (X hea condcii y(k K K K K [s dq / d s K C > K > K heefoe X C consan] X X Q K [s Q, K and ae consan] J ISION d SOUION (ae: 9

49 / 7 So (b e he condciiy of each od is K y considein he ods and C ae in paae, effecie hea condciiy of and C wi be K Now wih he hep of ien foa K K epeae of ineface K K K K 9 8 o 6 C K K 8 So (c ae of fow of hea aon Q dq d Q K dq ae of fow of hea aon Q d (i K s Q Effecie condciiy fo seies cobinaion of wo ods of sae enh So dq d Q K s Eqain (i and (ii 9 So (a KK K K K K KK (ii K K K K K KK K K K = 9K, = 8 c, = 6 c, = o C, = o C epeae of he jncion So (c o C 9 o C K 9K K 8 6 9K K 8 6 K K K K K K K 8 / o 7 C We can conside his aaneen as a paae cobinaion of wo aeias hain diffeen hea condciiies K and K Fo paae cobinaion K K K = ea of coss-secion of inena cyinde =, = ea of coss-secion of oe cyinde = ( ( = So (b K K K K K K K epeae of ineface K K Sbsiin So (c K and e he hea esisance of each od is K we e Effecie hea esisance beween and D = epeae of ineface So (b ccodin o Inen Hasz, So (d K K 9 K o C ie eqied in inceen of hickness fo y o y ( y y K In fis condiion y =, y = c hen ( In second condiion y = c, y = c hen ( C D o C o C D = = 7 = hs J ISION d SOUION (ae:

50 J ISION d SOUION (ae: So (b ccodin o Wien's aw and fo he fie ( ( ( heefoe > > 6 So (d ea nde ce epesens he eissie powe of he body 6 E E (ien (i fo Sefan's aw E E E (ii Fo (i and (ii 6 = K 7 So (c s he epeae of body inceases, feqency coespondin o ai eney in adiaion ( inceases his is shown in aph (c 8 So (c consan ( ( ( ( = 8 9 So (a ccodin o Wien's aw waeenh coespondin o ai eney deceases When he epeae of back body inceases ie consan / Now accodin o Sefan's aw 8 6 E E So (b Wien's aw o inceases wih epeae So he aph wi be saih ine So (a Q = Q Q : 6 6 So (d Eissie powe of a body ( in a sondin (, ( E o ( Q ( (67 ( (7 ( ( Q Q So (c Q [If = consan] So (d Eney adiaed by body pe second Q o b Q [ea = b] b b E E 6 / ( / ( b b E E 6 8 So (c c e d d ( ( 6 ( c a a e Fo he sae fa in epeae, ie a d c c a a d d d = d = sec = sec [s = 6a and = = a ] 6 So (b ( c e d d ae of cooin accodin o pobe = / aio of ae of cooin /

51 7 So (c Q = ( If,, and ae sae fo boh bodies hen Q Q sphee cbe sphee (i 6a cbe accodin o pobe, oe of sphee = oe of cbe / a a Sbsiin he ae of a in eqaion (i we e Q Q sphee cbe 6 8 So (d 6a / : 6 / 6 / ccodin o Newon's aw of cooin Fo fis condiion and fo second condiion 6 6 (ii y soin (i and (ii we e = 8 sec ~ sec 9 So (b (i ccodin o Newon's aw of cooin Fo fis condiion (i and fo second condiion 6 6 y soin (i and (ii we e = sec So (d d d ( c (ii If he iqids p in eacy siia caoiees and idenica sondin hen we can conside and consan hen d ( (i d c If we conside ha eqa asses of iqid ( ae aken a he sae epeae hen d d c So fo sae ae of cooin c shod be eqa which is no possibe becase iqids ae of diffeen nae ain fo (i eqaion d d ( c d ( d c Now if we conside ha eqa oe of iqid ( ae aken a he sae epeae hen d d c So fo sae ae of cooin ipicaion of c fo wo iqid of diffeen nae can be possibe So opion (d ay be coec So (b Fo fis condiion = K [ ] Fo second condiion 8 = K (6 6 6 (i (ii Fo (i and (ii we e = o C So (b eaion beween Cesis and Fahenhei scae of epeae is C F 9 y eaanin we e, C = 9 F 6 9 y eqain aboe eqaion wih sandad eqaion of ine 6 y c we e and c 9 9 ie Sope of he ine is 9 So (c epeae on any scae can be coneed ino ohe X F scae by = Consan fo a scaes UF F X C C 6 X = = 98 So (b o J ISION d SOUION (ae:

