METHOD OF IMPULSE AND MOMENTUM

Transcription

1 J-Phsis MTHOD OF IMPULS ND MOMNTUM Until now, we hae studied kineatis of partiles, Newton s laws of otion and ethods of work and energ. Newton s laws of otion desribe relation between fores ating on a bod at an instant and aeleration of the bod at that instant. Therefore, it onl helps us do analze what is happening at an instant. The work kineti energ theore is obtained b integrating equation of otion ( F a ) oer a path. Therefore, ethods of work and energ help us to in eploring hange in speed oer a position interal. Now, we diret our attention on another priniple priniple of ipulse and oentu. It is obtained when equation of otion ( F a ) is integrated with respet to tie. Therefore, this priniple failitates us with ethod to eplore what is happening oer a tie interal. Ipulse of a Fore Net fore applied on a rigid bod hanges oentu i.e. aount of otion of that bod. net fore for a longer duration ause ore hange in oentu than the sae fore ating for shorter duration. Therefore duration in whih a fore ats on a bod together with agnitude and diretion of the fore deide effet of the fore on the hange in oentu of the bod. Linear ipulse or sipl ipulse of a fore is defined as integral of the fore with respet to tie. If a fore F ats on a bod, its ipulse in a tie interal fro t to t is gien b the following equation. i f I p t f Fdt t i If the fore is onstant, its ipulse equals to produt of the fore etor F and tie interal t. I F t p For one-diensional fore, ipulse equals to area between fore-tie graph and the tie ais. In the gien figure is shown how a fore F along -ais aries with tie t. Ipulse of this fore in tie interal t i to t f equals to area of the shaded portion. If seeral fores F, F, F... 3 F at on a bod in a tie interal, the total ipulse I n p equals to ipulse of the net fore. F t i of all these fores t f t t f t f t f t f I F dt F dt... F dt F F... F dt p t i t n n i t i ti NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65 Ipulse is easured in newton-seond. Diensions of ipulse are MLT a ple alulate ipulse of fore F 3t ˆi t ˆj ˆ k N oer the tie interal fro t = s to t = 3 s. ti t f 3 3 I p Fdt 3 3 I 3 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ p t i t j k dt t i t t j tk i t j t t k t ˆi 6ˆj 4 ˆ k N-s

2 J-Phsis a ple one-diensional fore F aries with tie aording to the gien graph. alulate ipulse of the fore in following tie interals. (a) Fro t = 0 s to t = 0 s. (b) Fro t = 0 s to t = 5 s. () Fro t = 0 s to t = 5 s. For one-diensional fore, ipulse equals to area between foretie graph and the tie ais. (a) 00 I = rea of trapaziu O = 75 N-s (b) 05 I = rea of triangle D = 5 N-s () 05 I = rea of trapaziu O rea of triangle D = 50 N-s Ipulse Moentu Priniple F (N) 0 F (N) 0 O onsider bod of ass in translational otion. When it is oing with eloit, net eternal fore ating on it is F. quation of otion as suggested b Newton s seond law an be written in the for Fdt d ( ) If the fore ats during tie interal fro t i to t f and eloit of the bod hanges fro i to f, integrating the aboe equation with tie oer the interal fro t i to f, t we hae D t (s) t (s) tf ti Fdt f i Here left hand side of the aboe equation is ipulse I p of the net fore F in tie interal fro t i to t, f and quantities and on the right hand side are linear oenta of the partile at instants t i f i and t. f If we denote the b sbols p and p, the aboe equation an be written as i f I p p p f i The idea epressed b the aboe equation is known as ipulse oentu priniple. It states that hange in the oentu of a bod in a tie interal equals to the ipulse of the net fore ating on the bod during the onerned tie interal. For the ease of appliation to phsial situations the aboe equation is rearranged as p I p i p f This equation states that ipulse of a fore during a tie interal when added to oentu of a bod at the beginning of an interal of tie we get oentu of the bod at the end of the interal onerned. Sine ipulse and oentu both are etor quantities, the ipulse oentu theore an be epressed b there salar equations aking use of artesian oponents. p I p p, p I p p p, I p z p, z z The ipulse oentu priniple is dedued here for a single bod oing relatie to an inertial frae, therefore ipulses of onl phsial fores are onsidered. If we are using a non-inertial referene frae, ipulse of orresponding pseudo fore ust also be onsidered in addition to ipulse of the phsial fores. NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65

3 How to appl Ipulse Moentu Priniple J-Phsis The ipulse oentu priniple is dedued here for a single bod, therefore it is reoended at present to use it for a single bod. To use this priniple the following steps should be followed. (i) Identif the initial and final positions as position and and show oenta p and p of the bod at these instants. (ii) Show ipulse of eah fore ating on the bod at an instant between positions and. (iii) Use the ipulse obtained in step (ii) and oenta obtained in step (i) into equation p i I p p f. onsider a partile oing with oentu p in beginning. It is ated upon b two fores, whose ipulses in a tie interal are I p and I p. s a result, at the end of the tie interal, oentu of the partile beoes p. This phsial situation is shown in the following diagra. Suh a diagra is known as ipulse oentu diagra. I p p p I p p I I p p a ple partile of ass kg is oing with eloit ˆ 3ˆ onstant fore 3 ˆ 4ˆ p p I f i p i j /s in free spae. Find its eloit 3 s after a o F i j N starts ating on it. F t f o Substituting gien alues, we hae ˆ i 3ˆj 3ˆ i 4ˆj 3 3ˆ i 6ˆj f f 6.5ˆ i 3ˆj /s NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65 a ple partile of ass kg is oing in free spae with eloit ˆ 3ˆ ˆ 3 i j k /s is ated upon b fore F ˆi ˆj ˆ k N. Find eloit etor of the partile 3 s after the fore starts ating. p p I f i p F t f o Substituting gien alues, we hae ˆi 3ˆj ˆ k ˆi ˆj ˆ k 3 0ˆi 3ˆj 4 ˆ k f f 5ˆi.5ˆj ˆ k /s o

