Topper’s Package Physics - XI Circular Motion & Rotational

Topper’s Package Physics - XI Permutation and CombinationsCircular Motion & Rotational Motion

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1. CIRCURLAR MOTION

1. A mass m is attached to one end of a rigidrod of negligible mass. The other end ispivoted in such a manner that the rod canrotate in a vertical plane about a horizontalaxis. When a minimum horizontal velocity vis given to the mass m, it moves in a completevertical circle. Then v is

(a) 5gL

v m

L (b) 4gL

(c) 3gL

(d) 2gL

2. A particle describes a horizontal circle ofradius r in a funnel type vessel of frictionlesssurface with half cone angle (as shown infigure). If mass of the particle is m, then indynamical equilibrium the speed of theparticle must be

(a)tanrgv

(b) tanv rg

N

mg

r

(c)1tan

vrg

(d) tanv rg

3. The maximum velocity at the lowest point, sothat the string just slack at the highest pointin a vertical circle of radius l is(a) g (b) 3g

(c) 5g (d) 7g

4. Heavy mass is attached to a thin wire andis whirled in a vertical circle. The wire ismost likely to break(a) when the mass is at the highest point of

the circle(b) when the mass is at the lowest point of

the circle

(c) when the wire is horizontal(d) at an angle of 1cos (2/3) from the

upward vertical

5. A tube of length L is filled completely with anincompressible liquid of mass M and closedat both ends. The tube is then rotated ina horizontal plane about one of its end witha uniform angular velocity . Then theforce exerted by the liquid at the other endis(a) 2M L (b) 21/2 M L(c) 21/4 M L (d) 22 M L

6. A car of mass m when passes over a convexbridge of radius of curvature r, with a velocityv, then the normal force exerted by the bridgeon the car is(a) zero (b) mg

(c) mg + r

mv2(d) mg –

rmv2

7. A particle of mass m, attached with a stringof length l is moving in a vertical circle. Ifthe particle is just looping the loop withoutslackening of the string and , ,A B Dv v v arespeeds at positions A, B, D shown in figure,then A

B

D m

T

l

(a) B D Av v v

(b) tension in the string at D is 3 mg(c) 3Dv gl

(d) all of these

8. A small body of mass ‘m’ is placed on the topof a hemisphere of radius r. Then thesmallest horizontal velocity v that should begiven to the body so that it may leave thehemispherical surface and not slide down, is

Circular Motion andRotational Motion

Unit 6


72

V

r

m

(a) v gr (b) 2v gr

(c) 3v gr (d) 5v gr

9. A car is moving along a circular path of radius500 m with a speed of 30 m/s. If at someinstant its speed increases at the rate of 2m/s per second, then at that instant themagnitude of its resultant acceleration isnearly(a) 21.8 /m s (b) 22 /m s(c) 22.7 /m s (d) 23.8 /m s

10. The relation between the linear velocity andangular velocity is(a) v r

(b) v r

(c) v r (d) r v

11. The angular velocity of minute hand of awatch, if the length of needle is 2cm

(a) secrad/1600π

(b) secrad/3000π

(c) secrad/30π

(d) secrad/1800π

12. A particle is given an initial speed u insidea smooth spherical shell of radius 1R mthat it is just able to complete the circle.Acceleration of the particle when its velocityis vertical is

(a) 10g

Ru

(b) g

(c) 2g

(d) 3g13. A hemisphere of radius R and of mass 4 m

is free to slide with its base on a smoothhorizontal table. A particle of mass m is placedon the top of the hemisphere. The angularvelocity of the particle relative to hemisphereat an angular displacement when velocityof hemisphere has become v is

(a)5cos

vR

(b)2cos

vR

(c)3sin

vR

(d)5sin

vR

14. A particle of mass m moving with a speed vhits elastically another stationary particle ofmass 2 m on a smooth horizontal circulartube of radius r. The time in which the nextcollision will take place is equal to

r

vm

2m

(a)2 rv

(b)4 rv

(c)32

rv

(d)r

v

15. Two particles A and B are situated at adistance 2d m apart. Particle A has avelocity of 10 /m s at an angle of 60° andparticle B has a velocity v at an angle 30° asshown in figure. The distance d between Aand B at the instant shown in figure isconstant. The angular velocity of B withrespect to A is

d m = 2A B

u m s = 10 /

v

60° 30°

(a) 5 3 /rad s (b)5 /3

rad s

(c) 10 3 /rad s (d)10 /

3rad s

2. MOMENT OF INERTIA

16. The moment of inertia of a thin uniform rodof mass M and length l about an axisperpendicular to the rod, through its centreis I. The moment of inertia of the rod aboutan axis perpendicular to the rod through itsend point is –(a) /4I (b) /2I(c) 2I (d) 4I

17. Four particles each of mass m are placed atthe corners of a square of side length l. Theradius of gyration of the system about an axis


73

perpendicular to the square and passingthrough centre is(a) / 2l (b) /2l(c) l (d) l)2(

18. Three rings each of mass P and radius Q arearranged as shown in figure. The moment ofinertia of the arrangement about YY will be

(a) 272

PQ

(b) 225

PQ

(c) 252

PQ

P P Q Q

Q

1 2

3 P

y

y (d) 227

PQ

19. From a disc of radius R, a concentriccircular portion of radius r is cut out so asto leave an annular disc of mass M. Themoment of inertia of this annular disc aboutthe axis perpendicular to its plane and passingthrough its centre of gravity is

(a) 2 21 ( )2

M R r (b) 2 21 ( )2

M R r

(c) 4 41 ( )2

M R r (d) 4 41 ( )2

M R r

20. The moment of inertia of a thin square plateABCD (as shown in figure) of uniformthickness about an axis passing through thecentre O and perpendicular to the plane ofplate is