52 Fo a consan oe as heoee epeae in ceniade is ien as c C ( 6 c C ( So (a Since a consan diffeence in enh of c beween an ion od and a coppe cyinde is eqied heefoe o c (i Fe C Fe C O Fe C ie, inea epansion of ion od = inea epansion of coppe cyinde Fe Fe C Fe C Fe Fo (i and (ii 6 So (d 7 C C Fe C 7 8c, 8 c Fe C (ii he ape of he isoscees iane o eain a a consan disance fo he knife ede DC shod eains consan befoe and afe heain efoe epansion : In iane DC DC (i ( fe epansion : ( DC [ ( ] ( (ii Eqain (i and (ii we e [ ( ] ( [Neecin hihe es] ( ( 7 So (b he ass od and he ead od wi sffe epansion when paced in sea bah enh of bass od a C ' bass ( bass / / D C bass = 8[ 8 6 ] and he enh of ead od a C ' ead ( = 8[ 8 ] ead ead Sepaaion of fee ends of he ods afe heain = = 8[8 8] ' ' ead bass 8 8 So (c c 8 6 De o incease in epeae, adis of he sphee chanes e and ae adis of sphee a o C and o C [ ] Sqain boh he sides and neecin hihe es [ ] y he aw of conseaion of ana oen I I M M [ ] So (c [ ] De o oe epansion of boh iqid and esse, he chane in oe of iqid eaie o conaine is ien by = [ S ] Gien = cc, = / C / C / C = [8 ] = cc So (b In anoaos epansion, wae conacs on heain and epands on cooin in he ane C o C heefoe wae pipes soeies bs, in cod conies So (a s wih he ise in epeae, he iqid ndeoes oe epansion heefoe he facion of soid sbeed in iqid inceases Facion of soid sbeed a C f = oe of dispaced iqid (i ( and facion of soid sbeed a C f = oe of dispaced iqid (ii Fo (i and (ii f f f f f f ( J ISION d SOUION (ae:

53 So (b, d On heain, he sip ndeoes inea epansion So afe epansion enh of bass sip C ( and enh of coppe sip ( Fo he fie and c/adis] C Diidin (i by (ii d ( d (i c (ii [s ane = d ( ( C C = ( ( = ( d C ( o C C C d ( [Usin inoia heoe and neecin hihe es] So we can say ( C and So (c On heain he syse;,, d a inceases, since he epansion of isoopic soids is siia o e phooaphic enaeen So (c Iniia diaee of ye = ( 6 = 99, so iniia adis of ye and chane in diaee D = 6 so fe inceasin epeae by ye wi fi ono whee Inceen in he enh (cicfeence of he ion ye = [s ] and = 97 ] o C 6 d 97 C [s = So (a oss of ie de o heain a pend is ien as = = ( C 86 pe C 86 6 So (d De o heain he enh of he wie inceases onidina sain is podced Easic poenia eney pe ni oe E = Sess Sain = E = o E = [s 7 So (b Y (Sain Y Y Y = 8 and = (ien] Y hea capaciy = Mass Specific hea De o sae aeia boh sphees wi hae sae specific hea aio of hea capaciy : 8 8 So (c Hea eqied o aise he epeae of of sbsance by d is ien as dq = c d Q c d o aise he epeae of of sbsance fo C o C is Q 9 So (a ( d = 8 caoie Wok done in conein of ice a C o sea a C = Hea sppied o aise epeae of of ice fo C o C [ c ice ] + Hea sppied o cone ice ino wae a C [ ice] + Hea sppied o aise epeae of of wae fo C o C [ c wae ] J ISION d SOUION (ae:

54 + Hea sppied o cone wae ino sea a C [ apo] = [ c ice ] + [ ice] + [ c wae ] + [ apo] = [ ] [ 8] [ ] [ ] = 7 caoie 7 J So (b Iniiay ice wi absob hea o aise i's epeae o o C hen i's ein akes pace If = Iniia ass of ice, ' = Mass of ice ha es and w = Iniia ass of wae y aw of ie wae c ( ' = w c w [ ] Hea ain by ice = Hea oss by ( ' 8 = ' = k So fina ass of wae = Iniia ass of wae + Mass of ice ha es = + = 6 k So (b When wae is cooed a o C o fo ice hen 8 caoie/ (aen hea eney is eeased ecase poenia eney of he oeces deceases Mass wi eain consan in he pocess of feezin of wae So (a Sae aon of hea is sppied o coppe and wae so c c c = So (c c c ccc c c Hea os by = Hea ained by C c c ( Since ( and epeae of he ie ( = 8 C c c ( 8 c (8 : c So (b Hea os by ho wae in beake + Hea absobed by beake wae = Hea ained by cod (9 = ( + ( = 68 C So (b Hea eqied by k wae o chane is epeae fo C o 8 C in one ho is Q = ( c wae = ( (8 6 caoie In condensaion (i Sea eease hea when i ooses i's epeae fo o C o o C [ c sea ] (ii o C i cones ino wae and ies he aen hea [ ] (iii Wae eease hea when i ooses i's epeae fo o C o 9 o C [ s wae ] If sea condensed pe ho, hen hea eeased by sea in conein wae of 9 C Q = ( c ( s = sea sea wae [ ( ( 9] = 6 caoie ccodin o pobe Q = Q 6 ca = 6 ca = = k 6 So (c Since in he eion epeae is consan heefoe a his epeae phase of he aeia chanes fo soid o iqid and (H H hea wi be absob by he aeia his hea is known as he hea of ein of he soid Siiay in he eion CD epeae is consan heefoe a his epeae phase of he aeia chanes fo iqid o as and (H H hea wi be absob by he aeia his hea as known as he hea of apoisaion of he iqid 7 So (b Wok done by he syse = ea of shaded poion on - diaa ( 6 ( J and diecion of pocess is anicockwise so wok done wi be neaie ie W = J 8 So (a Wok done = ea encosed by iane C 9 So (c C C ( ( ea encosed beween a and f is ai So wok done in cosed cyces foows a and f is ai 6 So (b Incease in inena eney U C 96 ( ca 6 So (a ea encosed by ce < ea encosed by ce < ea encosed by ce Q Q Q [s U is sae fo a ces] J ISION d SOUION (ae:

55 6 So (d Q U W U U J 6 So (d U Q C C 6 So (a /( /( 7 / y adjoinin aph W and W C 8 [ ] J WC W WC J Now, Q C Q Q C 6 8 J Fo fis aw of heodynaics Q U W 8 U C C U C 6 J 6 So (c C In ea ases an addiiona wok is aso done in epansion de o ineoeca aacion 66 So (a Wok done in an isohea pocess W oe f i 8 ( 8 oe J 67 So (a y ande Waa s eqaion Wok done, W d n e n o ( n n n oe n n 68 So (a Gien 69 So (b 7 C n n n d n n n fo adiabaic pocess C C p Fo an adiabaic pocess consan d So, / K 67 7 K 7 So (c Fo an adiabaic pocess 7 So (b consan / 8 ( / Q U W, In an adiabaic pocess Q = U W In epansion W = posiie U = neaie Hence inena eney ie epeae deceases 7 So (b Sope of adiabaic ce oiciy of he is inesey popoiona o aoiciy of he as [s 66 So as fo onoaoic as, = fo diaoic as and = fo iaoic non-inea as] Fo he aph i is cea ha sope of he ce is ess so his shod be adiabaic ce fo diaoic as (ie O Siiay sope of he ce is oe so i shod be adiabaic ce fo onoaoic as (ie He 7 So (d Fo an adiabaic pocess 7 So (c / / s we know ha sope of isohea and adiabaic ces ae aways neaie and sope of adiabaic ce is aways eae han ha of isohea ce so is he ien aph ce and ce epesens adiabaic and isohea chanes especiey 7 So (c Fo isohea pocess ' ' (i Fo adiabaic pocess (ii Since heefoe ' J ISION d SOUION (ae: 6

56 76 So (b Fo an adiabaic pocess / Fina pesse = 8 77 So (c ( 8 Eney spen in oecoin ine oeca foces U Q W (67 6 Q ( caoie 78 So (b s we know, wok done J 79 So (c ( Q p C p 7 7 ( Q p 7 7 caoie [s = oe and = fo H] 8 So (a W Q Q U Q C p C C p ie pecenae eney iised in doin eena wok = = % 8 So (d ocess CD is isochoic as oe is consan, ocess D is isohea as epeae consan and ocess is isobaic as pesse is consan 8 So (d Hea ien Q ca 8 J Wok done W = J [s pocess is anicockwise] y fis aw of heodynaics U Q W 8 ( J 8 So (a Fo a cycic pocess oa wok done W W C W C ( W [W C = since hee is no chane in oe aon C] C J J W W C J 8 So (b C Wok done din pocess is posiie whie din pocess i is neaie ecase pocess is cockwise whie pocess is anicockwise aea encosed by - aph (ie wok done in pocess is sae so, ne wok done wi be neaie 8 So (c ocess is isochoic, ocess C is isohea ocess C is isobaic W C 86 So (a W W C ( n is isobaic pocess, C is isohea pocess, CD is isoeic pocess and D is isohea pocess hese pocess ae coecy epesened by aph (a 87 So (b Fo an adiabaic epansion ien pocess = consan consan and fo he I is aso an adiabaic epansion and din adiabaic epansion he as is cooed 88 So (c Fo he ien diaa, we can see ha In pocess, qaniy of he as eains sae In pocess C, C, = consan esse is consan (s = consan and in pocess hese pocesses ae coecy epesened on diaa by aph (c W 89 So (d Q W Q 6 8 =6 J (6 9 So (c Coefficien of pefoance K 9 So (d Q W So (b 7 J 7 W Q K 7C W Q J ISION d SOUION (ae: 7