4 J-Phsis a ple bo of ass = kg resting on a fritionless horizontal ground is ated upon b a horizontal fore F, whih aries as shown. Find speed of the partile when the fore eases to at. p p I f i p t f f i Fdt ti F 0 N 0 N s 4 s t = 0 /s a ple Two boes and of asses and M interonneted b an ideal rope and ideal pulles are held at rest as shown. When it is released, bo aelerates downwards. Find eloities of bo and as funtion of tie t after sste has been released. We first eplore relation between aelerations a and a of the boes and, whih an be written either b using onstrained relation or ethod of irtual work or b inspetion. T T T =...(i) ppling ipulse oentu priniple to bo p p I M 0 Tt gt...(ii) p, zero Tt gt ppling ipulse oentu priniple to bo p p I 0 Mgt Tt...(iii) p, Fro equations (i), (ii) and (iii), we hae M M 4 gt M and 4 gt M 4 zero Tt Mgt Ma NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65

5 Ipulsie Motion Soeties a er large fore ats for a er short tie interal on a partile and produes finite hange in oentu. Suh a fore is known as ipulsie fore and the resulting otion as ipulsie otion. When a batsan hits a ball b bat, the ontat between the ball and the bat lasts for a er sall duration t, but the aerage alue of the fore F eerted b the bat on the ball is er large, and the resulting ipulse Ft is large enough to hange oentu of the ball. J-Phsis During an ipulsie otion, soe other fores of agnitudes er sall in oparison to that of an ipulsie fore a also at. Due to negligible tie interal of the ipulsie otion, ipulse of these fores beoes negligible. These fores are known as non-ipulsie fores. ffet of non-ipulsie fores during an ipulsie otion is so sall that the are negleted in analzing ipulsie otion of infinitel sall duration. Non-ipulsie fores are of finite agnitude and inlude weight of a bod, spring fore or an other fore of finite agnitude. When duration of the ipulsie otion is speified, are has to be taken in negleting an of the non-ipulsie fore. In analzing otion of the ball for er sall ontat duration (usuall in ili-seonds), ipulse of the weight of the ball has to be negleted. Unknown reation fores a be ipulsie or non-ipulsie; their ipulse ust therefore be inluded. a ple 00 g ball oing horizontall with 0 /s is struk b a bat, as a result it starts oing with a speed of 35 /s at an angle of 37 aboe the horizontal in the sae ertial plane as shown in the figure. (a) Find the aerage fore eerted b the bat if duration of ipat is 0.30 s. (b) Find the aerage fore eerted b the bat if duration of ipat is 0.03 s. () Find the aerage fore eerted b the bat if duration of ipat is s. (d) What do ou onlude for ipulse of weight of the ball as duration of ontat dereases? 35 /s 0 /s The ipulse oentu diagra of the ball is shown in the figure below. Here F, g, and t represent the aerage alue of the fore eerted b the bat, weight of the ball and the tie interal. p i = gt =.0t P f = p f =. Ft F t F t p f =.8 NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65 ppling priniple of ipulse and oentu in - diretion, we hae pf pi Ip, Ft.8 F 4.8 N t 5...(i) ppling priniple of ipulse and oentu in - diretion, we hae pf pi Ip, 0.0 F t.0t. F..0 t N...(ii)

6 J-Phsis (a) Substituting t = 0.30 s, in equations (i) and (ii), we find F 6ˆ i 8ˆj N (b) Substituting t = 0.03 s, in equations (i) and (ii), we find F 60ˆi 7ˆj N () Substituting t = s, in equations and, we find F 600i ˆ 70ˆj N (d) It is lear fro the aboe results that as the duration of ontat between the ball and the bat dereases, effet of the weight of the ball also dereases as opared with that of the fore of the bat and for suffiientl short tie interal, it an be negleted. Moentu and Kineti nerg a ple oing partile possesses oentu as well as kineti energ. If a partile of ass is oing with eloit, agnitude of its oentu p and its kineti energ K bear the following relation. p K p n objet is oing so that its kineti energ is 50 J and the agnitude of its oentu is 30.0 kg-/s. With what eloit is it traeling? p 50 K p 0.0 /s 30.0 Internal and eternal Fores and Sste of interating Partiles odies appling fores on eah other are known as interating bodies. If we onsider the as a sste, the fores, whih the appl on eah other, are known as internal fores and all other fores applied on the b bodies not inluded in the sste are known as eternal fores. onsider two bloks and plaed on a fritionless horizontal floor. Their weights W and W are ounterbalaned b noral reations N and N on eah of the fro the floor. Push F b the hand is applied on. The fores of noral reation N on- and N on- onstitute Newton s third law ation-reation pair, therefore are equal i n agnitude and opposite i n diretion. ong these fores weights W and W applied b the earth, noral reations N and N applied b the ground and the push F applied b the hand are eternal fores and noral reations N on- and N on- are internal fores. 6 F W N on- If the bloks are onneted b a spring and the blok is either pushed or pulled, the fores W, W, N and N still reain eternal fores for the two blok sste and the fores, whih the spring applies on eah other are the internal fores. Here fore of graitational interation between the being negligible has been negleted. We an oneie a general odel of two interating partiles. In the figure is shown a sste of two partiles of asses and. Partile attrats with a fore F and attrats (or pulls) with a fore F. These fores F and F are the internal fores of this two- partile sste and are equal in agnitude and opposite in diretions. Instead of attration a repeal eah other. Suh a sste of two partiles repealing eah other is also shown. N F F N on- N F Partiles attrating eah other F Partiles repealing eah other W NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65

