Cengage MECHANICS 1

7.60 Physics for IIT-JEE: Mechanics I T,,a,ngent l ( \"Sol. 1. Tl III Ws Fig. 7,278 Fig. 7.276 2, The blocks move with a constant speed, so there is no net force on block A; the tension in the rope connecting A and 8 must be equal to the frictional force on block A, b. the normal force N by the wire track. Normal force on the ILk = (0.35) (25.0 N) = 9 N ring could be either radially outwards or radially inwards depending on whether the ring presses against the inner 3, The weight of block C will be the tension in the rope con- necting 8 and C. This is found by considering the forces on surface or outcr surface of the track. To ascertain whether block 8. The components of force along the ramp arc the tension in the first rope [9 N, from part (i)], the component of normal force is inward or outward assume that, to begin the weight along the ramp, the friction on block B, and the tension in the second rope. Thus, the weight of block C is F mawith. it is inwards, then from I: = find the value of We = 9 N + wB(sin 36.9' + ILk cos 36.9\") normal force, if it is +ve it is inward and if it is -ve, it is = 9N + (25J1N)[sin 36.9\" + (0.35) cos 36.9\"] = 31.0 N outwards. c. Force of the spring kx. Tn the given physical situation, the spring is extended, it will pull the ring. So the spring force /0; is along the spring towards O. 2, Length of the spring in the position shown = R. or 31 N to two figures. The intermediate calculation of the first tension may be avoided to obtain the answer in terms of (CP = CO = R; LCOP = LOPC = 60\"; !J, COP is equilateral) the cOIllmon weight w of blocks A and B, (~) ~)Change in length of the spring = R - We = W[ILk + (sin e + ILk cos ell, R= ( giving the same result. /0;= (\";) (~) = ~g 4, Applying Newton's second law to the rcmaining masses (8 Now from F, = ma,. (~g)cos 30\" + mg cos 30\" and C) gives 5v\"3 +a = g(Wc-p_kwBcosO-w-B\"\"s''i'\"n'O\"'')- = 1.54 mls2 = maraT = -8-g (WB w,) Blocks A, B, and C are placed as shown m\"In the Fig, 7,279 1n2, and Mare 20 in and connected by ropes of negligible mass. Both kg,S and 50 kg, respectively. The coefficient of fraction A and B weigh 25.0 N each, and the coefficient of kinetic between M and ground is zero, The coefficient of friction friction between each block and the surface is 0.35. Block C between ml and M and that between In, and ground is 0.3. descends with a constant velocity. The pulleys and the spring are massless. The string is per- fectly horizontally between PI and P, and also between P2 1, Draw two separate FBDs showing the forces acting ou A and B. and mz, The string is perfectly vertical between PI and P,. An external horizontal force F is applied to mass M. Take ~,,, 'B~ \"\"\"\"••\"\"\".~:~ g = 10 mIs' between P, and P,. An external horizontal force F is applied to mass M, Take g = 10 m/sz '~\\;A ',''3r',. ~ Fig. 7.277 Fig. 7,279 2, Find the tension in the rope connecting blocks A and B, 1. Draw a free-body diagram of mass m, clearly showing all 3, What is the weight of block C? the forces, 4: If the rope connecting A and B were cut, what would be 2. Let the magnitude of the li)fce of the friction between Inl the acceleration of C'? (IIT-JEE 1981) and M be II and that between m, and ground be j\" For a particular F it is found that !J = 21\" Find !J and j,.

Write downeqnations of motion of all the masses. Find F, Newton's Laws of Motion 7.61 tension in the string and the accelerating of the masses. '>A (IIT·JEE, 2000) 801. <::ji ;:>1 I. Free body diagram is shown in the Fig. 7.280. ~'l; 45\" Fig. 7.282 N ///45\" F mg Fig. 7.283 Fig. 7.280 0.89 0.79 a,l = ../2 and an = ../2 2. Supposing all the blocks are in motion {(AR is relative acceleration of A W.l'.t. 13 = {fA - an a. f\"m\" = 1-'1 N, = /LIIHlg = 0.3 x 20 x 10 = 60 N L = J2 m =;> L = ~J2f11aAIJlI2 rwhere L is the relative distance between A and Bl =and f,mm = I-'zNz = f.i211l2g = 0.3 x 5 x 10 15 N Given that 11 = 2/2 fl = 2 x 15 - 30 N The block In I cannot move. or o = -2_L. = - 2-10 - b. Let all the blocks arc at rest I\" F - fI = 0 and T - fl -;, 0 and T - h = 0 aAiR aA - (lIJ which gives It = f2 which does not satisfy the' given con- Putting values we get, t 2 = 4 or t = 2 $, dition. Distance moved by 13 during thal time is given hy c. Since m I cannot more over the block M, thercCore all the S= I2 = -10-.7-9 x4= .2._x 0-.7 x 1O=7vr2;; m blocks move together and 11 = 30 Nand h = 15 N. ..·alll 2 2 ../2../2 Similarly i()r A = 8../2 m. ,~7' ~~I.all A circular disc with a groove along its h lS N diameter is placed horizontally. A block of mass 1 kg is placed as shown. The coefficient of friction between the bl,!ck and Fig. 7.281 all surfaces of groove in contact is It = 2/5. The disc has an acceleration of 25 m/s2 (see Fig. 7.284). Find the acceleration 30 - T = 20a (i) of the block with respect to disc. (IIT-JEE, 2(06) F - 30 = 50\". and (ii) and T-15=5a (iii) After solving these equations, we get a = ,~m/s2, T = 18N. F = 60 N. Two block A and B of eqnal masses are placed on rough inclined plane as sbown in Fig. 7.282. When cos () '\"' 4/5 sin $= 3/5 and where will the two blocks come on the same line on the Fig. 7.284 inclined plane if they are released simultaneously? Initially the block A is../2m behind the block B. Coefficient of kinetic Sol. Normal reaction in vertical direction NJ = mg Normal reaction from side to the groove N2 = ma sin 37° friction for the blocks A and Bare 0.2 and 0.3, respectively Therefore, acceleration of block with respect to disc = e e(g 10 mm/sg2)s•m. - fLkmg cos (IIT-JEE,20()4) 801. a = Clr = ma cos 37\" - I-'NI - I-'N2 111 aA = g sin e - e,I-'k.Ag cos and (i) m an = g sin 0 - jLLBg·cos (1 (ii) Substituting the values we get, . Putting values we get {/r=-lOmjs2

7.62 Physics for IIT-JEE: Mechanics I EXERCISES Subjective Type Solutions'bn piige,!i118 A 1. Block B, with mass mE, rests on block A, with mass rnA, F which in turn is on a horizontal tabletop (as shown in Fig. 7.285). The coefficient of kinetic friction between block Fig. 7.288 A and the tabletop is 1'\" andthe coefficient of static friction 5. A block is placed 011 an inclined plane moving towards right between block A and block B is 1'.,. A light string attached to block A passes over a frictionless, massless pulley and block horizontally with an acceleration ao = g. The length of .the C is suspended [rom the other end of the string. What is the plane AC = 1 m. Friction is absent everywhere. Find the largest mass me that block C can have so that blocks A and B time taken by the block to reach from C to A. still slide together when the system is released from the rest? A Fig. 7.285 H C 2. A block of mass m = 4 kg is placed over a rough inclined Fig. 7.289 plane as shown in Fig. 7.286. The coefficient of friction between the block and the plane is I\" = 0.5. A force F = 10 N 6. You are designing all elevator for a hospital. The force is applied on the block at an angle of 30°. Find the contact exerted on a passenger by the floor of the c~cvator is not force between the block and the plane. to exceed l.60 times the passenger's weight. The elevator accelerates upward with a constant acceleration for a Fig. 7.286 distance of 3.0 m and then starts to slow down. What is the 3. A block of mass In = 2 kg is resting on a rough inclined plane maximum speed of the elevator? of inclination 30\" as shown in Fig. 7.287. The coefficient of friction between the block and the plane is I' = 0.5. What 7. A block A of mass m is placed over a plank B of mass 2 m. minimum force F should be applied perpeudieularly to the Plank B is placed over a smooth horizontal surface. The co- plane on the block, so that block does not slip on the plane'? efficient of friction between A and B is '/2. Block A is given a F velocity Vo towards right. Find acceleration of B relative to A. 30\" Fig. 7.290 8. Block A as shown in Fig. 7.291 has a mass of 4.00 kg and block Ii has mass 12.0 kg. The eoefficient of kinetic friction between block B and the horizontal surface is 0.25. Fig. 7.287 A c 4. Block A as shown in Fig. 7.288 weighs 1.40 N and block Fig. 7.291 B weighs 4.20 N. The coefficient of kinetic friction between· all the surfaces is 0.30. Find the magnitude of the horizontal force necessary to drag block B to the left at a constant speed if A and B are connected by a light, flexible cord passing around a fixed, frictionless pulley.

Newton's Law of Motion 7.63 a. What is the mass of block C if block B is moving to the Fig. 7.293 right and speeding up with an acceleration 2.00 mJs2? 14. With what force F a man pulls a rope in order to support b. What is the tension in each cord when block B has this the platform on which he stands, if the mass of man is 60 kg acceleration? and that at platform is 20 kg. With what forces N does the man press the platform. What is the maximum weight of the 9. Two blocks, with masses Tn! and m2, are stacked as shown platform that the man can support? in Fig. 7.292 and placed on a frictionless horizontal surface. There is a friction between the two blocks. An external force 15. A monkey A (mass = 6 kg) is climbing up a rope tied to a of magnitude F is applied to the top block at an angle? rigid support. The monkey B (mass = 2 kg) is holding on the below the horizontal. tail of monkey A (see Fig. 7.294). If the tail can tolerate a a. If the two blocks move together, find their acceleration. maximum tension of 30 N what force should monkey A apply b. Calculate the maximum value of force so that the blocks on the rope in order to carry monkey B with it? (g = 10 m/s2) will move toget.her. p Fig. 7.292 Monkey 10. A hot-air balloon consists of a basket, one passenger, and some cargo. Let the total mass be M, Even though there is Fig. 7.294 an upward lift force on the balloon, the balloon is initially 16. Figure 7.295 represent a painter in a crate which hangs accelerating downward at a rate of g/3. a. Draw an FBD for the descending balloon. alongside a building. When the painter of mass 10 kg pulls h. Find the upward lift force in terms of the initial total weight Mg. the rope, the force exerted by him on the noor of the crate is c. The passenger notices that he is heading straight for a waterfall ami dL:cidcs he needs to go up. What fraction 450 N, If the crate weighs 25 kg, find the acceleration and of the total weight must he drop overboard so that the tension in the rope (g = 10 m/s2). balloon accelerates upward at a rate of g12? Assume that the upward lift force remains the same, • 11. A student tries to raise a, chain consisting of three identical Fig. 7.295 links. Each link has a mass of 300 g. The three-piece chain 17. A smooth ring A of mass m can slide on a fixed horizontal is connected to a string and then suspended vertically with the student holding the upper end of the string and pulling rod X Y. A string tied to the ring passes oyer a fixed pulley upward. Because of the student's pull, an upward force of B and carries a block C of mass M (= 2m) as shown in 12 N is applied to the chain by the string. Fig. 7.296. At an instant the string between the ring and a. Draw a free body diagram for each of the links in the chain and also for the entire chain considered as a single body. epulley makes an angle with the rod. Now b. Usc the resulis of part (a) and Newton's laws to find (i) tbe acceleration of the chain; and (ii) the force exerted by a. show that if the ring slides with speed v, the block the top link on the middle link. descents with speed v cos e. ] 2. Two men of masses lV! and M + m start simultaneously from the ground and climb with uniform accelerations up from the free ends of ? massless inextensible rope which pm;ses over a smooth pulley at a height h from the ground. a. Which man reaches the pulley lirst? b. If the man who reaches first takes time I to reach the pulley, then find the.distance of the second man from the pulley at this instant. 13. Block A has a mass of 30 kg and block B a mass of 15 kg. The coefficients of friction between all surfaces of contact are 0.' = 0.15 and ILk = 0.10 (see Fig. 7.293). Knowing that 9 = 30\" and that the magnitude of the force F applied to block A is 250 N, determine (I) acceleration of block A, and (2) the tension in the rope.

7.64 Physics for IlT-JEE: Mechanics I A B A M X --E<-----r-l~-- y I-- ~ ~ m f) v l - :::) ~ c c f) Fig. 7.299 1?ig.7.296 the cube begin to slide towards the other end of the block? In what time will the cube fall from the block if the length b. with what acceleration will the ring start moving if the of the latter is l? esystem is released from rest with = 30° . B 18. A rod A B oflength 2 m is hinged at point A and B is attached ,-A__\",_/_L9:--1 F to a platform on which a block of man In is kept. Rod rotates +----1 • eabout point A maintaining angle = 30° with the vertical in Fig. 7.300 such a way that platform remains horizontal and revolves on the horizontal circular path (see Fig. 7.297). If the coefficient 22. Two bars I and 2 are placed on an inclined plane forming of static friction between the block and platform is I\" = 0.1 an angle a with the horizontal (sec Fig. 7.301). The masses than find the maximum angular velocity in rad/s of rod so of the bars are equal to In! and 1n2, and the coefficient of friction between the plane and the bars arc equal- to k1 and that block does not slip on the platform. [g = 10 m/s2J k2 respectively. with k, > k,. Find: a. the force of interaction of the bars in the process of motion, B b. the minimum value of a at which the bars start sliding down. e A .2 Fig. 7.297 Fig. 7.301 23. A block A, of weight W. slides down an inc:lined plane S of 19. A block weighing 20 kg is placed on a smooth surface A weight of 2 kg is mounted on the block. The coefficient of slope 37° at a constant velocity while the plank B, also of friction between the block and the weight is 0.25. Calculate the acceleration of the block and the weight and also the weight W; rests on top of A (Fig. 7.302). The plank B is at- frictional force between the block and the weight when a horizontal force of 2 N is applied to the weight as shown in tached by a cord to the top of the plane. If the coefficient ofki- the Fig. 7.298 What will these quantities be if the horizontal netic friction is the same between the surfaces A and B and be- force is 20 N? (g = 10 ms· 2). tween the surfaces A and Band S and A, determine its value. Fig. 7.298 20. In Fig. 7.299, find the acceleration of In assuming that B there is friction between m and M, and all other surface A are smooth and pulleys light and,\" = coefficient of friction between m and M. S L--L-_ _- - ' 21. A block A of mass M rests on a smooth horizontal surface Fig. 7.302 over which it can move without friction. A cube B of mass m lies on the block at one edge. The coefficient of hiction 24. A particle A of mass 2 m is held on a smooth horizontal table between the block and the cube is k (see Fig. 7.300). At what and is attached to one end of an inelastic string which runs [oree F applied to the block in the horizontal direction will

Newlon's Law of Malian 7.65 A k----l--+I B Fig. 7.303 over a smooth light pulley at the edge of the table, At the other Fig, 7.306 end of the string there hangs another particle B of mass m, the distance from A to the pulley is I (Fig, 7.303), The particle pulley as shown in Fig, 7,307, The coefficient of static A is then projceted towards the pulley with velocity u, Find friction is J1s and the coefficient of kinetic friction is fJ.-k. 3. the time before the string becomes taut; and show that a after the string becomes taut, the initial velocity of A and B is 4u13, Fig. 7.307 b. the common velocity when A reaehcs the pulley (assume that B has not yet rcached the ground), a. Find the mass In, for which block In, moves up the plane 25. The masses of the blocks A and B arc In and M, Between A at constant speed once it is set in motion. and B there is a constant frictional force F, but B can slide frictionlessly on the horizontal surface (Fig, 7,304), A is set b. Find the mass m2 for which block nIl moves down the in motion with velocity Va while B is at rest. What is the plane at constant speed once it is set in motion, --distance moved by A relative to B before they move with c. For what range of values of In, will the blocks remain at rest if they are released from rest? the same velocity? Objective Type Solutions on page 7.125 Vo 1. When a body is stationary A 1m a. there is no force acting on it BM b. the forces acting on its are not in contact with it Fig. 7.304 c. the combination of forces acting on it balance each 26. A smooth pulley A of mass Mo is lying on a frictionless table, other A massless rope passes round the pulley and has masses M! and M2 tied to its ends, the two portions of the string being d. the body is in vacuum perpendicular to the edge of the table so that the masses hang vertically (see Fig, 7.305), Find the acceleration of the pulley, I\" M M 2. A block of metal weighing 2 kg is resting 011 a frictionless plane, It is struck by a jet releasing water at a rate of I Fig. 7.305 kgls and a speed of 5 mis, The initial acceleration of the block will be 27. Two blocks of masses In and M are connected by a chord passing around a frictionless pulley which is attached to a. 2.5 mis' b. 5 m/s2 a rotating frame, which rotates about a vertical axis with an angular velocity w (see Fig, 7,306), If the coefficient of c. 10 m/s2 d. 20 mis' friction between the two masses and the surface be fL! and fL2, respectively, determ'inc the value of w, at which the 3. Two persons are holding a rope of negligible weight block starts sliding radially (M > lit), tightly at its ends so that it is horizontal. A 15 kg weight is attached to the rope at the mid point which how no m,28. A block with mass is placed on an inclined plane with longer remains horizontaL The minimum tension required to completely straighten the rope is a slope angle ex and is connected to a second hanging block with mass m2 by a cord passing over a small, frictionless a. 15 kg b. 15/2 kg c. 5 kg d. Infinitely large