57 9 So (b W Q Q Q Q whee Q hea absobed, Q hea ejeced / W Q 9 So (a 9 So (a a Q Q Q Q 7 7 and 7 7 W Q Q Q Q Q Q Q ( So eqied aio 77 7 % So 6% efficiency is ipossibe 96 So (d Kineic eney E = esse 97 So (d E 7 J, oe = ie = Mean sqae eociy of oece k 6 N/ Fo as, coponen of ean sqae eociy of oece w k Mean sqae eociy w (i Fo as ean sqae eociy k (ii Fo (i and (ii 98 So (a, w so w 6 N, k, s / s 6 N s 88 N/ 99 So (c ( Fo idea as eqaion we e K 7 C 6 So (b s O eans oe heefoe 8 O eans / oe ie So fo So (a we e o Mass of wae k, Moeca wof wae 8 k = 7 K and Fo N/ (S So (d M M Fo M o ; Hee epesens he sope of ce dawn on oe and epeae ais M Fo fis condiion sope aph is D (ien in he pobe Fo second condiion sope M M / ie sope becoes fo ie so aph is coec in his condiion So (a Specific as consan Uniesa as consan ( Moeca weih of as ( M Joe/oe-K So (d 8 Nbe of oes in fis esse and nbe of oes in second esse If boh esses ae joined oehe hen qaniy of as eains sae ie ( J ISION d SOUION (ae: 8

58 So (a Idea as eqaion, in es of densiy consan op oo 6 So (a s s M op oo oo op C / C M / M s 7 So (a M M ien O Fo oyen O and M O Fo hydoen H ccodin o pobe M O O H M H M H H H K 8 So (d s M O O 7 76 H So (b N s s 8 M H / so So (c E H E E E E ( So (a 9 86 K C Kineic eney pe deee of feedo s diaoic as possess wo deee of feedo fo oaiona oion heefoe oaiona K E k k In he pobe boh ases (oyen and nioen ae diaoic and hae sae epeae ( K heefoe aio of aeae oaiona kineic eney wi be eqa o one So (d E So (c 8 7 k Joe eae ansaiona kineic eney does no depends pon he oa ass of he as Diffeen ases wi possess sae aeae ansaiona kineic eney a sae epeae So (d E b s ie if epeae becoes wice hen eney wi becoes wo ie ie 6 = J s speed wi becoe ies ie 8 68 /s So (d If hen % of = Fo oye s aw = consan Faciona chane in oe ecenae chane in oe % % 76% ie oe decease by 76% 6 So (b consan epeae fo he ien oe of as k 7 he qaniy of as aken o of he cyinde = = 7 k J ISION d SOUION (ae: 9

59 7 So (c 8 So (d cc ccodin o oye s aw ipicaion of pesse and oe wi eains consan a he boo and op If is he aospheic pesse a he op of he ake and he oe of bbbe is hen fo h ( h 9 So (c h Fo a ien pesse, oe wi be oe if epeae is oe (Chae s aw Fo he aph i is cea ha > > So (c Fo Chae s aw So (a s Facion of as coes o So (c In he ien aph ine hae a posiie sop wih X-ais and neaie inecep on Y-ais So we can wie he eqaion of ine y = c (i ccodin o Chae s aw, by ewiin 7 his eqaion we e 7 7 (ii y copain (i and (ii we can say ha ie is epesened on Y-ais and oe in X-ais So (c Fo idea as eqaion = (i o (ii Diidin eqaion (ii by (i we e (ien So he aph beween and wi be ecana hypeboa So (b f Eney of oe of as whee f = Deee of feedo f Monoaoic o diaoic boh ases posses eqa deee of feedo fo ansaiona oion and ha is eqa o ie f = E hoh oa eney wi be diffeen, Fo onoaoic as Fo diaoic as So (c Gien eociy of sond Eoa [s f = ] Eoa [s f = ] k, oic pesse s, Densiy of as sec Sbsiin hese ae in sond we e Now fo 6 So (b s we e f f k d ies pesse wi becoe haf 7 So (b Mean fee pah N ie by inceasin wo 8 7 nbe of oeces pe ni oe n 7 pe Sbsiin hese ae in we e nd d Å 9 J ISION d SOUION (ae: