Let us consider equation (6.16) once again. We have, can be found by the following equation

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1 41 Le us consider equaion (6.16) once again. We hae, dp d Therefore, d p d Here dp is he change in oenu caused by he force in he ie ineral d. Change in oenu caused by he force for a ie ineral 1, can be found by he following equaion p 1 dp p d...(6.3) The change in oenu caused by a force, defined by equaion (6.3), is called as ipulse, J, of ha force. Ipulse is a ecor quaniy and has sae unis and diensions as ha of oenu. If force is he only force acing on he syse, hen, change in oenu conribued by i is he ne change in oenu of he syse. Therefore, for such a case, equaion (6.3) reduces o he following for: p 1 d f i p p d 1 p p d f i...(6.4) or a consan force,, equaion (6.3) reduces o he following for: p...(6.5) where is he lengh of he ie ineral for which he ipulse of he force is being calculaed. I is obious fro equaion (6.3) ha for a force consan in is direcion, is ipulse is area under force-ie graph and he direcion of he ipulse is along he direcion of he force only, as suggesed by equaion (6.5). Now, we hae one ore alernae way o eplain he conseraion of linear oenu. If ne force on a syse is zero, hus, ipulse obained by he syse is zero and hence, i oenu reains unchanged. O d J

2 4 ro his poin of iew also i is clear ha inernal forces can no change he oerall oenu of a syse. Ipulse conribued by any force for any ie ineral is negaie of ha conribued by is reacion force for he sae ie ineral. Therefore, for any ie ineral, cobined ipulse due o of he acion-reacion pair is zero. Using he concep of ipulse we can esablish an epression for he ie-aerage of a force. If a force acs for he ie ineral 1,, hen, for his ineral, aerage alue of he force,, would ipar he sae ipulse as he force iself. Therefore, a a. lengh of he ie ineral 1 d d...(6.6) ro aboe discussion i is obious ha if ore han one force ac on a syse, hen, for a ie ineral, aerage alue of he ne force can be gien as ne, a p sys...(6.7) 1 I could also obained using equaion (6.19). body of ass is hrown a an angle o he horizonal wih he iniial elociy u. ssuing he air drag o be negligible, find: (a) he change in oenu, p, ha he body acquires oer he firs seconds of oion (b) he odulus of he oenu change, p, during he oal ie of fligh, T. soe ie during is fligh he body is shown in figure 6.7. During he enire fligh of he body, he body is eperiencing only he graiaional force, g, which is consan. Hence, for he firs seconds, he change in oenu of he body, u g p ipulse ipared by graiy g or he oal ie of fligh, he change in oenu of he body is p g T

3 43 soe ie a paricle of ass kg has a elociy (iˆ ˆj ) /s. fer 10 seconds is elociy becoes (4iˆ 6 ˆj ) /s. ind aerage ne force on he paricle for hese 10 seconds. Soluion: erage ne force on he paricle, a change in oenu of he paricle, lengh of he ie ineral, p f pi f i ( f i p ( kg) (iˆ ˆj ) (4iˆ 6 ˆj ) /s 10 sec (4iˆ 8 ˆj ) kg /s sall ball of ass his a rigid wall perpendicularly wih a speed u. If he ball coes o res iediaely afer colliding wih he wall and duraion of collision be, find he agniude of he aerage force eered by he wall on he ball. collision, hen, we hae, If a be he agniude of he aerage force eered by he wall on he ball for he duraion of a p p p f i p i p i [ p f 0] p i is he agniude of he oenu of he ball jus before hiing he wall u

4 44 cannon of ass M sars sliding freely down a sooh inclined plane a an angle o he horizonal. fer he cannon coered he disance, l, a sho was fired, he shell leaing he cannon in he horizonal direcion wih a oenu p. s a consequence, he cannon sopped. ssuing he ass of he shell o be negligible, as copared o ha of he cannon, deerine he duraion of he sho. : When he cannon slides freely on he inclined plane wih angle of inclinaion, is acceleraion is g sin down he incline. fer coering a disance l on he inclined surface is speed becoes g sin l. Siuaions jus before he sho, during he sho and jus afer he sho are shown in figures 6.8(a), (b) and (c), respeciely. y y y O N O O P Mg Mg = gl sin In figure 6.8(b) is he force applied by he cannon on he shell a soe insan during he sho and is reacion applied by he shell on he cannon. N is he noral conac force acing on he cannon fro he inclined surface a he sae insan and g sin any g cos are coponens of weigh of he cannon along he inclined surface and perpendicular o he inclined surface, respeciely. If we consider he shell and he cannon as a single syse, henduring he sho, eernal forces acing on he syse are weigh of he cannon and he noral conac force on he cannon fro he inclined surface, as shown in figure 6.8(d). I is obious ha only hese forces are responsible for he change in he oerall oenu of he syse. If he duraion of he sho be, hen, change in -coponen of he oenu of he syse is P P P, fin, in P p cos M...(i) [Using figures 6.8(a) and (c)] where p is he agniude of he oenu of he shell jus afer he sho. or he duraion of sho, as g sin is he only eernal force along he -direcion, we can wrie Mg N Mg y O

5 45 ro equaions (i) and (ii), we hae, g sin p cos M p cos Mg sin M where gl sin, Here you should ry o noice ha during he sho noral conac force on he cannon, N, fro he inclined surface has a uch greaer agniude han g cos, alhough is agniude before and afer he sho is Mg cos. In he las eaple find he aerage alue of N for he duraion of sho. Using figure 6.8(d), we hae change in oenu of he syse along Y direcion = aerage force along Y-direcion P ( N Mg cos ) y a N a P y Mg cos P y,fin P y,in Mg cos p sin 0 ( p cos M) sin Mg cos Using resul of las eaple for and figures 6.8(a) and (c) for P and P Mg y,in y,fin p Mg sin p cos M Mg cos where, gl sin. or he duraion of sho he only eernal forces on he syse are graiy, Mg, and noral conac force, N. Therefore, we hae, p p Mg N f i N Mg M p

6 46 The only force acing on a.0-kg objec oing along he ais is shown. If he elociy is.0 /s a = 0, wha is he elociy a = 4.0 s? (N) (s) (a).0 /s (b) 4.0 /s (c) 3.0 /s (d) +1.0 /s (e) +5.0 /s 3.0 kg ball wih an iniial elociy of (4i + 3j) /s collides wih a wall and rebounds wih a elociy of ( 4i + 3j) /s. Wha is he ipulse eered on he ball by he wall? (a) +4 i Ns (b) 4 i N s (c) +18 j N s (d) 18 j Ns (e) +8.0 i N s 1.-kg objec oing wih a speed of 8.0 /s collides perpendicularly wih a wall and eerges wih a speed of 6.0 /s in he opposie direcion. If he objec is in conac wih he wall for.0 /s, wha is he agniude of he aerage force on he objec by he wall? (a) 9.8 kn (b) 8.4 kn (c) 7.7 kn (d) 9.1 kn (e) 1. kn n asronau ouside a spaceship haers a loose rie back in place. Wha happens o he asronau as he swings he haer? (a) (b) (c) (d) (e) Nohing. The space ship akes up he oenu of he haer. He oes away fro he space ship. He oes owards he space ship. He oes owards he space ship as he pulls he haer back and oes away fro i as he swings he haer forward. He oes away fro he space ship as he pulls he haer back and oes oward i as he swings he haer forward. 000-kg ruck raeling a a speed of 6.0 /s akes a 90 urn in a ie of 4.0s and eerges fro his urn wih a speed of 4.0 /s. Wha is he agniude of he aerage resulan force on he ruck during his urn? (a) 4.0 kn (b) 5.0 kn