7 J-Phsis In siilar wa we a oneie a odel of a sste of n interating partiles haing asses,,... i......and j n respetiel. The fores of interation F ij and F ji between i and j are shown in the figure. Siilar to these other partiles a also interat with eah other. These fores of utual interation between the partiles are internal fores of the sste. n of the two interating partiles alwas appl equal and opposite fores on eah other. Here fore sipliit i Fij F ji j n 3 Sste of n interating partiles. onl the fores F ij and F ji are shown. Priniple of onseration of linear oentu The priniple of onseration of linear oentu or sipl onseration of oentu for two or ore interating bodies is one of guiding priniples of the lassial as well as the odern phsis. To understand this priniple, we first disuss a sste of two interating bodies, and then etend the ideas deeloped to a sste onsisting of an interating bodies. onsider a sste of two partiles of ass and. Partile attrats with a fore F and attrats with a fore F. These fores hae equal agnitudes and opposite diretions as shown in the figure. If the bodies are let free i.e. without an eternal fore ating on an of the, eah of the oe and gain oentu equal to the ipulse of the fore of interation. Sine equal and opposite interation fores at on both of the for the sae tie interal, the oenta gained b the are equal in agnitude and opposite in diretion resulting no hange in total oentu of the sste. F F F Partiles attrating eah other F Partiles repealing eah other Howeer, if an eternal fore ats on an one of the or different fores with a nonzero resultant at on both of the, the total oentu of the bodies will ertainl hange. If the sste undergoes an ipulsie otion, total oentu will hange onl under the ation of eternal ipulsie fore or fores. Internal ipulsie fores also eist in pairs of equal and opposite fores and annot hange the total oentu of the sste. Non-ipulsie fores if at annot hange oentu of the sste b appreiable aount. For eaple, grait is a non-ipulsie fore, therefore in the proess of ollision between two bodies near the earth the total oentu reains onsered. NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65 The total oentu of a sste of two interating bodies reains unhanged under the ation of the fores of interation between the. It an hange onl if a net ipulse of eternal fore is applied. In siilar wa we a oneie a odel of a sste of n interating partiles haing asses,,... i......and j n respetiel. The fores of interation F ij and F ji between i and j are shown in the figure. Sine internal fores eist in pairs of equal and opposite fores, in an tie interal of onern eah of the hae a finite ipulse but their total ipulse is zero. Thus if the sste is let free, in an tie interal oentu of eer indiidual partile hanges but the total oentu of the sste reains onstant. 7 i F ij F ji j n 3 Sste of n interating partiles. It an hange onl if eternal fores are applied to soe or all the partiles. Under the ation of eternal fores, the hange in total oentu of the sste will be equal to the net ipulse of all the eternal fores.

8 J-Phsis Thus, total oentu of a sste of partiles annot hange under the ation of internal fores and if net ipulse of the eternal fores in a tie interal is zero, the total oentu of the sste in that tie interal will reain onsered. p initial p final The aboe stateent is known as the priniple of onseration of oentu. It is appliable onl when the net ipulse of all the eternal fores ating on a sste of partiles beoes zero in a finite tie interal. It happens in the following onditions. When no eternal fore ats on an of the partiles or bodies. When resultant of all the eternal fores ating on all the partiles or bodies is zero. In ipulsie otion, where tie interal is negligibl sall, the diretion in whih no ipulsie fores at, total oponent of oentu in that diretion reains onsered. Sine fore, ipulse and oentu are etors, oponent of oentu of a sste in a partiular diretion is onsered, if net ipulse of all eternal fores in that diretion anishes. a ple Two bloks of asses and M are held against a opressed spring on a fritionless horizontal floor with the help of a light thread. When the thread is ut, the saller blok leaes the spring with a eloit u relatie to the larger blok. Find the reoil eloit of the larger blok. M When the thread is ut, the spring pushes both the blok, and ipart the oentu. The fores applied b the spring on both the blok are internal fores of the two-blok sste. ternal fores ating on the sste are weights and noral reations on the bloks fro the floor. These eternal fores hae zero net resultant of the sste. In addition to this fat no eternal fore ats on the sste in horizontal diretion, therefore, horizontal oponent of the total oentu of the sste reains onsered. Veloities of both the objets relatie to the ground (inertial frae) are shown in the adjoining figure. M Sine before the thread is ut sste was at rest, its total oentu was zero. Priniple of onseration of oentu for the horizontal diretion ields n i p horizontal 0 M ( u ) 0 u M 8 u NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65

9 a ple a ple shell fired ertiall up, when reahes its highest point, eplodes into three fragents, and of asses = 4 kg, = kg and = 3kg. Iediatel after the eplosion, is obsered oing with eloit = 3 /s towards north and with a eloit = 4.5 /s towards east as shown in the figure. Find the eloit of the piee. North J-Phsis 3 /s 4.5 /s plosion takes negligible duration; therefore, ipulse of grait, whih is a finite eternal fore, an be negleted. The piees fl off aquiring aboe-entioned eloities due to internal fores deeloped due to epanding gases produed during the eplosion. The fores applied b the epanding gases are internal fores; hene, oentu of the sste of the three piees reains onsered during the eplosion and total oentu before and after the eplosion are equal. ssuing the east as positie -diretion and the north as positie -diretion, the oentu etors p and p of piees and beoe ˆ p j kg-/s and p ˆ i 9 kg-/s efore the eplosion, oentu of the shell was zero, therefore fro the priniple of onseration of oentu, the total oentu of the fragents also reains zero. p p p 0 p (9ˆ i ˆj ) Fro the aboe equation, eloit of the piee is p (3ˆ 4 ˆ i j ) = 5 /s, 53 south of west. ast NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65 In free spae, three idential partiles oing with eloities ˆ o i, 3 ˆ o j and 5 ˆ o k ollide suessiel with eah other to for a single partile. Find eloit etor of the partile fored. Let be the ass of a single partile before an of the ollisions. The ass of partile fored after ollisions ust be 3. In free spae, no eternal fores at on an of the partiles, their total oentu reains onsered. ppling priniple of onseration of oentu, we hae a ple p p ˆ 3 ˆ 5 ˆ i j k 3 initial final o o o 3 ˆi 3ˆj 5 ˆ k /s bullet of ass 50 g oing with eloit 600 /s hits a blok of ass.0 kg plaed on a rough horizontal ground and oes out of the blok with a eloit of 400 /s. The oeffiient of frition between the blok and the ground is 0.5. Neglet loss of ass of the blok as the bullet pieres through it. o /s (a) In spite of the fat that frition ats as an eternal fore, an ou appl priniple of onseration of oentu during interation of the bullet with the blok? (b) Find eloit of the blok iediatel after the bullet pieres through it. () Find the distane the blok will trael before it stops. lok

