E EST (JP ) & ANOOP (EP) NO. 2

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DPP DPPDPPDAILY PRACTICE PROBLEMS

NO. 2

DPP Syllabus : Circular Motion, Centre of Mass, Rigid Body Dynamics.

REVISION DPP No. # 2

Total Marks : 300 Max. Time: 180 min. Single correct Objective ('–1' negative marking) Q.1 to Q.60 (4 marks) [240] Integer type Questions (‘0’ negative marking) Q.61 to Q.75 (4 marks) [60]

SECTION - I Straight Objective Type

This section contains 60 multiple choice questions. Each question has 4 choices (1), (2), (3) and (4) for its answer, out of which ONLY ONE is correct.

1. A particle moves along a circle of radius R with a constant angular speed .Its displacement

(only magnitude) in time t will be

(1) t (2) 2 R cos t (3) 2 R sin t (4) 2R sin ω t

2

2. A conical pendulum of length L makes an angle with the vertical as

shown in the figure. The time period will be

(1) L cos

2g

(2)

L2

g cos

(3) L tan

2g

(4)

L2

g tan

3. A particle O of mass m is attached to a vertical rod with two inextensible

strings AO and BO of equal lengths . If be the distance between the

points of suspension on the vertical rod. if the set up rotation with angular

velocity , than

(1) Tension is thread AO is greater (2) Tension is thread BO is greater (3) Tension in the two threads are equal. (4) Tension in AO or BO is greater according as is anticlockwise or clockwise.




4. A weightless rod of length 2 carries two equal masses 'm', one secured at

lower end A and the other at the middle of the rod at B. The rod can rotate in vertical plane about a fixed horizontal axis passing through C. What horizontal velocity must be imparted to the mass at A so that it just completes the vertical circle.

(1) g5

48 (2)

45g

5

(3) 40

g5

(4) 35

g5

5. A ball suspended by a thread swings in a vertical plane so that its acceleration in the extreme position

and lowest position are equal in magnitude. Angle of thread deflection in the extreme position will be:

(1) 2 tan11

2 (2) tan1

1

2 (3) tan1 2 (4) tan1 2

6. A body of mass m moving with velocity V makes a head-on elastic collision with another body of mass

2m which is initially at rest. The ratio of kinetic energies of colliding body before and after collision will be -

(1) 9 : 1 (2) 1 : 1 (3) 4 : 1 (4) 2 : 1 7. In the figure shown find out the distance of centre of mass of a system

of a uniform circular plate of radius 3 R from O in which a hole of radius R is cut whose centre is at 2R distance from centre of large circular plate. (1) R/4 (2) R/5 (3) R/2 (4) none of these

8. A particle A of mass m is situated at highest point of wedge B of mass 2

m is released from rest. Then

distance travelled by wedge B (with respect to ground) when particle A reaches at lowest position. Assume all surfaces are smooth.

(1) 4/3 cm (2) 8/3 cm (3) 2/3 cm (4) none of these 9. A solid sphere is placed on a horizontal surface. An horizontal impulse is applied as shown in the

figure. If soon after the impact sphere starts rolling (without slipping) the ratio of h/r is :

(1) 1/2 (2) 1/5 (3) 2/5 (4) 1/4




10. A ring of radius R rolls without slipping on a rough horizontal surface with a constant velocity. The radius of curvature of the path followed by any particle of the ring at the highest point of its path will be : (1) 1 R (2) 2 R (3) 4 R (4) none of these

11. Two blocks of mass m and 2m are fixed to the ends of a spring. The spring is initially compressed &

then the system is released in air. Neglecting the air resistance after time t (1) the momentum of the system will be zero (2) the momentum of the system will be 3 m

g

t

(3) the momentum of the system will be m g

t

(4) will depend on the value of spring constant.