7 47 The agniude of he force (in newons) acing on a body aries wih ie (in icroseconds) as shown in he fig., C and CD are sraigh line segens. The agniude of he oal ipulse of he force on he body fro = 4 s o = s is...n-s. C orce N E D Tie s body of ass 3 kg is aced on by a force which aries as shown in he graph. The oenu acquired is gien by: (a) Zero (b) 5 N-s (c) 30 N-s (d) 50 N-s. 0.4-kg paricle iniially oing a 0 /s is sopped by a consan force of 50 N, which lass for a shor ie. (a) Wha is he ipulse of his force? (b) ind he ie ineral. girl can eer an aerage force of 00 N on her achine gun. Her gun fires 0-g bulles a 1000 /s. How any bulles can she fire per inue? consan force 4iN ˆ is applied a = 0 o he wo-paricle syse of Eercise 5. (a) ind he elociy of he cener of ass a = 5 s. (b) If he cener of ass is a he origin a = 0, where is i a = 5 s? Two blocks of asses 10 kg and 4 kg are conneced by a spring of negligible ass and placed on fricionless horizonal surface. n ipulse gies elociy of 14/s o he heaier block in he direcion of he ligher block. The elociy of he cenre of ass is: (a) 30 /s (b) 0 /s (c) 10 /s (d) 5 /s srea of glass beads coes ou of a horizonal ube a 100 per second and srikes a balanced pan, as shown in figure. They fall a disance of 0.5 o he balance and bounce back o he sae heigh. Each bead has ass 0.5 g. How uch ass us be placed in he oher pan of he balance o keep he poiner reading zero?

8 ( ui c ) ( ui c ) PHYSICS LOCUS 48 In any cases when we eaine only he relaie oion of paricles wihin a syse, bu no he oion of his syse as a whole, i is os adisable o chose he reference frae in which he cenre of ass is a res. Then we can significanly siplify boh he analysis of he phenoena and he calculaions. The reference frae rigidly fied o he cenre of ass of a gien syse of paricles and ranslaing wih respec o inerial fraes is referred o as he frae of he cenre of ass, or, C frae. In C frae he cenre of ass is always a res. Therefore, r c...(6.8(a)) or, for a one diensional analysis...(6.8(b)) s he elociy of he cenre of ass in he C frae is always zero, he oerall linear oenu of he syse in his frae is Psys M sysc P [ c 0]...(6.9) Hence, s he acceleraion of he cenre of ass of a syse of paricles in is C frae is always zero, ne eernal force on he syse, e M sysac C e 0 [ a c 0]...(6.30) Hence, Le us find he relaionship beween he alues of he echanical energy of a syse in soe frae, le say k frae, and C frae. Le us begin wih kineic energy, K.E., of he syse in k frae. The elociy of he i h paricle in k frae ay be represened as i ui c where u i is he elociy of ha paricle in he C frae and c is he elociy of he C frae wih respec o he k frae. Now, we can wrie, kineic energy of he syse in k frae, K E.. ii

9 49 u u i i c i i i c Since, in he C frae iu i (ne oenu of he syse) = 0, he preious epression reduces o he K E u u.. i i i c Vc K E K. E...(6.31) where K. E. is he oal kineic energy of he paricles in C frae and is he oal ass of he syse. Equaion (6.31) can be rearranged o boain, K. E. K. E. P where P is he agniude of he oal oenu of he syse in he K frae....(6.3) Thus, he kineic energy of a syse of paricles coprises he oal kineic energy in he C frae and he kineic energy associaed wih he oion of he syse of paricles as a whole. This iporan conclusion will be repearedly uilized hereafer (specifically, in he ne opic, roaional oion, which basically deals wih he oion of solid bodies). I follows fro equaion (6.31) ha he kineic energy of a syse of paricles is iniu in he C frae, anoher disincie feaure of ha frae. In C frae c 0, and equaion (6.31) gies K. E. K. E.. Le us consider he following eaple: Two blocks and of asses and 1, respeciely, conneced by a assless spring of spring consan k, are placed on a sooh horizonal surface, as shown in figure 6.9. lock is gien a horizonal speed u owards he block when he spring has is naural lengh. Now, we hae o analyze he subsequen oion of he wo blocks. In eaple 15, we hae already considered a case ery siilar o his one in uch deail. u een hen for he sake of clariy we will discuss his case firs in he frae of he horizonal surface on which he blocks are placed and hen only we would analyze is oion fro is C frae. When he block is gien a speed owards righ, he spring sars geing copressed due o which a he sae ie he spring also sars eering a force on he block owards righ and a force on he block owards lef. Iniially his force acceleraes he block and reards he oion of he block, as shown in figure The agniude of he spring force on each block increases wih he copression in he spring. fer sae ie speeds of he wo blocks becoe equal and hence copression in he spring becoes aiu. Thereafer, due o spring force, speed of he block becoes greaer han he speed of he block and hence spring sars geing relaed. u when we consider he wo blocks and he spring as a single syse, he echanical energy of he syse and he linear oenu of he syse us be consered because here is no ne eernal force is acing upon he syse. Hence, he cenre of ass of he syse oes wih a consan 1u (0) 1u elociy 1 1 owards righ and he poenial energy sored in he spring and he kineic energy of k k he syse keeps arying as he syse oes bu heir su reains consan. Here poenial energy in he spring increases wih he copression in he spring bu a he cos of kineic u K

