mechanics practice papers 2

8/17/2019 Mechanics Practice Papers 2

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ANDHERI / BORIVALI / DADAR / CHEMBUR / THANE / MULUND/ NERUL / POWAI

IIT – JEE: 2016 CRASH COURSE ASSINGNMENT-2 DATE: 17/11/2015

TIME: 2 Hrs. TOPIC: CENTRE OF MASS, COLLISION & MOMENTUM,

ROTATION DYNAMICS

SECTION–I (Single Choice Questions)

This section contains 20 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for

its answer, out which ONLY ONE is correct.

1. A car speeds up with constant magnitude of tangential acceleration in a circular path moving inanticlockwise direction. Which of the following figure represents acceleration of the car.

(A) (B)

(C) (D)

2. A skier plans to ski a smooth fixed hemisphere of radius R. He starts from rest from a curved smoothsurface of height (R/4). The angle at which he leaves the hemisphere is

(A) 1cos 2 / 3 (B) 1cos 5 / 3 (C) 1cos 5 / 6 (D) 1cos 5 / 2 3

3. A small bead of mass m can moves on a smooth circular wire (radius R) under

the action of a force2

KmF

r

directed (r = position of bead from P & K =

constant) towards a point P with in the circle at a distanceR

2 from the centre.

What should be the minimum velocity of bead at the point of the wire nearestthe centre of force (P) so that bead will complete the circle

(A)3K

R (B)

8K

3R (C)

6K

R (D) none of these


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4. A particle of mass m is projected at an angle of o60 with a velocity of 20 m/s relative to the groundfrom a plank of same mass m which is placed on smooth surface. Initially plank was at rest. The

minimum length of the plank for which the ball will fall on the plank itself is (g = 10 m/ 2s )

(A) 40 3 m (B) 20 3 m (C) 10 3 m (D) 60 3 m

5. Five identical balls each of mass m and radius r are strung like beads at random and rest along asmooth, rigid horizontal thin rod of length L, mounted between immovable supports. Assume 10r <L and that the collision between balls or between balls and supports are elastic. If one ball is struckhorizontally so as to acquire a speed v, the average force felt by the support is

(A)

2

5 mvL 5r

(B)

2

mvL 10r

(C)

2

5mvL 10r

(D)

2

mvL 5r

6. A block of mass M is tied to one end of a massless rope. The other end of the rope is in the hands ofa man of mass 2M as shown in the figure. The block and the man are resting on a rough plank ofmass M as shown in the figure. The whole system is resting on a smooth horizontal surface. The man

pulls the rope. Pulley is massless and frictionless. What is the displacement of the plank when the block meets the pulley. (Man does not leave his position on plank during the pull)

(A) 0.5 m (B) 1 m (C) zero (D) 2/3 m

7. Figure shows a square lamina with a disc of radiusL

2removed from it which is now placed

symmetrically over upper right quarter. What is location of centre of mass of system relative toorigin shown in figure.

(A) L ˆ î j12

(B) L ˆ î j

8

(C) L ˆ î j

4

(D) L ˆ î j

16


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8. In the figure shown the cart of mass 6 m is initially at rest. A particle of mass m is attached to the endof the light rod which can rotate freely about A. If the rod is released from rest in a horizontal

position shown, determine the velocity relv of the particle with respect to the cart when the rod is

vertical. Assume frictionless surface.

(A)7

g3

(B)7

g6

(C)14

g3

(D)8

g3

9. Suppose a golf ball is hurled at a heavy bowling ball initially at rest and bounces elastically from the bowling ball. After the collision.(A) the golf ball has the greater momentum and the greater kinetic energy(B) the bowling ball has the greater momentum and the greater kinetic energy(C) the golf ball has the greater momentum but has the smaller kinetic energy(D) the bowling ball has the greater momentum but has the smaller kinetic energy

10. A mass ‘m’ moves with a velocity ‘v’ and collides inelastically with another identical mass at rest.

After collision the 1st mass moves with velocity

3

in a direction perpendicular to the initial

direction of motion. Find the speed of the 2nd mass after collision.

(A)2

3

(B)3

(C) (D) the situation of the problem is not possible without external impulse.