12. Two masses ‘ m ’ and ‘ 2 m ’ are placed in a fixed horizontal circular hollow tube as shown. The mass ‘ m ’ is moving with speed ‘u’ and the mass ‘ 2 m ’ is stationary. After their first collision, the time elapsed for next collision is : (coefficient of restitution e = 1/2)

(1) π2 r

u (2)

π4 r

u (3)

π3 r

u (4)

π12 r

u

13. A solid homogeneous cylinder of height h and base radius r is kept vertically on a conveyer belt moving

horizontally with an increasing velocity v = a + bt2. If the cylinder is not allowed to slip then the time when the cylinder is about to topple, will be equal to

(1) rg

bh (2)

2rg

bh (3)

2bg

rh (4)

rg

2 bh

14. A uniform ball of mass ‘m’ rolls without sliding on a fixed horizontal surface. The velocity of the lowest

point of the ball with respect to the centre of the ball is V. The total kinetic energy of the ball is :

(1) 7

10 mv2 (2) mv2 (3)

1

5 mv2 (4)

4

3 mv2

15. A solid sphere, a hollow sphere and a disc, all having same mass and radius, are placed at the top of

an incline and released. The friction coefficients between the objects and the incline are same and not sufficient to allow pure rolling. Least time will be taken in reaching the bottom by

(1) the solid sphere (2) the hollow sphere (3) the disc (4) all will take same time.

16. Suppose in gravity free space a disc of mass m

0 rotates freely about a fixed horizontal axis through its

centre. A thin cotton pad is fixed to its rim, which can absorb water. The mass of water dripping onto the

pad is kg per second. After what time will the angular velocity of the disc get reduced to half of its initial value?

(1) 2m

0/ (2) 3m

0/ (3) m

0/ (4) m

0/2




17. A particle performs uniform circualar motion with an angular momentum L. If the frequency of particle's motion is doubled and its kinetic energy halved, the angular momentum becomes-

(1) 2L (2) 4L (3) 2

L (4)

4

L

18. A flywheel is making 3000

revolutions per minute about its axis. If the moment of inertia of the

flywheel about that axis is 400 kgm2, its rotational kinetic energy is- (1) 2 × 106 J (2) 3 × 103 J (3) 5002 J (4) 12 × 103 J 19. Three thin uniform rods each of mass M and length L are placed along the three axis of a cartesian

coordinate system with one end of each rod at the origin. The M.I. of the system about z-axis is-

(1) 2ML

3 (2)

22ML

3 (3)

22ML

6 (4) ML2

20. represents the moment of inertia of the solid sphere about an axis parallel to the diameter of the solid

sphere and at a distance X from it. Which one of the following represents the variations of I with X ?

(1)

X

I

(2)

X

I

(3)

X

I

(4)

R

I

21. A particle is revolving in a circle of radius R with initial speed u. It starts retarding with constant

retardation

2u

4 R. The number of revolutions it makes in time

8 R

u

is :

(1) 3 (2) 4 (3) 2 (4) none of these 22. From a uniform disc of radius 2R, a semi circular disc of radius R is

cut and removed. Find out the distance of centre of mass of this part from the point ‘O’ as shown in the figure is:

(1) 3R

28 (2)

3R

4 (3)

4R

21 (4) none of these

23. In the fig. shown a cart moves on a smooth horizontal surface due to

an external constant force of magnitude F. The initial mass of the cart is M

0 and velocity is zero. Sand falls on to the cart with negligible

velocity at constant rate kg/s and sticks to the cart. The velocity of the cart at time t is:

(1)0

Ft

M t (2) 0

0

F m tn

m

(3)

0

Ft

M (4)

0

Ft

M t et

24. A thin, uniform square plate ABCD of side 'a' and mass m is

suspended in vertical plane as shown in the figure. AE and BF are two massless inextensible strings. The line AB is horizontal. The tension in the string AE just after BF is cut, will be

(1) 2mg

5 (2) mg

(3) 0 (4) None of these

A B

CD a

a

E F




25. Two identical uniform rectangular blocks (with longest side L) and a solid sphere of radius R are to be balanced at the edge of a heavy table such that the centre of the sphere remains at the maximum possible horizontal distance from the vertical edge of the table without toppling as indicated in the figure.