7.66 Physics for IIT-JEE: Mechanics I 4. Three equal weights A, B. and C of mass 2 kg each arc a. F sine eh. F / sin hanging on a string passing over a fixed frictionless pul- c. Fcose ed. Flcos ley as shown in the Fig. 7.308. The tension in the string connecting weights Band C is 8. Two bodies of mass 4 kg and 6 kg are attached to the ends of a string passing over a pulley (see Fig. 7.311). The 4 a. zero b. ]3 N kg mass is attached to the table top by another string. The tension in this string TI is equal to (take g = 10 m/s2) c. 3.3 N d. 19.6 N Fig. 7.308 Fig. 7.311 S. A block of mass M is pulled along a horizontal frictionless a.20N b. 25 N c. 1O.6N d. ION surface by a rope of mass m, Force P is applied at one end of rope. The force which the rope exerts on the block 9. In the Fig. 7.312, the pulley PI is fIxed and the pulley 1'2 is is movable. If WI = W, = 100 N, what is the angle AP,PI ? The pulleys are frictionless P P a. b. M(m + M) (M - m) d. PM _ _c. (M - m) -_..PM . (m + M) 6. The elevator shown in Fig. 7.309 is descending with an Fig. 7.312 acceleration of 2 ms-·2 . The mass of the block A = 0.5 kg. The force exerted by the block A on the block B is (take g = 10 m/s2) a. 30\" c, 90\" 10. A man sits on a chair supported by a rope passing over a ffictionless fixed pulley. The man who weighs 1,000 N exerts a force of 450 N on the chair downwards while pulling the rope on the other side. If the chair weighs 250 N, then the acceleration of the chair is Fig. 7.309 a. 0.45 mis' h. 0 a.2N b.4N c.6N d. 8 N c. 2 mis' d. 9125 mis' 7. A mass M is suspended by a rope from a rigid support at 11. In the Fig. 7.313, the ball A is released from rest, when the A as shown in Fig. 7.310. Another rope is tied at the end spring is at its natural (unstretched) length. For the block B and it is pulled horizontally with a force F.Ifthe rope B of mass M to leave contact with ground a~ some stage, AB make an angle f) with the vertical, then the tension in the minimum mass of A must be the string A B is A e A rB--+ F B M Fig. 7.313 Fig. 7.310

a.2M b. M Newton's Law of Motion 7.67 c. MI2 d. MI4 17. A body of mass 2 kg has an initial velocity of 3 mls along 12. Two skaters weighing in the ratio 4 : 5 and 9 m apart are o E and it is subjected to a force of 4 N in a direction skating on a smooth frictionless surface. They pull on a rope stretched between them. The ratio of the distance perpendicular to OE (see Fig. 7.316). The distance of body from 0 after 4 s will be covered by them when they meet each other will be s a.5:4 b.4:5 t=4scc S2 ace!. = a c. 25: 16 d. 16: 25 s, 13. Three forces arc acting on a particle of mass In initially in \"'---'--~- - - -- --- equilibrium. If the first 2 forces (II, and II,) arc perpen- oE dicular to each other and suddenly the third f(lree (R3) is Fig. 7.316 removed, then the acceleration of the particle is a. 12 m b.20m c.8m d. 48 m a. R3 ~ R, + R, c. 18. In order to raise a mass of 100 kg a man of mass 60 kg m fastens a rope to it and passes the rope over a smooth m pulley. He climbs the rope with acceleration 5 gl4 relative d. R, to the rope (sec Fig. 7.317). The tension in the rope is m (take g = 10 m/s'). 14. n balls each of mass m impinge elastically each second on a surface with velocity u, The average force experienced by the surface will be 3. mnu h. 2mnu c. 4mnu d. mnul2 15. A ball of mass In moving with a velocity u rebounds from a walL The collision is assumed to be elastic and the force of interaction between the ball and wall varies as shown in t.he Fig. 7.314. Then the value of Po is F Frot·_-7 Fig. 7.317 ~---~!~t a. 1432 N b. 928 N 0.5 T T c. 1219 N d. 642 N Fig. 7.314 19. A plumb bob is hung from the ceiling of a train compart- ment. The train moves on an inclined track of inclination a. miliT b. 2muiT 30e) with horizontal. Acceleration of train up the plane is a = gl2. The angle which the string supporting the bob e. 4muiT d. mul2T makes with normal to the ceiling in equilibrium is 16. A unidirectional force F varying with time t as shown in b. tan-' (21.)3) the Fig. 7.315 acts on a body initially at rest for a short duration 2T. Then the velocity acquired by the body is F e. tan-' (.J312) d. tan-'(2) Fa --/-~-~, 20. Two particles A and B, each of mass m, are kept stationary by applying a horizontal force F =mg on particle B as shown in Fig. 7.318. Then F{) -------------- 0, Fig. 7.315 ,, \" T, A, T, rr Pi, T rrFoT a. b. Fig. 7.318 4m 2m FoT c. d. zero 4m

7.68 Physics for lIT-J EE: Mechanics I a. 2tan f! = tan\" b. 2TJ = 5T, d. None of these. ·c.TJ=T, 21. A lift is going up, the total mass of the lift and the passen- gers is 1500 kg. The variation in the speed of lift is shown in Fig. 7.319. Then the tension in the rope at 1= 1 swill be v (m,':;) Fig. 7.321 t (.,)'--_\"-..J.....J....L~_..L.._I.. b. if M > 2m 2 4 6 8 10 12 c. if M > m/2 Fig. 7.319 d. For any value of M (Neglect friction and masses of a. 17400 N pulley, string and spring) b. 14700 N c. 12000 N 26. A trolley T of mass 5 kg on a horizontal smooth surface d. None of the above is pulled by a load of 2 kg through a uniform rope ABC 22. In the above problem the tension in the rope will be least of length 2 m and mass I kg (see Fig. 7.322). As the load falls from BC =0 to BC =2 m, its acceleration (in m/s') changes from at A B T a. I = Is b. 1=4 s c. f = 9 s d. I = II s 23. A block is placed on a rough horizontal plane attached 2 kg with an clast.ic spring as shuwn in Fig. 7.320 Fig. 7.322 a a. -20 to -30- 20 30 66 Fig. 7.320 b.-to- Initially spring is unscratchcd. If the plane is gradually R8 lifted from () = 0° 10 0 = 9(P, then the graph showing extension in the spring (x) versus angle (8) is 20 30 d. none of these c. ~- to -- 56 27. Two wooden blocks are moving on a smooth horizontal surface such that. the mass In remains stationary with re- spect to block of mass M as shown in the Fig. 7.323. The magnitude of force P is •p m M n. h. c. d. Fig. 7.323 24. A balloon of mass M is descending at a constant accel- a. (M + /Il)g tan f! eration ex. When a mass In is released from the balloon it starts rising with the same acceleration a. Assuming that b. Ii tan f! its volume does not change, what is the value of m? c. mg cos f3 a. - -\" M b. -2-\"M d. (M + m)g coseef! a+g a+g c. -a+-gM a+g 28. A bead of mass III is attached to one enel of a spring of d. - - M ct . eV3+I)mg 2a natural length Rand spnng constant K = R. 25. The system shown in Fig. 7.321 is released from rest. The spring gets elongated The other end of the spring is fixed at a point A on asmooth vertical ring of radius R as shown in the Fig. 7.324. The 3. if Ill> 111 normal reaction at B just after it is released to move is

B Newton's Law of Motion 7.69 A 1\"'--'-'-----_-1 ,,,,,,A Fig. 7.324 Fig. 7.327 a. mgl2 h. v'3mg a. 0 b. mg c; 3v'3 mg 3../3mg c. J'img d. mglJ'i d. 2 29. An inclined plane makes an angle 30° with the horizontal. 33. Blocks A and C start from rest and move to the right with acceleration UA = I'2l Ill/52 and ac = 3 l111s2. Here t is in A groove (OA) of length 5 m cut, inthe plane makes an seconds. The time when block B again comes to rest is angle 30\" with 0 X. A short smooth cylinder is free to slide down the influence of gravity. The time taken by the a.2s b.Is cylinder to reach from A to 0 is (g = 10 m/s') c. 3/2 s d. 112 s o LL'-.L_ _ _..L\"::'::\"-... X Fig. 7.325 a. 4 s h. 2 s c. 2 s d. 1 s Fig. 7.328 30. A man is raising himself and the crate on which he stands 34. In the given Fig. 7.329 the mass In, starts with velocity Vo with an acceleration of 5 m/s2 by a massless rope-and- and moves with constant velocity on the surface. During motion the normal reaction between the horizontal surface pulley arrangement. Mass of the man is 100 kg and that and fixed triangle block m! is N. Then during motion of the crate is 50 kg. If g = 10 mis', then the tension in the rope is eA Fig. 7.326 Fig. 7.329 a. N = (ml + m2)g h. N = illIg c. N < (11'11 +m2)g d. N > (ml +m2)g a. 2250 N h.1125N 35. Figure 7.330 shows an arrangement in which three identi- c. 750 N d: 375 N cal blocks arc joined together with an inextensible string. 31. In question 30, contact force between man and the crate All the surfaces are smooth and pulleys are massless. IfilA. {lB. and ac are the respective accelerations of the blocks is A, B, and C. then the value of all in terms of a A and ac is a. 2250 N. h. 1125 N b. -aA---e-tc c. 750 N d. 375 N a. aAtaC c. aA +ac 2 32. Two objects A and B each of mass m arc connected by a light inextensible string. They are restricted to move on a 2 d. ai\\ +ac frictionless ring of radius R in a vertical plane (as shown in Fig. 7.327). The objccts arc released from rest at the 36. In the Fig. 7.33l shown, blocks A and B move with ve- positic)I\\ shown. Then, the tension in the cord just after release is locities v! and V2 along horizontal direction, The ratio of VI . V,-IS

7.70 Physics for IlT-JEE: Mechanics I c, b1 +2b] d. None of these Ac 39. If the blocks A and B arc moving towards each other with acceleration a and b as shown in the Fig. 7.334. Find the net acceleration of block C, Fig. 7.330 I) 1:1~JI~ Fig. 7.334 v, a. a1- 2(a +h)] Fig. 7.331 b. --(a +b)] c. a7 - (a +b)] d. None of these a. sin e, b. sinez 40, The small marble is projected with a velocity of 10 m/s in a direction 4SC' from the horizontal y-direction on the sina2 sinOI smooth inclined plane, Calculate the magnitude v of its velocity after 2 s. cos(h d. COSBl a. 1OV2 m/s b. 5 m/s C. COS O2 c. 10 m/s d. 5V2 m/s cos OJ 37. Assuming that the bloek is always remains horizontal. hence the acceleration of B is 41. Two masses each equal to m arc lying on X-axis at. (-a, 0) and (+a. 0). respectively, as shown in Fig. 7.335. They are connected by a light string. A force F is ap- plied at the origin along vertical direction. As a result, the masses move towards each other without loosing contact with ground. What is the acceleration of cach mass? As- sume the instantaneous position of the masses as (-x, 0) and (x, 0), respectively ,,,,y F Fig. 7.332 a. 6 m/sz b. 2 m/s2 (·a, 0) (a, 0) c. 4 m/s2 d. None of these m m --. x 38. If the block B moves towards right with acceleration b, Fig. 7.335 then the net acceleration of block A is 21\". j(~-.;2) 21\" x a. mx b. In J(a2 _~;2) FX FX c. Ii.. rl. rr:; 2m V(a 2 - x2) .7 B In V(a 2 - x2) L, A b 42. A light string passing over a smooth light pulley connects two blocks of masses 11\" and 1112 (vertically). If the ac- celeration of the system is (g/8), then the ratio of masses Fig. 7.333 is a. 5: 3 b. 4: 3 a. h1 + 4b] b. b1 +b] c. 9: 7 d. 8: 1

Newton's Law of Motion 7.71 43. A lift is moving down with an acceleration a. A man in 48. A block of mass m is placed on a smooth inclined plane the lift drops a ball inside the lift. The acceleration of the baIl as observed by the man in the lift, and a man standing eof inclination with the horizontal. The force exerted by stationary on the ground are, respectively the plane on the block has magnitude a. a, g h. (g - a);g a. mg tan () h. rng cos (j d. mg ec. mgj cos C. a, a d. g, g 49. A wooden block of mass M resting on a rough horizontal floor is pulled with a force F at an angle ¢ with the hor- 44. Block B has a mass m and is released from rest when it is izontaL If )l is the coefficient of kinetic friction bet ween on top of wedge A, which has a mass 3 m (see Fig. 7.336). the block and the surface. then acceleration of the block Determine the tension in cord CD needed to hold the is wedge from moving while B is sliding down A. Neglect frictioTI. F. a. M sm¢> h. F + I\" . </J) -ILg M(cos¢> sm DC 1\"1' e. -cos¢> eA M F d. M(cos¢> - I\" sin(p) - I\"g Fig. 7.336 50. A particle of small mass m is joined to a very heavy body by a light string passing over a light pulley. Both bodies a. 21ng cose b. -m-g case arc frcc to move. The total downward force on the pulley mg 2 is c. Tsin2e d. mgsin28 a. »mg h.4mg 45. A particle of mass 2 kg moves with an initial velocity of c.2mg d. mg v = (41 + 4]) m/s. A constant force of F =-20., N is 51. A light string passing over a smooth light pulley connects applied on the particle. ,Initially, the particle was at (0, two blocks of masses 1111 and m2 (vertically). If the ac- 0), The x-coordinate of the particle when its y-coordinate celeration of the system is (gI8), then the ratio of masses again hecomes zero is given by is a. 1.2 m h. 4.8 m a. 8: I b. 9: 7 c. 6.0 m d.3.2m c. 4: 3 d. 5: 3 46. Three blocks A, B, and C are suspended as shown in 52. An object is suspended from a spring balance in a lift. Fig. 7.337. Mass of each of blocks A and B is m. If system The rcading is 240 N when the lift is at rest. If the spring is in equilibrium, and mass of C is M then balance reading now changes to 220 N, then the lift is moving a. downward with constant speed h. downward with decreasing speed c. downward with increasing speed d. upward with increasing speed AB 53. When force Fl, F2, and F3 arc acting on a particle of mass 111 such that F2 and F3 arc mutually perpendicular, Fig. 7.337 then the particle remains stationary. If the force FJ is now removed, then the acceleration of the particle is a. M > 2rn h. M=2m c. M <2m d. None of these a. Fl h. Pi 47. A balloon with mass M is descending down with an ac- c. 111 celeration a (a < g). What mass m be detached from it, m so that it starts moving up with an acceleration a. 54. In Fig. 7.338, the system is initially at rest. A 5 kg block Ma h. 2Ma is now released. Assuming the pulleys and string to be a. g+a massless and smooth, the acceleration of block C will be g+(I 2Ma d. a. zero h. 2.5 m/s2 2Mg c. g e. 10/7 mis' d. 5/7 mis' a

7.72 Physics for IIT-JEE: Mechanics I Fig. 7.338 m 55. As shown in the Fig. 7.339, if acceleration of M with Fig. 7.342 respect to ground is 2 m/s2, then 59. A man pulls himself up the 300 incline by the method sin 37° 3/5 shown in Fig. 7.343.lfthe combined mass of the man and cos 37° = 4/5 cart is 100 kg, determine the acceleration of the cart. if the man excrts a pull of 250 N on the ropc. Neglect all friction and the mass of the rope, pulleys and wheels. Fig. 7.339 Fig. 7.343 a. Acceleration of m with respect to M is 5 m/s2 . b. Acceleration of m with respect to ground is 5 m/s2 . a. 4.5 m/s2 b. 2.5 m/s2 c. Acceleration of m with respect M is 2 m/s2 . e. 3.5 m/s2 d. 1.5 mis' d. Acceleration of In with respect to ground is 1() rn/52. 60. A painter of mass M stands on a platform of mass 111. and 56. A force-time graph for the motion of a body is shown in pulls himself up by two ropes which hang over pulley the Fig. 7.340. The change in the momentum of the body as shown in Fig. 7.344. He pulls each rope with force F between zero and lOs is and moves upward with a uniform acceleration Q, Find a neglecting the fact that no one could do this for long time. y 4 F (N) f---:+---;=-:-,- x 5 101 (s) \"----' Fig. 7.340 a.15kgm/s h. 4 kg mls e. 3 kg mls d. 5 kg mls 57. A block A has a velocity of 0.6 mls to the right, determine Fig. 7.344 the velocity of cylinder Ii. 4F+(2M+m) b. 4F+(M+m)g CJ t2~W~~'~#11.'//~Y/m-///'/''/, w///),0; a. ,.. B M+2m M+2m 4F -(M +111) Fig. 7.341 4F - (M +m) rl. e. a. 1.2 m/s h. 2.4 mls 2M+m e. 1.8 mls d. 3.6 m/s M+m 58. For the pulley system shown in Fig. 7.342, each of the 61. An object is resting at the bottom ofthc two strings which cables at A and Ii is given a velocity of 2 mls in the arc inclined at an angle of 1200 with each other. Each string direction of the arrow. Determine the upward velocity v can withstand a tension of 20 N, The maximum weight of thc load m. of the object that can be sustained without breaking the string is a. 1.5111/8 b. 3 mls a. JON b.20N C. 6 mls rl. 4.5 m/s e. 20.J2 N d. 40 N

