Note : The average speed of a particle gives the overall rapidity with which the particle

Transcription

1 Dynaics 4 Intrdctin Dynaics is that branch echanics which deals with the stdy laws vernin tins aterial syste nder the actin iven rces () Displaceent : The displaceent a vin pint is its chane psitin T knw the displaceent a vin pint, we st knw bth the lenth and the directin the line jinin the tw psitins the vin pint Hence the displaceent a pint invlves bth anitde and directin ie, it is a vectr qantity () Speed : The speed a vin pint is the rate at which it describes its path The speed a vin particle r a bdy des nt ive s any idea its directin tin; s it is a scalar qantity In MKS r SI syste, the nit speed is etre per secnd (/s) verae speed : The averae speed vin particle in a tie-interval is deined as the distance travelled by the particle divided by the tie interval I a particle travels a distance s in tie-interval t, then verae speed = t s Nte : The averae speed a particle ives the verall rapidity with which the particle ves in the iven interval tie Instantanes speed : The instantanes speed r siply speed a vin particle is deined as the rate chane distance aln its path, straiht r crved I s is the distance travelled by a particle aln its path, straiht r crved, in tie t, then ds ds Instantanes speed = r speed at tie t Nte : The averae speed is deined r a tie interval and the instantanes speed is deined at a particlar instant 4 Velcity and cceleratin () Velcity : The velcity a vin pint is the rate its displaceent It is a vectr qantity Let a particle startin r the ixed pint and vin aln the straiht line X describes the distance P = x in tie t I the particle x x X describes a rther distance PQ x in tie t, then is x P Q t called the averae velcity the particle in tie t x dx nd li v is called the instantanes velcity (r siply velcity) the particle t0 t at tie t Nte : verae velcity in tie t is the ean the initial and inal velcity () cceleratin : The rate chane velcity a vin particle is called its acceleratin It is a vectr qantity

2 b vt Dynaics The acceleratin a vin pint at tie t at distance x is iven by, v v dv d x v t dv dx In MKS r SI syste, the nit acceleratin is /sec It shld be nted that a 0 i v increases with tie and a 0 i v decreases with tie neative acceleratin is called retardatin Clearly, retardatin eans decrease in the velcity Exaple: The averae speed a bicycle ver a jrney 0 k; i it travels the irst 0 k at 5 k/hr and the secnd 0 k at 0 k/hr, is k/hr (b) 0 k/hr (c) 5 k/hr (d) Nne these 0 Sltin: Tie taken by the bicycle t travel the irst 0 k = hr = hr Tie taken by the bicycle t travel the secnd 0 k = hr = hr 0 Ttal distance travelled = = 0 k Ttal tie taken = 5 hr 3 3 verae speed bicycle = Distance 0 60 = k/hr Tie 5 / Exaple: I a particle ves in a straiht line accrdin t the rla, x t t 5 t, then the tie interval drin which the velcity is neative and acceleratin is psitive, is [0, 5] (b) (, ) (c) (, 5) (d) Nne these 3 6 Sltin: (c) We have x t t 5 t dx 3 d x v t t 5 and a 6t Nw v 0 and a 0 3( t 4t 5) 0 and 6( t ) 0 t 5 and t t (,5) Exaple: 3 particle ves in a ixed straiht path s that S t I v is the velcity at any tie t, then its acceleratin is 3 v (b) Sltin: We have s t Exaple: 4 ds t d s 4( t) 3 v (c) 3 / 4 3 t v (d) 3 3 v d s ds d s 3 v 4 Tw particles are vin with nir velcities and v respectively aln X and Y axes, each directed twards the riin I the particles are at distances a and b r the riin, the tie at which they will be nearest t each ther will be eqal t a v b (b) v a bv (c) v a (d) bv Sltin: (c) Let the tw particles be at and B respectively at t 0 and at tie t their psitins be P and Q respectively Then, P t and BQ vt P a t and Q b vt Nw, PQ P Q PQ ( a t) ( b vt) Y B vt Q P t X

3 Dynaics 3 d d ( PQ ) ( a t) v( b vt) and ( PQ ) ( v ) Fr axi r ini vale d ( PQ ) 0 PQ, we st have a bv ( a t) v( b vt) 0 t v d a bv Clearly, ( PQ ) 0 r all t Hence, PQ is least at t v 43 Resltant and Cpnents Velcities () Resltant velcities : The resltant tw velcities pssessed by a particle is iven by the law parallelra velcities Parallelra law velcities : I a vin particle has tw siltanes velcities represented in anitde and directin by the tw sides a parallelra drawn r an anlar pint, then their resltant is represented in anitde and directin by the dianal the parallelra passin thrh that pint Let a particle at has siltanesly tw velcities and v represented in anitde and directin by and B respectively the parallelra CB The dianal C represents the resltant velcity Let R be the resltant velcity I B and C, then the anitde their resltant velcity is R v v cs and, the directin this resltant velcity akes an anle with the directin sch that v sin tan v cs Case I : When 0, then R v ax Case II : When, then R v in Case III : When /, ie, and v are at riht anle t each ther, then R v tan v and Case IV : When v, then R cs and Nte : The anle ade by the directin the resltant velcity with the directin v is iven by sin tan v cs v B R C

4 Dynaics 4 I the directin resltant velcity R akes an anle with the directin, then v sin sin and R v cs cs R () Cpnents a velcity in tw iven directins : I cpnents a velcity V in tw iven directins akin anles and with it, then by the law parallelra velcities the reqired cpnents V are represented by B and C = BD aln X and Y respectively B BD D ie, sin DB sin BD sin BD sin B D and sin( ) B BD D sin sin sin( ) sin BD D sin( ) Hence, the cpnents velcity V in the directins akin anle and are V sin respectively sin( ) Iprtant Tips V sin sin( ) I cpnents velcity V are tally perpendiclar, then 90 Ths cpnents V are V cs aln X and V sin aln Y C Y V (+) B D X and Exaple: 5 Sltin: (b) Drps water allin r the r the tnnel sees r the windw the train t be allin r an anle akin tan r the hrizntal It is knwn that their velcities are 4 decieter/sec ssin the resistance air neliible, what will be the velcity the train 4 decieter/sec (b) 48 decieter/sec (c) 45 decieter/sec (d) 44 decieter/sec Let the velcity train is v decieter/sec Let the real velcity drps in anitde and directin ay be represented by B and the velcity train in ppsite directin is represented by D, then B 4, D BC v Cpletin parallelra BCD, C will represent the relative velcity drps I the drps are seen in a directin akin an anle with the hrizntal, ie, CD, then tan r tan r ct D C v Directi n 90 4 B Train In BC, r sine rla, B BC sin sin(90 ) v 4 ct 4 48 decieter/secnd r 4 v r sin cs Exaple: 6 I and v be the cpnents the resltant velcity w a particle sch that v w, then the anle between the velcities is 60 (b) 50 (c) 0 (d) Sltin: Let be the anle between the cpnents the resltant velcity w 30

5 Dynaics 5 Then w v vcs ( cs ) 4 cs cs 60 Exaple: 7 an swis at a speed 5 k/hr He wants t crss a canal 0 etres wide, in a directin perpendiclar t the directin lw I the canal lws at 4 k/hr, the directin and the tie taken by the an t crss the canal are 3 tan, 4 in 4 3 (b) tan,44 sec (c) tan,00 sec 4 (d) Nne these Sltin: (b) Sppse the an swis in a directin akin an anle with the directin crrent Since the an wants t crss the canal in a directin perpendiclar t the directin lw Therere, 5 sin tan 4 5 cs cs 4 cs 5 D 4 cs 4 cs 5 5 Let V be the resltant velcity, then V cs 4 V V 3k/ hr 5 tan B C V 4 k/hr Tie taken by the an t crss the canal = hr hr in 4 in 44 sec Relative Velcity Let and B are tw bdies in tin The velcity relative t B is the velcity with which appears t ve as viewed by B ie, velcity relative t B = Resltant velcity and reversed velcity B pparent velcity as seen r B eans velcity relative t B The relative velcity with respect t B is als called the velcity relative t B and is dented by V B and V B V V B I tw bdies and B are vin with velcities anitdes each ther, then V and V B V B respectively inclined at an anle with V V B V V B cs I the directin the relative velcity with respect t B akes an anle with the directin velcity B, then tan V B V sin V cs Case I : When the particles and B ve parallel t each ther in the sae directin with velcities and v respectively Here 0, then ( v) in the directin V B V B ( v ) in the directin B V V B B V B V B V = V B = v

6 Dynaics 6 Case II : When the particles and B ve parallel t each ther in ppsite directins with velcities and v respectively Here, then ( v) in the directin V B V B ( v ) in the directin B Nte : Tre velcity = Resltant the relative velcity with respect t B and the tre velcity B V = B V B = v ie, V V B V B [Since V B V V B ] Exaple: 8 train lenth 00 travellin at 30 /sec vertakes anther lenth 300 travellin at 0 Sltin: (b) 0) = 0 /sec /sec The tie taken by the irst train t pass the secnd is 30 sec (b) 50 sec (c) 0 sec (d) 40 sec Distance cvered by the irst train t pass the secnd = ( ) = 500 etres Since bth the trains are travellin in the sae directin Velcity irst train relative t secnd = ( Tie taken by the irst train t pass the secnd = 50 secnd 0 Exaple: 9 Sltin: (c) T a an rnnin at a speed 0 k/hr, the rain drps appear t be allin at an anle the vertical I the rain drps are actally allin vertically dwnwards, their velcity in k/hr is 0 3 (b) 0 (c) 0 3 (d) 40 Velcity rain relative t an = ctal velcity rain Velcity an Reslvin hrizntally and vertically, V cs 30 and V sin = 0 ie, 0 V k/ hr, V 40 0(Rev 0(ct 30 cta 30 r V(Rel) Exaple: 0 an is walkin at the rate 3 k/hr and the rain appears t hi allin vertically with a velcity 3 3 k/ hr I the actal directin the rain akes an anle with vertical, then = 5 (b) 30 (c) 45 (d) Sltin: (d) Let V M dentes the velcity an Then, V V 3k/ hr It appears t the an that rain is allin vertically at the rate velcity rain with respect t an is M 3 3k/ hr This eans that the relative perpendiclar t the velcity the an Let relative velcity rain with respect t an and velcity rain M 3 3k/ hr, in a directin V RM dente the V R be the 60 V R=33K/ V = 3k/hr V R Let V R, B V RM Cplete the parallelra CB Then, the dianal C represents tre velcity R M RM M RM V R the rain V V V V V cs 90 [Usin w v v cs] 3 (3 3) k/ hr V R B C

7 Dynaics 7 Exaple: Let be the anle between the directin velcity an and velcity rain Then, tan VRM sin tan 3 VM VRM cs90 3 Ths, the actal velcity the rain is 6 k/hr in a directin akin an anle tin the an 60 with the directin The particles start siltanesly r the sae pint and ve aln tw straiht lines, ne with nir velcity and the ther r rest with nir acceleratin Let be the anle between their directins tin The relative velcity the secnd particle wrt the irst is least ater a tie [IEEE 003] sin Sltin: (d) ter t, velcity = t V B V B t ( ) t (b) t cs cs d Fr ax and in, ( VB ) t cs 0 cs t 45 Rectilinear tin with nir cceleratin (c) sin (d) cs pint r a particle ves in a straiht line, startin with initial velcity, and vin with cnstant acceleratin in its directin tin I v be its inal velcity at the end tie t and s be its distance at the instant, r its startin pint, then the eqatins describin the tin the particle are, B t = 0 Velcity = s t P X (i) v t (ii) s t t (iii) v s (iv) I s n is the distance travelled by the particle in n th secnd, then s n (n ) Iprtant Tips I a particle ves in a straiht line with initial velcity /sec and cnstant acceleratin /sec, then distance travelled in t secnds is iven by, s t t [t t v ] [ ( t)] t = t, where v t Exaple: Sltin: (d) = (verae velcity) t {verae velcity = v } bdy is vin in a straiht line with nir acceleratin It cvers distances 0 and in third and rth secnds respectively Then the initial velcity in /sec is (b) 3 (c) 4 (d) 5 Let initial velcity is /sec and acceleratin is / sec 5 7 S, ( 3 ) 0 r 0 (i) and ( 4 ) r (ii)

8 Dynaics 8 Exaple: 3 Sltin: (d) Sbtractin (i) r (ii), we et 0 r / sec 5 Sbstittin vale in eqatin (i), 0 r 5 0 r 5 / sec Tw trains and B, 00 ks apart, are travellin t each ther with startin speed 50 k/hr r bth The train is acceleratin at 8 k/hr and B is deceleratin at 8 /h The distance where the enines crss each ther r the initial psitin is 50 ks (b) 68 ks (c) 3 ks (d) 59 ks Let enine train travel x k and crss enine train B in t hrs 50k/ hr, cceleratin 8 k / hr, s s 50 t 8 t, s 50 t 9t (i) S distance travelled by enine train B will be ( 00 s) in t hr 50 k / hr, cceleratin = 8 k/hr 00 s 50 t 8 t 00 s 50 t 9t (ii) ddin (i) and (ii), we et t, t hr Fr (i), s ks Exaple: 4 Sltin: (c) x ( q r) y( r p) z( p q) 0 Exaple: 5 Sltin: (d) particle is vin with a nir acceleratin I drin its tin, it ves x, y and z distance in p th, q th and r th secnds respectively, then ( q r) x ( r p) y ( p q) z (b) ( q r) x ( r p) y ( p q) z (c) ( q r) x ( r p) y ( p q) z 0 (d) ( q r) x ( r p) y ( p q) z 0 Let be the initial velcity and be the acceleratin the particle Then s pth (p ) x, s qth (q ) y, s rth (r ) z Nw, ltiplyin (i) by ( q r), (ii) by ( r p), (iii) by ( p q) and addin, we et bdy travels a distance s in t secnds It starts r rest and ends at rest In the irst part the jrney, it ves with cnstant acceleratin and in the secnd part with cnstant retardatin r The vale t is iven by [IEEE 003] s (b) r s r (c) s( r) (d) s r Prtin, B crrespnds t tin with acceleratin and retardatin r respectively rea B s and B t Let L t, LB t and L v v s s B L = t v and v t v v s v v s ls,, t and r, t v t t t r tr s s t t t ; t tr s t r t s t r t L t B t 46 Mtin Under Gravity When a bdy is let all in vac twards the earth, it will ve vertically dwnward with an acceleratin which is always the sae at the sae place n the earth bt which varies slihtly r place t place This acceleratin is called acceleratin de t ravity Its vale in MKS syste is 98 / sec, in CGS syste 98 c / sec and in FPS syste 3 t / sec It is always dented by () Dwnward tin: I a bdy is prjected vertically dwnward r a pint at a heiht h abve the earth s srace with velcity, and ater t secnd its velcity beces v, the eqatin its tin are v t,

9 Dynaics 9 h t t, v h, s t (t ) In particlar, i the bdy starts r rest r is siply let all r drpped, then, v t, h t, v h ( 0 ) () Upward tin : When a bdy be prjected vertically pward r a pint n the earth s srace with an initial velcity and i the directin the pward tin is rearded as +ve, the directin acceleratin is ve and it is, therere, dented by The bdy ths ves with a retardatin and its velcity radally beces lesser and lesser till it is zer Ths, r pward tin, the eqatins tin are, v t, h t t, v h, s t (t ) (3) Iprtant dedctins (i) Greatest heiht attained : Let H be the reatest heiht Fr the reslt, v h We have, 0 H H Hence reatest heiht ( H) (ii) Tie t reach the reatest heiht : Let T be the tie taken by the particle t reach the reatest heiht Fr the reslt, v t We have, 0 T ie, T Therere tie t reach the reatest heiht ( T) (iii) Tie r a iven heiht : Let t be the tie taken by the bdy t reach at a iven heiht h Then, h t t t t h 0 Hence, t t pint h h and Clearly t and t are real; i h I h, then t t, which is the tie taken t reach the hihest S, let h The lesser vale t ives the tie when the bdy is in p ie, it is tie r t P and the larer vale t ives the ttal tie taken by the bdy t reach the hihest pint and then cin back t the iven pint, ie, it is the tie r t and then t P (iv) Tie liht : It is the ttal tie taken by the particle t reach the reatest heiht and then retrn t the startin pint aain When the particle retrns t the startin pint, h 0 Therere, r the reslt, h t t We have, 0 t t r t = 0 r t = 0 crrespnds t the instant with the bdy starts, and t = / crrespnds t the tie when the particle ater attainin the reatest heiht reaches the startin pint Tie liht t h P

10 Exaple: 6 Dynaics 0 stne is drpped r a certain heiht which can reach the rnd in 5 sec I the stne is stpped ater 3 secnds its all and then tie taken by the stne t reach the rnd r the reainin distance is [MNR 985; UPSET 000] secnds (b) 3 secnds (c) 4 secnds (d) Nne these 5 Sltin: (c) Let distance travelled in 5 sec is h etre h r h Exaple: 7 Sltin: 9 Let distance travelled in 3 secnds is h etre S, h r h S reainin distance h h h r h r h 8 Let tie taken by the stne t reach the rnd r the reainin distance 8 is t secnd 8 0 t r t 6, t 4 sec an in a balln, risin vertically with an acceleratin 49 /sec releases a ball secnds ater the balln is let r the rnd The reatest heiht abve the rnd reached by the ball is 47 (b) 96 (c) 98 (d) 45 Velcity ater secnds = (49) = 98 /sec Distance cvered in secnds = t (49)() 9 8 ain ater the release ball, velcity the ball = 98 /sec v 0 ; Usin v h, we et 0 (98) 98 (98) h h 4 9 etre Hence reatest heiht attained = = 47 Exaple: 8 particle is drpped nder ravity r rest r a heiht h( 98 / sec ) and then it travels a distance 9h in the last secnd The heiht h is 5 00 etre (b) 5 etre (c) 45 etre (d) 675 etre Sltin: (b) S n (n ) (n ) ( 0, ) 9h (n ) and h n n (n ) n (n ) 9n 50n n 50n n ; n 5 r 9 5 Since, rejectin this vale We have n 5 ; h (98) 5 5 etre 9 47 Laws Mtin () Newtn s laws tin (i) First law : Every bdy cntines in its state rest r nir tin in a straiht line except when it is cpelled by external ipressed rces t chane that state (ii) Secnd law : The rate chane ent is prprtinal t the ipressed rce, and takes place in the directin the straiht line in which the rce acts d dv dv ( v) F F KF F ( r K ) Frce = ass acceleratin The SI nit rce is Newtn and CGS nit rce is dyne

11 Dynaics Newtn = k-/sec, Dyne = -c/sec (iii) Third law : T every actin there is an eqal and ppsite reactin, r the actins and reactin are always eqal and ppsite Nte : Newtn = 0 5 Dynes The actin and reactin d nt act tether n the sae bdy r the sae part the bdy () Weiht : The weiht a bdy is the rce with which it is attracted by the earth twards its centre Fr a bdy ass, the weiht W is iven by W (By Newtn's secnd law tin) Nte : Let W and W be the weihts tw bdies asses and respectively at a place n the earth then, and W W W W wt = dynes = 98 dynes k wt = Newtns = 98 N (3) Ment a bdy : It is the qantity tin in a bdy and is eqal t the prdct its ass () and velcity (v) with which it ves Ths, ent the bdy is v The nits ent are -c/sec r k-/sec Ment = Mass Velcity = v (4) Iplse a bdy : The iplse a rce in a iven tie is eqal t the prdct the rce and the tie drin which it acts The iplse a rce F actin r a tie t is therere Ft Exaple: 9 Sltin: (c) Exaple: 0 Sltin: (b) Exaple: train weihin W tns is vin with an acceleratin t/sec When a carriae weiht w tns is sddenly detached r it, then the chane in the acceleratin the train is W W w w t / sec (b) t / sec (c) t / sec (d) t / sec W w W w W w W w Mass train = W 40 lbs Pll the enine = W 40 Pndals When a carriae ass W tns is detached, ass the train = (W w) tns = (W w) 40 lbs Pll is the sae as bere ie, W 40 Pndals W 40 New acceleratin = = ( W w)40 W t / sec W w W W W w Chane in acceleratin = = = W w W w w W w hckey stick pshes a ball at rest r 00 secnd with an averae rce 50 N I the ball weihs 0 k, then the velcity the ball jst ater bein pshed is 35 /sec (b) 5 /sec (c) 5 /sec (d) 45 /sec I v /sec is the velcity the ball jst ater bein pshed, then = 0 v[ Iplse = chane v = 5 /sec ass is acted pn by a cnstant rce P lbwt, nder which in t secnds it ves a distance x eet and acqires a velcity v t/sec Then x is eqal t

12 Dynaics Sltin: (d) Exaple: P t (b) v P Frce = P lbwt= P pndals and ass = lbs F P t / sec Nw initial velcity = 0, inal velcity = v t/sec Distance = x and tie = t (c) v v 0 v Hence x = = P P n enine and train weiht 40 tns and the enine exerts a rce 7 tns I the resistance t tin be 4 lbs wt per tn, then the tie, the train will take t acqire a velcity 30 /hr r t P rest is [BIT Ranchi 994] (b) 6 (c) 8 (d) 3 (d) v P Sltin: Eective rce = Pll the enine Resistance ls acceleratin = = 7 tns 40 4 lbs = 7 40 lbs 5880 lbs = 9800 lbs = Pndals P t / sec ain initial velcity = 0 and inal velcity = 30 ph = 44 t/sec Use v t, we et 44 0 t 3 t 3 secnd = inte; 60 t inte = inte 5 48 Mtin a Bdy released r a Balln r a Lit () When a lit is ascendin with nir acceleratin /sec and ater t secnd a bdy is drpped r it, then at the tie when the bdy is drpped: (i) Initial velcity the bdy is sae as that the lit and is in the sae directin S, the velcity the bdy is t /sec (ii) Initial velcity the bdy relative t the lit = Velcity the bdy Velcity the lit = t t = 0 directin (iii) cceleratin the bdy = /sec in dwnward directin (iv) cceleratin the lit = /sec in pward directin (v) cceleratin the bdy relative t the lit = cceleratin the bdy cceleratin the lit = ( ) = + in dwnward () When a lit is ascendin with nir acceleratin is thrwn vertically pward with velcity / sec and ater t secnd a bdy v / sec, then at that tie, we have the llwin : (i) Initial velcity the bdy = v + velcity lit = v + t, in pward directin (ii) Initial velcity the bdy relative t the lit = Velcity the bdy Velcity lit = (v + t) t = v /sec

13 /sec Dynaics 3 (iii) cceleratin the bdy relative t the lit in vertically dwnward directin is ( + ) (3) When a lit is descendin with nir acceleratin drpped r it Then at that tie, we have the llwin / sec and ater tie t a bdy is (i) Velcity the bdy = Velcity the lit = t /sec in dwnward directin (ii) cceleratin the bdy relative t the lit in dwnward directin = cceleratin the bdy cceleratin lit = / sec 49 pparent weiht a Bdy restin n a vin Hrizntal plane r a Lit Let a bdy ass be placed in a lit vin with an acceleratin and R is the nral reactin, then () When the lit is risin vertically pwards : Eective rce in pward directin = S the external rces in the sae directin R M R ( ) Clearly, the pressre R exerted by the bdy n the plane is reater than the actal weiht the bdy This pressre is als knwn as apparent weiht I a an ass is standin n a lit which is vin vertically pwards with an acceleratin, then R ( ) R R = (weiht the an) R = Weiht the an + (weiht the an) R Ths, the apparent weiht the an is ties re than the actal weiht () When the lit is descendin vertically dwnwards : Eective rce in dwnward directin = S the rces in the sae directin R R R ( ) (ii) Clearly, the pressre exerted by the bdy n the plane is less than its actal weiht when the plane ves vertically dwnwards I a an ass in is standin n a lit, which is vin vertically dwnward with an acceleratin, then the pressre R Sand

14 Dynaics 4 R ( ) = R = weiht the an (weiht the an) Ths, the apparent weiht the an is ties less than his actal weiht Nte : Eective rce stppin a allin bdy = F I R be the resistance sand n a bdy ass allin in sand, then eective rce = R Iprtant Tips I the plane ves vertically pward with retardatin eqal t ie,, then r R ( ), we et R 0 Ths there is n pressre the bdy n the plane when the plane rises vertically with retardatin eqal t I the plane ves dwn reely nder ravity ie, with acceleratin eqal t, then r R ( ), we et R 0 Ths there is n pressre the bdy n the plane, when it ves vertically dwnwards with an acceleratin eqal t Exaple: 3 paracte weihin lbs wt allin with a nir acceleratin r rest, describes 6 t in the irst 4 sec Then the resltant pressre air n the parachte is 85 lbs wt (b) 95 lbs wt (c) 05 lbs wt (d) 5 lbs wt Sltin: (c) Let be the nir acceleratin Here 0, s 6, t 4 sec Since s t t, (4) 6 0 t / sec Eqatin tin is R (Dwnward) Exaple: 4 R ( ) = (3 ) = 30 Pndals 30 lbs wt = 05 lbs wt 3 an alls vertically nder ravity with a bx ass n his head Then the reactin rce between his head and the bx is (b) (c) 0 (d) 5 Sltin: (c) Let reactin be R Since the an alls vertically nder ravity Exaple: 5 Fr R, we have R R 0 an ass 80 k is travellin in a lit The reactin between the lr the lit and the an when the lit is ascendin pwards at 4 /sec is 4648 N (b) 7848 N (c) 9598 N (d) 048 N Sltin: (d) When the lit is ascendin, we have R a ( a) = 80(98 4) 80(38) = 048 N Exaple: 6 particle ass alls r rest at a heiht 6 and is brht t rest by penetratin int 6 the rnd I the averae vertical thrst exerted by the rnd be 388 k wt, then the ass the particle is k (b) 3 k (c) 4 k (d) 8 k Sltin: (c) Let / sec be the retardatin prdced by the thrst exerted by the rnd and let v / sec be the 0 velcity with which the ass penetrates int the rnd Then, v 6 [ v h] v / sec Since the ass penetrates thrh Therere, its inal velcity is zer 6 in the rnd /6 =0 6 v=0

15 Dynaics 5 S, 0 v s 0 (4 96) / sec (388 ) [ / sec ] Hence, the ass the particle is 4 k Mtin tw particles cnnected by a Strin () Tw particles hanin vertically : I tw particles asses and ( ) are sspended reely by a liht inextensible strin which passes ver a sth ixed liht plley, then the particle ass ) will ve dwnwards, with cceleratin Tensin in the strin ( ( ) T T T 4 Pressre n the plley T () ne particle n sth hrizntal table : particle ass attached at ne end liht inextensible strin is hanin vertically, the strin passes ver a plley at the end a sth hrizntal table, and a particle ass placed n the table is attached at the ther end the strin Then r the syste, cceleratin Tensin in the strin T Pressre n the plley T (3) ne particle n an inclined plane : particle ass attached at ne end a liht inextensible strin is hanin vertically, the strin passes ver a plley at the end a sth plane inclined at an anle t the hrizntal, and a particle ass placed n this inclined plane is attached at the ther end the strin I the particle ass ves vertically dwnwards, then r the syste, ( sin ) cceleratin Tensin in the strin ( sin ) T T T sin T T

16 Dynaics 6 Pressre n the plley {( sin )} T 3 / ( sin ) Exaple: 7 The s tw weihts an twd's achine is 6 lbs The heavier weiht descends 64 t in 4 secnds The vale weihts in lbs are 0, 6 (b) 9, 7 (c) 8, 8 (d), 4 Sltin: Let the tw weihts be and (lbs) Exaple: 8 Sltin: (d) Exaple: 9 ( ), 6 (i) Nw, the distance ved by the heavier weiht in 4 secnds r rest is 64 t I acceleratin =, then 64 0 (4) 8 t / sec Bt (ii) 6 Fr (i) and (ii), 0 lbs, 6 lbs Tw weihts W and W are cnnected by a liht strin passin ver a liht plley I the plley ves with an pward acceleratin eqal t that ravity, the tensin the strin is WW W W (b) WW W W (c) 3WW W W (d) 4WW W W Sppse W descends and W ascends I is the acceleratin the weihts relative t the plley, then since the plley ascends with an acceleratin, the actal acceleratin W and W are ( ) (dwnwards) and ( ) (pwards) respectively Let T be tensin in the strin T T W T W( ) and T W W( ) r and W W Sbtractin, We et T W T WW r T 4 k wt W W W T ne end a liht strin passin ver a sth ixed plley is attached a particle ass M, and the ther end carries a liht plley ver which passes a liht strin t whse ends are attached particles asses and The ass M will reain at rest r will ve with a nir velcity, i [MNR 993] (b) M (c) M Sltin: (c) I the ass M has n acceleratin, then M T Bt T T 4 M r 4 Ipact Elastic Bdies 4 4 M 4 (d) M 8 M () Elasticity : It is that prperty bdies by virte which they can be cpressed and ater cpressin they recver r tend t recver their riinal shape Bdies pssessin this prperty are called elastic bdies () Law cnservatin ent : It states the ttal ent tw bdies reains cnstant ater their cllisin any ther tal actin Matheatically v v T T T

17 where the irst bdy Dynaics 7 ass the irst bdy, initial velcity the irst bdy, v inal velcity, v Crrespndin vales r the secnd bdy, (3) Ceicient restittin : This cnstant rati is dented by e and is called the ceicient restittin r ceicient elasticity The vales e varies between the liits 0 and The vale e depends pn the sbstances the bdies and is independent the asses the bdies I the bdies are perectly elastic, then e and r inelastic bdies e 0 (4) Ipact : When tw bdies strike aainst each ther, they are said t have an ipact It is tw kinds : Direct and bliqe (5) Newtn's experiental law ipact : It states that when tw elastic bdies cllide, their relative velcity aln the cn nral ater ipact bears a cnstant rati their relative velcity bere ipact and is in ppsite directin I and be the velcities the tw bdies bere ipact aln the cn nral at their pint cntact and v and v be their velcities ater ipact in the sae directin, v v e( ) Case I : I the tw bdies ve in directin shwn in diara iven belw, then v v e( ) and (b) v v v v B Case II : I the directin tin tw bdies bere and ater the ipact are as shwn belw, then by the laws direct ipact, we have v v e ( )] r v v e ) [ ( and (b) v v ( ) r v v v v Case III : I the tw bdies ve in directins as shwn belw, then by the laws direct ipact, we have ( v ) e[ ( r v v e( ) v )] and (b) v v ) ( ) ( ) r v v ) ( ( v v B (6) Direct ipact tw sth spheres : Tw sth spheres asses and vin aln their line centres with velcities and (easred in the sae sense) ipine directly T ind their velcities iediately ater ipact, e bein the ceicient restittin between the v v

18 Dynaics 8 Let v and v be the velcities the tw spheres iediately ater ipact, easred aln their line centres in the sae directin in which and are easred s the spheres are sth, the iplsive actin and reactin between the will be aln the cn nral at the pint cntact Fr the principle cnservatin ent, v v (i) ls r Newtn s experiental law ipact tw bdies, v v e[ ] (ii) Mltiplyin (ii) by and sbtractin r (i), we et ) v ( e ) ( ) ( e v ( e ) ( e) (iii) (7) bliqe ipact tw spheres : sth spheres ass, ipines with a velcity bliqely n a sth sphere ass vin with a velcity I the directin tin bere ipact ake anles and respectively with the line jinin the centres the spheres, and i the ceicient restittin be e, t ind the velcity and directins tin ater ipact, Let the velcities the sphere ater ipact be v and v in directins inclined at anles and respectively t the line centres Since the spheres are sth, there is n rce perpendiclar t the line jinin the centres the tw balls and therere, velcities in that directin are naltered v sin sin (i) v sin sin (ii) nd by Newtn s law, aln the line centres, v cs v cs e( cs cs ) (iii) ain, the nly rce actin n the spheres drin the ipact is aln the line centres Hence the ttal ent in that directin is naltered v cs v cs cs cs (iv) The eqatins (i), (ii), (iii) and (iv) deterine the nknwn qantities (8) Ipact a sth sphere n a ixed sth plane : Let a sth sphere ass vin with velcity in a directin akin an anle with the vertical strike a ixed sth hrizntal plane and let v be the velcity the sphere at an anle t the vertical ater ipact v v Since, bth the sphere and the plane are sth, s there is n chane in velcity parallel t the hrizntal plane v sin sin (i) nd by Newtn s law, aln the nral CN, velcity separatin = e(velcity apprach) v cs 0 e cs v cs e cs (ii) N v c

19 Dividin (i) by (ii), we et ct e ct Particlar case : I 0 then r (i), v sin 0 sin 0 0 ; v 0 and r (ii) v e Dynaics 9 Ths i a sth sphere strikes a sth hrizntal plane nrally, then it will rebnd aln the nral with velcity, e ties the velcity ipact ie velcity rebnd = e(velcity bere ipact) Rebnds a particle n a sth plane : I a sth ball alls r a heiht h pn a ixed sth hrizntal plane, and i e is the ceicient restittin, then whle tie bere the rebndin ends h e e e nd the ttal distance described bere inishin rebndin h e Exaple: 30 ball is drpped r a heiht h n a hrizntal plane and the ceicient restittin r the ipact is e, the velcity with which the ball rebnds r the lr is eh (b) eh (c) e h (d) e h Sltin: (d) Let be the velcity with which the ball strikes the rnd Then, h h (i) Sppse the ball rebnds with velcity v, then V e( 0) v e v e h, [r (i)] Exaple: 3 sphere ipines directly n a siilar sphere at rest I the ceicient restittin is, the velcities ater ipact are in the rati : (b) : 3 (c) : 3 (d) 3 : 4 Sltin: (c) Fr principle cnservatin ent, Exaple: 3 v (i) v Fr Newtn's rle r relative velcities bere and ater ipact v v e( ) (ii) Here, e and let 0 Then r (i) v v r v v (iii) Fr (ii), v v (iv) ddin (iii) and (iv), v r v (v) 4 3 Fr (iii) and (v), v Hence v : v : ball ipines directly pn anther ball at rest and is itsel redced t rest by the ipact I hal the KE is destryed in the cllisin, the ceicient restittin, is 4 (b) 3 (c) 4 3 (d) Sltin: (d) Let asses the tw balls be and By the iven qestin, we have 0

20 v 0 v 0 0 v r v (i) and 0 v e(0 ) ie, v e (ii) ain r the iven cnditin, KE (bere ipact) = KE ater ipact 4 Prjectile Mtin 0 0 v (iii) r Bt r (i), v, v ie, v Pttin in (ii), we et v e v, v Dynaics 0 e When a bdy is thrwn in the air, nt vertically pwards bt akin an acte anle with the hrizntal, then it describes a crved path and this path is a Parabla The bdy s prjected is called a prjectile The crved path described by the bdy is called its trajectry The path a prjectile is a parabla whse eqatin is Its vertex is x y x tan cs sin cs, sin sin cs X Fcs is S, L B () Se iprtant dedctins abt prjectile tin: Let a particle is prjected with velcity in a directin akin anle with X Let the particle be at a pint P ( x, y) ater tie t and v be its velcity akin an anle with X sin (i) Tie liht (ii) Rane liht ie, Hrizntal Rane sin R Maxi hrizntal Rane and this happens, when 4 (iii) Greatest heiht sin sin Tie taken t reach the reatest heiht (iv) Let P ( x, y) be the psitin the particle at tie t Then x ( cs ) t, dx (v) Hrizntal velcity at tie t cs dy Vertical velcity at tie t sin t y ( sin ) t t (vi) Resltant velcity at tie t t sin t nd its directin is sin tan t cs Y v sin v v cs

21 Dynaics (vii) Velcity the prjectile at the heiht h sin h is tan cs () Rane and tie liht n an inclined plane Y h and its directin R (R cs,r sin X Let X and Y be the crdinate axes and be the inclined plane at an anle t the hrizn X Let a particle be prjected r with initial velcity inclined at an anle x with the hrizntal X The eqatin the trajectry is y x tan cs (i) Rane and tie liht n an inclined plane with anle inclinatin are iven by cs sin( ) R and axi rane p the plane, where cs ( sin ) sin( ) (ii) Tie liht T cs (iii) Rane dwn the plane cs sin( ) cs (iv) Maxi rane dwn the plane, where ( sin ) sin( ) (v) Tie liht, dwn the plane cs (vi) Cnditin that the particle ay strike the plane at riht anles is ct tan( ) Exaple: 33 Sltin: (d) Tw stnes are prjected r the tp a cli h etres hih, with the sae speed s as t hit the rnd at the sae spt I ne the stnes is prjected hrizntally and the ther is prjected at an anle t the hrizntal then tan eqals h h R ( cs ) t ; Nw (b) h ( sin ) t t, t r (i) (c) h h t (i) cs sin h h h h, h tan h sec cs cs h h tan h tan h, tan tan 0 ; h h (d) tan h Y h h R X

22 Exaple: 34 n prjects a ball at an anle ball will rise t Dynaics 45 with the hrizntal I the hrizntal rane is 39, then the 98 (b) 49 (c) 45 (d) 96 sin sin90 Sltin: Hrizntal rane = Exaple: 35 sin Greatest heiht attained = 39 = sin 45 = ( 96) 98 [MU 999] I the rane any prjectile is the distance eqal t the heiht r which a particle attains the velcity eqal t the velcity prjectin, then the anle prjectin will be Sltin: (b) Rane = ls, 60 (b) 75 (c) sin, where is the anle prjectin h h 36 (d) 30 Exaple: 36 Sltin: (c) By the iven cnditin, 5 r sin sin, 30 r Hence, reqired anle prjectin = particle is thrwn with the velcity v sch that its rane n hrizntal plane is twice the axi heiht btained Its rane will be v (b) 3 Rane = (axi heiht) v sin v sin sin sin 4v (c) v (d) 5 v 7 sin cs sin 0 sin ( cs sin ) 0 sin 0 r tan Bt 0, sin 0, tan Nw v sin v tan R tan v 4v 4 5 Exaple: 37 Let be the velcity prjectin and v be the velcity strikin the plane when prjected s that Sltin: (b) the rane p the plane is axi, and v be the velcity strikin the plane when prjected s that the rane dwn the plane is axi, then is the M v,v (b) GM v,v (c) HM v,v (d) Nne these Let be the velcity prjectin and be the inclinatin the plane with the hrizn Let R be the axi rane p the plane Then, R ( sin ) The crdinates the pint where the prjectile strikes the plane are ( R cs, R sin ) It is iven that v is the velcity strikin the plane when prjected s that the rane p the plane is axi v R sin v sin ( sin ) sin v (i) sin Replacin by in (i), we et sin v (ii) sin

23 Dynaics 3 Fr (i) and (ii), we et 4 v v v v ie, is the GM v,v 43 Wrk, Pwer and Enery () Wrk : Wrk is said t be dne by a rce when its pint applicatin nderes a displaceent In ther wrds, when a bdy is displaced de t the actin a rce, then the rce is said t d wrk Wrk is a scalar qantity Wrk dne = rce displaceent bdy in the directin rce W F d W F d cs, where is the anle between F and d () Pwer : The rate din wrk is called pwer It is the ant wrk that an aent is capable din in a nit tie watt = 0 7 ers per sec = jle per sec, HP = 746 watt (3) Enery : Enery a bdy is its capacity r din wrk and is tw kinds : (i) Kinetic enery : Kinetic enery is the capacity r din wrk which a vin bdy pssesses by virte its tin and is easred by the wrk which the bdy can d aainst any rce applied t stp it, bere the velcity is destryed K E v (ii) Ptential enery : The ptential enery a bdy is the capacity r din wrk which it pssesses by virte its psitin cniratin P E h Exaple: 38 Tw bdies asses and 4 are vin with eqal ent The rati their KE is [MNR 990; UPS : 4 (b) 4 : (c) : (d) : Sltin: (b) Ment irst bdy =, where is the velcity the irst bdy Ment secnd bdy = 4v, where v is the velcity the secnd bdy By the iven cnditin, 4v 4v (4 ) Rati KE = v 4 Hence, the rati KE is 4 : 4v 4v Exaple: 39 bwler thrws a bper with a speed 4 / sec The ent the ball tches the rnd, it lsses its enery by 5 k I the weiht the ball is 5, then speed the ball at which it hits the bat is [UPSET 000] /sec (b) /sec (c) 444 /sec (d) 444 /sec Sltin: (b) Weiht the ball = w 5 = 05 k Velcity the ball = v = 5 /sec Lss enery = 5 k Exaple: 40 KE the ball bere it tches the rnd = wv 05 (5) 77 k 98 KE the ball ater bpin = KE bere tchin the rnd lss KE = 77 5 = 567 k Let v be the velcity the ball at which it hits the bat Nw KE the ball at the tie hittin the bat v 494 v / sec w v v 98 particlar starts r rest and ve with cnstant acceleratin Then the rati the increase in the KE in th and ( ) secnd is Sltin: (c) Reqired rati = th : (b) : (c) : (d) Nne these Increase in KE in Increase in KE in ( ) th secnd th secnd = Wrk dne aainst the rce in Wrk dne aainst the th rce in ( ) secnd th secnd d F

24 Dynaics 4 = ) ) (( 0 ) ( 0 P P =