If the mass of each block is M and of the sphere is M/2, then the maximum distance x that can be achieved is

(1) 8L/15 (2) 5L/6 (3) (3L/4 + R) (4) (7L/15 + R) 26. A horizontal force F is applied to the lower end of a uniform thin rod

of mass 4kg and length L = 50 cm as shown in the figure. The rod undergoes only translational motion along the smooth horizontal

surface. If F = 60 N determine the angle for translation motion of

the rod. (g = 10 m/s2)

(1) 1 2tan

3

(2) 1 1tan

2

(3) 1 1sin

5

(4) 1 1sin

4

27. A particle is moving in a circular path. The acceleration and momentum vectors at an instant of time are

= 2 i + 3 j m/s2 and P

= 6 i – 4 j kgm/s. Then the motion of the particle is

(1) uniform circular motion (2) circular motion with tangential acceleration

(3) circular motion with tangential retardation (4) we cannot say anything from a

and P

given here. 28. A force of constant magnitude F acts on a particle moving in a plane such that it is perpendicular to the

velocity v

( | v |

= v) of the body, and the force is always directed towards a fixed point. Then the angle

turned by the velocity vector of the particle as it covers a distance S is :(take mass of the particle as m)

(1) 2

FS

2mv (2)

2

2FS

mv (3)

2FS

mv (4)

2

FS

mv

29. A body moves along an uneven surface with constant speed at all

points. The normal reaction of the road on the body is : (1) maximum at A (2) maximum at B (3) minimum at C (4) the same at A, B & C

30. A particle is moving along an elliptical path with a constant speed. As

it moves from A to B, magnitude of its acceleration : (1) continuously increases (2) continuously decreases (3) Remains constant (4) first increases and then decreases

y

B

Ax




31. A smooth and vertical cone-shaped funnel is rotated with an angular velocity in such a way that an object on the inner wall of the funnel is at rest w.r.t. the funnel. If the object is slightly displaced along the slope from this position and released : (1) it will be in equilibrium at its new position. (2) it will execute SHM (3) it will oscillate but the motion is not SHM (4) none of these

32. Six nails are driven through a smooth plank placed in a horizontal plane. The positions of the nails are

the vertices of a hexagon of side length L/6 each. A bob attached to a light inextensible string of length L is held straight as shown in the figure in the same plane and is projected with with constant velocity V. Then the time taken by the bob to hit the nail A is :

L/6

L

V

A

(1) L

V

(2)

2 L

3V

(3)

3 L

2V

(4) None of these

33. Construction workers don'ts always follow the blueprints precisely. After one "hard night", they read the

plan incorrectly and built a road curve tilted to the other side that it is usually done. What is the minimum speed, with which a car may cross the curve without slipping? The radius of the curve is r, the angle by which it's tilted is and the static friction coefficient between a tyre and the road surface is .

(1) cos sin

v grcos sin

(2)

sin cosv gr

cos sin

(3) sin cos

v grcos sin

(4)

cos sinv gr

cos sin

34. A small toy car moves along a circle in horizontal surface. The coeficient of friction between the toy car

and the horizontal surface is = 0

r1

R

where 0 & R are constant & r is radius of circle. Then

radius of circle at which the toy car can move with maximum constant speed :

(1) r = R

4 (2) r =

R

2 (3) r =

R

3 (4) r = R

35. Two cars A and B start racing at the same time on a flat race

track which consists of two straight sections each of length 100 and one circular section as in fig. The rule of the race is that each car must travel at constant speed at all times without ever skidding (1) car A completes its journey before car B (2) both cars complete their journey in same time (3) velocity of car A is greater than that of car B (4) car B completes its journey before car A.




36. A small block slides with velocity 0.5 gr on the horizontal frictionless

surface as shown in the Figure. The block leaves the surface at point C. The angle in the Figure is :

(1) cos1 (4/9) (2) cos1(3/4)

(3) cos1(1/2) (4) none of the above 37. A particle of mass m oscillates between P1 and P2 inside a fixed smooth

spherical shell of radius R. At any instant the kinetic energy of the particle is E. Then the force exerted by the particle on the shell at this instant is :

(1) E

R (2)

E

2R

(3) 2E

R (4)

3E

R

38. The ‘ y ’ co-ordinate of the centre of mass of the system of three rods of

length ‘2a‘ and two rods of length ‘a’ as shown in figure is : (Assume all rods to be of uniform density)

(1) 9a

8 3 (2)

9a

16 3

(3) zero (4) 8a

3

39. In the figure shown a semicircular area is removed from a uniform

square plate of side ‘’ and mass (before removing) ‘m’. The x-coordinate of centre of mass of remaining portion is (The origin is at the centre of square)