Newton's Law of Motion 7.73 62. A block is lying on the horizontal frictionless surface. One 67. Two persons are holding a rope of negligible weight end of a uniform rope is fixed to the block which is pulled in the horizontal direction by applying a force F at the tightly at its endsso that it is horizontal. A IS kg weight in other end. If thc mass of the rope is half the mass of the block. the tension in the middle of the rope will be attached to rope at the mid point which now no more re- mains horizontal. The minimum tension required to com- pletely straighten the rope is a. F b. 2FI3 a. ISON b.7SN c. 3F1S d. SFl6 c. SON d. infinitely large 63. A 60 kg man stands on a spring scale in a lift. At some 68. A balloon of fixed volume containing mass M is coming instant, he finds that the scale reading has changed from 60 kg to 50 kg for a while and then comes back to original down with an acceleration of a towards earth. How much mark. What should be concluded? a. The lift was in constant motion upwards. mass should be released from the balloon so that it starts rising with acceleration a. b. The lift was in constant motion downwards. 2Ma Ma a. b. c. The lift while in dnwnward motion suddenly stopped. g-a g+a Ma c. d. 2Ma g-a g+a d. The lift while in upward motion suddenly stopped. 69. If bloek is moving with an acceleration of 5 m/s2 (see Fig, 7.346), the acceleration of B w.r.t. ground is 64. The Fig. represents a light inextensible string ABC DE in which AB = BC = CD = DE and to which are at- .j tached masses M. II! and M at the points B. C and D. respectively. The system hangs freely in equilibrium with ] ends A and E of the string fixed in the same horizon- tal line (see Fig. 7.345). It is given that tan a = 3/4 and tan fi = 12/5. Then the tension in the string BC is m IIIIII//i//IIIIIII////i//IIIIIIII/, 1// , A E, Fig. 7.346 a. 5 m/s2 b. s.)2 m/s2 d. 10 mis' c. sv\"5 m/s2 Fig. 7.345 70. l\\Vo particles A and B, each of mass In, are kept stationary by applying a horizontal forcc F = mg on particle B as shown in Fig. 7.347. Then fia o a.2mg b. (13/10) mg c. (3/10) mg d. (20111) mg 65. A monkey of mass 40 kg climbs on a massless rope of , breaking strength 600 N. The rope will break if the mon- ,,,/3 key F~mg a. Climbs up with a uniform speed of 5 m/s. Fig. 7.347 b. Climbs lip with an acceleration of 6 m/52, a. 2tanfi = tana b. 2T, = 512 c. Climbs down with an acceleration of 4 m/s2. d. None of these c. T,.)2 = T, v\"5 d. Climbs down with a uniform speed of S m/s. 71. A block placed on a horizontal surface is being pushed 66. Three light strings are conneeted at the point P. A weight eby a force F making an angle with the vertical. The W is suspended from one ofthe strings. End A of string AP and end B of string P B are fixed as shown. In equilibrium coefficient of friction bctween block and surface is f.L The P B is horizontal and P A makes an angle of 60\" with the force required to slide the block with uniform velocity on horizontal. If the tension in P B is 30 N then the tension the floor is in P A and weight Ware respectively given by a. I L m gh(. -,s..:.'in::..::..e_-_fi.c:...::.c:.::os:.::8..:.) a. 60N; 30N (sin e - fi. cos 8) I,mg b. 60/vSN; 30/vS N c. f-Lmg d. None of these c. 60 N ; 30vS N 72. A block slides with velocity of 10 mls on a rough hori- zontal surface. It comes to rest after covering a distance d.60vSN;30vSN of SO m. If g is 10 m/s'. then the coefficient of dynamic friction between the block and the surface is

7.74 Physics for IIT-JEE: Mechanics I a. 0.1 b. 1 c. 10 d.5 78. Two blocks of mass M, and M, are connected with a string which passes over a smooth pulley. The mass M! is placed 73. A block of mass 1 kg is at rest on a horizontal table. on a rough incline plane as shown in the Fig. 7.350. The The coemeient of static friction between the block and coefficient of friction between the block and the inclined the table is 0.50. If g = 10 ms- 2, then the magnitude of a plane is 11.. What should be the minimum mass M2 so that force acting upwards at an angle of60° from the horizontal the block M, slides upwards? that will just start the block moving is: a. 5 N b. 5.36 N c. 74.6 N d. 10 N 74. A heavy uniform chain lies on a horizontal table top. If Mr the coefficient of friction between the chain and the table surface is 0.25, then the maximum fraction of the length e of the chain, that can hang over one edge of the tab19 is a.20% b.25% Fig. 7.350 c. 35 % d. 15 % ea. M2 = M, (sin + IkOOS 0) 75. In the Fig. 7.348, a block of weight 60 N is placed on b. M2 = Mr (sin e - IJ. cos 0) a rough surface. The coefficient of friction between the block and the surfaces is 0.5. What should be the weight M2 = -- M, W such that the block does not slip on the surface? sin -+ --- eC. f1. cos (i M, d. M, = sin e - efJ. cos c 79. A box of mass 8 kg is placed on a rough inclined plane Tr of inclination e. Its downward motion can be prevented w by applying an upward pull F and it can be made to slide upwards by applying a force 2F. The coefficient offriction between the box and the inclined plane is a. ('an 8)/3 b. 3[31l 0 Fig. 7.348 c. (tan 0)/2 d. 2tan 0 a.60N b. -6-0N 80. A block of mass 15 kg is resting on a rough inclined plane c. 30N as shown in Fig. 7.351. The block is tied by a horizontal .Ji string which has a tension of 50 N. The coefficient of friction between the surfaces of contact is 30 d. - N .Ji 76. A suitcase is gently dropped on a conveyor belt moving at LM a velocity of 3 m/s. If the coefficient of friction between 45' the belt and the suitcase is 0.5, find the displacement of the suitcase relative to conveyor belt before the slipping =between the two is stopped (g 10 m/s2 ) a. 2.7 m b. 1.8 m Fig. 7.351 c. 0.9 m d. 1.2 m a. 112 b. 2/3 c. 3/4 d. 1/4 77. A horizontal i(lfee of 10 N is necessary to just hold a 81. A horizontal force, just sufficient to move a body of mass block stationary against a wall. The coefficient of friction between the block and the wall is 0.2 (Fig. 7.349). The 4 kg lying on a rough horizontal surface, is applied on weight of the block is it. The coefficient of static and kinetic friction between the body and the surface are 0.8 and 0.6, respectively. If ----+ the force continues to act even after the hlock has started moving, the acceleration of the block in m/s2 is (g = ]() ION m/s2) a. 114 b. 1/2 c. 2 d. 4 Fig. 7.349 82. Blocks A and B in the Fig. 7.352 are connected by a bar a.2N b.20N ofnegligihle weight. Mass of each block is l70 kg and !.LA c. 50 N d. 100 N = 0.2 and If..[f = 0.4, where jf..A and f1.B are the coeftkients of limiting friction between blocks and plane. Calculate the itlfCe devel')ped in the bar (g = I() m/sec2 ).

15 Newton's Law of Motion 7.75 Fig. 7.352 86. Two blocks of masses MI and M2 are connected with a string passing over a pulley as shown in Fig. 7.355. The 3. 150 N b. 75 N blo\"ck Ml lies on a horizontal surface. The coefficient of c. 200 N d. 250 N friction between the block M 1 and the horizontal surface is p.... The system accelerates. What additional mass m should be placed on the block Ml so that the system docs not accelerate? m MJ 83. The upper half of an inclined plane with inclination ¢ is Fig. 7.355 perfectly smooth while the lower half is rough. A body staliing from rest at the top will again come to rest at the bottom if the coefficient of friction for the lower half is given by 3. 2tnn ¢ b. tan ¢ a. M2 -M1 b~ -M2- M J c. 2sin ¢ d. 2cos ¢ /1- /1- d. (M, - M 1),\" 84. A block of mass m is placed on another block of mass Ml M which itself is lying on a horizontal surface (see c. M, - - - Fig. 7.353). The coet1icient of friction between two block is /1-J and that between the block of block M and horizon- /1- tal surface is /h2. What maximum horizontal force can be applied to the lower block so that thc two blocks move 87. A block of mass m is placed on the top of another hlock without separation? of mass M as shown in the Fig. 7.356. The coefficient of friction between them is fL. What is the maximum accel- eration with which the block M may move so that m also moves along with it? Fig. 7.353 R~. . .·•.·./I.' 3. (M + m)(/1-, - /1-1)g Fig. 7.356 b. (M - m)(/1-2 - /1-1)g b. g/fJ, 88. The system is pushed by a force F as shown in Fig. 7.357. c. (M - m)('\"2 + /1-Ilg d. (M + m)(/1-2 + /1-1)g All surfaces arc smooth except between Band C. Friction coefficient between Band C is /1-. Minimum value of F 85. Two blocks A and B of masses 6 kg and 3 kg rest on a to prevent block B from down ward slipping is smooth horizontal surface as shown in the Fig. 7.354. If coefficient of friction between A and B is 0.4, the maxi- A Bc mum horizontal force which can make them without sep- aration is 7777777//777777//77 Fig. 7.357 3 kg B b. (:/1-)m g 6kg F (Dd. lung A 89. A body of mass M is resting on a rough horizontal plane Fig. 7.354 surface, the coefficient of friction being equal to /1-. At a.72N b.40N t = 0 a horizontal force F = Fot starts acting on it, where c. 36 N d.20N r)P is a constant. Find the time T at which the motion starts?

7.76 Physics for IIT-JEE: Mechanics I a. I'Mg/ Fa b. Mg/I'Fo •/~ e. I'Fo/Mg d. None of these 90. The maximum value of mass of block C so that neither A A nor B moves is (sec Fig. 7.358) (Given that mass of A is 100 kg and that of B is 140 kg. Pulleys are smooth and () () friction coefficient between A and B and between Band Fig, 7,361 = =horizontal surface is It 0.3.) Take g 10 m/s2 a.210kg b.190kg G)a. tan-I e. 185 kg d. 162 kg b. tan-I (3) e, tan-I (,,12) d,tan,I(2) c 94. Two block M and In arc arranged as shown in the Fig. 7.358 Fig. 7.362. The coefficient of friction between the blocks I \"'I = 0.25 and between the ground and M be IL2 = -. If 3 M = 8 kg then find the value of In so that the system will remain at rest. 91. A block A of mass 2 kg is placed over another block B of M mass 4 kg which is placed over a smooth horizontal floor, Fig. 7,362 The coefficient of friction between A and B is 004. When a horizontal force of magnitude ION is applied on A, the acceleration of blocks A and Bare ~F ~ a. -4 kg b, -8 kg 9 Fig, 7.359 3 c. 1 kg rl. :85 kg a. 1 ms- 2 and 2 ms-- 2, respectively. 95, Find the minimum force required to pull the lower block. b, 5 ms- 2 and 2.5 ms-2, respectively. If the coefficient of friction between the blocks is 0.1 and between the ground and 2 kg block is 0.2. e, Both the blocks will moves together with acceleration a.IN b.5N e.7N d. ION I13 m5\"2 d. Both the blocks will move together with acceleration 96, A body of mass nt is launched up on a rough inclined plane making an angle 4Y' with horizontal. {fthe time of ascent 5/3 ms-2 is half of the time of descent, the frictional coefficient 92. Two blocks m and M tied together with an inextensible between plane and body is string are placed at rest DI1 a rough horizontal surface with 2 b. -3\" 3 d, ~ coefficient of friction \",. The block m is pulled with a a. - 5 e. 5 evariable force F at a varying angle with the horizontal. 5 4 eThe value of at which the least value of F is required to 97. A wooden block of mass M resting on a rough horizontal move the blocks is given by floor is pulled with a force F at an angle ¢ with the hor- izontal. If f.L is the coefficient of kinetic friction between M ofFn m e the block and the surface, then acceleration of the hlock /):77TJ//////////;~- ---- Rough surface (It) Fig, 7.360 is F ea. = tan-I\", h. () > tan-'\" t,L ee, < tan-I\", a, M(cos¢ - \",sin¢) - \",g d. Insufficient data ,\"Fb, - cos¢ 93. A trolley A has a simple pendulum suspended from a M frame fixed to its desk. A block B is in contact on its ver- t.ical slide. The trolley is on horizontal rails and accelerates c, I\" + . -I'g towards the right such that the block is just prevented from M(cos¢ \",sm¢) falling. The value of coefficient offriction between A and B is 0.5 (see Fig. 7.361). The inclination of the pendulum I\" . to the vert.ical is d, M sm¢

98. A given object takes 11 times more time to slide down 4SO Newton's Law of Motion 7.77 rough inclined plane as it takes to slide down a perfcctly Fig. 7.364 b. The contact force between block and incline is of smooth 45\\) incline. The coefficient of kinetic friction be- magnitude m(g + CI). tween the object and the incline is c. The contact force between block and incline is per- ~a. ) I /12 h. )1 _I pendicular to the incline. I /1 2 I d. The contact force is of magnitude mg cos e. c. 1 -/1-2 d. lOS. A block of mass 5 kg is at rest on a rough inclined plane 2 - n2 as shown in the Fig. 7.365. The magnitude of net force 99. A passenger is travelling in a·train which is moving at 40 exerted by the surface on the block will be m/s. His suitcase is kept on the berth. The driver of the train applies breaks such that the speed of the train decreases at a constant rate to 20 mls in 5 s. What should be the minimum coefficient of friction between the suitcase and the berth if the suitcase is not to slide during retardation of the train? a. 0.3 h. 0.5 c. 0.1 d. 0.2 100. Starting from rest, a body slides down a 45° inclined plane Fig. 7.365 in twice the time it takes to slide the same distance in the absence of friction. What is the coefficient of friction a.25N h. 50 N c. 10 N d.30N between the body and the inclined plane? a. )3(2 h. 3/4 c. 1/2 d. 1/4 106. A box of mass 8 kg is placed on a rough inclined plane of inclination 45° . Its downward motion can be prevented 101. The tension in rope (rope is light) by applying an upward pull F and it can be made to slide upwards by applying a force 2F. The coefficient offriction m between the box and the inclined plane is !II> lv! I I d. ~ e Rough inclined 3. 2\" c. 2J2 3 Fig. 7.363 107. A block of mass m ~ 3 kg is placed on the top of another block of mass M ~ 5 kg as shown in the Fig. 7.366. The a, (M+m)gsine coefficient of friction between them is /..L = 0.4. What is the maximum acceleration with which the block M may e - eh. (M+ m)g sin {Lmg cos move so that I'll also moves along with it? (M is on fric- tionless surface.) c. zero In d, (M+m)gcos8 102. There is a chain of length 6 m and coefficient of friction Ma 2I:' What will be the maximum length of chain which can be held outside of table without sliding L2m ~4m ~3m d:lm Fig. 7.366 103. A given object takes n times more time to slide down 45° a. 2 m/s2 b. 1 m/s2 rough inclined plane as it takes to slide down a perfectly c. 3 m/s2 d. 4 m/s2 smooth 45\" incline. The coefficient of kinetic ffiction be- tween the object and the incline is lOS. Two blocks A (2 kg) and B (5 kg) rest one over the other on a smooth horizontal plane (see Fig. 7.367). The coefficient 1 of static and dynamic friction between A and B is the same h. 1 -/1-2 and is equal to 0.80. The maximum horizontal force that can be applied to B in order that both A and B do not have p ;d• -1 --n-2 relative motion is 104. A block of mass III is at rest with respect to a rough incline kept in elevator moving up with acceleration a (Fig. 7.364). Which offollowing statement is correct'? a. The contact force between block and incline is par- Fig. 7.367 allel to the incline. a. 1.2N b.42N c.4.2N d.56N

7.78 Physics for IIT-JEE: Mechanics I 113. A block of metal weighing 2 kg is resting on a frictionless plane. It is struck by ajet releasing water at a rate of 1 kgls 109. The minimum acceleration that must be imparted to the and at a speed of5 m/s. The initial acceleration of the block catt in the Fig. 7.368 so that the block A will not fall (given is fl. = 0.5 is the coefficient of friction between the surfaces of block and cart) is given by L Block p;gJ//#///#//#/// Fig. 7.372 Fig. 7.368 a. :53 m/s2 b. 25 m/s2 - c. (25fm1s2 4 a. 2 mls2 b. 20 m/s2 d. 25: m/s2 c. 5 mis' d. 7.5 mis' 110. A block of mass m, lying on a horizontal plane, is acted 114. Springs of spring cosnstant K, 3K, 9K, 27K, \"',00 upon by a horizontal force P and another force Q, inclincd are connected in series. Equivalent spring constant of the eat an angle to the vertical. The block will remain in combination is equilibrillm, if the coefficient of friction between it and the surface is 3K b. K a. 2 , 2 d. 00 '8 Q 2K c. 3 (I) . . .115. A block of mass m is lying on a wedge having inclination 5\"'angle a = tan~ 1 Wedge IS mOVIng WIth a constant Fig. 7.369 acceleration a = 2m/s2 • The minimum value of coeffi- cient of friction fl., so that m remains stationary w.r.t. to a. (P sin e - Q)/(mg - cos e) wedge is b. (P-QsinO)/(mg+Qsine) a. 2/9 b. 5/12 c. 115 d. 2/5 c. (P cose + Q)/(mg - Q sine) d. (P+ QsinO)/(mg+ Qcose) 111. A horizontal force of 25 N is necessary to just hold a block stationary against a wall the coefficieni of friction between the block and the wall is 0.4. The weight of the block is --+ 116. In the given Fig. 7.374 the blocks are at rest and a force 25 N of ION acts on the block of 4 kg mass. The coefficient Fig. 7.370 of static friction and the,coefficient of kinetic friction are 1\", = 0.2 and fl.k =0.15 for both the surfaces in contact. The magnitude of friction force acting between the surface of contact between the 2 kg and 4 kg block in this situation is a.2.5N b.20N c. ION d.5N 112. A solid block of mass 2 kg is resting inside a cube as Fig. 7.374 shown in Fig. 7.371. The cube is moving with a velocity a. 3 N b.4N =v 51 + 2) m/s. If the coefficient of friction between the c. 3.33 N d. 0 surfacc of cube and block is 0.2. Then the force of friction between the block and cube is /: . /] I/',1----C.E--ll-----1/-0 117. An ideal liquid of density p is pushed with velocity v through the central limb of the tube shown in the Fig. 7.371 Fig. 7.375. What force does the liquid exert on the tube? The cross-sectional areas of the three limbs are equal to a. JON b.4N c. 14 N d.O A each. Assume stream-line flow.