10 50 Here we see ha when we analyze he wo blocks fro he frae of he horizonal surface, heir indiidual oion is quie coplicaed alhough heir cenre of ass oes wih unifor elociy. Now, le us analyze he oion of he wo blocks fro he C frae of he syse. When we oe he C frae we subrac he elociy of he cenre of ass of he syse fro he elociies of he each block o ge heir c u c u K u u K u p = p = u u elociy in he new frae, as shown in figure In his case i is suggesed ha we should change he frae a he iniial oen, when he block is se in oion, only. In his way i is relaiely easier o carry ou he foraliies of frae change due o following wo reasons: (a) (b) speed of each block is known; he block is a res. In cases when he cenre of ass has an acceleraion, we will hae o also apply pseudo forces when we oe o he C frae, which was no required here. You should noice ha equaions 6.8, 6.9 and 6.30 hold rue in his frae and i is relaiely easier o predic he subsequen oion of he blocks. Each block iniially copresses he spring and loses is own speed. When one block coes o res, second one also coes o res because ne oenu of he syse has o be zero in his frae. Thereafer, due o he spring force he blocks oe away fro each oher. Eenually, in C frae he wo blocks oscillae abou heir ean posiions. Deailed analysis is ery siilar o wha we did in he eaple 15. u we should no forge ha he C frae iself is oing owards righ wih a consan speed in he frae aached wih he horizonal surface. ind he aiu copression in he spring during he subsequen oion of he syse gien in figure 6.9. ssue l 0 as he naural lengh of he spring. Le he block was se in oion was se in oion a = 0, hen a soe ie, if be he copression in he spring, and be he speeds of he wo blocks, as shown in figure 6.3, we hae, rae of change of disance beween he blocks = speed of he blocks speed of he block dl d d ( l0 ) d d d u l l

11 51 d d (i) i.e., he wo blocks hae coon speed or we can say heir relaie speed is zero. gain, using principle of conseraion of linear oenu, we hae P in P fin u 1 1 u [Using...(i)] 1 1 1u 1...(ii) pplying, conseraion of echanical energy, we hae, E in E fin Kin Uin K fin U fin u 0 ( 1 ) k 0...(iii) where, 0 i he required aiu copression in he spring. Soling equaion (ii) and (iii), we can ge epression for 0 : When he copression in he spring is aiu, relaie speed of he wo blocks is zero can also be proed using 1 equaion 6.31, which is K. E. K. E.. sys c When copression in he spring is aiu, he poenial energy sored in he spring is also aiu, hence, a ha oen kineic energy of he syse us be iniu because echanical energy of he syse is consered. In he frae aached wih horizonal surface kineic energy is iniu when K. E. is zero because in he aboe 1u epression c is equal o 1 and is always here. When kineic energy in C frae K. E., is zero, boh he blocks us be a res in ha frae. Hence, a ha insan heir elociies in he frae aached wih horizonal surface are equal o he elociy of heir cenre of ass. Therefore, when copression in he spring is aiu, boh he blocks oe wih he sae elociy of agniude c 1u 1 wih respec o he horizonal surface. Now, using principle of conseraion of echanical energy, as we did in las ehod, we can sole for aiu copression 0.