(1) –( 2)

2 (8 )

(2)

( 2)

2 (8 )

(3) – ( 2)

8

(4) None of these

x

y

40. A uniform solid right circular cone of base radius R is joined to a

unifrom solid hemisphere of radius R and of the same density , so as to have a common face. The centre of mass of the composite solid lies on the common face. The height of the cone is :

(1) 1.5 R (2) 3 R

(3) 3 R (4) 2 3 R

41. A shell of mass 2

m projected with a speed '

u

' at an angle to the horizontal explodes at the highest

point of its motion into two pieces of mass ' m

' each. If one piece whose initial speed is zero, falls

vertically, the distance at which the other piece will fall from the gun is given by:

(1) 23 u sin 2

g

(2)

3

2

2u sin 2

g

(3)

2u sin 2

g

(4) none of these




42. Two particles of masses m and 2m has initial velocity 1ˆ û 2i 3 j

m/s and 2

ˆ û 4i 3 j

m/s

respectively. These particles have constant acceleration 1ˆ â 4i 3 j

(m/s2) and 2

ˆ â 4i 2 j

(m/s2).

Path of the centre of mass of this two particle system will be : (1) Straight line (2) Circular (3) Parabolic (4) Helical 43. A train of mass M is moving on a circular track of radius ' R ' with constant speed V. The length of the

train is half of the perimeter of the track. The linear momentum of the train will be

(1) zero (2) 2MV

(3) MVR (4) MV

44. The centre of mass of two masses m & mmoves by distance x

5 when mass m is moved by distance x

and m is kept fixed. The ratio m

m

is

(1) 2 (2) 4 (3) 1/4 (4) None of these 45. A man of mass ‘m’ is standing in a lift of the same mass ‘m’ which is

balanced on a pulley by a block of mass 2m. If the man jumps suddenly with a velocity V0 upwards with respect to ground, then the speed of the man relative to lift the just after it jumps would be :

(1) Vo (2) oV

3

(3) o2V

3 (4) o4V

3

46. AB is a long friction less horizontal surface. One end of an ideal

spring of spring constant K is attached to a block of mass m, which is being moved left with constant velocity v, and the another end is free. Another block of mass 2m is given a velocity 3v towards the spring. Work done by external agent in moving m with constant velocity v in long time will be :

(1) –5 mv2 (2) –8mv2 (3) –4mv2 (4) None of these 47. Particle A makes a perfectly elastic collision with another stationary particle B. They fly apart in opposite

directions with equal speeds. The mass ratio will be:

(1) 1

3 (2)

1

2 (3)

1

4 (4)

1

3

48. Which one of the following four graphs best depict the variation with x of the moment of inertia I of a

uniform triangular lamina about an axis parallel to its base at a distance x from it :

x

h

(1)

h x

I

(2)

h x

I

(3)

h x

I

(4)

h x

I




49. Consider two masses with m1 > m2 connected by a light inextensible string that passes over a pulley of radius R and moment of inertia about its axis of rotation. The string does not slip on the pulley and the pulley turns without friction. The two masses are released from rest separated by a vertical distance 2h. When the two masses pass each other, the speed of the masses is proportional to.

(1) 1 2

1 2 2

m m

m mR

(2) 1 2 1 2

1 2 2

(m m )(m m )

m mR

(3)

1 2 2

1 2

m mR

m m

(4)

2

1 2

Rm m

50. A hollow tilted cylindrical vessel of negligible mass rests on a horizontal plane as known. The diameter

of the base is a and the side of the cylinder makes an angle with the horizontal. Water is then slowly poured into the cylinder. The cylinder topples over when the water reaches a certain height h, given by.

a

h

(1) h = 2a tan (2) h = a tan2 (3) h = a tan (4) h = a

2tan

51. A solid cube of wood of side 2a and mass M is resting on a horizontal surface as shown in the figure. The cube is free to rotate about a fixed axis AB. A bullet of mass m (<< M) and speed v is shot horizontally at the face opposite to ABCD at a height of 4a/3 from the surface to impart the cube and angular speed . It strikes the face and embeds in the cube. Then is close to (note : the moment of inertia of the cube about an axis perpendicular to the face and passing through the center of mass is

22Ma

3

A

B

C

D

(1) Mv

ma (2)

Mv

2m a (3)

m v

Ma (4)

m v

2Ma

52. The moments of inertia of a non-uniform circular disc (of mass M and radius R) about four mutually

perpendicular tangents AB,BC CD, DA are I1, I2, I3 and I3 respectively (the square ABCD circumscribes the circle.) The distance of the center of mass of the disc from its geometrical center is given by.