11. When you pull a tablecloth out from under a set of dishes, it’s important to pull the cloth as fast as possible because(A) the work done on the dishes by the cloth is proportional to the time during which the cloth pulls

on them.(B) the weight of the dishes on the cloth is proportional to the time during which the cloth is

moving.(C) the force of sliding friction that the cloth exerts on the dishes is proportional to the time during

which the cloth is moving.(D) the momentum transferred to the dishes is proportional to the time during which the cloth pulls

on them.

12. Two identical carts constrained to move on a straight line, on which sit two twins of same mass, aremoving with same velocity. At some time snow begins to drop uniformly vertically downward. Ram,sitting on one of the trolleys, throws off the falling snow sideways and in the second cart shyam isasleep. (Assume that friction is absent)(A) Cart carrying Ram will speed up while cart carrying shyam will slow down(B) Cart carrying Ram will remain at the same speed while cart carrying shyam will slow down(C) Cart carrying Ram will speed up while cart carrying shyam will remain at the same speed(D) Cart carrying Ram as well as shyam will slow down


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13. A particle of mass 15 kg has an initial velocityi

ˆ ˆi 2i

m/s. It collides with another body and the

impact time is 0.1s, resulting in a velocity f ˆ ˆ ˆ6i 4 j 5k m / s

after impact. The average force of

impact on the particle is

(A) ˆ ˆ ˆ150 5i 6j 5k (B) ˆ ˆ ˆ15 5i 6 j 5k (C) ˆ ˆ ˆ150 5i 6j 5k (D) ˆ ˆ ˆ15 5i 6j 5k

14. Two particles are inter connected by an ideal spring (see figure). The spring iscompressed and system is projected in air under gravity. If the acceleration of

1m is a

find acceleration of 2m .

(A) g

(B) g a

(C) 1 1

2

m g m a

m

(D) 1 2 1

2

m g m g m a

m

15. A particle of mass m is moving along the x-axis with speed v when it collides with a particle ofmass2 m initially at rest. After the collisions, the first particle has come to rest, and the second

particle has split into two equal-mass pieces that move with equal velocities at equal angles 0 with the x-axis, as shown in the figure. Which of the following statements correctly describes the

speeds of the two pieces?

(A) Each piece moves with speed v

(B) One of the pieces moves with speed v , the other moves with speed less than v (C) Each piece moves with speed / 2 (D) Each piece moves with speed greater than / 2

16. The ring shown in figure has a mass m and radius r. A point mass m is attached at point B. System inreleased from rest find normal reaction between ring and horizontal surface at the instant ring is

released from rest. Given that is no slipping.

(A) mg (B)3mg

4 (C)

7mg

4 (D) 2mg

17. Choose the correct statements:(i) The resultant torque on a rigid body is zero but the resultant force is not zero. Then only the

translational kinetic energy of body changes(ii) The resultant torque on a rigid body is zero but the resultant force is not zero. Then only the

rotational kinetic energy of body changes(iii) The resultant force acting on a rigid body is zero but the resultant torque is not zero. Then only

the rotational kinetic energy of body changes(iv) The resultant force acting on a rigid body is zero but the resultant torque is not zero. Then only

the translational kinetic energy of body changes(A) (i) & (ii) are correct (B) (i) &(iii) are correct(C) (ii) & (iv) are correct (D) none of these


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Paragraph for Question Nos. 18 to 20

A uniform board of mass m is placed on two rollers of negligible masses as shown in figure. The board can move on the rollers without slipping and rollers roll on horizontal ground without slipping.Initially board makes an angle with horizontal and distance between centers of wheels is 10R. Thesystem is now released from rest. Answer the following question based on this information:

18. Motion of board as seen from ground reference frame will be:(A) Pure translation (B) Pure rotation(C) Rotation with translation (D) None of these

19. Centre of mass of the board will move at an angle with horizontal where is equal to:

(A) zero (B)2

(C) (D) varies with time

20. After the release, acceleration of centre of mass of the board will be:

(A) zero (B) gsin (C) gsin2

(D) Variable

SECTION-II (Multiple Choice Questions)This section contains 13 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for

its answer, out which ONE OR MORE is/are correct.