Newton's law of Motion 7.79 ---+ Vo S M Fig. 7.378 Fig. 7.375 121. Two Small rings 0 and 0' are put on two vcrtical sta- tionary rods AB and A' B', rcspectively (see Fig, 7,379). a. -9pAv? b. 5_pAv2 One end of an inextensible thread is tied at point A', The 8 4 thread passes through ring 0' and its other end is tied to ring O. C. -3p A V 2 d. pAv2 Assuming that ring 0' moves downwards at a constant 2 velocity V2 of the ring 0, when LAOO' = a 118. Two uniform solid cylinders A and B each of mass 1 kg A A' are connected by a spring of constant 200 Nlm at their axles and are placed on a fixed wedge as shown in the Fig, 7,376, The coefficient of friction between the wedge v, o and the cylinders is 0,2, The angle made by the line AB ta ~ with the horizontal, in equilibrium, is o S S' Fig. 7.379 b. v, [ -2C-O;s,2-a-/'z-] a/2]2 sma a. V, [2sin COSet a. 0° b. 15' c. VI [ a/2]3 COS2 d. None of these c. 30\" d. None of these . SIll ex 119. Velocity of point A on the rod is 2 mls (leftward) at the 122. A fixed U-shaped smooth wire has a semi-circular bend- instant shown in the Fig, 7.377. The velocity of the point ing between A and B as shown in the Fig, 7.380. A bead B on the rod at this instant is of mass m moving with uniform speed v through the wire , enters the semicircular bend at A and leaves at B, The average force exerted by the bead on the part A B of the wire is A Fig. 7.377 2 b. 1 mls Fig. 7.380 a. - mls 4mv2 v'3 a.O b. rrd 1 d. 2v'3 mls c. M mls c. d. None of these 123. Find frictional force on block 30 kg (Fig, 7.381) 2'13 JON 120. The masses of the blocks A and Bare m and M. Between J1.s =: 0.5 A and B there is a constant frictional force F, and Bean 1', ~ 0,3 slide frictionlessly on horizontal surface (see Fig, 7.378), A is sct in motion with velocity while B is at rest. What is the distance moved by A relative to B before they move with the same velocity? mM b. 2F(m - M) a. F(rn - M) mMv5 d. 2 F(rn + M) Fig. 7.381 c. F(m + M) a. ZON b.30N c.40N d.50N

7.80 Physics for IIT-JEE: Mechanics I 127. Three blocks A. B, and C arc of equal mass III and are placed one over other on a frictionless surface (table) as 124. Two identical particles A and B, each of mass tn, are shown in the Fig, 7.385, Coefficient of friction between interconnected by a spring of stiffness k, If the particle B any blocks A, Band C is fL, The maximum value of mass experiences a force F and the elongation of the spring is of block MD so that the block A, B, and C move without x, the acceleration of particle B relative to particle A is slipping over each other is equal to C B A Fig. 7.382 a. F F -kx Fig. 7.385 2m b. a. 3mlt b. 3m(1 - fL) F-2kx m I-' + I fL c. 3m(l + It) d. kx 3mfJ, In e. --- In m, In,125. The system shown in the Fig. 7,383 is in equilibrium. d. (1 - fL) Masses and are 2 kg and 8 kg, respectively, Spring fL constants k, and k, are 50 N/m and 70 N/m, respectively, 128. Two blocks of masses 0.2 kg and 0,5 kg, which are placed If the compression in second spring is O,5m. What is the 22 m apan on a rough horizontal surface (fL = 0,5), are compression in first spring? (Both .springs have the same acted upon by two forces of magnitude 3 N each as shown natural length,) in Fig, 7.386 at time I = 0, Then, the time I at which they collide each other is 0.2 kg 0.5 kg n=+--~ nI\"\"O.5 • •22m Mm Fig. 7.386 Fig. 7,383 a. sec b• .j2sec c. 2 sec d. None a. 1.3 m b. -0.5 m 129. In an arrangement shown below in Pig. 7.387, the accel- c. 0.5 m d.O,9m eration of blo.ck A and B are given 126. In Fig, 7,384, the block of mass M is at rest on the floor. The acceleration with which a boy of mass m should climb along the rope of negligible mass so as to lift the block from the floor is Fig. 7.387 M a. g13, gl6 b; g16, gl3 c. g12, gl2 d. 0,0 Fig, 7,384 b. (~ - 1) 130. A block of mass 1 kg lying on the floor is subjected to a a. (M _1) g m M horizontal force given by f = 2 sin wI. The coefficient of M d. >-g c. -g friction between the block and the floor is 0.25, m m a. Acceleration of the block is positive and uniform, b. Acceleration of the block depend on val\"e of W, c. The block always remains at rest. d. Acceleration of the block is always zero,

131. A solid block of mass 2 kg is resting inside a cube as Newton's Law of Motion 7.81 shown in Fig. 7.388. The cube is moving with a velocity c.T,=T, d. Data insufficient v = 51 + 21m!S. If the coefficient of friction between the 135. In the situation shown in Fig. 7.392 all the string are light surface of cube and block is 0.2. Thcn the force offriction and inextensible and pullics are light. There is no fric- tion at any surface and all block are of cuboidal shape. A between the block and the cube is horizontal force of magnitude F is applied to right most free end of string in both the cases of Fig. 7.392(a) and a. ION 0 \"9 c. 14N d. 0 Fig. 7.392(b) as shown. At the instant shown, the tension Fig. 7.388 in all strings are non zero. Let the magnitude of the accel- b.4N eration of large blocks (of mass M) in Fig. 7.392(a) and a, a\"Fig. 7.392(b) arc and respectively. Then 132. A block of metal weighing 2 kg is resting on a frictionless F plane. It is struck by a jet releasing watcr at a rate of I kg!s and at a specd of:) m!s. The initial acceleration of Smooth horizontal surface the block is (a) b\"\",w,,;w Block '\" : 0'. ~ C 2 0', F Fig. 7.389 JEJ1IIIII~I Smooth horizontal surface 5, b. 25 mis' (b) a. -3 m/s. 4 c. -285 m/5. 2 d. 5 m/s2 - 2 133. All surfaces are smooth and pulleys are ideal. The string is pulled with force F. mass of A = mass of B = -m. a, aa,, Fig. 7.392 F_-,,\", a. a, = f 0 = 0 b. = c. al > a2 d. al < a2 Fig. 7.390 136. In Fig. 7.393, a person wants to raise a block lying on the ground to a height h. In both the cases if time required a. ClA = ae is same then in which case he has to exert more force. Assume pulleys and stings lights. b. aA = 0, aB f 2aB m,134. A block of mass lies on top of fixed wedge as shown in Fig. 7.391(a) and another block ofmass m2 lies on top of wedgc which is frce to move as shown in Fig. 7.391(b). At time t = 0, both the blocks are released from rest from a vertical height h above the respective horizontal surface on which the wedge is placed as shown. There is no friction between the block and the wedge in both the figures. Let T, and T2 be the timc taken by block in Fig. 7.391(a) and block in Fig. 7.391 (b) respectively to just reach the (i) (ii) horizontal surface, then Fig, 7.393 i m] a, (i) b. (ii) 1 Fixed G 8 c. Same in both wedge d, Cannot be determined Horizontal surface Smooth horizontal surface 137. A chain of length L is placed on a horizontal surface as (,) (b) shown in Fig. 7.394. At any instant x, the length of chain on rough surface and the remaining portion lies on smooth Fig, 7.391 surface. Initially, x = O. A horizontal force P is applied te the chain. In the duration x changes from x = 0 to x = L a.T,>T, for chain to move with a constant speed. b,T,<T2

7.82 Physics for IIT-JEE: Mechanics I Fig. 7.394 142. A coin is placed at the edge of a horizontal disc rotating about a vertical axis through its axis with a unifonn angu- lar speed 2 rad/s. The radius of the disc is 50 em. Find the minimum coefficient of friction between disc and coin so that the coin does not slip (g = 10 ms-2 ). a. The magnitude of P should increase with time. a, 0.1 b. 0.2 b. The magnitude of P should decrease with time. c. 0.3 d. 0.4 c. The magnitude of P should increase first and then 143. Mark out the incorrect statement decrease with time. -, d. The magnitude of P should decrease first and then a. Second law of motion is a local relation, i.e., if a at increase with time. Fa point charges at any time t, then has to change at the same point at same time t. l38. A 1.5 kg box is initially at rest on a horizontal surface h. In Newton's third law, action and reaction start acting Fwhen at t = 0 a horizontal force = (1.81) 1N (with t at the same instant. in seconds), is applied to the box. The acceleration of the 0 _ _c. If pseudo force acting on an object in a non-inertial box as a function of time t is given by frame is F, then its reaction on inertial frame is - F, G= 0 for 0 :s I :s 2.85 144. Friction force can be reduced to a great extent by G = (1.21 - 2.4)1 m/s2 for t > 2.85 The coefficient of kinetic friction between the box and the surface is a. Lubricating the two moving parts. a. 0.12 b,0.24 b. Using ball bearings between two moving parts. c. 0.36 d, 0.48 c. Introducing a thin cushion of air maintained between two relatively moving surfaces. 139. A vehicle is moving with a velocity v Oil a curved road of width b and radius of curvature R. For counteracting the d. AII of the above. centrifugal force on the vehicle, the difference in elevation required in between the outer and inner edges of the rod 145, An intersteller spacecraft far away from the influence of is any star or planet is moving at high speed under the in- fiuence of fusion rockets (due to thrust exerted by fusion Outer edge rockets, the spacecraft is accelerating). Suddenly the en- gine malfunctions and stops. The spacecraft will Inner edge h a, immediately stops, throwing all of the occupants to \\. the front b, begins slowing down and eventually comes to rest 8 c. keep moving at constant speed for a while, and then b begins to slow down Fig. 7.395 d. keeps moving forever-with constant speed a. v2bl Rg b, vbl Rg 146. Three arrangement of a light spring balance are shown in c. vb21Rg d, vbl R2g the Fig. 7.396,7.397, and 7.398 below. The readings of the spring scales in three arrangements are, respectively 140, A circular road of radius 1,000 m has a banking angle of 45\". What will be the maximum safe speed (in m/s) of a ear whose mass is 2,000 kg and the coefficient of friction between the tyre and the road is 0.5 a. 172 b. 124 c.99 . d. 86 141. A circular table of radius 0.5 m has a smooth diametrical 20kg 20 kg groove. A ball of mass 90 g is placed inside the groove along with a spring of spring constant 102 N/cm. One end (a) 40g b. 20 g, 20 g, -3-- of the spring is tied to thc edge of the table and the other Fig. 7.396 end to the ball. The ball is at a distance of 0.1 m from 40g a. 20 g, 20 g, 10 g d. zero, 20 g, - - the centre when the table is at rest. On rotating the table with a constant angular frequency of 102 rad-s- t , the ball 3 moves away from the centre by a distance nearly equal to a, lO-t m b. lO-2 m c. zero, 20 g, 10 g C, 10-3 In d. 2x 10--' m

Newton's Law of Motion 7.83 d. N, = N, > N3 20 kg 150. Consider a 14-tyre truck, whose only rear 8 wheels are power driven (means only these 8 wheels can produce ac- celeration). These 8 wheels are supporting approximately half of the entire load. If coefficient of friction between rod and each of the tyres is 0.6, then what could be the maximum attainable acceleration by this truck? (b) a. 6 mis' b. 24 mis' c. 3 mis' d. 10 mis' Fig. 7.397 151. A house is built on the top of a hill with 45\" slope. Due to sliding of the material and the sand from the top to . the bottom of the hill the slope angle has been reduced. If the coefficient of static friction between the sand par- ticles is 0.75, what is the final angle attained by the hill? [tan-\"(0.75 ::: 37')J 10 kg (e) 10 kg Fig. 7.398 147. Mark out the most appropriate statement. Fig. 7.400 a. The normal force is the same thing as the weight. a. 8° b. 45\" c.31' d. 30' h. The normal force is different from the weight, but 152. A block of mass 4 kg is pressed against the wall by a force always has the same magnitude. of 80 N as shown in the Fig. 7.401. Determine the value off'rietion force and block's acceleration, (Take IL., = 0.2. c. The normal force is different from the weight, but ILk = 0.15). the two form an action-reaction pair according to (he Newton's third law. d. The normal force is different from the weight. but the two may have same magnitude in certain cases. 148. A system of two blocks, a light string and a light and frictionless pulley is atTanged as shown in Fig. 7.399. The coefficient of frietion between fixed incline and 10 kg block is given by IL, = 0.25 and ILk = 0.20. If the system is released from rest. then find the acceleration of IO kg block? Fig. 7.401 4 kg a. 8 N, 0 mis' h. 32 N, 6 m/s2 c. 8 N, 6 m/s2 d. 32 N, 2 m/s2 37' 153. For the situation shown in the Fig. 7.402, the block is Fig. 7.399 stationary w,r,t. incline fixed in an elevator. The elevator a. 0 b. 0.114 mis' has an acceleration of ,J5ao whose components arc shown c. 0.228 m/s2 d. 2.97 mis' ,in the figure. The surface is rough and coefficient of static 149. A wooden box is placed on a table. The normal force on friction between the incline and block is Ms. Determine the box from the table is N,. Now another identical box is kept on first box and the normal force on lower block the magnitude of force exerlcd by incline on the block. due to upper block is N, and normal force on lower block by the table is N3. For this situation mark out the correct eITake ao = ~ and = 37°, 11., = O.6J statement(s) mg 9mg a. N, = N, = tV3 h. N, < N, = N3 a. h. 10 25 v'I3mg c. -3m-g x v44T1 d. 25 2

7.84 Physics for IlT-JEE: Mechanics I Fig. 7.402 a. 4a m/s2 154. A block of mass In is placed on a rough table. which is b. a ..;r.l~7--~8-c-·(-)s-a m/s2 kept in a gravity free hall. Coefficient of friction between c. ,fj7a m/s2 block and table is lL and a horizontal force F is applied to d. v'l7 cos 'e2x x a m/s2 the block. The reaction force exerted by the table on the 158. In the arrangement shown in the Fig. 7.406 below at a particular instant the roller is coming clown with a speed of 12 m/s2 and C is moving up with 4 mis, At the same instant it is also known that W.r.t. pulley P, block A is moving down with speed 3 m/s. Determine the motion of block B (velocity) w.r.t. ground. block is a. mg b.mg~ c. F d. (JI + IL2) F ISS. If coefficient of friction between all surfaces (seee t? A Fig. 7.403) is 0.4. then find the minimum force F to have equilibrium of the system. 12 m/s c Fig. 7.406 a. 4 m/s in downward direction F b. 3 mls in upward direction c. 7 m/s in downward direction d. 7 111111/5 in upward direction Fig. 7.403 159. A professor holds an eraser against a vertical chalkboard by pushing horizontally on it. He pushes with a force that is a. 62.5 N b. 150 N much greater than it required to hold the eraser. The force of friction exerted by the board on the eraser increases if c. 135 N d. 50 N he a. pushes eraser with slightly greater foree. 156. In the Fig. 7.404 shown below. if acceleration of B is a-+, then find the acceleration of A. b. pushes eraser with slightly less force. Fig. 7.404 c. raises his elbow so that the force he exerts is slightly downward but has same magnitude. 3. a sina b. acote c. atane d. asinacote d. lowers his elbow so that the force he exerts is slightly upward but has the same magnitude. 157. If the acceleration of wedge in the Fig. 7.405 shown is a 111/52 towards left, then at this instant acceleration of t.he 160. A particle of mass 2 kg moves with an intial velocity of block (magnitude only) would be (41 + 27) mls on the x-y plane. A force 'jo = (21 - 8]) N acts on the particle. The initial position of the particle is (2 m. 3 m). Then for y = 3 m a. The possible value of x is only x = 2 m. b. The possible value ofx is not only x = 2 m, but there exists some other value of x also. c. Time taken is 2 s. d. All of the abo\"e. Fig. 7.405 161. Find the least horizontal foree P to start motion of any part of the system of the three blocks resting upon one another as shown in Fig. 7.407. The weights of blocks are A = 300 N.B = JOO Nand C= 200 N. Between Aand B, eoefficient of friction is 0.3. between Band C is 0.2 and between C and the ground is 0.1.

