NEWTON S LAWS OF MOTION

Transcription

1 NEWON S LAWS O MOION 3.1 Inerti (1) Inherent property of ll the bodies by virtue of which they cnnot chne their stte of rest or uniform motion lon striht line by their own is clled inerti. () wo bodies of equl mss possess sme inerti becuse it is fctor of mss only. 3. Liner Momentum. (1) Liner momentum of body is the quntity of motion contined in the body. () It is mesured s the product of the mss of the body nd its velocity i.e., Momentum mss velocity. If body of mss m is movin with velocity v then its liner momentum p is iven by p m v (3) It is vector quntity nd it s direction is the sme s the direction of velocity of the body. (4) Units: k m/sec [S.I.], cm/sec [C.G.S.] (5) Dimension: [ML 1 ] 3.3 Newton s irst Lw A body continue to be in its stte of rest or of uniform motion lon striht line, unless it is cted upon by some externl force to chne the stte.

2 (1) If no net force cts on body, then the velocity of the body cnnot chne i.e. the body cnnot ccelerte. () Newton s first lw defines inerti nd is rihtly clled the lw of inerti. Inerti re of three types: Inerti of rest, Inerti of motion, Inerti of direction 3.4 Newton s Second Lw. (1) he rte of chne of liner momentum of body is directly proportionl to the externl force pplied on the body nd this chne tkes plce lwys in the direction of the pplied force. () If body of mss m, moves with velocity v then its liner momentum cn be iven by p m v nd if force is pplied on body, then 3.5 orce dp or (K 1 in C.G.S. nd S.I. units) dt m (orce mss ccelertion) (1) orce is n externl effect in the form of push or pulls which: (i) Produces or tries to produce motion in body t rest. (ii) Stops or tries to stop movin body. (iii) Chnes or tries to chne the direction of motion of the body. () Dimension: orce mss ccelertion [] [M][L ] [ML ] (3) Units: Absolute units: (i) Newton (S.I.) (ii) Dyne (C.G.S) (4) Grvittionl units: (i) Kilorm-force (M.K.S.) (ii) Grm-force (C.G.S) m formul is vlid only if force is chnin the stte of rest or motion nd the mss of the body is constnt nd finite.

3 (5) If m is not constnt ( ) d dv dm mv m + v dt dt dt (6) No force is required to move body uniformly lon striht line. m \ 0 As 0 ( ) (7) When force is written without direction then positive force mens repulsive while netive force mens ttrctive. (8) Out of so mny nturl forces nucler force is stronest while rvittionl force wekest. (9) Centrl force : If position dependent force is lwys directed towrds or wy from fixed point it is sid to be centrl otherwise non-centrl. (10) Conservtive or non conservtive force : If under the ction of force the work done in round trip is zero or the work is pth independent, the force is sid to be conservtive otherwise non conservtive. Exmple: Conservtive force: Grvittionl force, electric force, elstic force. Non conservtive force: rictionl force, viscous force. (11) Common forces in mechnics: (i) (ii) (iii) (iv) Weiht : Weiht of n object is the force with which erth ttrcts it.(wm) Rection or Norml force : When body is plced on riid surfce, the body experiences force which is perpendiculr to the surfces in contct. hen force is clled Norml force or Rection. ension : he force exerted by the end of tut strin, rope or chin inst pullin (pplied) force is clled the tension. he direction of tension is so s to pull the body. Sprin force: Every sprin resists ny ttempt to chne its lenth. his resistive force increses with chne in lenth. Sprin force is iven by Kx ; where x is the chne in lenth nd K is the sprin constnt (unit N/m).

4 3.6 Equilibrium of Concurrent orce. (1) If ll the forces workin on body re ctin on the sme point, then they re sid to be concurrent. () A body, under the ction of concurrent forces, is sid to be in equilibrium, when there is no chne in the stte of rest or of uniform motion lon striht line. (3) he condition for the equilibrium of body is tht the vector sum of ll the forces ctin on the body must be zero. 3.7 Newton s hird Lw. o every ction, there is lwys n equl (in mnitude) nd opposite (in direction) rection. If force exerted on body A by body B (Action) nd force exerted AB on body B by body A (Rection). hen ccordin to Newton s third lw of motion BA Exmple: AB BA (i) A book lyin on tble exerts force on the tble which is equl to the weiht of the book. his is the force of ction. (ii) Swimmin is possible due to third lw of motion. (iii) When un is fired, the bullet moves forwrd (ction). he un recoils bckwrd (rection) 3.8 rme of Reference. (1) A frme in which n observer is situted nd mkes his observtions is known s his rme of reference. It is ssocited with co-ordinte system. () rme of reference re of two types: (i) Inertil frme of reference (ii) Non-inertil frme of reference. (i) Inertil frme of reference :