(1) I I I I2 23 3 2 4

1( ) ( )

4MR (2) I I I I2 2

3 3 2 4

1( ) ( )

12MR

(3) I I I I2 21 2 3 4

1( ) ( )

3MR (4) I I I I2 2

1 3 2 4

1( ) ( )

2MR




53. A block of mass m slides on a frictionless table. It is constrained to move inside a ring of radius which is fixed to the table. At t = 0, the block is moving along the inside of the ring (i.e., in the tangential direction) with velocity v0. The coefficient of friction between the block and the ring is . The velocity of the block at time t.

v0

(1) 0

2

0

v

v t1

(2) 0v

t1

(3 ) 0

0

v

v t1

(4) 0

0

v

v t1

54. A particle is moving on a circular path such that at any instant its position vector, linear velocity, angular

velocity, angular acceleration with respect to centre are r

, v

,

,

respectively. Net acceleration of the particle is

(1) (

× v

) – ( r

×

) (2) (

× v

) + ( r

×

) (3) ( v

×

) + ( r

×

) (4) ( v

×

) – ( r

×

)

55. A roller coaster car travels down the helical path at constant speed such that it's parametric coordinates varies as x = c sin (kt), y = c cos (kt), z = h – bt where c, h, k and b are constants, then the magnitude of it's acceleration is:

(1) 0 (2) ck2 (3) 2 2c k

h (4) bk2

56. A sphere of radius R made up of Styrofoam (light polystyrene material) has a cavity of radius R/2. The centre of the cavity is situated at a distance of R/2 from the centre of the Styrofoam sphere. The cavity is filled with a solid material of density five times that of Styrofoam. Now, the centre of mass is seen to be located at a distance x from the centre of Styrofoam sphere, therefore x is.

(1) R/2 (2) R/3 (3) R/4 (4) R/6 57. Two identical particles move towards each other with velocity 2v and v respectively. This velocity of

centre of mass is – (1) v (2) v/3 (3) v/2 (4) zero 58. Consider the following two statements : A. Linear momentum of a system of particles is zero B. Kinetic energy of a system of particles is zero, Then, (1) A does not imply B and B does not imply A (2) A implies B but B does not imply A (3) A does not imply B but B implies A (4) A implies B and B implies A 59. A body A of mass M while falling vertically downwards under gravity breaks into two parts; a body B of

mass 1

3M and, a body C of mass

2

3M. The centre of mass of bodies B and C taken together shifts

compared to that of body A towards: (1) Depends on height of breaking (2) does not shift (3) Shift towards body C (4) shift towards body B




60. The block of mass M moving on the frictionless horizontal surface collides with the spring of spring constant k and compresses it by length L. The maximum momentum of the block after collision is :

M

(1) Mk L (2) 2kL

2M (3) zero (4)

2ML

k

SECTION-II : (INTEGER TYPE QUESTIONS)

This section contains Fifteen (15) questions. The answer to each question is NUMERICAL VALUE with two digit integer and decimal upto two digit.

61. A flywheel rotates with a uniform angular acceleration. Its angular velocity increases from 20 rad/s to

40 rad/s in 10 seconds. If Number of rotations did it make in this period are 10 x then the value of x is ?