21. A ball of mass 1 kg bounces against the smooth ground as shown in the figure.

The velocity just before collision is 25 m/s and the velocity after just after hitting the ground is 15 2

m/s. Select the correct alternative(s)(A) Magnitude of Impulse = 5 Ns (B) Magnitude of Impulse = 35 Ns(C) Coefficient of restitution, e = 0.75 (D) Coefficient of restitution, e = 0.25

22. Two blocks A and B of the same mass are connected to a light spring and placed on a smooth

horizontal surface. B is given velocity 0v (as shown in the figure) when the spring is in naturallength. In the subsequent motion,

(A) the maximum velocity of B will be 0v

(B) as seen from ground, A can move towards right only

(C) the spring will have maximum extension when A and B both stop(D) the spring will be at natural length again when B is at rest


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23. Two identical blocks of mass 2M are joined by means of a light spring of spring constant k. A manof mass M is standing on one of the block as shown in the diagram. If man jumps horizontally with avelocity V relative to block and horizontal surface is smooth, then

(A) the maximum compression in the spring is2M V

k 3

(B) man lands at horizontal distance2h

Vg

from initial position of the block

(C) right block loses contact with wall when the elongation in spring is maximum

(D) velocity of centre of mass of two blocks after 2M loses contact with wall isV

6

24. Consider a block of mass 10 kg which rests on a smooth surface and is subjected to a horizontalforce of 6 N for 4s. If observer A is in a fixed frame x and B is in an inertial reference frame moving

with velocity 2 m/s as shown in figure. Mark the correct option(s).

(A) The final speed of the block is 7.4 m/s, if it has initial speed of 5 m/s measured from fixedframe.

(B) Same momentum change of block will be observed by observer B in 4s.(C) Principle of impulse and momentum is valid for observers in any inertial reference frame.(D) Momentum of a body is reference frame dependent.

25. Illustrated below is a uniform cubical block of mass M and side a. Mark the correct statement(s).

(A) The moment of inertia about axis A, passing through the centre of mass is 2A1

I Ma6

(B) The moment of inertia about axis B, which bisects one of the cube faces is 2B5

I Ma12

(C) The moment of inertia about axis C, along one of the cube edges is 2C2

I ma3

(D) The moment of inertia about axis D, which bisects one of the horizontal cube faces is27

Ma12


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26. The uniform solid cylinder rolls without slipping in the system shown. If the maximum compressionin spring is 15cm, the possible friction force acting on the cylinder during its motion is:

(A) 4 N (B) 10N (C) 12N (D) 15N

27. Two equal uniform rods P and Q move with the same velocity v as shown in the figure. The secondrod has an angular velocity (< 6v/l) (clockwise) about G' in addition to v.

(A) If the ends A and A ' are suddenly fixed simultaneously both rods will rotate with the sameangular velocity

(B) If the ends A and A ' are suddenly fixed simultaneously, the rod Q will rotate with greaterangular velocity

(C) If the ends B and B ' are suddenly fixed simultaneously both rods will rotate with the sameangular velocity

(D) If the ends B and B ' are suddenly fixed simultaneously, the rod P will rotate with greaterangular velocity

28. A disc is given an angular speed 0 and released from a certain height (asshown in figure). Motion of disc is observed after collision with the roughsurface. Velocity of centre of mass of ball and direction of is shown infigure after the collision. Mark possible path(s), disc CANNOT follow after

the collision.

(A) (B)

(C) (D)

29. A thin bar of mass M and length L is free to rotate about a fixed horizontal axis through a point at itsend. The bar is brought to a horizontal position and then released. The angular velocity when itreaches the lowest point is(A) directly proportional to its length and inversely proportional to its mass(B) independent of mass and inversely proportional to the square root of its length(C) dependent only upon the acceleration due to gravity and the length of the bar(D) directly proportional to its length and inversely proportional to the acceleration due to gravity


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30. A disc of mass m and radius r is gently placed on another disc of mass 2m & radius

r. The disc of mass 2m is rotating with angular velocity 0 initially. The disc is placed such that axis of both are coincident. The coefficient of friction is forsurfaces of contact. Assume that pressure on disc is uniformly distributed. Find thecorrect statement.

(A) Loss in kinetic energy of system 2 201

K mr 3

.