5 () A frme of reference which is t rest or which is movin with uniform velocity lon striht line is clled n inertil frme of reference. (b) In inertil frme of reference Newton s lws of motion holds ood. (c) Idelly no inertil frme exist in universe. or prcticl purpose frme of reference my be considered s inertil if it s ccelertion is neliible with respect to the ccelertion of the object to be observed. Exmple: he lift t rest, lift movin (up or down) with constnt velocity, (ii) Non inertil frme of reference: () Accelerted frme of references re clled non-inertil frme of reference. (b) Newton s lws of motion re not pplicble in non-inertil frme of reference. Exmple: Cr movin in uniform circulr motion, lift which is movin upwrd or downwrd with some ccelertion, plne which is tkin off. 3.9 Impulse (1) When lre force works on body for very smll time intervl, it is clled impulsive force. An impulsive force does not remin constnt, but chnes first from zero to mximum nd then from mximum to zero. In such cse we mesure the totl effect of force. () Impulse of force is mesure of totl effect of force. (3) I t t1 dt. (4) Impulse is vector quntity nd its direction is sme s tht of force. (5) Dimension: [ML 1 ]

6 (6) Units : Newton second or K ms 1 (S.I.) nd Dyne second or m cm sr l (C.G.S.) (7) orce-time rph- Impulse is equl to the re under t curve. orce t ime I Are between curve nd time xis 1 t (8) If v is the vere mnitude of the force then I t t dt dt t v t1 t1 v (9) rom Newton s second lw or t p t1 p1 dp dt dt d p I p - p p 1 i.e. he impulse of force is equl to the chne in momentum. his sttement is known s Impulse momentum theorem. (10) Exmples. Hittin, kickin, ctchin, jumpin, divin, collision etc. In ll these cses n impulse cts. I dt v. t p constnt So if time of contct At is incresed, vere force is decresed (or diluted) nd vice-vers. (i) In ctchin bll plyer by drwin his hnds bckwrds increses the time of contct nd so, lesser force cts on his hnds nd his hnds re sved from ettin hurt. (ii) Chin wres re wrpped in strw or pper before pckin.

7 3.10 Lw of Conservtion of Liner Momentum. If no externl force cts on system (clled isolted) of constnt mss, the totl momentum of the system remins constnt with time. (l) Accordin to this lw for system of prticles dp dt In the bsence of externl force 0 then p constnt i.e., p p 1 + p + p constnt. () Lw of conservtion of liner momentum is independent of frme of reference thouh liner momentum depends on frme of reference. (3) Prcticl pplictions of the lw of conservtion of liner momentum (i) When mn jumps out of bot on the shore, the bot is pushed slihtly wy from the shore, (ii) A person left on frictionless surfce cn et wy from it by blowin ir out of his mouth or by throwin some object in direction opposite to the direction in which he wnts to move, (iii) Recoilin of un-or bullet nd un system, the force exerted by trier will be internl so the momentum of the system remins unffected ree Body Dirm In this dirm the object of interest is isolted from its surroundins nd the interctions between the object nd the surroundins re represented in terms of forces. 3.1 Apprent Weiht of Body in Lift. When body of mss in is plced on weihin mchine which is plced in lift, then ctul weiht of the body is m. his cts on weihin mchine which offers rection R iven by the redin of weihin mchine. he rection exerted by the surfce of contct on the body is the pprent weiht of the body.