10 J-Phsis (a) There is no net eternal fore in the ertial diretion and in the horizontal diretion, onl eternal fore frition is non-ipulsie, therefore oentu of the bullet-blok sste during their interation reains onsered. (b) Let us denote eloities of the bullet before it hits the blok and iediatel after it pieres through the blok b bo and b, eloit of the blok iediatel after the bullet pieres through it is and asses of the bullet and the blok b and M respetiel. These are shown in the adjaent figure. bo b M M Iediatel before the bullet hits the blok Iediatel after the bullet pieres the blok () ppling priniple of onseration of oentu for horizontal oponent, we hae M bo b bo b M Substituting the gien alues, we hae = 0 /s To alulate distane traeled b the blok before it stops, work kineti energ theore has to be applied. K M Mg K = 0 M F k = Mg M N = Mg During sliding of the bo on the ground onl the fore of kineti frition does work. W K K Mg M 0 g Substituting gien alues, we hae = 0 a ple allisti Pendulu : ballisti pendulu is used to easure speed of bullets. It onsists of a wooden blok suspended fro fied support. wooden blok of ass M is suspended with the help of two threads to preent rotation while swinging. bullet of ass oing horizontall with eloit o hits the blok and beoes ebedded in the blok. Reeiing oentu fro the bullet, the bullet-blok sste swings to a height h. Find epression for speed of the bullet in ters of gien quantities. When the bullet hits the blok, in a negligible tie interal, it beoes ebedded in the blok and the bulletblok sste starts oing with horizontall. During this proess, net fore ating on the bullet-blok sste in ertial diretion is zero and no fore ats in the horizontal diretion. Therefore, oentu of the bullet-blok sste reains onsered. 0 o NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65

11 J-Phsis efore the bullet hits the blok Iediatel after the bullet beoes ebedded in the blok o p Let us denote oentu of the bullet-blok sste iediatel after the bullet beoes ebedded in the blok b p and appl priniple of onseration of oentu to the sste for horizontal oponent of oentu. p = o Using equation K = p /, we an find kineti energ K of the bullet-blok sste iediatel after the bullet beoes ebedded in the blok. K o M During swing, onl grait does work on the bullet-blok sste. ppling work-kineti energ theore during swing of the bullet-blok sste, we hae Iediatel after the bullet beoes ebedded in the blok K = 0 K p h o M gh 0 W K K M Rearranging ters, we hae NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65 Ipat between two bodies M o gh Ipat or ollision is interation of er sall duration between two bodies in whih the bodies appl relatiel large fores on eah other. Interation fores during an ipat are reated due to either diret ontat or strong repulsie fore fields or soe onneting links. These fores are so large as opared to other eternal fores ating on either of the bodies that the effets of later an be negleted. The duration of the interation is short enough as opared to the tie sale of interest as to perit us onl to onsider the states of otion just before and after the eent and not during the ipat. Duration of an ipat ranges fro 0 3 s for ipats between eleentar partiles to illions of ears for ipats between galaies. The ipats we obsere in our eerda life like that between two balls last fro 0 3 s to few seonds. entral and entri Ipat The oon noral at the point of ontat between the bodies is known as line of ipat. If ass enters of the both the olliding bodies are loated on the line of ipat, the ipat is alled entral ipat and if ass enters of both or an one of the olliding bodies are not on the line of ipat, the ipat is alled eentri ipat.

12 J-Phsis oon Tangent oon Noral or Line of Ipat oon Tangent oon Noral or Line of Ipat en tra l I pat en tri I pa t entral ipat does not produe an rotation in either of the bodies whereas eentri ipat auses the bod whose ass enter is not on the line of ipat to rotate. Therefore, at present we will disuss onl entral ipat and postpone analsis of eentri ipat to oer after studing rotation otion. Head on ( Diret) and Oblique entral Ipat If eloities etors of the olliding bodies are direted along the line of ipat, the ipat is alled a diret or head-on ipat; and if eloit etors of both or of an one of the bodies are not along the line of ipat, the ipat is alled an oblique ipat. u u n u t u n en tra l I pat O b liq u e I pat In this hapter, we disuss onl entral ipat, therefore the ter entral we usuall not use and to these ipats, we all sipl head-on and oblique ipats. Furtherore, use of the line of ipat and the oon tangent is so frequent in analsis of these ipats that we all the sipl t-ais and n-ais. Head on (Diret) entral Ipat To understand what happens in a head-on ipat let us onsider two balls and of asses and oing with eloities u and u in the sae diretion as shown. Veloit u is larger than u so the ball hits the ball. During ipat, both the bodies push eah other and first the get defored till the deforation reahes a aiu alue and then the tries to regain their original shape due to elasti behaiors of the aterials foring the balls. u u u u Instant when ipat starts Deforation Period Instant of aiu deforation Restitution Period Instant when ipat ends The tie interal when deforation takes plae is alled the deforation period and the tie interal in whih the ball tr to regain their original shapes is alled the restitution period. Due to push applied b the balls on eah other during period of deforation speed of the ball dereases and that of the ball inreases and at the end of the deforation period, when the deforation is aiu both the ball oe with the sae eloit sa it is u. Thereafter, the balls will either oe together with this eloit or follow the period of restitution. During the period of restitution due to push applied b the balls on eah other, speed of the ball derease further and that of ball inrease further till the separate fro eah other. Let us denote eloities of the balls and after the ipat b and respetiel. NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65

13 J-Phsis quati on of Ipulse and Moentu duri ng ipat Ipulse oentu priniple desribes otion of ball during deforation period. u Ddt u u Ddt u...(i) Ipulse oentu priniple desribes otion of ball during deforation period. u Ddt u u Ddt u...(ii) Ipulse oentu priniple desribes otion of ball during restitution period. u Rdt u Rdt...(iii) Ipulse oentu priniple desribes otion of ball during restitution period. u Rdt u Rdt...(i) onseration of Moentu duri ng ipat u u u...() Fro equations, (i) and (ii) we hae Fro equations, (iii) and (i) we hae Fro equations, () and (i) we obtain the following equation. u...(i) u u...(ii) The aboe equation eluidates the priniple of onseration of oentu. NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65 oeffiient of Restituti on Usuall the fore D applied b the bodies and on eah other during period differs fro the fore R applied b the bodies on eah other during period of restitution. Therefore, it is not neessar that agnitude of ipulse Ddt of deforation equals to the agnitude of ipulse Rdt restitution. The ratio of agnitudes of ipulse of restitution to that of deforation is alled the oeffiient of restitution and is denoted b e. e Rdt Ddt Now fro equations (i), (ii), (iii) and (i), we hae e u u...(iii)...(i) oeffiient of restitution depend on arious fators as elasti properties of aterials foring the bodies, eloities of the ontat points before ipat, state of rotation of the bodies and teperature of the bodies. In general, its alue ranges fro zero to one but in ollision where kineti energ is generated its alue a eeed one. Depending on alues of oeffiient of restitution, two partiular ases are of speial interest. 3

14 J-Phsis Perfetl Plasti or Inelasti Ipat Perfetl lasti Ipat For these ipats e =. For these ipats e = 0, and bodies undergoing ipat stik to eah other after the ipat. a ple Strateg to sole prob les of head- on ipat Write oentu onseration equation Write rearranging ters of equation of oeffiient of restitution Use the aboe equations and. u u...() e u u...() ball of ass kg oing with speed 5 /s ollides diretl with another of ass 3 kg oing in the sae diretion with speed 4 /s. The oeffiient of restitution is /3. Find the eloities after ollision. Denoting the first ball b and the seond ball b eloities iediatel before and after the ipat are shown in the figure. u = u = 4 Iediatel before ipat starts Iediatel after ipat ends ppling priniple of onseration of oentu, we hae u u (i) ppling equation of oeffiient of restitution, we hae e u u a ple (ii) Fro equation (i) and (ii), we hae = 4 /s and = 4.67 /s ns. blok of ass 5 kg oes fro left to right with a eloit of /s and ollides with another blok of ass 3 kg oing along the sae line in the opposite diretion with eloit 4 /s. (a) (b) If the ollision is perfetl elasti, deterine eloities of both the bloks after their ollision. If oeffiient of restitution is 0.6, deterine eloities of both the bloks after their ollision. Denoting the first blok b and the seond blok b eloities iediatel before and after the ipat are shown in the figure. u = u = 4 Iediatel before ipat starts ppling priniple of onseration of oentu, we hae u u (i) 4 Iediatel after ipat ends NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65

15 J-Phsis ppling equation of oeffiient of restitution, we hae e u u e 4 (a) (b) 6e...(ii) For perfetl elasti ipat e =. Using this alue in equation (ii), we hae = 6 Now fro equation (i) and (iia), we obtain...(iia) =.5 /s and = 3.5 /s For alue e = 0.6, equation is odified as = 3.6 Now fro equation (i) and (iib), we obtain (iib) =.6 /s and =.0 /s lok reerse bak with speed.6 /s and also oe in opposite diretion to its original diretion with speed.0 /s. a ple Two idential balls and oing with eloities u and u in the sae diretion ollide. oeffiient of restitution is e. (a) (b) Dedue epression for eloities of the balls after the ollision. If ollision is perfetl elasti, what do ou obsere? quation epressing oentu onseration is u u...() quation of oeffiient of restitution is (a) eu eu...() Fro the aboe two equations, eloities and are e e u u...(i) NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65 (b) e e u u...(ii) For perfetl elasti ipat e =, eloities and are = u...(iii) = u...(i) Idential bodies ehange their eloities after perfetl elasti ipat. onseration of kineti energ in perfetl elasti i pat For perfetl elasti ipat equation for onseration of oentu and oeffiient of restitution are u u...() u u...() Rearranging the ters of the aboe equations, we hae u u u u 5

16 J-Phsis Multipling LHS of both the equations and RHS of both the equations, we hae u u Multipling b ½ and rearranging ter of the aboe equation, we hae u u In perfetl elasti ipat total kineti energ of the olliding bod before and after the ipat are equal. In inelasti ipats, there is alwas loss of kineti energ. Oblique entral Ipat In oblique entral ipat, eloit etors of both or of an one of the bodies are not along the line of ipat and ass enter of bodies are on the line of ipat. Due to ipat speeds and diretion of otion of both the balls hange. In the gien figure is shown two balls and of asses and oing with eloities u and u ollide obliquel. fter the ollision let the oe with eloities and as shown in the nest figure. u u Iediatel before Ipat Iediatel after Ipat To analze the ipat, we show oponents of eloities before and after the ipat along the oon tangent and the line of ipat. These oponents are shown in the following figure. u n t u n n n t n n u t u u u t t t Iediatel before Ipat Iediatel before Ipat oponent along the t-ais oponent along the n-ais If surfaes of the bodies undergoing ipat are sooth, the annot appl an fore on eah other along the t-ais and oponent of oentu along the t-ais of eah bodies, onsidered separatel, is onsered. Hene, t-oponent of eloities of eah of the bodies reains unhanged. t = u t and t = u t...() For oponents of eloities along the n-ais, the ipat an be treated sae as head-on entral ipat. The oponent along the n-ais of the total oentu of the two bodies is onsered n n u n u n...() onept of oeffiient of restitution e is appliable onl for the n-oponent eloities. 6 e u u...() n n n n The aboe four independent equation an be used to analze oblique entral ipat of two freel oing bodies. NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65

17 a ple disk sliding with eloit u on a sooth horizontal plane strikes another idential disk kept at rest as shown in the figure. If the ipat between the disks is perfetl elasti ipat, find eloities of the disks after the ipat. (a) u J-Phsis We first show eloit oponents along the t and the n-ais iediatel before and after the ipat. angle that the line of ipat akes with eloit u is 30. t 30 u u n t u t n t n n n Iediatel before Ipat Iediatel after Ipat oponent along t-ais oponent along n-ais oponents of oentu along the t-ais of eah disk, onsidered separatel, is onsered. Hene, t-oponent of eloities of eah of the bodies reains unhanged. u t u t and u 0 t t...(i) The oponent along the n-ais of the total oentu of the two bodies is onsered u u n n n n u 3 n n 0 u 3 n n...(ii) onept of oeffiient of restitution e is appliable onl for the n-oponent eloities. e u u n n n n u 3 n n...(iii) NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65 a ple Fro equations (ii) and (iii), we hae 0 and 7 n n u 3...(i) Fro equations (i) and (i) we an write eloities of both the disks. ball ollides with a fritionless wall with eloit u as shown in the figure. oeffiient of restitution for the ipat is e. (a) Find epression for the eloit of the ball iediatel after the ipat. (b) If ipat is perfetl elasti what do ou obsere? (a) Noral Let us onsider the ball as the bod and the wall as the bod. Sine the wall has infinitel large inertia (ass) as opared to the ball, the state of otion of the wall, reains unaltered during the ipat i.e. the wall reain stationar. Now we show eloities of the ball and its t and n-oponents iediatel before and after the ipat. For the purpose we hae assued eloit of the ball after the ipat. u

18 J-Phsis t t n u t u n u n ' n t oponent along t-ais Iediatel before Ipat Iediatel after Ipat oponents of oentu along the t-ais of the ball is onsered. Hene, t-oponent of eloities of eah of the bodies reains unhanged. u u sin...(i) t t oponent along n-ais onept of oeffiient of restitution e is appliable onl for the n-oponent eloities. e u u eu n n n n n n eu os...(ii) Fro equations (i) and (ii), the t and n-oponents of eloit of the ball after the ipat are (b) If the ipat is perfetl elasti, we hae u sin, u sin and '= t u sin and eu sin The ball will rebound with the sae speed aking the sae angle with the ertial at whih it has ollided. In other words, a perfetl elasti ollision of a ball with a wall follows the sae laws as light follows in refletion at a plane irror. n t n n Oblique entral Ipat when one or both the olliding bodies are onstrained in otion In oblique ollision, we hae disussed how to analze ipat of bodies that were free to oe before as well as after the ipat. Now we will see what happens if one or both the bodies undergoing oblique ipat are onstrained in otion. oponent along the t- ais Moentu onserati on If surfaes of the bodies undergoing ipat are sooth, the t-oponent of the oentu of the bod that is free to oe before and after the ipat reain onsered. If both the bodies are onstrained, the t-oponent of neither one reains onsered. We a find a diretion in whih no eternal fore ats on both the bodies. The oponent of total oentu of both the bodies along this diretion reains onsered. oeffiient of restituti on onept of oeffiient of restitution e is appliable onl for the n- oponent eloities. 8 e u u n n n n NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65

19 a ple 50 g ball oing horizontall with eloit 0.0 /s strikes inlined surfae of a 70 g sooth wedge as shown in the figure. The wedge is plaed at rest on a fritionless horizontal ground. If the oeffiient of restitution is 0.8, alulate the eloit of the wedge after the ipat. 0 /s 37 J-Phsis Let us onsider the ball as the bod and the wedge as the bod. fter the ipat, the ball bounes with eloit and the wedge adanes in horizontal diretion with eloit. These eloities and their t and n-oponents are iediatel before and after the ipat are shown in the following figures. n n t u t u n t t 37 n u Iediatel before Ipat n n Iediatel after Ipat oponent along t-ais The ball is free to oe before and after the ipat, therefore its t-oponent of oentu onsered. Hene, t-oponent of eloities of the ball reains unhanged. u 0 os 37 8 /s...(i) t t Moentu onseration In the horizontal diretion, there is no eternal fore on both the bodies. Therefore horizontal oponent of total oentu of both the bodies reain onsered. u os 37 os 53 t n 4 3 n (ii) NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65 oeffiient of restitution onept of oeffiient of restitution e is appliable onl for the n-oponent eloities. 3 n n e u n un n Fro equations (i), (ii) and (iii), we obtain =.0 /s 9...(iii)