62. A particle is moving with constant angular acceleration = /25.4 in a circular path of radius 3 m . At

t = 0, it was at rest and at t = 1 sec, the magnitude of its acceleration becomes 6 m/s2, then is :

63. A body of mass 1kg strikes elastically with another body at rest and continues to move in the same

direction with one fourth the in initial velocity. If the mass of the other body is m/100 then the value of m is -

64. A ring of mass 2 kg and of radius 0.25 m is making 300 rpm about an axis through its centre perpendicular to its plane. The tension (in newton's) developed in the ring is 10x then the value of x is

(take 2 = 10) 65. A peg is placed directly below the pendulum’s point of support at a

distance = 0.5m from the lowest point of the pendulum swing as shown in the fig. If minimum height h from which the pendulum can be released such that the string remains taut for at least one full revolution around the peg. the value of 10h is

66. A plate, of uniform thickness and uniform density, has shape in the x-y plane defined by the lines

x = 0, y = 2, and curve y =1

2 x2. The plate lies in first quadrant of x-y plane. If the x-coordinate of centre

of mass of this plate is n/100 then the value of n is 67. A solid cylinder P and hollow sphere Q are of equal masses and

same radii. There is no slipping at any surface. If ratio of rotational kinetic energy of P and Q is x/100 then the value of x is :

68. A truncated cone is rolling without slipping on a table in such a way, that a point on the perimeter of its smaller base moves at a speed v1 = 1.0 ms–1 with respect to the base's centre and a point on the perimeter of its greater base moves at a speed v2 = 1.5 ms–1 (again, with respect to the base's centre).

The slant height is = 0.1 m. If x/10 sec. will it take the cone to return to the point from which it started rolling. Then the value of x is ?




69. A mass ‘m’ moves with a velocity v= 310 m/s and collides inelastically with another identical mass at

rest. After collision the 1st mass moves with velocity v

3 in a direction perpendicular to the initial

direction of motion. If the speed (in m/s) of the 2nd mass after collision is

70. An insect of mass m = 3 kg is inside a vertical drum of radius 2 m that is rotating with an angular velocity of 5 rad/sec. The insect does not fall off, If the minimum co-efficient of friction required is x/100 then the

value of x is :

71. A car is moving on circular path of radius 100m such that its speed is increasing at the rate of 5m/s2. At

t=0 it starts from rest, what is the radial acceleration (in m/s2) of car at the instant it completed one round trip?

72. A bomb of mass 16 kg at rest explodes into two pieces of masses of 4 kg and 12 kg. The velocity of the 12 kg mass is 4 ms–1. The kinetic energy of the other mass is 10x Jule then the value of x is:

73. A shell of mass 4 kg moving with a velocity 10 m/s vertically upward explodes into three parts at a height 50 m from ground. After three seconds, one part of mass 2 kg reaches ground and another part of mass 1 kg is at height 40 m from ground. The height of the third part from the ground is 10x then the value of x is [ g = 10 m/s2]

74. A small ring of mass m is connected with a particle of same mass by an ideal string & the whole system is released as shown in figure. Coefficient of friction between ring A and wire is 3/5, ring A will starts sliding when connecting string will make an angle with the vertical, then (in degree) will be : (particle is free to move and ring can slide only)

75. A rocket with a lift-off mass 3.5 × 104 kg is blasted upwards with an intial acceleration of 10 m/s2. If the initial thrust of the blast is 2n × 104 then the valueo of n is.

ANSWER KEY OF REVISION DPP No. # 2

1. (4) 2. (1) 3. (1) 4. (1) 5. (1) 6. (1) 7. (1)

8. (1) 9. (3) 10. (3) 11. (2) 12. (2) 13. (1) 14. (1)

15. (4) 16. (4) 17. (4) 18. (1) 19. (2) 20. (4) 21. (3)

22. (3) 23. (1) 24. (1) 25. (1) 26. (1) 27. (4) 28. (4)

29. (1) 30. (2) 31. (4) 32. (1) 33. (4) 34. (2) 35. (4)

36. (2) 37. (4) 38. (2) 39. (4) 40. (2) 41. (2) 42. (3)

43. (2) 44. (2) 45. (4) 46. (2) 47. (1) 48. (1) 49. (1)

50. (3) 51. (4) 52. (1) 53. (4) 54. (1) 55. (2) 56. (4)

57. (3) 58. (3) 59. (2) 60. (1) 61. 15.00 62. 25.40 63. 60.00

64. 25.00 65. 12.50 66. 75.00 67. 18.80 68. 12.60 69. 20.00 70. 20.00

71. 62.80 72. 28.80 73. 10.00 74. 45.00 75. 35.00

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E EST (JP ) & ANOOP (EP) NO. 2

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