(B) Loss in kinetic energy of system

2 2

0

1

K mr 6 .

(C) The common angular velocity is 02

.3

(D) The common angular velocity is 04

.3

31. Three particles, each of mass m are placed at the points 1 1 1x , y , z , 2 2 2x , y , z and 3 3 3x , y , z on

the inner surface of a paraboloid of revolution obtained by rotating the parabola,2

x 4ay about they-axis. Neglect the mass of the paraboloid. (y-axis is along the vertical)

(A) The moment of inertia of the system about the axis of the paraboloid is 1 2 3I 4ma y y y .

(B) If potential energy at O is taken to be zero, the potential energy of the system is

1 2 3mg y y y .

(C) If the particle at 1 1 1x , y , z slides down the smooth surface, its speed at O is 12 gy

(D) If the paraboloid spins about OY with an angular speed , the kinetic energy of the system will be 2ma 21 2 3y y y .

32. In the figure the block on the smooth table is set into motion in acircular orbit of radius r around the centre hole. The hanging mass isidentical to the mass on the table and remains in equilibrium.

Neglect friction. The string connecting the two blocks is masslessand inextensible. Select the correct answer.

(A) The angular speed of the block in its circular motion isg

r

(B) Kinetic energy as function of r is given bymgr

2

(C) Angular momentum about the hole is conserved even if hanging block is pulled down.(D) The block on table is in equilibrium.


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33. A bobbin of mass m and moment of inertia I relative to its own axis is being pulled along ahorizontal surface by the light string tightly wrapped as shown in figure. There is no slipping on thesurface throughout the motion.

(A) The angular velocity of bobbin when string is pulled horizontally with velocity V isV

3r

(B) The angular velocity of bobbin when string is pulled horizontally with velocity V isV

r

(C) If string is pulled by horizontal acceleration a. Then tension in string is2

a IT 4m

9 r

(D) If string is pulled by horizontal acceleration a. Then tension in string is2

I aT 2m

r 2

SECTION III: (INTEGER CORRECT TYPE)This section contains 09 Questions. The answer to each question is a Single digit integer, ranging from 0 to

9 (both inclusive)

34. One side of a spring of initial, unstretched length 0 1ml , lying on a frictionless table, is fixed, theother one is fastened to a small puck of mass m = 0.1 kg. The puck is given velocity in a direction

perpendicular to the spring, at an initial speed 0 11v m/s. In the course of the motion, the maximum

elongation of the spring is 0 /10l l . If k is the force constant of the spring (in SI units), find k/30?

35. Consider an “L-shaped rod” of uniform mass density, hinged at point ‘O’ is held at rest initially. Themass of the rod is ‘m = 1 kg’. The earth’s gravitational field is assumed to be uniform and directed

downwards. Assume no friction at hinges. The rod is now released. Find the maximum magnitude of

angular speed of the rod (in rad/s). (Take : L 1m, 10 19 / 6 )

36. A spool, initially at rest, is kept on a frictionless incline making an angle o37 with the horizontal.The mass of the spool is 2kg and it is pulled by a string shown with a force of T = 10N. The stringconnecting the spool and the pulley is initially horizontal. Find the initial acceleration of the spool on

the incline. Express your answer in 2m/s .


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37. A 200 kg beam 2.0 m in length slides broadside down the ice with a speed of 16 m/s (figure). A 50

kg small Fevicol Can at rest sticks at one end as it goes past, both go spinning down the ice. Assumefrictionless motion. What is the angular velocity of combined mass in rad/s.

38. A uniform rod of length 1 m is free to move and rotate in gravity-free space. When an impulse isgiven to one end of the rod, perpendicular to its length, its centre of mass moves with velocity v = 1m/s. What will be its angular velocity (in rad/s) about its centre of mass.

39. A uniform rod AB of length 4m and mass 12 kg is thrown such that just after the projection thecentre of mass of the rod moves vertically upwards with a velocity 10 m/s and at the same time it is

rotating with an angular velocity2

rad/sec about a horizontal axis passing through its mid point.

Just after the rod is thrown it is horizontal and is as shown in the figure. Find the acceleration (in m/2sec ) of the point A in m/ 2s when the centre of mass is at the highest point.