20 J-Phsis SYSTM OF PRTILS Stud of kineatis enables us to eplore nature of translation otion without an onsideration to fores and energ responsible for the otion. Stud of kinetis enables us to eplore effets of fores and energ on otion. It inludes Newton s laws of otion, ethods of work and energ and ethods of ipulse and oentu. The ethods of work and energ and ethods of ipulse and oentu are deeloped using equation F a together with the ethods of kineatis. The adantage of these ethods lie in the fat that the ake deterination of aeleration unneessar. Methods of work and energ diretl relate fore, ass, eloit and displaeent and enable us to eplore otion between two points of spae i.e. in a spae interal whereas ethods of ipulse and oentu enable us to eplore otion in a tie interal. Moreoer ethods of ipulse and oentu proides onl wa to analze ipulsie otion. The work energ theore and ipulse oentu priniple are deeloped fro Newton s seond law, and we hae seen how to appl the to analze otion of single partile i.e. translation otion of rigid bod. Now we will further inquire into possibilities of appling these priniples to a sste of large nuber of partiles or rigid bodies in translation otion. Sste of Partiles the ter sste of partiles, we ean a well defined olletion of seeral or large nuber of partiles, whih a or a not interat or be onneted to eah other. s a sheati representation, onsider a sste of n partiles of asses,,... i... j... and n respetiel. The a be atual partiles of rigid bodies in translation otion. Soe of the a interat with eah other and soe of the a not. The partiles, whih interat with eah other, appl fores on eah other. The fores of interation fij and fji between a pair of i th and j th partiles are shown in the figure. Siilar to these other partiles a also interat with eah other. These fores of utual interation between the partiles of the sste are internal fores of the sste. 0 i f ij j n 3 Sste of n interating partiles. These internal fores alwas eist in pairs of fores of equal agnitudes and opposite diretions. It is not neessar that all the partiles interat with eah other; soe of the, whih do not interat with eah other, do not appl utual fores on eah other. Other than internal fores, eternal fores a also at on all or soe of the partiles. Here b the ter eternal fore we ean a fore that is applied on an one of the partile inluded in the sste b soe other bod out-side the sste. In pratie we usuall deal with etended bodies, whih a be deforable or rigid. n etended bod is also a sste of infinitel large nuber of partiles haing infinitel sall separations between the. When a bod undergoes deforation, separations between its partiles and their relatie loations hange. rigid bod is an etended bod in whih separations and relatie loations of all of its partiles reain unhanged under all irustanes. Sste of Partiles and Mass enter Until now we hae deal with translation otion of rigid bodies, where a rigid bod an be treated as a partile. When a rigid bod undergoes rotation, all of its partiles do not oe in idential fashion, still we ust treat it a sste of partiles in whih all the partiles are rigidl onneted to eah other. On the other hand we a hae partiles or bodies not onneted rigidl to eah other but a be interating with eah other through internal fores. Despite the ople otion of whih a sste of partiles is apable, there is a single point, known as enter of ass or ass enter (M), whose translation otion is harateristi of the sste. The eistene of this speial point an be deonstrated in the following eaples dealing with a rigid bod. onsider two disks and of unequal asses onneted b a er light rigid rod. Plae it on a er sooth f ji NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65

21 J-Phsis table. Now pull it horizontall appling a fore at different points. You will find a point nearer to the heaier disk, on whih if the fore is applied the whole assebl undergoes translation otion. Furtherore ou annot find an other point haing this propert. This point is the ass enter of this sste. We an assue that all the ass were onentrated at this point. In eer rigid bod we an find suh a point. If ou appl the fore on an other point, the sste oes forward and rotates but the ass enter alwas translates in the diretion of the fore. F F F Fore a pplied on the ass en ter Fore not a pplied on the as s enter Fore not a pplied on the as s enter In another eperient, if two fores of equal agnitudes are applied on the disks in opposite diretions, the sste will rotate, but the ass enter reains stationar as shown in the following figure. F F od rota tes b u t the a ss enter re a ins s ta tionar u nd er a tion of eq u a l a nd oppos ite fores. If the aboe eperient is repeated with both disks and of idential asses, the ass enter will be the id point. nd if the eperient is repeated with a unifor rod, the ass enter again is the id point. F NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65 od rota tes and the as s en ter tra ns la tes u n d er ation of u nb a la ned fores a pplied at d ifferent poin ts. s another eaple let us throw a unifor rod in air holding it fro one of its ends so that it rotates also. Snapshots taken after regular interals of tie are shown in the figure. The rod rotates through 360. s the rod oes all of its partiles oe in a ople anner eept the ass enter, whih follows a paraboli trajetor as if it were a partile of ass equal to that of the rod and fore of grait were ating on it. F

22 J-Phsis Thus ass enter of a rigid bod or sste of partiles is a point, whose translation otion under ation of unbalaned fores is sae as that of a partile of ass equal to that of the bod or sste under ation the sae unbalaned fores. nd if different fores haing a net resultant are applied at different partiles, the sste rotates but the ass enter translates as if it were a partile of the ass sae as that of the sste and the net resultant were applied on it. onept of ass enter proides us a wa to look into otion of the sste as a whole as superposition of translation of the ass enter and otion of all the partiles relatie to the ass enter. In ase of rigid bodies all of its partiles relatie to the ass enter an oe onl on irular paths beause the annot hanges their separations. The onept of ass enter is used to represent gross translation of the sste. Therefore total linear oentu of the whole sste ust be equal to the linear oentu of the sste due to translation of its ass enter. enter of Mass of Sste of Disrete Partiles sste of seeral partiles or seeral bodies haing finite separations i between the is known as sste of disrete partiles. Let at an instant i partiles of suh a sste,,., i and n are oing with eloities,,, i, and at loations n r, r, r i, and r respetiel. For the n sake of sipliit onl i th partile and the ass enter are shown in the figure. The ass enter loated at r is oing with eloit at this instant. z O r s the ass enter represents gross translation otion of the whole sste, the total linear i.e. su of linear oenta of all the partiles ust be equal to linear oentu of the whole ass due to translation of the ass enter.... i i... n n M We an write the following equation in ters of asses and position etors as an analogue to the aboe equation. This equation on differentiating with respet to tie ields the aboe equation therefore an be thought as solution of the aboe equation. r r... iri... nrn Mr If M i denotes total ass of the sste, the aboe two equations an be written in short as M i i r Mr i i () () NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65

23 J-Phsis The aboe equation suggests loation of ass enter of a sste of disreet partiles. r r... iri... nrn iri r M M artesian oordinate (,, z ) of the ass enter are oponents of the position etor r (3) of the ass enter. i i ; M i i ; M z z i i (4) M a ple enter of Mass of Two Partile Sste (a) Find epression of position etor of ass enter of a sste of two partiles of asses and loated at position etors r and r. (b) press artesian oordinates of ass enter, if partile at point (, ) and partile at point (, ). () If ou assue origin of our oordinate sste at the ass enter, what ou onlude regarding loation of the ass enter relatie to partiles. (d) Now find loation of ass enter of a sste of two partiles asses and separated b distane r. (a) onsider two partiles of asses and loated at position etors r and r. Let their ass enter at position etor r. Fro eq., we hae iri r M r r r (b) Fro result obtained in part (a), we hae and NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65 () If we assue origin at the ass enter etor r anishes and we hae r r 0 Sine either of the asses and annot be negatie, to satisf the aboe equation, etors r and r ust hae opposite signs. It is geoetriall possible onl when ass enter lies between the two partiles on the line joining the as shown in the figure. If we substitute agnitudes r and r of etors r and r equation, we hae r = r, whih suggest r r 3 in the aboe r r Now we onlude that ass enter of two partile sste lies between the two partiles on the line joining the and diide the distane between the in inerse ratio of asses of the partiles.