Newton's Law of Motion 7.85 P (kg mls) P (kg 111/s) .. 80 -------- 80 --------- 40 b. 40 Fig. 7.407 t (s) t (s) 24 24 a.60N b.90N c. SON d.70N P(kg m/s) P (kg m/s) 162. A person is drawing himself up and a trolley on which he 40 -------- 40 --------- stands with some acceleration. Mass of the person is more than the mass of the trolley. As the person increases his <. 20 d. 20 force on the string, the normal reaction between person and the trolley will I (s) I (s) 24 24 165. The system shown in Fig, 7.411 is released from rest when the spring was in its normal length. On releasing, the spring starts elongating Fig. 7.408 a. increase b. decrease c. remain same Fig. 7.411 d. cannot be predicted as data is' insufficient a.IfM>m 163. A block of mass III is attached with a mass less spring of b.IfM>2m force constant k. The block is placed over a rough inclined c. If M < .I_n.- surface for which the coemcient of friction is 0.5, M is released from fcst when the spring was unstretchcd (see 2 Fig, 7.409). The minimum value of M required to move d. For any value of M the block III up the plane is (neglect mass of string and pulley and (i'iction in pulley) 166. In Fig. 7.412 string c10es not slip on pulley P, but pulley P is free to rotate about its own axis. Block A is displaced towards left, then pulley P Fig. 7.409 Fig. 7.412 a. ml2 b. m/3 a. rotates clockwise and translates c. ml4 d. None of these h. rotates anticlockwisc and translates c. only translates 164. Figure 7.410 shows the variation of force acting on a body, d. only r~)tates (clockwise or anticlockwise) with time. Assuming the body to start from rest, the vari- ation of its momentum with time is best represented by 167. In the system in Fig. 7.413, the friction coefficient be- which plot? F(N) tween and bigger block fl.. There is no friction between both the blocks. The string connecting both the blocks is 20 ------- light; all three pulleys arc light and frictionless. Then the minimum limiting value of /l. so that 'the system remains in equilibrium is a. I 3 - b. - 22 0 I(S) 23 2 d. 3 c. - Fig. 7.410 32

7.86 Physics for IIT-JEE: Mechanics I a, 30\" b,45° d. None of these c. 60\" 2m 171. Two blocks A and B each of mass m arc placed one over another on an incline as shown in Fig. 7.417. When the Fig. 7.413 system is released from rest the block slides down with constant velocity while block B 'rests on top of A. If the 168. Two blocks A of 6 kg and B of 4 kg are placed in contact coefficient of friction betwcen A and B and between B with each other as shown in Fig. 7.414. and incline are same, then value of coefficient of friction There is no friction between A and ground and between would be both the blocks. CoetJicient of friction between Band ground is 0.5. A horizontal force F is applied on A. Find fixed the minimum and maximum value of F which can be incline applied so that both blocks can move combinely without any relative motion between them. 31'\" smooth l>'ig.7.417 AB a. - b. 3 6 kg 37\" <I kg 45 Fig. 7.414 4 c. d, Information a. 10 N, 50 N b. 12 N, 50 N c. 12 N, 75 N d, None of these 5 insufficient 169. Two blocks are resting on ground with masses 5 kg and 172. Two blocks of masses 3 kg and 2 kg are placed side by 7 kg. A string connects them which goes over a massless side on an incline as shown in the Fig. 7.418. A force, pulley A. There is no friction between pulley and string. F = 20 N is acting on 2 kg block along the incline. The A force F = 124 N is applied on pulley A. The acceleration coefficient of friction betwecn the block and the incline of centre of mass of 7 kg block and 5 kg block in vertical is same and equal to 0.1. find the normal contact force direction is exerted by 2 kg block on 3 kg block. /1- 2 kg fixed 3 kg incline F= !24 N Fig. 7.418 a, J8N b. 30N c. l2N d. 27.6N 173. Determine the time in which the smaller block reaches Fig. 7,415 other end of bigger block in the Fig. 7.419 a, 0 b, 5 ms- 2 ION_E;:::'k 03 ~u 0.0 g c. 2.5 ms-2 d. 1 ms-2 IIII 7777177777 170, Two masses vS m and .,fi m tied by a light string are 14 _I L 3.0 rn placed on a wedge of mass 4 m. The wedge is placed on a smooth horizontal surface (see Fig. 7.416). Find out Fig. 7.419 ethe value of so that the wedge does not move after the a, 4 s b. 8 e,2.19s d. 2.13s system is set free from the state of rest. 174, A body of mass m is held at rest at a height h on two smooth wedges of mass M each which are themselves at. rest on a horizontal frictionless surface (sec Fig, 7.420), When the mass In is released, it moves down, pushing aside the wedges. The velocity with which the wedges recede from each other, when m reaches the ground is 8 8mgh 40mgh x 4 b. Fig. 7,416 a. m+2M 5m+6M

Newton's Law of Motion 7.87 m Ai 37\" Fig. 7.420 32mgh x 4 d. None of these c. Fig. 7.422 32M +9m 178. Figure 7.423 shows two blocks, each of mass m. The sys- 175. For the system shown in Fig. 7.421. m, > mz > m3 > tem is released form rest. If accelerations of blocks A and m4. Initially, the system is at rest in equilibrium condition. If the string joining m4 and ground is cut, then just after a,B at any instant (not initially) are and a2, respectively, the string is cut: then Statement I: m!, tn2, m3 remain stationary. Statement II: the value of acceleration of all the 4 blocks can be determined. Statement III: Only tn4 remains stationary. Statement IV: Only m4 accelerates. Statement V: All the four blocks remain stationary. Now, choose the correct options. Fig. 7.423 3. al =a2cosf} b. {l2 = (11 cose c. al = a2 d. None of these 179. A small block of mass m rests on a smooth wedge of angle f). With what horizontal acceleration a should the wedge be pulley, as shown in the Fig. 7.424, so that the block falls freely. <-~n \" .e. Fig. 7.421 Fig. 7.424 a. All the statement are correct. a.geose b. g sin e b. Only I, II and IV are correct. c. Only II and V are correct. c.geote d.gtane d. Only II and IV are correct. 180. In the given Fig. 7.425, man A is standing on a movable 176. A particles is moving in x-y plane. At certain instant .of plank while man B is standing on a stationary platform. time, the components of its velocity and acceleration arc Both are pulling the string down such that the plank moves as follows: Vx = 3 ms-· j , Vy = 4 ms·-I, ax = 2 ms-2 , and slowly up. As a result of this, the string slips through the Q y = 1ms-2 , The rate.ofchange ofspeed at this moment is. hands of the men. Find the ratio of length of the string that slips through the hands of A to B. a. Jilims-2 b. 4ms-2 d. 2 ms-'z c. J5 ms-2 177. If block A is moving horizontally with velocity VA then 2 find the velocity of block B at the instant as shown in the Fig. 7.422. A+-------, a. hVA X VA B 2JxZ + h2 b. Jx2 + h2 Fig. 7.425 X VA d. h VA C. 2JxZ + h 2 JxZ + hZ

7.88 Physics for IIT-JEE: Mechanics I a. 312 b. 3/4 c. 4/3 d. 2/3 181. A uniform chain is placed at rest on a rough surface of Fig. 7.429 base length I and height h on an irregular surface as shown in Fig. 7.426. Then, the minimum coefficient of friction between the chain and the sUlface must be equal to h. ma2 c. rna] d. Data insufficient 185. In Fig. 7.430 all the surfaces are frictionless while pulley and string are massless. Mass of block A is 2 m and that of Fig. 7.426 block B is m. Acceleration of block B immediately after h h system is released from rest is a. f.L = 21 b. f.L=- 3h 1 c. f.L =-- d. 2h 21 I\" =3-1 182. In the Fig. 7.427, the block of mass III is at rest relative to the wedge of mass M and the wedge is at rest with respect to ground. This implies that Fig. 7.430 Lo M a. gl2 b. g c. gl3 d. None of these /7//1/7 186. In two pulley-particle Figs. 7.431 (i) and (ii), the acceler- Fig. 7.427 ation and force imparted by the string on the pulley and a. Net force applied by III on M is mg. tension in the strings are, (aI, a2), (N I. N,), and (1'1, 7,) b. Normal force applied by m on M is mg. respectively. Ignoring friction in all contacting surfaces c. Force of friction applied by Tn on M is mg. Study the following stat.ements d. None of the above of the above. (i) (ii) 183. A triangular prism of mass M with a block of mass In placed on it is released from rest on a smooth inclined • e.plane of inclination The block does not slip on the prism. Then m, L~smooth Fig. 7.431 b. -1'1 < I 1', Fig. 7.428 d. at < I a. The acceleration of the prism is g cos g. (-/, b. The acceleration of the prism is g tan g. Now mark correct. answer (i) Relation (ii) and (iii) always follows. c. The minimum coefficient of friction between the (ii) Relation (ii) and (iv) always follows. block and the prism is J-tmin = cot e. (iii) Relation (i) only always follows. (iv) Relation (iv) always follows. d. The minimum coefficient of friction between the 187. A block of mass III is pressed against a vertical wall with a block and the prism is JLmin = tan e. horizontal force F = mg (see Fig. 7.432). Another force 184. Two trolleys 1 and 2 are moving with accelerations Qj and \"m\"\"g2 , F' = is acting vertically upon the block. If the co- Q2, respectively, in the same direct. A bock of mass m on trolley I is in cquilihri.um from the frame of observer sta- efficient of friction between the block and wall is 2l :' the tionary with respect to trolley 2. The magnitude of friction force on block due to trolley is (assume that no horizontal friction between them is force other than friction force is acting on block)

F,=mg Newton's Law of Motion 7.89 2 Cilr ---F\"\"'mg --> Fig. 7.432 Fig. 7.435 mg mg a. mJg2+ a2 a'Tup b. Tdown h. +m)g2 a2 - ma c. mg up d. 0 c. 111/g2 + a2 + ma v188. A particle has initial velocity, = 37 + 4] and a constant d.m(g+a) force 'f' = 47 - 3) acts on the particle, The path of the particle is 192. Inside a horizontally moving box, an experimenter finds that when an object is placed on a smooth horizontal table a. strai~ht line b. parabolic and is released, it moves with an acceleration of 10 I11/S2. In this box of I kg body is suspended with a light string, c. circular d. elliptical the tension in the string in equilibrium position, (w.r.t. 189. In the Fig. 7.433 shown the acceleration lif A is, =experimenter) will be (take g 10 m/s') aA = 157 + IS} then the acceleration of B is (A remains a. 10 mis' b. 10)2 m/s2 c. 20m/s' d. zero in contact with II) tv 193. Two blocks A and II each of mass m are placed on a smooth horizontal surface. Two horizontal force F and L<. 2F arc applied on both the blocks A and B, respeetively, as shown in Fig. 7.436. The block A does not slide on Fig. 7.433 bloek II. Then the normal reaction acting between the two blocks is a. 67 b. -157 c. -107 d. -57 iI B 190. Two blocks A and B of masses m and 2m (Fig. 7.434), r '11/ respectively, arc held at rest such that the spring is in nat- ural length. Find out the accelerations of both the blocks just after release. JiiiiLtLiiiil,tiiiUJ.IiiiiL!'/ Fig. 7.436 a. F b. Fl2 F d.3F c. J3 Spring 194. In the given Fig. 7.437. by what acceleration the boy must go up so that 100 kg block remains stationary on m 2m the wedge. The wedge is fixed and friction is absent ev- Fig. 7.434 erywhere (take g = 10 m/s2) a.gt,gt b. !?, t, !?, t c. 0,0 33 d. g,j., c 191. A bob is hanging over a pulley inside a car through a string. IOOkg m \"\"'50 kg The second end of the string is in the hand of a person 53° standing in the car. The car is moving with constant ac- b. 4 m/s2 celeration a directed horizontally as shown in Fig. 7.435. Fig. 7.437 d. 8 mis' Other end of the string is pulled with constant acceleration a vertically. The tension in the string is equal to a. 2 m/s2 c. 6 m/s2

7.90 Physics for IIT-JEE: Mechanics I 195. A system is shown in the Fig. 7.438. A man standing on l!(~r (ccn\"c) \\jA ju the block is pulling the rope. Velocity of the point of string in contact with the hand of the man is 2 mls downwards. /) The velocity of the block will be [assume that the block Fig. 7.441 does not rotate1 199. A plank is held at an angle a to the horizontal (Fig. 7.442) on two fixed supports A and B. The plank can slide against the supports (without friction) because of the weight Mg. Acieration and direction in which a man of mass m should move so that the plank does not move. Fig. 7.438 a. 3m/s b. 2m1s e. 112 mls d. 1 mls 196. In the Fig. 7.439 shown the velocity of lift is 2 mls while A string is winding on the motor shaft with velocity 2 m/s and block A is moving downwards with velocity of 2 mIs, then find out the velocity of block B. Fig. 7.442 ~ t 2 m/s (1a. g sin a + : ) down the incline 2 m/s ~ b. (I ~)g sin a + down the incline m) .. (I I'c. g sm ex + M up the me me Fig. 7.439 d. (I M)g sina + up the incline a. 2m/s t b. 2m/s t In c. 4m/s t d. None of these 200. Object A and B each of maSs m are connected by light inextensible cord, They are constrained to move on a fric- 197. A system is shown in the Fig. 7.440. Assume that cylinder tionless ring to a vertical plane as shown in Fig. 7.443. remains in contact with the two wedges hence the velocity The objects are released form rest at the positions shown. of cylinder is The tension in the cord just after release will be A Cylinder Fig. 7.440 a. }19 - 4,fl ~m/s mub. m/ s mg 2 c. ,flu mls d• .j7 u m/s Fig. 7.443 a. mg.,fi mg 198. Two beads A and B move along a semicircular wire frame b. ..fi as shown in Fig. 7.441. The beads are connected by an mg d. mg inelastic string with always remains tight. At as instant the c. 2 4 speed of A is u. t.BAC = 45\" and t.BOC = 75°. where 201. A pendulum of mass m hangs from a support fixed to a o is the centre of the semicircular arc. The speed of bead trolley. The direction of the string when the trolley rolls B at that instant is up a plane of inclination Q:' with acceleration ao is a. .,fiu b. u a. () = tate l a u b. e = tan-I (; ) c. 2.,fi

Newton's Law of Motion 7.91 206. In the arrangement shown in the Fig_ 7.447, the block of mass m = 2 kg lies on the wedge of mass M = 8 kg. The initial acceleration of the wedge if the sUlfaces are smooth Fig. 7.444 m 60\" a)d. e = tan--I (~-\"--+=-3 sin Fig. 7.447 g cos a .,f3g 2 b. 3.,f3g m/s2 202. Find the acceleration of the block. B in the Fig. 7 -445, assuming that the surfaces and the pulleys P, and P2 are a. 23m/s 23 all smooth. 3g 2 d. JL m/s' c. 23 m/s 23 2m 2m 207. An object moving with a constant acceleration in a non- P:a::~~/t;Lifl inertial frame a. must have non-zero net force acting on it. Fig. 7.445 b. may have zero net force acting on it. c. may have no force acting on it. F F d. this situation is practically impossible, (The pseudo force acting on tbe object has also to be considered) a. b. 4m 6m 208. An object moving with constant velocity in a non-inertial 3F frame of reference F a. must have non-zero net force acting on it. ~ ~ 17m 2m 203. In the Fig. 7-446 shown all blocks are of equal mass m_ All surfaces are smooth_ The acceleration of the block A with respect to the ground is b. may have zero net force acting on it. c. must have zero net force acting on it. d. may have non-zero net force acting on it (Consider only the real forces)_ 209. In the Fig, 7.448 if II, h. and T be the frictional forces on 2 kg block, 3kg block % tension in string, respectively, then their values are Fig. 7.446 b. e4g sin' I\"'~'l ~N , 4g sin e I + 3 sin e IN ~~ ;~ .'1,~{),2 a. 1+3 sin' e 1IIIlJ 4g sin2 e d. None of these Fig. 7.448 c. 204. In the question 203 the acceleration B wx_L ground is a. 2 N, 6 N, 3_2 N 2g sin e 4g sin e b.2N,6N,ON c. IN, 6 N, 2 N a. I + 3 sin' e b. I + 3 sin' e d. Data insufficient to calculate the required value c. 2g sin e 4g sin 0 210. A rope of length 4 m having mass L5 kg/m lying on a ~ I + 3 sin' e d. Ji + 3 sin' e 205. In the question 203 the acceleration C wx_t. ground is horizontal frictionless surface is pulled at one end by a 2g sin e cos e eg sin 8 cos force of 12 N_ What is the tension in the rope at a point eb. I + 3 sin' a. I + 3 sin' 8 1.6 m from the other end? g sin 28 d. -~g7SI:;n-:;8~co:;s~e a.5N b. 4_8 N c. /1 + 3 sin'e c.72N d.6N ~l + 3 sin' 0

7.92 Physics for IIT-JEE: Mechanics I Fig. 7.451 211. A block of mass 5.0 kg slides down from the top of an Multiple Garrect inclined plane of length 3 m. The first 1m of the plane is Answers Type smooth and the next 2 m is rough. The block is released from rest and again comes to rcst at the bottom of the 1. Which of the following arc correct? plane. If the plane is inclined at 300 with the horizontal (Fig. 7.449), find the coefficient of friction on the rough a. A parachute of weight W strikes the ground with his portion. legs and comes to rest with an upward acceleration 212. A body of mass m starting from rest slides down a friction- of magnitude 3 g. Force exerted on him by ground less inclined surface of gradient tan ex fixed on the floor during landing is 4 W. of a lift accelerating upward with acceleration a. Taking width uf inclined plane as W, the time taken by body to b. Two massless spring balances are hung vertically in slide from top to bottom of the plane is series from a fixed point and a mass M kg is attached to the lower end of the lower spring balance. Each spring balance reads M kg -f. c. A rough vertical board has an acceleration a along the .horizontal direction so that a block of mass m passing against it docs not falL The coeffIcient of friction between the hlock ami the board is greater than gla. d. A man is standing at a spring platform. If man jumps away from the platform t.he reading of the spring bal- ance first increases and then decreases to zero. 2. A man tries to remain in equilibrium by pushing with his hands and feet against two parallel walls. For equilibrium, a I' w '1 Fig. 7.450 )!a. ( 2W (g + a) sin 0' !4W Fig. 7.452 b. a. The forces of friction at the two walls must be equal. eg -a) sina) h. Friction must be present on both walls. )!c. ( 4W c. The coefficient of friction must be the same between both walls and the man, (g + a) sin 2a d. None of the above. )!d. ( W 3. Two blocks of masses III I and 1~2 are connected through a (g+a) sinla massless inextensible string. Block of mass III 1 is placed 213. A rope is stretched between two boats at rest. A sailor at the fixed rigid inclined surface while the block of mass in the first boat pulls the rope with a constant force of m2 hanging at the other end of the string, which is pass- 100 N. First boat with the sailor has a mass of 250 kg ing through a fixed massless frictionless pulley shown in whereas the mass of second boat is double of this' mass Fig. 7.453. The coet1icicnt of static friction between the (see Fig. 7.451). If the initial distance between the boats block and the inclined plane is O.S. The system of masses was 100 m, the time taken for two boats to meet each other m 1 and m2 is released from rest. is (neglect water resistance between boats and water) a. The tension in the string is 20 N after releasing the a. 13.8 s b. 18.3 s system. c.3.18s d.31.8s

Newton's law of Motion 7.93 ~g=!Om/s2 /111 =4 kg !1l2= 2 kg Fixed '. Fig. 7.453 o Fig. 7.456 b. The contact force by the inclined surface on the block c. AceeIeratI.On 0f man I.S Mg ) is along normal to the inclined surface. (M+m c. The magnitude of contact force by the inclined sur- d. Measured mass of man is M, face on the block In, is 20./3N. 7. The string shown in the Fig. 7.457 is passing over small d. None of these. smooth pulley rigidly attached to trolley A. If speed of trolley is constant and equal to VA' Speed and magnitude 4. Figure 7.454 shows two blocks, each of mass In. The sys- of acceleration of block B at the instant shown in figure tem is released from rest at the position, shown in fig. If is initial acceleration of blocks A and B are a} and a2, re- spectively and during the motion velocities of A and B are VI and V2 respectively, then Ifr----~~___________ ~ x=3cm B Fig. 7.454 Fig. 7.457 a. al = a2 cose b. a2 = al cose a. Vn=VA,aB=O b. a8=O d. V2 = Vj case c. Vj = V2COSe 3 16vA c. VB = -VA d. an = 125 5. A body of mass 5 kg is suspended by the strings mak- ing angles 60\" and 3~\" with the horizontal as shown in 5 Fig. 7.455 (g = 10 m/s2) 8. Figure 7.458 shows a block of mass m placed on a smooth wedge of mass M, Calculate the value of M' and tension in the string, so that the block of mass In will move vertically downward with acceleration 10 m/s2 T, In T, M 5 kg Smooth Fig. 7.455 a. T, =25N b.T2=25N Fig. 7.458 c. T, = 25./3 N d. T2 = 25./3 N Mcote 6. In Fig. 7.456, a man of true mass M is standing on a a. the value of M' is -:----:: weighing machine placed in a cabin. The cabin is joined 1- cote by a string with a body of mass m. Assuming no friction, and negligible mass of cabin and weighing machine, the Mtane measured mass of-man is (normal force between the man b. the value of M' -:----c---::: and the machine is proportional to the mass) 1 -tane Mm a. Measured mass of man is M .)' c. the vaIue 0 f tenSI.O'n.1ll the str.mg.IS -M-ge tan ( +m mg d. the value of tension is f.'g b. Acceleration of man is -----). cote (M+m 9. The ring shown in the Fig. 7 .459 is given a constant hori- zontal acceleration (aD = g / ./3). Maximum deflection of the string from the vertical is eo, then

7.94 Physics for IIT-JEE: Mechanics I F Light pulley ,111 g'\"'IOm/s2 smooth horizontal rail 10 kg 6 kg 1particle Ground m Fig. 7.46,2 Fig. 7.459 13. A 20 kg block is placed on top of 50 kg block as shown a. 80 = 300 in Fig. 7.463, A horizontal force F acting on A causes an acceleration of 3 rn/s2 to A and 2 m/s2 to B as shown in b. 80 = 60\" Fig. 7.464. For this situation mark out the correct state- c. At maximum deflection, tension in string is equal to ment (s). mg. d. At maximum deflection, tension in string is equal to Rough --+ 2m/s2 2mg ~'L_ _ _ _L.:;----J> 3 m/s2 y'3' A 50 kg 10. In the Fig. 7.460 small block is kept on m then Smooth Fig. 7.463 a, The friction force between A and B is 40 N. b. The net force acling on A is 150 N. a. The acceleration of m w.r,t, ground is ~. c. The value of F is 190 N. m d. The yalue of F is 150 N. h. The acceleration of m \\V.r.t. ground is zero. 14. A block is pressed against a vertical wall as shown in Fig. 7.464. 2~_nc. The time taken by m to separate from M is ) . )21Md • ThC ti'me takeno by rn to separate t.ram .M IS F ' 11. In the Fig. 7.461, if F = 4 N, m =2kg, M = 4 kg then Fig. 7.461 Fig. 7.464 a. The acceleration of m w.r.t. ground is ~m/s2. a, It is most easier to slide the block along 4. b. It. is most diffi~ult to slide the block along L b. The acceleration of m w.r.t. ground is 1.2 m/s2 . c. It is equally easier or difficult to slide the block in any c. Thc acceleration of M is 0.4 mlg2 direction. d, It is most difficult to slide the block along 3. d. The acceleration of m w.r.t. ground is ~ m/s2, 15. Two blocks A and B masses mA and mB velocity v and 3 2v, respectively, at a given instant (see Fig. 7.465). A horizontal force F acts on the block A. There is no friction 12. A force F is applied vertically upward to the pulley and it between ground and block B and there is cocfficient of is observed that the pulley in the Fig. 7.462 moves upward friction between A and B is 1\". The friction with a uniform velocity of 2 mls. The possible value(s) of F islare (in newtons) _/F7_J;: ;~A;;):;-;-;-~-v~ VO a. 150 b. 120 e. 75 d. 400 Fig. 7.465

a. on A oppose its motion. Newton's law of /'tlotion 7.95 b. on B oppose its motion relative to A. c. To change the momentum (given), the force required is independent of time. c. on A and E is !\"mAg. d. The same force acting on two different bodies for the d. on the block E is JLtnB F. same time causes the same change in momentum for different bodies. (rnA + mB) 19. Some statements are given below, which are in one way 16. A block of mass m is placed in contact with onc end of or other can be explained by Newton's laws of motion. a smooth tube of mass M (see Fig. 7.466). A horizontal Mark the correct statement(s). force F acts on the tube in each case (i) and (ii). Then, a. In a tug of war, the team that pushes the ground harder, F_II m I M WInS. I (i) b. In a tug of war, the team that pushes the ground harder F-Im M (horizontally), wins. I (ii) c. Observers win two different inertial frames will mea- sure the same acceleration of a moving object then Fig. 7.466 the velocity of the object wrt two observers would be also same. .F d. A horizontal force acts on a body that is free to move. a. am = 0 and aM = - in (i) M Can it produce an acceleration if this force is equal to half of the weight of that body? F b. am = aM = - - - - in (i) 20. Suppose a body that is acted on by exactly two forces is accelerated. For this situat.ion mark the incorrect state- M+m mentes). c. am = F a. The body cannot move with constant speed. aM = ---- in (ij) h. The velocity can never be zero.' M+m c. The sum of two forccs cannot be zero d. Force on m is ~ in (ii) d. The two forces must act in the same line, M+m 21. Mark the correct statement(s) regarding friction. 17. Two rough block A and 13, A placed over E, move with ac- a. Friction force can be zero, even though the contact an,celerations ilAand velocities VA and VB by the action of surface is rough. horizontal forces FA and FB , respectively (see Fig. 7.467). b. Even though there is no relative motion between sur- facps, frictional force may exist between them. When no friction exists between the blocks A and B e. The expressions /E. = !\"., N or /K = I\"k Narc ap- a. VA = Vn and aA = an prosimate expressions, h. a\" = an d. Does the expression !L = !\"., N tells that direction of c. both (a) and (b) ft.. and N are the same. FA FB 22. A3kgblockofwoodisonalevelsurfacewhere!\", = 0.25 d. - = - and VA = VB and!\", = 0.2. A force on N is being applied horizontally to the block Mark the correct statcment(s) regarding this inA ntlJ situation. a. If the bock is initially at rest, it will remain at rest and 18. Which of the following statement(s) can be explained by friction force will be about 7 N. Newton's second law of motion? b. If the block is initially moving, then it will continue its motion for forever if force applied is in direction a. To stop a heavy body (say truck), much greater force of motion of the block. is needed than to stop a light body, (say motorcycle) c. If the block is initially moving and direction of ap- in the: same time, if they are moving with the same plied force is same as that of motion. of block then speed. block moves with an acceleration of 113 m/s2 along its initial direction of m9tiOli. b. For a given body, the greater the speed, the greater d. If the block is initially moving and direction of ap- is the opposing force needed to stop the body in a plied force is opposite to that of initial motion of certain time. block then block decelerates, comes to a stop and starts moving in the opposite direction.

7.96 Physics for I1T-JEE: Mechanics I 26. A block of mass In is placed on a wedge. The wedge can be accelerated in four manners marked as (1), (2), (3), and 23. Seven pulleys are connected with the help of three light (4) as shown in Figs. 7.470 and 7.471. If the normal reac- strings as shown in the Fig. 7.468 given below. Consider tions in situation (I), (2), (3), and (4) are NI, N 2, NJ, and P3, P4 , and Ps as light pulleys and pulleys P6 and P7 have N4, respectively, and acceleration with which the block masses m each. For this arrangement mark the correct slides on the wedge in situations are hi, b2, b3• and /;4, statement(s). respectively, the,n am m a 3T (2) (I) Fig. 7.470 Fig. 7.468 a a. Tension in the string connecting Ph P3, and P4 is m zero. m b. Tension in the string connecting Pl. P3, and P4 is 3T a mg/3. (3) (4) Fig. 7.471 c. Tension in all the 3 strings is same and equal to zero. d. Acceleration of P6 is g downward and that of P7 is g a. N J > NI > N2 > N4 upward. b. N4 > NJ > NI > N2 \" c. b2 > b3 > b4 > hI 24. A force F (larger than the limiting friction force) is applied d. b2 > h3 > hI > h4 to the left to an object moving to the right on a rough horizontal surface. Then 27. Two blocks A and B of mass 5 kg and 2 kg, respectively. a. thc object would be slowing down initially. connected by a spring of force constant = 100 N/m are placed on an inclined plane of inclination 30G as shown ~ in Fig. 7.472. If the system is released from rest then b. for some time F and friction force will act in same Fig. 7.472 direction and for remaining time they act in opposite directions. a. There will be no compression or elongation in the 'c. the object comes to rest for a moment and after that spring if all the surfaces are smooth. its motion is accelerating in the direction of '\"/, b. There will be elongation in the spring if A is rough d. the object slows down and finally comes to rest. and B is smooth. c. Maximum elongation in the spring is 35 cm if all 25. A block is resting over a rough horizontal floor. At t = 0, a time varying force starts acting on it, the force is described surfaces are smooth. by equation F = kt, where k is constant and t is in seconds. d. There will be elongation in the spring if A is smooth Mark the correct statement(s) for this situation. and B is rough. )' 3 4 2 \"'-_-1.'-_ _ _-. X Fig. 7.469 a. Curve J shows acceleration-time graph. b. Curve-2 shows acceleration-time graph. c. Curve-3 shows velocity-time graph. d. Curve 4 shows displacement-time graph.

Newton's'Law of Motion 7.97 28. Force exerted by the floor of a lift on the foot of a boy 32. Assume right side to be positive. The coefficient offriction standing on it is more than the actual weight of the boy if the lift is moving between the blocks and ground is shown in Fig. 7.475. The correct options are a. down and speed is increasing 0.2 kg 0.5 kg h. up and speed is increasing =-y7] tl \"\" 0.5 [7}+- c. up and speed is decreasing i\"-- 22 m-----.j d. down and speed is decreasing Fig. 7.475 29. The acceleration of a particle as observed from two dif- ferent fi'amcs SI and S2 have equal magnitudes 0[2 ms~2. a. Acceleration of block A is 10 m/s2 • b. Acceleration of block B is -I m/s2• a. The relative acceleration of the frame may either be c. The time (/) at which they collide to each other is 2 s. oor 4 rnls2 d. None of the above, b. Their relative acceleration may have any value between 0 and 4 m/s2 33. A 10 kg block is placed on the top of a 40 kg block as shown in Fig. 7.476. A horizontal force F acting on B c. Both of the frames may be stationary with respect to causes an acceleration of 2 m/s2 to B. For this situation earth, mark out the correcl statement (8). d. The frames may be moving with same acceleration in same direction. 30. Coefficient of friction between the two blocks is 0.3. Whereas the surface A B is smooth A B 40 kg T, B 10 kg Smooth Fig. 7.473 Fig. 7.476 a. Acceleration of the system of masses is 88115 m/s'. a. The acceleration of A may also be 2 m/s2 b. Net force acting on 3 kg mass is greater than that on h. The acceleration of A must also be 2 m/s2 . 2 kg mass. c. The coefficient of friction between the blocks may be c. Tension T2 > TJ. 0.2. d. Since 10 kg mass is accelerating downward, so net d. The coefficient of friction between the blocks must force acting on it should be greater than any of the be 0.2 only. two blocks shown in Fig. 7.473. Assertion-Reasoning .. 31. In the Fig. 7.474, the blocks A, B. and C of mass m each Type Splutions (m page ;7,}55 have acceleration ai, (12, and (13, respectively. Fl and F2 arc Some questions (Assertion-Reason type) are given below. Each external forces of magnitude 2 mg and mg, respectively, question contains Statement I (Assertion) and Statement II (Rea- then son). Each question has 4 choices (a), (b), (e), and (d) out of which only one is correct. So select the COITect choice. AH m (a) Statement I is True, Statement II is True; Statement II is a correct explanation for Statement 1. Fig. 7.474 c (b) Statement 1: is True, Statement II is True; Statement II is NOT a,a. a, # # a3 2m a correct explanation for Statement 1. =b. ([1 G2 #- ([3 (e) Statement I: is True, Statement II is False. (d) Statement I: is False, Statement II is True. c. al > (12 > (13 1. Statement I: Rate of change of linear momentum is equal to d. at #- 02 = (13 external force. Statement II: There is equal and opposite reaction to every action,

7.98 Physics for IIT-JEE: Mechanics I 2. Statement I: A block of mass m is kept at rest on an inclined 13. Statement I: A body in equilibrium has to be at rest only. plane. the re8t force applied by the surface to the bloek will Statement II: A body in equilibrium may be moving with bemg. a constant speed along a straight line path. Statement II: Normal contact force is the resultant of normal contact force and friction force. 14. Statement I: Inertia is the property by virtue of which the body is unahle to change by itself the state of rest. 3. Statement I: The driver of a moving car sees a wall in front Statement II: The bodies do not change their state unless of him. To avoid collision, he should apply brakes ratherthan acted upon by an un-balanced external force. taking a turn away from the wall. Statement II: Friction force is needed to stop the c~r or taking 15. Statement I: Pulling (refer Fig. 7.477) is casier than pushing a turn on a horizontal road. [refer Fig. 7.477(b)] on a rough surface. Statement II: Normal reaction is less in pulling than is 4. Statement I: A bird alights on a stretched wire depressing it pushing. slightly. The increase in tension of the wire is more than the weight of the bird. Fig. 7.477 Statement II: The tension must be more than the weight as it is required to balance weight. 16. Statement I: A block is lying stationary as on inclined plane and coefficient of friction is I).. Friction on block is 5. Statement I: Newton's fIrst law is merely a special case jJ.1ng cos o. (a = 0) of the second law. Statement II: Contact force on the block is mg (Fig. 7.478). Statement II: Newton's first law defines the frame from '\" a,where Newton's second law; f' = In F' representing the 8 net real force acting on a body; is applicable. Fig. 7.478 6. Statement I:A uniform rope of mass In hangs freely from a ceiling. A monkey of mass M climbs up the rope with an 17. Statement I: Static frictional force is always greater than acceleration a. The force exerted by the rope on the ceiling the kinetic frictional force. isM(a+g)+mg. Statement II: (Coefficient of static friction) IL., > ILk Statement II: Action and reaction force are acting on two (coefficient of kinetic friction). eli f[crent bodies. 18. Statement I: Two particles are moving towards each other 7. Statement I: According to Newton's second law of motion due to mutual gravitational attraction. The momentum of action and reaction forces are equal and opposite. each particle will increase. Statement II: Rate of change of momentum depends upon Stntement II: Action and reaction forces never cancel out each other because they are acting on different objects. r~xt· 8. Statement I: A block of mass III is placed on a smooth in- 19. Statement I: A concept of pseudo forces is valid both for clined plane of inclination f) with the horizontal. The force inertial as well as non-inertial frame of reference. exerted by the plane on the block has a magnitude mg cos O. Statement II: A frame accelerated with respect to an Statement II: Normal reaction always acts perpendicular to inertial frame is a non-inertial frame. the contact surface. 20. Statement I: For all bodies the momentum always remains 9. Statement I: A particle is found to be at rest when seen the same. from a frame S I and moving with a constant velocity when Statement II: If two bodies of different masses have same seen ii-om another frame S'2. We can say both the frames are momentum the lighter body posseses greater velocity. inertial. Statement II: All frames moving uniformly with respect to 21. Statement I: In Fig. 7.479 the ground is smooth and the an ineltial frame are themselves inertial, masses of both the blocks are different. Net force acting on each of the block is not same. 10. Statement I: Coefficient of friction can be greater than unity. Statement 11: Acceleration of the bloeks both will be Statement II: Force of friction is dependent on normal reac- different. tion and ratio of force-of fhction and normal reaction cannot exceed unity. F BA 11. Statement I: In high jump, it hurts less when an athlete lands Smooth on a heap of sand. Statement II: Because of greater distance and hence greater Fig. 7.479 time over which the motion of an athlete is stopped, the athlete experience less force when lands on heap of sand. 12. Statement I: I'or a boy i( is difficult to run at high speed on a rainy day. Statement II: Coefficient of friction /1 is decreased due to rain.

Newton's Law of Motion 7.99 22. Statement I: When static friction acts bctween two bodies. rest by thread BC. Now thread BC is burnt. Answer the follow- there is no loss of mechanical energy. ings: Statement II: When kinetic friction acts between two bodies, there is loss of mechanical energy. 1. Before burning the thread, what are the tensions in spring and thread BC. respectively? 23. Statement I: Greater is the rate of the change in the momentum vector, the grater is the force applied. Fig. 7.481 a. mIg, m2f? ~ b. in,g,ln,g-I1l,g ~ dp Statement II: Newton's second law is F = - . c. m,g, mig dt d. m,g,l1l,g+m,g 24. Statement I: Frictional heat generated by the moving ski 2. Just after burning the thread, what is, the tension in the is the chief factor which promotes sliding in skiing while spring? waxing the ski makes skiing more easy. Statement II: Due to friction energy dissipates in the form a. mu? b. rn,g of heat as a result it melts the snow below it. Wax is water repelient. c. 0 d. can't say 25. Statement I: The acceleration of a part.icle as seen from an inertial frame is zero. 3. Just after burning the thread, what is the acceleration of Statement II: No force is acting on the particle. m2? 26. Statement I: A reference frame attached to the earth is an inertial frame of reference. a. (m'I-ni,n')g Statement II: Newton's laws can be applied in this frame of reference. C.O 27. Statement I: If a body is trying to slip over a surface then friction acting on the body is necessarily equal to the limiting For Problems 4~ friction. Three blocks A, 13, and C have masses I kg, 2 kg, and 3 kg, Statement II: Static friction can be less than the limiting respectively are arranged as shown in Fig. 7.482. The pulleys friction force. P and Q are light and frictionless. All the blocks are resting 28. Statement I: Blocks A is moving on horizontal surface on a horizontal floor and the pulleys are held such that strings towards right under action of force F. All surfaces are remain just taut. At moment t = 0 a f{lfce F = 401 N starts smooth. At the instant shown the force exerted by block A acting on pulley P along vertically upward direction as shown on block B is equal to net ii)rce on block B. in Fig. 7.482 (take g = 10 m/s'). Statement II: From Newton's third law of motion, the force exerted by block A on 13 is equal in magnitude to force F=40t exerted by block B on A (Fig. 7.480). p Fig. 7.480 29. Statement I: A man standing in a lift which is moving upwards, will feel his weight to be greater than when the lift was at rest. Statement II: If the acceleration of the lift is a upward, then the man of mass m shall fecI his weight to be equal to normal reaction (N) exerted by the lift given by N = m (g + a) (where g is accyleration due to gravity). 30. Statement I: Greater is the mass, greater is the force required to change the state of body at rest or in uniform motion. Statement II: The rate of change of momentum is the measure of t.he force. Comprehensive Fig. 7.482 Type 4. Regarding the time when the blocks lose contact with In,For Problems 1-3 ground, which of the following is correct? a. A looses contact at 1 = 2 s In the system shown in Fig. 7.481, > rn,. System is held at

7.100 Physics for IlT-JEE: Mechanics I b. C looses contact at t = 1.5 s m,13. Find force exerted by on the incline. c. A and B loose the contact at the same time a, (m) + m2)g sine d. All three blocks loose the contact at the same time eb, (2m, + m2),J? sin 5. Find the velocity of A when B loses contact with ground. c. (m, + 2m2) g cose a. 5 mls b. 5/4 mls c. 4 mls d. 7/3 mls ed. (m) + m2)g cos 6. What is the magnitude of relative acceleration of A with For Problems 14-16 respect to B just after both have lost contact with ground? A ball of mass 200 gm is thrown with a speed 20 ms- I The ball strikes a bat and rebounds along the same line at a speed of a. 15 m/s2 b. 5 m/s2 40 ms~l. Variation in the interaction force. as long as the ball remains in contact with the bat, is as shown in Fig. 7.484. c. 20 mls2 d. IO m/s2 F For Problems 7-9 Fo ---------- A body hangs from a spring balance supported from the roof of an elevator. o 4ms 6ms 7. If the elevator has an upward acceleration of 2 m/s2 and Fig. 7.484 balance reads 240 N, what is the true weight of the body? a.20N b.200N 14. Maximum force Fa exerted by the bat on the ball is c. lOON d.300N 8. Under what acceleration will the balance read 160 N? a. 4,000 N b, 5,000 N a. 2 m/s2, up b. 2 m/s2, down c, 3,000 N d. 2,500 N c. 4 mls2 , up d, 4 m/s2, down 15. Average force exerted by the bat on the ball is 9. What will the balance read if the elevator cable breaks? a. 5,000 N b. 2.000 N d. 6,000 N a. 100 N b. 200 N c. 2.500 N c. zero d. 300 N 16. What is the speed of the ball at the instant the force acting For Problems 10-13 on it is maximum? m,Figure 7.483 shows a block of mass sliding on a block of a. 40 mls b. 30 mls mass m2, with In I > m2. Friction is absent everywhere. c, 20 mls d. 10 mls }<'or Problem 17-20 In the system shown in Fig. 7.485, mA = 4 m, mn = 3 m, and me = 8 m. Friction is absent everywhere, and the string isinex- tensible and light. If the system is released from rest, then find the following. Fig. 7.483 10. Find the acceleration of each block. c a. (m, + m2)g sin 9 (m) - m2) g sin e Fig, 7.485 (m, -m2) 17. The tension in the string is b. (m, + m,) e(m, - 1n2) sin a. 1.5 mg (m) - m2)gsine c. 4.7 mg +c. (m, 2m2) +d. 18. The acceleration of Block C is (2m, m2) a. 1/4 mis' 11. Find tension in the string. c. 5/4 m/s2 b. 5.8 mg 19. The acceleration of Block B is d. 3.2 mg +a. 2m]m2g sine e2mlm2 g sin a. 20 mls2 b. 217 mis' (m, 2m2) b. (2m, + m,) c, 15 mis' d. 113 m/s2 +c. 2m]fll2g sine mlm2 sin 9 b. 5 m/s2 d, 10 mis' (m, m2) d. (m) + m2) 12. Find force exerted by m, On 1n2. a. m,g tan e b. m)gcose c. m,g tan e ed. m2gcos

Newton's Law of Motion 7.101 20. The acceleration of Block A in horizontal direction.is 70 kg 35kg r~-;;-~le ~-l -B I P.=JOON a. 5/4 mis' b. 2112 m/s2 c. 11/7 rn/s2 d. 2715 m/s2 For Problems 21-23 Fig. 7.488 =Two blocks of mass m 10 kg and M = 20 kg are connected by 27. What is the tension in the string? a string passing over a pulley B as shown in Fig, 7.486. Another a. 700 b, -45N0 string connects the centre of pulley Ii and passes over another -N II pulley A as shown. An upward force F is applied at the centre of II pulley A. Both the pulleys are massless. When F is 600 N then 500 d. 900 21. Tension T in string over B connecting the blocks m and c. -N ---N 11 11 Mis F 28, What is the acceleration of Block Ii? A a, 30177 rn/s2 b. 60/77 m/s2 c. 80/77 m/s2 d. 120177 m/s2 29. What is the acceleration of Block C? a. 60/77 mis' b. 80177m/s2 c. 180177 m/s2 d. 60177 mlsl Fig. 7.486 For Problems 30-33 A monkey of mass m clings to a rope slung over a fixed pulley a. 25 N b. SON c.75N d. ISON (see Fig_ 7.489). The opposite end of the rope is tried to a weight of mass M lying on a horizontal plate. The coefficient of friction 22. The acceleration of mass m is b. 5 mis' between the weight and the plate is p... Find the acceleration of d. 0 mis' the weight and the tension of the rope for two cases. a. 2.5 mls2 c. 10 mis' 23. The acceleration of mass M is b. 1.5 m/s2 AI a. 20 m/s2 d. 5 mis' c. 0 m/s2 For Problems 24-26 lOON m ) Fig. 7.489 Fig. 7.487 Case 1 The monkey does not move with respect to the rope. Block Ii rests on a smooth surface (Fig. 7,487). Trthe coefficient of static friction between A and Ii is IL= 0,4, determine the 30. The acceleration of the weight is acceleration of each. When F = 30 N then (m - ILM) (m - ILM) g b, -- a. m+M g m-M 24. The acceleration of upper block is (M -Iun) d. -('2-:M-,----,1--1-m-')- g a. 3/2 m/s2 b. 6/7 mis' M+2m c. M+m g c. 4/3 mis' d. 3/7 mis' 31. The tension of rope is 25. The acceleration of lower block is mMg (3 + 1-') b. -2m-M-g-(2-+-IL-) a. 3/7 m/s2 b. 4/3 m/s2 a. M+m c. 6/7 mls2 d. 3/2 m/s2 2M+m d, mMg (l + IL) =26. When F 250 N, the acceleration of lower block is mMg (1 -IL) M+m c. M-m a. 4/5 m/s2 b. 3/2 mls2 Case 2 The monkey moves downwards with respect to the rope with c. 3/5 m/s2 d. 8/5 m/s2 an acceleration b. For Problems 27-29 32. The acceleration of the weight is Study the following diagram (Fig. 7.488) and answer Ihe following questions accordingly. Neglect all the friction and a. 2m (g + b) - I-'Mg the masses of the pulleys. M+2m

7.102 Physics for I1T-JEE: Mechanics I h. m(g+b)-I\"Mg 36. What is the mass of block C (approximately) if block Ii is 2(M+m) moving to the right and speeding up with an acceleration 2.00 mls2? meg +b) - 31\"Mg c. a. 20 kg b. 32 kg M+3m c. 25 kg d. 13 kg d. meg - b) -1,Mg 37. When block B has this acceleration. What is the tcnsion 72 (approximately)? M+m 33. The tension of the rope is Mmg(/l+ 1+b) a. 102 N h. 200N a. M-m c. 48 N d. 60 N 2Mmg (I' - 2 + b) b. 38. What is thc tension T, (approximately)? M+m Mmg(1\" + 1- b) c. a. 102 N h. 200 N M+m c. 48 N d. 60 N 2Mmg (I\" -- 1 + b) d. For Problems 39-40 M-m 39. As shown in Fig. 7.492 Block A weighs 60.0 N. The coef- For Problems 34--35 tident of static friction between the block and the surface In Fig. 7.490 the weight w is 60.0 N. on which it rests is 0.25. The weight w is ] 2.0 N and the system is in equilibrium. Find the friction force exerted on block A. 45.0 A Fig. 7.490 34. The tension in the diagonal string is Fig. 7.492 a. 60 N b.90N a. 5.7N b. 7.5N c. 9.0N d. 12.0N c. 85 N d. 100 N 40. Find the maximum weight w for which the system will 35. Find the magnitudes of the horizontal forces F, and F2 remain in equilibrium. that must be applied to hold the systcm in the position shown. a. 5N b.7N c. 15N d. 17N a. 75 N, 90 N, respectively For Problems 41-43 h. 60 N. 60 N. respectively Block A weights 4 N and block Ii weights 8 N (Fig. 7.493). The coefficient of kinetic friclion is 0.25 for all surfaces. Find c. 90 N, 90 N, respectively the force F to slide B at a constant speed when d. 45 N, 90 N, respectively OnA ~I For Problems 36-38 ~.~~A .. As shown in Fig. 7.491 Block A has a mass of 4.00 kg and block B has mass of 12.0 kg. The coefficicnt of kinetic friction s s S between block B and the horizontal surface is 0.25. (e) (a) (b) Fig. 7.493 41. A rests on B and moves with it. a.2N b. 3 N c. 4 N d. 5 N 42. A is held at rest. a. 2 N b. 3 N c. 4 N d. 5 N 43. A and B arc connected by a light cord passing over a smooth pulleys. Fig. 7.491 L2N ~3N ~4N ~5N

Newton's Law of Motion 7.103 For Problems 44-46 c. v'1664 N d. ../8'f5 N Block A of mass m and block B of mass 2m are placed on a fixed triangular wedge by means of a massless inextensible For Problems 49-50 string and a frictionless pulley as shown in Fig. 7.494. The Block A has a mass of 40 kg ancl block B has a mass of 15 wedge is inclined at 45'\" to the horizontal on both sides. The kg, and F of 500 N is applied parallel to smooth inclined plane coefficient of friction between block A and the wedge is 2/3 (see Fig. 7.496). The system is moving together. and that between block B and the wedge is 1/3. If the system of A and B is released from rest, find the following. B A B Fig. 7.496 A 49. The acceleration of the system is 2m 'II a. l4T5 m/s2 b. -23 ill1S2 II Fig. 7.494 c. -13 mls2 d. _8 m/s2 7 44. The acceleration of A is 3 a. Ii b. zero 50. The least coefficient of friction between A and B is 3v'2 g a. 5v'2 b9.v-'3 g 53 c. d. 2/3 12 ../7 c. 9v'2 5v'3 28 d. 18 45. The tension in the string is 3 5 For Problems 51-52 a. -~m'ft b. 3v'2 mg A 10 kg block rests on a 5 kg bracket as shown in the Fig. 7.497. 2.Jimg The 5 kg bracket rests on a horizontal frictionless surface. The c. d. mg coefficients of friction between the 10 kg block and the bracket 3 on which it rests are IL.,. = 0.40 and fl., = 0.30. 3 46. The magnitude and direction of the force of friction acting 5 kg: 1',' 0.4 on A is I .-\"'-+F ,Uk 0.3 a. mg, clown the plane b. 2mg' up the plane mg Fig. 7.497 c. v'2' up the plane mg 51. The maximum force F that can be applied if the 10 kg d. M' down the plane block is not to slide on the bracket is 3'12 a. 32 N h. 24 N For Problems 47-48 A block of mass 10 kg is kept on a rough floor. Coefficients c. 18 N d. 48 N of friction between floor and block are fl.., = 0.4 and fl.k = 0.3. 52. If 10 kg block does not slide on the bracket, the COlTe- Forces F, = 5 Nand F2 = 4 N arc applied on the block as sponding acceleration of the 5 kg bracket is shown in Fig. 7.495. a. 1.6 mls2 b. O.S m/s2 c. 1.2 m/s2 d. 2.4 m/s2 Fig. 7.495 For Problems 53-56 Block B rests on a smooth surface (Fig. 7.498). !fthe coefficient 47. The magnitude of friction force is of static friction between A and B is J-1= 0.4, determine the acceleration of each, When F = 30N a. .j3] N b.~N c. ffi N d.~N 1',48. If P, = 5 Nand = a N, for what maximum value of a the motion of block impends? a. v'1575 N b. v'1225 N Fig. 7.498

7.104 Physics for IIT-JEE: Mechanics I 53. The acceleration of upper the block is c. d. \" a. 3/7 mis' b. 4/3 mis' ,,\" Block PlanK c. 6/7 mis' d. 312 mis' ,I Plank ,! Blot;k 54. The acceleration of the lower block is 12 s 12 s a. 312 mis' b. 6/7 mis' 59. The average acceleration of the plank in the time interval c. 4/3 mis' rl. 3/7 mis' oto 15 s in figure will be a. 0.20 mls2 b. 0.30 mls 55. When F = 2.50 N the acceleration of the upper block is c. 0040 mis' rl. 0.60 mis' a. 15 mis' b. ]() mis' For Problems 60-62 c. 25 mis' rl. 21 mis' Three blocks A, /3 and C of mass 3M, 2M, and M, respectively, arc suspended vertically with the help of springs PQ and TU 56. The acceleration of the lower block is and a string RS as shown in Fig. 7.500. If acceleration of blocks A, Band Care a1, a2 and a3, respectively. a. 8/5 mis' b. 3/5 mis' c. 312 mis' d. 4/5 mis' For Problem 57-59 A sufficiently long plank of mass 4 kg is placed on a smooth horizontal surface. A small block of mass 2 kg is placed over the plank and is being acted upon by a time varyi!1g horizontal force F = (0.5 t), where F is in Nand t is in second as shown in Fig. 7.499(a). The coefficient offriction between the plank and the block is given as /1., = fkk = fk. At time t = 12 s, the relative slipping between the plank and the block is just likely to occur. Fig. 7.500 60. The value of acceleration a3 at the moment spring PQ is cut is a. g downward Ca) b. g upward c. more than g downward b/~: J'kg~;H~n ;~7Tk ~t2l d. zero Cb) 61. The value of acceleration a 1 at the moment string RS is cutis Fig. 7.499 a. g downward If the force F acting on 2 kg block is removed and the system h. g upward c. more than g downward (plank + block) is given horizontal velocity Vo, as shown in Fig. 7.499, this system strikes a massless spring of spring d. zero constant k = 120 N/m fixed at the end of the horizontal place 62. The value of acceleration Q2 at the moment spring TU is and it is observed that during compression of the spring no relative slipping occurs between the plank and the block. cut is 57. The coefficient of friction fk is equal to a. gl5 upward b. g/5 downward c. gl3 upward d. zero a.O.lO b. 0.15 e. 0.20 d. 0.30 For Problems 63-65 In Fig. 7.501 all the pulleys and strings are massless and all the 58. The acceleration (a) versus time (t) graph for the plank surfaces are frictionless. Small block of mass In is placed on and the block as shown is correctly represented in fixed wedge (take g = 10 ms-,2). a. \" Block a Block 63. The tension in the string attached to m is ~ j Plank I Pklilk b. a. 40N b. ION c. 20N d.5N ~~ 12 s 12 s 64. The acceleration of In is a. 4.5 m/52 down the incline

Newton's Law of Motion 7.105 the plank and the cube is It. The size of cube is very small in comparison to the plank (Fig. 7.503). F--\"80N m'~ IOkg Fig. 7.503 69. At what force F applied to the plank in the horizontal 30\" () direction will the cube begin to slide towards the other Fig. 7.501 end of the plank? b. 4.5 m/s2 up the incline a. F > /L(m + M)g c. 5 m/s2 down lhe incline d. 5 mls2 up the incline b. F >0.5{4(m + M)g 65. The acceleration of pulley P4 is e. F = O.5/L(m + M)g a. 2.25 mls2 towards left d. F = /L(m + M)g b. 2.25 m/s2 towards right c. 9 m/s2 towards left 70. In what time will the cube fall from the plank if the length d. 9 m/s2 towards right of the latter is 1 . For Problems 66-68 In Fig. 7.502 both the pulleys and the string are massiess and a. j F-p..g~~+m) all the surfaces are frictionless. / 2Ml Fig. 7.502 Yb. F _ /Lg(M +m) Given ml = 1 kg, m2 = 2 kg, m3 = 3 kg. b 240 N .7 e. / F + /Lg~~ + m) 66. The tension in the string is d. None of these I 2Ml 120 30 . a. - N b. - m/s2 d. VF + /Lg(M +m) 7 7 For Problems 71-74 130 d. None of these In the arrangement shown in Fig. 7.504, all pulleys are smooth c. - N and massless. When the system is released from the rest, accel- 7 eration of blocks 2 and 3 relative to 1 are Im/s2 downwards and 67. The acceleration of m I is 5m/s2 downwards. Acceleration of block 3 relative to 4 is O. a. 4_0m/s2 4 7 Fig. 7.504 e. 720 m/s2 71. Thc absolute acceleration of block 1 is 68. The acceleration of m3 is a. 2 mls2 upward b. 1 mls2 downward a. 740 m/s2 30 2 e. 3 mis' upward b. -7 m/s d. 1.5 m/s2 downward 20 d. None of these 72. The absolute acceleration of block 2 is e. - m/s2 a. 2 m/s2 downward b. 1 m/s2 upward 7 For Problems 69-70 A plank A of mass M rests on a smooth horizontal surface over which it can move without friction. A cube B of mass m lies on the plank at one edge. The coefficient of friction between

7.106 Physics for IIT-JEE: Mechanics I 78. The speed of the ball is c. 3 m/s2 upward a. ,j2gI' tan 8 b. ,jgrtan8 d. 1.5 m/s2 downward c. ,jgrcot8 d. ,jgr cose 73. The absolute acceleration of block 3 is For Problems 79-81 a; 2 m/s2 upward A small block of mass m is placed over a long plank of mass M. b. 1 mis' downward c. 3 m/s2 downward Coefficient of friction between them is f.L. Ground is smooth. d. 1.5 mis' upward At t = 0, m is given a velocity VI and M a velocity V2 (> vd as shown in Fig. 7.506. After this M is maintained at constant 74. The absolute acceleration of block 4 is a. 2 m/s2 upward acceleration a « ttg). b. 1 mis' downward c. 3 m/s2 downward Fig. 7.506 d. 1.5 mis' upward For Problems 75-76 Initially, there will be some relative motion between block and the plank. but after some time relative motion will cease and ',e velocities of both will become same. 79. Find the time to when velocities of both block and plank , become same. +h. V2 v! ,- - - - - -1- ____ _ a. V2 - v! fJ,.g -a oe------Jt fLg +a d. +V2 Vj V2 - Vj {Lg +a c. fLg - a Fig. 7.505 80. The -variation of velocity of block as a function of time is A sphere of mass 500 gm is attached to a string of length shown as .Ji m, whose other end is fixed to a ceiling. The sphere is made to a. describe a circle of radius 1 m in a horizontal plane (Fig. 7.505). 75. The period of revolution for the sphere is a. Jf vfJO s b. Jf co/S S c. C. 2Jf/ vfJO s d. Jf/ co/S S 76. The tension in the string is b. 10../2 N d. 10,;3 N a. 5../2 N c. 5,;3 N For Problems 77-78 81. The forward force acting on the plank before and after to A ball of mass m is suspended from a rope of length L. It respectively is a. Ma,(M+m)a describes a horizontal circle of radius r with speed v. The rope b. fLmg + Ma, (M + m)a emakes an angle with vertical. c. fLMg + ma, Ma d. (M + lIl)a, lung + Ma 77. The tension in the rope is For Problems 82--84 a. (mg)2+ (m- v')' Two blocks of masses mj and ln2 are connected with-a light 2r spring of force constant k and the whole system is kept on a frictionless horizontal surface. The masses are applied with (mv')'b. (mg)' - -}-. forces FJ and F, as shown in Fig. 7.507. At any time, the blocks c. (mg)' - (l-nV2)' have same acceleration ao but -in opposite directions. 2r Fig. 7.507 d. (mg)2+ (n-t,-V.')'

Newton's Law of Motion 7.107 Now answer the following questions. 87. For FI ~ ISDN and F, ~ lOON. the direction and magni- 82. The value of ao is tude of friction force acting on block are a. F1 - F2 b. F,- F2 (~)a. 90 N, making an angle of tan-I with the hori- ntj +m2 mj -m2 zontal in upward direction F, +F2 c. d. F, +Fi (~)b. 75 N, making an angle of tan ., with the hori- m1 -In2 nIl +111.2 83. The value of spring force is zontal in upward direction a. mlF2+ Fl m 2 b. m,F2 - Fj 1n 2 c. 107.7N.makingananglcoftan-· 1 (~)withthehor­ mj-ml fflj +ml izontal in upward direction +c. mjF1 Pj nt 2 d. zero d. mlF2 - Fjm2 fill +ml mj -In2 84. If FI is removed at this moment, then just after this accel- 88. For data of question no. 85, find the magnitude of accel- erat.ion of 1n2 is eration of block. a. zero F, b. ao+F-I b. 22.5 m/s2 a. - -ao nl2 c. 26.925 mis' d. 8.175 m/s2 m2 c. -F2 -an d. ao+F-i For Problems 89-93 ml nl2 A system of two blocks and a light string arc kept on two inclined For Problems 85-88 faces (rough) as shown in Fig. 7.509. All the required data arc A block of mass 4 kg is pressed against a rough wall by mentioned in the diagram. Pulley is light and frictionless. (take two perpendicular horizontal forces FJ and P2 as shown in g ~ 10 m/s2, sin 37\" = 3 -). 5 Fig. 7.508 below. The cocfficient of static friction bctween the block and floor is 0.6 and that of kinetic li'iction is 0.5. 5 kg JI,~ '-: 0.1 .\"''--->I'k 0.075 Fig. 7.509 Fig. 7.508 89. If the system is released from rest, then the acceleration of the system is 85. For FI ~ 300 Nand h ~ 100 N, the direction and magni- 7 b. zero a. - m/s2 tude of friction force acting on the block are IS c. 1~5 2 d. 12~5 , m/s m/s- a. 180 N, vertically upwards. b. 40 N, vertically upwards. 90. A system is initially moving in such a way that block of 10 kg is coming down the incline with a speed of 2 m/s. (~)c. 107.7 N making an angle oftan-' withthehor- Then how much time does the system take to come to a izontal in upward direction. rstop? Assume the length of incline to be large enough. J (~)d. 91.6 N making an angle of tan-I with the hori- a. 13.33 s zontal in upward direct.ion. b. 80 s 86. For the data given in question 8'5, the accelcration of block c. Infinitc is d. Question is irrelevant 3. zero 140 91. In the above question the motion of system would be best b. --;;- m/s2. upward described by one of the following. 180 a. The system first decelerates, comes to a stop and then c. --;;- m/s2, upward tocontinues move in the opposite direction, d. -107-.7 m/s2 at an angle of tan - J. (2- ) with the hori- b. The system will continuously move with constant 45 speed. zontal in upward direction c. The system first decelerates and then comes to a stop. d. The system accelerates and its speed increases with time.

7.108 Physics for IIT-JEE: Mechanics I 92. If the system is released from rest. the tension in the string 97. The coefficient of static friction between the block and the would be floor is a. 40:5 T :5 43 a. 0045 b.0:5 b.40:5t:560 c. 0.3 d, 1.45 c. 36:5 T :5 60 98. Which set of the readings of Experiment II is absolutely d. Cannot be determined wrong? a, b. 2 . 93. For above situation, the friction force between 10 kg block c. 3 d. None of these and the incline can be a, 24N b. ISN c. 21 N d. 15N For Problems 94-96 99. The speed of the block after 3 s (beginning from the start- A system of two blocks is placed on a rough horizontal surface ing of application of force) in set 2 for I1,d experiment as shown in Fig. 7.510. The coefficients of static and kinetic is friction at two surfaces are shown, A force F is horizontally a. 6 mls applied on the upper block. Let fl' f, represent the frictional forces between upper and h. 2 mls lower surtaces of contact, respectively, and aI, ([2 represent the acceleration of 3 kg and 2 kg block, respectivcly. c. 3 mls d. Information is insufficient ,us --~(l.,5 \" u.,.'·'0•3. For Problems 102-100 Two smooth blocks are placed at a smooth comer as shown in p, ~ 0.2. I'., -, 0.1 Fig. 7.511. Both the blocks ate having mass m. We apply a force F on the small block m. Block A presses the block B in the nor- Fig. 7.510 mal direction, due to which the pressing force on vertical wall will increase, and the pressing force on the horizontal wall de- . 94, If F is a gradually increasing force then which of the creases, as we increases F (8 = 3]0 with horizontal). As soon following statement(s) would be true? as the pressing force by block B on the horizontal wall becomes zero, it will lose the contact with the ground. If the value of F a, For a particular value of F ( < Fo) there is no motion further increases, the block B will accelerate in upward direction at any of the contact surface. and simultaneously the block A will move toward right. b. The value of Fo is 10 N. y c. As F increases beyond Fo, It increases and continues Ii to increase until it acquires its limiting value. d. All of the above 95. For F = 12 N, mark the COlTect option. smooth a. fl =7.S N, f, =7.8 N, GI = lA mis', G2 =0 m/s2 x h. fl =7.8 N, 12 =10 N, a, =G2 = 1.4 mis' Fig. 7.511 c. II =7.8N,f,=5N,GI =a,= lAmls2 Q,d. f1 =7.4 N, f, =5 N, G1 = = 1.2 m/s2 96. For relative motion to be there between two blocks, the 100. The minimum value of F, to lift block B from the ground minimum value of F should be is a.15N h,30N c.25N d. 32N a. -25mg b. 5 -mg 12 4 For Problems 97-99 3 A student performs two experiments to determine the coefficient c, -mg 4 4 of static and kinetic hietion between a block of mass 100 kg d. -mg and the horizontal floor. 3 r t Experiment: He applies a gradual increasing force on the 101. If both the blocks are stationary, the force exerted by block and is just able to slide the block when force is 450 N. ground on block A is II,d Experiment: He applies constant force of different magnitudes for the duration of 2 s and determine the distance a. mg +\"3'4F travelled by the block in this duration. h. mg -\"3'4F Set Force Distance 1. 300 N 0.5 m 4F 2. 600 N 2.0 m c. mg+- 3. 750 N 3.0 m Assume all the forces have been appplied horizontally. 3 d. mg- 34F\"

Newton's Law of Motion 7.109 102. !fthe acceleration of block A is from left to right, then the acceleration of block B will be a. 43a upwards 4a F---+ B c, S3a upwards b. - upwards 5 kg 3 Fig. 7.514 d. 4Sa upwards For Problems 103-104 106. If F = 50 N, the friction force acting between blo,k Band Two containers of sand arc arranged like the block as ,shown ground will be in Fig. 7.512. The containers alone have negligible mass; the a. ION b.20N sand in these containers has a total mass Mtot ; the sand in the d. None hanging container H has mass m. c. 30 N Massless 107. The force of friction acting on B varies with the applied r;:-:701-----~~ pulley force F according to curve Frictionicss Hanging surface block H Fig. 7.512 a. IL..----->r b. To measure the magnitude a of the acceleration of the system, a large number of experiments have been carried out where m varies from experiment to experiment but Mfat does not.; that is, sand is shifted between the containers before each trial. e', Fig. 7.513 T 103. Which of the curves in the graph correctly gives the ac· .--_--------,r------- __ , celeration magnitude as a function of the ratio m / M tot (vertical axis is for accelera~ion)? In -- .. --- __ ~_--- LI h2 ~3 ~4 0) . . Fig.7.51S 104. Which of the curves gives the tension in the connecting A string of length I is fixed at one end and can'ies a' mass m at cord (the vertical axis is for tension)? the other end. The string makes 2/n: rps around a vertical axis through the fixed end so that the mass moves in the horizontal LI h2 ~3 ~4 circle (Fig. 7.515). 108. What is the tension in string? For Problems 105-107 a. ml b. 16 ml Two bodies A and B of masses 10 kg and 5 kg are placed very c.4ml d. 2ml slightly separated as shown in Fig. 7.514. The coefficients of friction between the floor and the blocks arc as fl., = 0.4. Block A 109. What is the angle of inclination ofthe string with vertical? is pushed by an ex,ernal force F. The value of F can be changed. When the welding between block A and ground breaks, block Uf)a. cos 1 b. COS-I C~J A will start pressing block B and when the welding of B also breaks, block B will start pressing the vertical wall. (:Jc. COS-I d. COS-I (~) lOS. If F = 20 N, with how mueh force does block A press the 110. What is the linear speed of the mass? block B? a. ION b.20N a. 41 [I + ( 1161<)2JI c. 30N d. zero