12 PHYSICS LOCUS 5 I a ery sure ha ill now you us hae deeloped a good undersanding of such siuaions. u sill I would like o srech he discussion. Now, I will analyze he relaie oion of he wo blocks in he cenre of ass frae fro he iniial oen only. This approach would proide you a sar approach for een ore coplicaed siuaions. When he block is gien a elociy u, he elociy of he cenre of ass a ha oen is c u u 1 as, shown in figure u '1 + ' REST c = u K s he ne eernal force on he syse is zero, he cenre of ass of he syse always oes wih his elociy only. Hence, while analyzing he oion of he indiidual blocks fro he C frae, we us subrac he elociy of he cenre of ass, c, fro he elociies of he blocks wih respec o he horizonal surface, as we did in figure (6.31). We do so a he iniial oen only, hen, elociies of he blocks in C frae a his oen would be as shown in figure u u = + u u = + k REST REST k Maiu copression sae I is clear fro he figure 6.33 ha as he blocks oe along he direcions of heir iniial elociies, spring forces acing on he would reard heir oion and afer soe ie hey coe o res. When one block coes o res, oher one coes o res a he sae oen, because oenu of he syse in his frae us be zero all he ie. When he blocks coe o res, copression in he spring becoes aiu. If 0 be he aiu copression hen in he aiu copression sae each block is being aced upon by an ouward force of agniude due k0 (as shown in figure 6.35) due o which blocks sar oing in he ouward direcion. Hence, he spring acquires he aiu copression sae jus for a oen. In his frae, as here is no nonconseraie force is doing work on he syse, we hae Ein E fin

13 u u k 1 0 u 1u k u 1 1 k0 ( 1 ) k 1 1 u urher analysis of he siuaion would sugges you ha in he C frae boh he blocks oscillae abou heir 0 equilibriu posiion (when he ne force on he each block is zero) wih apliudes 0 1 (for ). 1 (for 1 ) and Maiu elongaion in he spring is 0 only. (This can be proed by applying he conseraion of energy in he C frae beween he oens when he blocks are a heir eree ends, as shown in figure 6.36). k = = k = k k = = + = + Suppose he gien syse has o be replaced by a spring of sae spring consan and a single block of equialen ass eq, ass shown in figure Now, if eq is gien he sae speed u. hen, we us ge he copression in he spring. In his case by applying conseraion of echanical energy beween he oens when he spring has is naural lengh and when he copression in he spring is aiu, we ge, eq 1 1 Naural lenh k u eq

14 eq 1 This equialen ass is saller han boh and 1, hence, i is defined as reduced ass of he syse. Hence, we hae...(6.33) boe resul is quie frequenly used o siplify he coplicaed syses before deailed analysis of he syse. Two blocks, of asses and 1, conneced by a weighless spring of spring consan K and naural lengh l 0 res on a sooh horizonal plane. consan force sars acing on one of he blocks as shown in figure ind he aiu disance beween he blocks during he subsequen oion of he syse. K s is he ne eernal force acing on he syse (weigh of he syse is balanced by he noral conac forces acing on he wo blocks fro he horizonal surface), he acceleraion of he cenre of ass of he syse wih respec o he horizonal surface is along he sae direcion as ha of, and has he agniude equal o ( 1 ), as shown in figure a = c + Now, le us oe o he C frae of he syse. s he cenre of ass of he syse is iniially a res, we need no o subrac anyhing fro he elociies of he wo blocks if we change he frae a he iniial oen only. Hence, in he C frae oo, he blocks are iniially a res. When we oe o he C frae, we us apply a pseudo force on each block, as shown in figure a c a c / + / + K K l l Now, fro figure 6.40, i is quie easy o predic he subsequen oion of he syse in his frae. I is quie obious ha equaions (6.8), (6.9) and (6.30) hold rue in his case. If sops afer coering a lengh 1 1, sops a he sae oen, as shown in figure K + K + + l

15 55 Till now if has coered a lengh, hen aiu elongaion in he spring is 1. pplying wnoncon U k beween he oens shown in figure 6.40 and figure 6.41, we ge, K( 1 ) 0 [0 0] ( 1 ) K( 1 ) 1 1 K( ) 1 1 K( ) 1 lile ore analysis of he siuaion, quie siilar o wha we did in he eaple 5 of he chaper WORK POWER ENERGY, would lead you o he fac ha in he C frae boh he blocks oscillae abou heir 1 ean posiions, which is when he elongaion in he spring is K( ) 1 The spring neer ges copressed during he subsequen oion, i.e., iniu elongaion in he spring during he oion of he syse is zero. Hence, iniu disance beween he blocks is l 0.