2 2Take g 10m / s and 10

40. A small ball is projected from point P on floor towards a wall as shown.

It hits the wall when its velocity is horizontal. Ball reaches point P after one bounce on the floor. Ifthe coefficient of restitution is the same for the two collisions, find the value of its reciprocal.[All surfaces are smooth]

41. A massless spring of force constant 1000 N/m is compressed a distance of 20cm between discs of8kg and 2kg, spring is not attached to discs. The system is given an initial velocity 3m/s

perpendicular to length of spring as shown in figure. What is ground frame velocity of 2kg block (inm/s) when spring regains its natural length.


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42. A small ball of mass 2g is attached to a string of length whose other end is fastened to an uprightvertical rod fixed on a wooden board resting on a horizontal table. The combined mass of the boardand rod is 1kg. The friction coefficient between the board and the table is . The ball is releasedfrom rest with the string in a horizontal position. While the ball swings, the board does not move.

What is minimum value that must have to prevent the board from moving to the left while the ball

swings down? Using necessary approximation, express your answer as 310 x and find x .

SECTION-IV (Matrix-Match Type Question)

43. A rigid cylinder is kept on a smooth horizontal surface as shown. If Column I indicates velocities of

various points on it shown, choose correct state of motion from column II. (Here point 3-centre ofcylinder, 2-top point, 4- bottom point, 1- on the level of 3)

Column I Column II

(A)1 2

ˆ ˆ ˆ, 2 v i j v i (P) Pure rotation about centre

(B)1 3

ˆ ˆ ˆ, v i j v i (Q) Pure rolling to left

(C)2 3

ˆ , 0 v i v (R) Pure rolling to right

(D)4 1

ˆ ˆ0 , v v i j (S) Not possible

44. A particle P strikes the uniform rod R normally, elastically at a distance x from hinge as shown. Therod of length l is suspended vertically with upper end hinged.

Column I Column II

(A)If

2

l x , elastic collision

(P) Linear momentum of (P + R) systemincreases

(B) If , x l elastic collision (Q) Linear momentum of (P + R) systemdecreases

(C)If

2

l x , P sticks to R

(R) KE of the particle P decreases

(D) If x l , P sticks to R (S) Angular momentum of the (P + R) system isconserved about hinge.


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45. Assume that 2 bodies collide head on. The graph of their velocities with time are shown in column-1match them with appropriate situation in column – 2

Column I Column II

(A) (P)

(B) (Q)

(C) (R)

(D) (S)

(T)

46. In the figure shown, upper block is given a velocity 6m/s & lower block a velocity 3m/s. Whenrelative motion between them stops. (Here block 2 Kg is very long)

Column I Column II

(A) Work done by friction on 1 kg block (P) 3 Joule

(B) Work done by friction on 2 kg block (Q) Negative

(C) Net work done by friction (R) Positive

(D) Loss in K.E. of system (2kg + 1kg block) (S) 7 Joule


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ANDHERI / BORIVALI / DADAR / CHEMBUR / THANE / MULUND/ NERUL / POWAI

CRASH COURSE ASSINGNMENT-2 (ANSWER KEY)

TOPIC: CENTRE OF MASS, COLLISION & MOMENTUM, ROTATION DYNAMICS

1. (C) 2. (C) 3. (B) 4. (A) 5. (B)

6. (A) 7. (D) 8. (A) 9. (D) 10. (D)

11. (D) 12. (D) 13. (A) 14. (D) 15. (D)

16. (C) 17. (B) 18. (A) 19. (B) 20. (C)

21. (BC) 22. (ABD) 23. (AD) 24. (ABCD) 25. (ABC)

26. (AB) 27. (BD) 28. (ACD) 29. (BC) 30. (BC)

31. (ABCD) 32. (ABC) 33. (AC) 34. (7) 35. (5)

36. (2) 37. (6) 38. (6) 39. (5) 40. (2)

41. (5) 42. (3)

43. A – R ; B – S ; C – P ; D – Q

44. A – QRS ; B – PRS, C – QRS, D – PRS

45. A – S ; B – R ; C – Q ; D – P

46. A – Q ; B – RS ; C – QP ; D – RP

mechanics practice papers 2

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