8 Condition iure Velocity Accelertion Rection Conclusion Lift is t rest LI R v0 0 R m0 :. R m Apprent weiht Actul weiht Sprin Blnce m Lift movin upwrd or downwrd with constnt velocity LI R v constnt 0 R m0 :. R m Apprent weiht Actul weiht Sprin Blnce m Lift ccelertin upwrd t the rte of LI R v vrible < R m m :.Rm( + ) Apprent weiht > Actul weiht Sprin Blnce m Lift ccelertin upwrd t the rte of LI R V vrible R m m Rm Apprent weiht Actul weiht Sprin Blnce m Lift ccelertin downwrd t the rte of LI R V vrible < m R m :. Rm( ) Apprent weiht < Actul weiht Sprin Blnce m Lift ccelertin downwrd t the rte of LI R V vrible m R m R0 Apprent weiht Zero (weihtlessness) Sprin Blnce m

9 Condition iure Velocity Accelertion Rection Conclusion Lift ccelertin downwrd t the rte of (>) LI R > V vrible > m R m R m m R ve Apprent weiht netive mens the body will rise from the floor of the Sprin Blnce m lift nd stick to the ceilin of the lift Accelertion of Block on Horizontl Smooth Surfce. (l) When pull is ctin t n nle (θ) to the horizontl (upwrd) R+ sin θ m R m sin θ nd cos θ m cos θ m sin θ R m m θ cos θ 3.14 Accelertion of Block on Smooth Inclined Plne (1) When inclined plne is t rest Norml rection R m cosθ orce lon inclined plne m sin θ m m sin θ sin θ () When inclined plne iven horizontl ccelertion b. Since the body lies in n ccelertin frme, n inertil cts on it in the opposite direction. Norml rection R m cos θ + mb sin θ nd m m sin θ mb cos θ. sin θ b cos θ m sin θ R θ θ m m cos θ R θ m sin θ mb cos θ θ θ m m cos θ b

10 3.15 Motion of Blocks in Contct. Condition ree body dirm Eqution orce nd ccelertion m f 1 f m1+ m B A m 1 f m f f m + m 1 A B f f f f m + m 1 m 1 f m + m 1 f 1 f 1 B C m 3 f 1 f f 1 f f 1 ( + ) m m3 m 3 f m 3 f m 3 f m 3

11 f 1 f 1 A C B m 3 f 1 f f f 1 m 1 f1 f m 3 m 3 f f ( + ) m1 m 3.16 Motion of Blocks Connected by Mss Less Strin. Condition ree body dirm Eqution ension nd ccelertion B A m + m 1 m m + m m 1 1 m m1 1 m + m 1 B A m m + m 1

12 Condition ree body dirm Eqution ension nd ccelertion C B A m m 3 m m 1 1 m 3 m 3 m 3 ( m1+ m) m1 A B C m m 1 3 m 1 ( + ) m m3 m 3 m3 m 3 m 3

13 3.17 Motion of Connected Block over???lley Condition ree body dirm Eqution ension nd ccelertion mm 1 1 m + m 1 P m A 4mm 1 1 m + m 1 B m m 1 m1+ m m m1 1 [ m + m3] 1 3 P + + mm 1 3 A B 4m 3 [ m + m3] m 3 m 3 m 3 m 3 3 m 3 C 3 ( ) 3 m m m + m3 m1 + + m

14 Condition ree body dirm Eqution ension nd ccelertion m1+ m m M + m + 1 M M m1 m + 1 M m1+ m + B orque ( )R 1α A α ( )R 1 R ( )R 1 MR R M M m m1+ M m1+ m riction [Introduction. If we slide or try to slide body over surfce the motion is resisted by bondin between the body nd the surfce. his resistnce is represented by sinle force nd is clled friction. he force of friction is prllel to the surfce nd opposite to the direction of intended motion ypes of riction (1) Sttic friction : he opposin force tht comes into ply when objects re t rest. (i) In this cse sttic friction P. R P (ii) Sttic friction is self-djustin force becuse it chnes itself in ccordnce with the pplied force. m

15 () Limitin friction- he mximum vlue of sttic friction upto which body does not move is clled limitin friction. (i) he mnitude of limitin friction between ny two bodies in contct is directly proportionl to the norml rection between them. 1 R or 1 u s R (ii) Direction of the force of limitin friction is lwys opposite to the direction in which one body is t the vere of movin (iii) Coefficient of sttic friction: () (b) µ s is clled coefficient of sttic friction. Dimension: [M L ] (c) Unit: It hs no unit. (d) Vlue of µ s lies in between 0 nd 1 (e) Vlue of µ depends on mteril nd nture of surfces in contct. (f) Vlue of µ does not depend upon pprent re of contct. (3) Kinetic or dynmic friction- If the pplied force sets the body in motion, the friction opposin the motion is clled kinetic friction. (i) Kinetic friction depends upon the norml rection. k R or k µ k R where µ k is clled the coefficient of kinetic friction (ii) Kinetic friction is lwys lesser thn limitin friction k < 1 :. µ k - < µ s hus we require more force to strt motion thn to mintin it inst friction. his is becuse when motion hs ctully strted, irreulrities of one surfce hve little time to et locked in into the irreulrities of the other surfce. (iv) ypes of kinetic friction: () Slidin friction (b) Rollin friction