16 56 1. assless spring of force consan 1 kn/ is copressed a disance of 0 c beween asses of 8 and kg. The spring is released on a sooh able. Since he asses are no aached o he spring, hey oe away wih speed 1 and. (a) Show ha he kineic energy of he asses when hey leae he spring can 1 be wrien Ek p (1/ 1 1/ ), where p is he oenu of eiher ass. Use conseraion of energy o find p. (b) ind he elociy of each ass as i leaes he spring. (c) ind he elociy of each ass if he syse is gien an iniial elociy of 4 /s perpendicular o he spring, as shown in figure. (d) Wha is he energy of cener-of-ass oion in his case? Wna is he energy of oion relaie o he cener of ass? (e) ind he elociy of each ass if he syse is gien in iniial elociy of 4 /s in he direcion along he spring, as shown in figure. Wha is he energy of cener-of-ass oion in his case? Wha is he energy of oion relaie o he cener of ass? kg 8 kg 4 /s kg 8 kg 4 /s (a) (b). In a reference frae in which he cener of ass is a res, he oal oenu of a syse is zero. 3. In a reference frae in which he cener of ass is a res, i is possible for all he kineic energy o be los in a collision kg body oes a 5 /s o he righ. I is chasing a second 3-kg body oing a 1 /s also o he righ. (a) ind he oal kineic energy of he wo bodies in his frae. (b) ind he elociy of he cener of ass. (c) ind he elociies of each body relaie o he cener of ass. (d) ind he kineic energy of oion relaie o he cener of ass. (e) Show ha your answer for par (a) is greaer han ha for par (d) by he aoun 1 M CM, where CM is he elociy of he cener of ass and M is he oal ass. Two heaenly bodies S 1 and S no far fro each oher are seen reoling in orbis: (a) round heir coon cenre of ass (b) Which are arbirary (c) Wih S 1 fied and S oing round S 1 (d) Wih S fied and S 1 oing round S. 6. closed syse consiss of wo paricles of asses 1 and which oe a righ angle o each oher wih elociies 1 and. ind: (a) he oenu of each paricle and (b) he oal kineic energy of he wo paricles in he reference frae fied o heir cenre of ineria. Two poin asses 1 and are conneced by a spring of naural lengh l 0. The spring is copressed such ha he wo poin asses ouch each oher and hen hey are fasened by a sring. Then he syse is oed wih a elociy 0 along posiie -ais. When he syse reaches he origin he sring breaks ( = 0). The posiion of he poin ass 1 is gien by 1 0 (1 cos ) where and are consans. y, R

17 57 Two bodies are conneced by a spring. Describe how he bodies can oe so ha (a) he oal kineic energy is jus he energy of cener-of-ass oion and (b) he kineic energy is enirely energy of oion relaie o he cener of ass. Two blocks of asses 5 kg and kg are placed on a fricionless surface and conneced by a spring. n eernal kick gies a elociy 14 /s o he heaier block in he direcion of ligher one. deduce (a) he elociy gained by he cenre of ass and (b) he separae elociies of he wo blocks in he cenre of ass frae jus afer he kick. Describe how a solid ball can oe so ha (a) is oal kineic energy is jus he energy of cener-of-ass oion and (b) i oal kineic energy is energy of oion relaie o he cener of ass. Two blocks of asses 1 and are conneced by a spring of force consan k. lock of ass 1 is pulled by a consan force 1 and oher block is pulled by a consan force. ind he aiu elongaion ha he spring wih suffer.

Chapter 12: Velocity, acceleration, and forces

Chapter 12: Velocity, acceleration, and forces To Feel a Force Chaper Spring, Chaper : A. Saes of moion For moion on or near he surface of he earh, i is naural o measure moion wih respec o objecs fixed o he earh. The 4 hr. roaion of he earh has a measurable