24 J-Phsis (d) onsider two partiles asses and at distane r fro eah other. There ass enter ust lie in between the on the line joining the. Let distanes of these partiles fro the ass enter are r and r. r r r Sine ass enter of two partile sste lies between the two partiles on the line joining the and diide the distane between the in inerse ratio of asses of the partiles, we an write r r and r r a ple Mass entre of seeral parti les Find position etors of ass enter of a sste of three partile of asses kg, kg and 3 kg loated at r 4ˆ i ˆj 3 ˆ k r ˆ i 4ˆj ˆ k r ˆi ˆj ˆ k respetiel. position etors, and Fro eq., we hae iri r M 4 ˆ ˆ ˆ ˆ i ˆ j 3k ˆ i 4 ˆ j k 3 ˆ i ˆ j k ˆ ˆ r ˆ i j k 3 3 enter of Mass of an tended od or ontinuous Distribution of Mass n etended bod is olletion of infinitel large nuber of partiles so losel loated that we neglet separation between the and assue the bod as a ontinuous distribution of ass. rigid bod is an etended bod in whih relatie loations of all the partiles reain unhanged under all irustanes. Therefore a rigid bod does not get defored under an irustanes. Let an etended bod is shown as a ontinuous distribution of ass b the shaded objet in the figure. onsider an infinitel sall portion of ass d of this bod. It is alled a ass eleent and is shown at position gien b position etor r. Total ass M of the bod is M d. The ass enter is assued at position gien b position etor r. Position etor of entre of ass of suh a bod is gien b the following equation. 3 d rd r M z O r artesian oordinate (,, z ) of the ass enter are oponents of the position etor r enter. d d ; M M ; z r (5) of the ass zd (6) M 4 NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65

25 J-Phsis a ple Mass entre of unifor setri al bodies. Show that ass enter of unifor and setri ass distributions lies on ais of setr. For sipliit first onsider a sste of two idential partiles and then etend the idea obtained to a straight unifor rod, unifor setri plates and unifor setri solid objets. Mass enter of a sste of two idential partiles Mass enter of a sste of two idential partiles lies at the idpoint between the on the lie joining the. Mass enter of a sste of a straight unifor rod onsider two idential partiles and at equal distanes r r/ r/ d d fro the enter of the rod. Mass enter of sste these two partiles is at. The whole rod an be assued to be ade of large nuber of suh sstes eah haing its ass enter at the id point of the rod. Therefore ass enter of the whole rod ust be at its id point. Mass enter of a sste of a unifor setri ured rod onsider two idential partiles and loated at equal distanes fro the line of setr. Mass enter of sste these two partiles is at. The whole rod an be assued to be ade of large nuber of suh sstes eah haing its ass enter at the id point of the joining the. Therefore ass enter of the whole rod ust be on the ais of setr. Line of setr Line of setr Mass enter of a uni for plate ( laina) onsider a setri unifor plate. It an be assued oposed of seeral thin unifor parallel rods like rod shown in the figure. ll of these rods hae ass enter on the line of setr, therefore the whole laina has its ass enter on the line of setr. NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65 Mass enter of a unifor setri solid ob jet unifor setri solid objet oupies a olue that is ade b rotating a setri area about its line of setr though 80º. onsider a unifor setri solid objet shown in the figure. It an be assued oposed of seeral thin unifor parallel disks shown in the figure. ll of these disks hae ass enter on the line of setr, therefore the whole solid objet has its ass enter on the line of setr. Mass enter of uni for bodi es 5 Line of setr Following the siilar reasoning, it an be shown that ass enter of unifor bodies lies on their geoetri enters.

26 J-Phsis a ple Mass enter of a sste of a segent of a unifor irular rod (ar) Find loation of ass enter of a thin unifor rod bent into shape of an ar. onsider a thin rod of unifor line ass densit (ass per unit length) and radius r subtending angle on its enter O. The angle bisetor OP is the line of setr, and ass enter lies on it. Therefore if we assue the angle bisetor as one of the oordinate aes sa -ais, -oordinate of ass enter beoes zero. Let two er sall segents and loated setri to the line of setr (-ais). Mass enter of these two segents is on P at a distane r os fro enter O. Total ass of these two eleents is d = rd. Now using eq., we hae r O r d P ros d M r os rd r r sin a ple Mass enter of a thin unifor ar shaped rod of radius r subtending angle at the enter lies on its angle bisetor at distane O fro the enter. r sin O Find oordinates of ass enter of a quarter ring of radius r plaed in the first quadrant of a artesian oordinate sste, with entre at origin. Making use of the result obtained in the preious eaple, distane O of the ass enter for the enter is r r oordinates of the ass enter (, ) are, / 4 r sin / 4 r O O /4 a ple Find oordinates of ass enter of a seiirular ring of radius r plaed setri to the -ais of a artesian oordinate sste. The -ais is the line of setr, therefore ass enter of the ring lies on it aking -oordinate zero. Distane O of ass enter fro enter is gien b the result obtained in eaple 4. Making use of this result, we hae r sin O / r sin / r 6 / O NOD6 : \Data\04\Kota\J-daned\SMP\Ph\Unit No-3\entre of ass & ollision\ng\theor.p65