16 Rollin friction is directly proportionl to the norml rection (R) nd inversely proportionl to the rdius (r) of the rollin cylinder or wheel. Rrollin R µ r r µ is clled coefficient of rollin friction. It would hve the dimensions of lenth nd would be mesured in metre. Rollin friction is often quite smll s compred to the slidin friction. In rollin the surfces t contct do not rub ech other. he velocity of point of contct with respect to the surfce remins zero ll the times. 3.0 Grph Between Applied orce nd orce of riction (1) Prt OA sttic friction ( s ). () At point A limitin friction ( 1 ). (3) Beyond A, the force of friction is seen to decrese slihtly. he portion BC kinetic friction ( K ). orce of friction s A l B k C (4) As the portion BC of the curve is prllel to x xis therefore kinetic friction does not chne with the pplied force. O Applied force 3.1 Anle of riction. Anle of friction my be defined s the nle which the resultnt limitin friction nd norml rection mkes with the norml rection S θ R P By definition nle 0 is clled the nle of friction tn θ R m

17 \ n θ µ [As we know R µ or θ tn 1 (µ) 3. Anle of Repose Anle of repose is defined s the nle of the inclined plne with horizontl such tht body plced on it is just beins to slide. If α is clled the nle of repose. α θ i.e. nle of repose nle of friction. α UNI - III - LAWS O MOION 1 Mrk Questions 1. A pssener sittin in cr t next, pusher the cr from within. he cr doesn t move, why?. Give the mnitude nd directions of the net force ctin on rin drop fllin with constnt speed. 3. Why the psseners in movin cr re thrown outwrds when it suddenly tkes turn? 4. Wht is the purpose of usin shockers in cr? 5. Why re tyres mde of rubber not of steel? 6. Wheels re mde circulr. Why? 7. A force is required to keep body in uniform motion in striht line. Comments. 8. On riny dy skiddin tkes plce lon curved pth. Why? 9. Why does un recoils when bullet is bein fired? 10. Why is it difficult to ctch cricket bll thn tennis bll even when both re movin with the sme velocity?

18 11. Clculte the impulse necessry to stop 1500 k cr movin t speed of 5ms 1. ( N S) 1. Lubricnts re used between the two prts of mchine. Why? 13. Wht provides the centripetl force to cr tkin turn on level rod? 14. Wht is inertil frme of reference? 15. An thlete runs certin distnce before tkin lon jump. Why? 16. Action nd rection forces do not blnce ech other. Why? 17. he wheels of vehicles re provided with mudurds. Why? 18. Chin wres re wrpped in strw pper before pckin? 0. Why is it difficult to wlk on snd? 1. he outer ede of curved rod is enerlly rised over the inner ede. Why?. Explin why the wter doesn t fll even t the top of the circle when the bucket full of wter is upside down rottin in verticl circle? 3. Why does speedy motor cyclist bends towrds the centre of circulr pth while tkin turn on it? 4. If the net force ctin upon the prticle is zero show tht its liner momentum remins constnt? Mrks Questions 5. A mn ettin out of movin bus runs in the sme direction for certin distnce. Comment. 6. he motion of prticle of mss m is described by h ut + 1/t. ind the force ctin on prticle. ( m) 7. It is difficult to push box full of clothes thn n empty box. Explin.

19 8. A prticle of mss 0.3 k is subjected to force of kx with k 15 Nm 1. Wht will be its initil ccelertion if it is relesed from point 0 cm wy from the oriin? ( 10 ms ) 9. A 50 bullet is fired from 10 k un with speed of 500 ms 1. Wht is the speed of the recoil of the un. (.5 ms 1 ) 30. A block of mss M is pulled lon horizontl frictionless surfce by rope of mss m by pplyin force P t the free end of the rope. ind the force exerted by the rope on the block. Ans : p M M + m 31. hree forces 1, nd 3 re ctin on the prticle of mss m which is sttionry. is perpendiculr to 1 if 1 is removed, wht will be the ccelertion of prticle? ( 1 /m) 3. A sprin blnce is ttched to the ceilin of lift. When the lift is t rest sprin blnce reds 49 N of body hn on it. If the lift moves :- (i) Downwrd (ii) upwrd, with n ccelertion of 5 ms (iii) with constnt velocity. Wht will be the redin of the blnce in ech cse? (4N, 74 N, 49N) 33. It is esier to pull roller thn to push it. Why? 34. A horse cnnot pull crt nd run in empty spce. Why? 35. A bob of mss 0.1 k hun from the ceilin of room by strin m lon is oscilltin. At its men position the speed of the bob is 1 ms 1. Wht is the trjectory of the oscilltin bob if the strin is cut when the bob is (i) At the men position (ii) At its extreme position. Ans : (Prbolic) Ans : (verticlly downwrds)

20 36. Define force of friction? How does bll berin reduce friction? 37. Is lrer surfce re brek on bicycle wheel more effective thn smller surfce re brke? Explin? 38. Define nle of friction nd nle of repose. 39. A block plced on rouh horizontl surfce is pulled by horizontl force. Let f be the force pplied by the rouh surfce on the block. Plot rph of f versus. 40. A mss of k is suspended with thred AB. hred CD of the sme type is ttched. o the other end of k mrs. (i) Lower end of the lower thred is pulled rdully, hnder nd hnder is the downwrd direction so s to pply force on AB. Which of the thred will brek & why? (ii) If the lower thred is pulled with jerk, wht hppens? B C A k 41. A block of mrs M is held inst rouh verticl wll by pressin it with finer. If the coefficient of friction between the block nd the wll is u nd the decelertion due to rvity is, clculte the minimum force required to be pplied by the finer to hold the block inst the wll? 3 Mrks Questions 4. A block of mss 500 is t rest on horizontl tble. Wht stedy force is required to ive the block velocity of 00 cms 1 in 4 s? 43. A force of 98 N is just required to move mss of 45 k on rouh horizontl surfce. ind the coefficient of friction nd nle of friction? (0., 1 4 ) 44. Clculte the force required to move trin of 00 quintl up on n incline plne of 1 in 50 with n ccelertion of ms. he force of friction per quintl is 0.5 N. (4000N0)

21 45. An eroplne requires to tke off speed of 80 km h 1 on runwy of 100m. Mss of the plne is k nd coefficient of friction between the plne nd the round is 0.. If the ccelertion of the plne is uniform durin tke off, Clculte the minimum force required by the enine for the tke off. (7.13 N) 46. A smooth block is relesed from rest on 30 incline nd trvels distnce d. If the time tken to slide on rouh 30 inclined surfce is n times lre to cover the sme distnce on smooth incline. ind the ( ) coefficient of friction? 1/ 3 ( 1 1/n ) 47. Wht is the ccelertion of the block nd trolly system s in fi., if the coefficient of kinetic friction between the trolly nd the surfce is 0.04? Also clculte tension in the strin. ke 10 ms, mss of strin is neliible. (0.957 ms, 7.13 N) 0 k 3 k f. 48. hree blocks of msses 10 k, 0 k nd m 3 30 k re connected by strins on smooth horizontl surfce nd pulled by force of 60 N. ind the ccelertion of the system nd tensions in the strin. M M M (1ms - 10N, 30N) 49. A hunter hs mchine un tht cn fire 50 bullets with velocity of 150 ms 1. A 60 k tier sprins t him with velocity of 10 ms 1 How mny bullets must the hunter fire into the tret so s to stop him in his trck?

22 Ans. Given, m mss of bullet 50 m k M mss of tier 60 k. v velocity of bullet 150 ms 1 V velocity of tier 10 ms A force of 100 N ives mss, n ccelertion of 10 ms nd of 0 ms to mss. Wht ccelertion must be iven to it if both the msses re tied toether? 51. he pulley rrnement of fi. re identicl. he mss of the rope is neliible. In () mss m is lifted up by ttchin mrs (m) to the other end of the rope. In (b), m is lifted up by pullin the other end of the rope with constnt downwrd force m. In which cse, the ccelertion of m is more? m m m m m m m () (b) 5. i. shows the position-time rph of prticle of mss 4 k. Wht is the () orce on the prticle for t < o, t> 4s, o < t < 4s? 3 A (b) Impulse t t o nd t 4s? (Consider one dimensionl motion only) o 4 t(s)

23 53. A bullet of Mss 0.01 k is fired horizontlly into 4 k wooden block t rest on horizontl surfce. he coefficient of kinetic friction between the block nd the surfce is 0.5. he bullets ets embeded in the block nd the combintion moves 0m before comin to rest. With wht speed did the bullet strike the block? 54. he ner side of truck is open nd box of 40 k mss is plced 5m wy from the open end. he coefficient of friction between the box nd the surfce below it is on striht rod, the truck strts from rest nd ccelertes with m/s. At wht distnce from the strtin point does the box fll off the truck (inore the size of the box) 55. wo msses of 5 k nd 3 k re suspended with help of mss less inextensible strins s shown. Clculte 1 nd when where system is oin upwrds with ccelertion m/s (use 9.8 m/s ) 5k 3k 56. here re few forces ctin t point P produced by strins s shown, which is t rest. ind the forces 1 nd. N 45 P 1N 45 1

24 57. A helicopter of mrs 000k rises with verticl ccelertion of 15m/s. he totl mss of the crew nd psseners is 500 k. Give the mnitude nd direction of the: (i) orce on the floor of the helicopter by the crew nd pssener. (ii) Action of the rotor of the helicopter on the surroundin ir (iii) orce on the helicopter due to the surroundin ir 58. A rectnulr box lies on rouh inclined surfce. he coefficient of friction between the surfce nd the box is u. Let the mss of the box be m. () At wht nle of inclintion. Q of the plne to the horizontl will the box just strt to slide down the plne? (b) Wht is the force ctin on the box down the plne, if the nle of inclintion of the plne is incresed to L > Q? (c) Wht is the force needed to be pplied upwrds lon the plne to mke the box either remin sttionry or just move up with uniform speed? (d) Wht is the force needed to be pplied upwrds lon the plne to mke the box move up the plne with ccelertion? 5 Mrks Questions 59. Define the principle of conservtion of liner momentum. Deduce the lw of conservtion of liner momentum from Newton s third lw of motion. 60. Why circulr rods re bnked? Derive n expression for nle of bnkin for sfe circulr turn? 6. Obtin n expression for minimum velocity of projection of body t the lowest point for Loopin verticl loop. 63. Show tht the re under the force-time rph ives the mnitude of the impulse of the iven force for the followin cse when (i) force is constnt (ii) vrible force. 64. Derive n expression for ccelertion of body down rouh inclined plne? (Slidin only)

25 UNI III LAWS O MOION solutions f A Ans.39. O - Upto point A, f (50 Lon s block is sttionry) - beyond A, when increses, block strts movin f remins constnt. Ans.40. (i) hred AB breks down (ii) CD will brek f Ans.41. Wll M or the block not to fll f M But f µr µ so µ M M µ

26 Ans49. Given m mss of bullet 50m 0.050k M mss of tier 60k v Velocity of bullet 150m/s V Velocity of tier 10m/s (.. it is comin from opposite direction n no. of bullets fired per second t the tier so s to stop it.) P i 0, before firin P f n(mv) + MV.....(i).....(ii).. rom the lw of conservtion of momentum, P i p f 0 n(mv) + MV n MV mv 60 ( 10) Ans.50. Suppose, ccelertion produced if nd re tied toether, 100 N. Let 1 nd be the ccelertion produced in nd, respectively. \ 1 10 ms, 0 ms (iven) Ain nd ms nd ms

27 So, m ms Ans51. Cse (), cse (b) 3 Ans5. Refer NCER Ans53. Velocity of the bullet before it ets embeded 3970 m/s Ans54. 0m (refer NCER) Ans N, 35.4N Ans N, 3 N Ans.57. () (b) (c) R 1 500( )N N R ( ) ( )N N R N (upwrds) Ans.58. () θ tn 1 µ (b) orce ctin on the box slm the plne m sin µ cos (c) orce needed m (sin + µ cos ) (d) orce needed m (sin + µ cos ) + m.