(a) 1 2I I

(b) 3 4I I

A B

C D

1

2

3

4

O

(c) all of the above(d) none of these

21. Two identical rods are joined to form an ‘X’.The smaller angle between the rods is . Themoment of inertia of the system about an axispassing through the point of intersection ofthe rods and perpendicular to their plane is(a) (b) 2sin (c) 2cos (d) independent of

22. A thin rod of length L and mass M is bent

at the middle point O at an angle of 60°, figure.The moment of inertia of the rod about anaxis passing through O and perpendicular tothe plane of the rod will be

L/2 L/2 60°

(a)2

6ML

(b)2

12ML

(c)2

24ML

(d)2

3ML

23. A square plate of side l has mass M. Whatis its moment of inertia about one of itsdiagonals?

(a)M 2

6(b)

M 2

12

(c)M 2

3(d)

M 2

424. The moment of inertia of a ring of mass M

and radius R about PQ axis (figure) will be

O

D M

P

Q

O

R

(a) 2MR (b)M R2

2

(c)32 MR2 (d) 22 MR

25. Three identical solid discs, each of mass Mand radius R, are arranged as shown in thefigure. The moment of inertia of the systemabout the axis AA will be

M

R 2

A

A

1

3

(a) 274

MR (b) 2114

MR


74

(c) 2154

MR (d) 2194

MR

26. In the adjoining figure along which axis theM.I. of the triangular lamina will bemaximum(a) AB

(b) BC

(c) CA

A

B C

(d) None of these27. Two rings of the same radius and mass are

placed such that their centres are at acommon point and their planes areperpendicular to each other. The moment ofinertia of the system about an axis passingthrough the centre and perpendicular to therings is (mass of the ring = m, radius = r)

(a) 212

mr (b) 2mr

(c) 232

mr (d) 22mr

28. For the given uniform square lamina ABCD,whose centre is O,

(a) 2AC EFI I

(b) 2 AC EFI I

A

CD E

F

O

B(c) 3AD EFI I

(d) AC EFI I29. Four point masses, each of value m, are placed

at the corners of a square ABCD of side l. Themoment of inertia of this system about anaxis through A and parallel to BD, is(a) 2ml (b) 22ml(c) 23ml (d) 23ml

30. One quarter section is cut from a uniformcircular disc of radius R. This section has amass M. It is made to rotate about a lineperpendicular to its plane and passing throughthe centre of the original disc. Its momentof inertia about the axis of rotation is

(a) 212

MR

(b) 214

MR

(c) 218

MR

(d) 22 MR31. From a circular disc of radius R and mass 9

M, a small disc of radius R/3 is removed fromthe disc. The moment of inertia of theremaining disc about an axis perpendicularto the plane of the disc and passing throughO is

(a) 24MR

(b) 2409

MR

(c) 210 MRR O

R/3

(d) 2379

MR

32. A solid sphere of radius R has moment ofinertia I about its geometrical axis. It ismelted into a disc of radius r and thicknesst. If it’s moment of inertia about the tangentialaxis (which is perpendicular to plane of thedisc), is also equal to I, then the value of ris equal to

(a)215

R

(b)25

R

I

r

(c)315

R

(d)3

15R

33. Moment of inertia I of a solid sphere aboutan axis parallel to a diameter and at adistance x from it varies as

(a)x

I

(b)x

I

(c)x

I

(d)x

I

34. A wire of length l and mass m is bent in the

form of a rectangle ABCD with 2ABBC

. The


75

moment of inertia of this wire frame aboutthe side BC is

(a) 211252

ml (b) 28203

ml

(c) 25136

ml (d) 27162

ml35. A wire of mass m and length l is bent in the

form of a quarter circle. The moment ofinertia of this wire about an axis passingthrough the centre of the quarter circle andperpendicular to the plane of the quarter circleapproximately(a) 20.6 ml (b) 2ml(c) 20.2 ml (d) 20.4 ml

36. A T joint is formed by two identical rods A andB each of mass m and length L in the XY planeas shown. Its moment of inertia about axiscoinciding with A is

(a)22

3mL (b)

2

12mL

(c)2

6mL (d) None of these

37. The moment of inertia of a thin uniform rodof mass M and length L about an axis passingthrough its midpoint and perpendicular to itslength is I0. Its moment of inertia about anaxis passing through one of its ends andperpendicular to its length is

(a) I0 + ML2 (b) 20 2

MLI

(c)2

0 4MLI (d) I0 + 2ML2

38. A uniform square plate has a small piece Qof an irregular shape removed and glued tothe centre of the plate leaving a hole behindin figure. The moment of inertia about thez-axis is then

(a) increase(b) decreases(c) the same(d) changed in unpredicted manner

39. Four thin rods of same mass M and samelength l, form square as shown in figure.Moment of inertia of this system about anaxis through centre O and perpendicular toits plane is

(a) 243

Ml (b)2

3Ml

(c)2

6Ml (d) 22

3Ml

40. ABC is a triangular plate of uniform thickness.The sides are in the ratio shown in the figureIAB, IBC, ICA are the moments of inertia ofthe plate about AB, BC, CA respectively. Whichone of the following relations is correct.

(a) ICA is maximum (b) IAB > IBC(c) IBC > IAB (d) IAB + IBC = ICA

41. A rod of length L and mass M is bent to forma semi-circular ring as shown in figure. Themoment of Inertia about XY is

(a)2

22ML

(b)2

2ML

(c)2

24ML


76

(d)2

22ML

42. Three identical spherical shells, each of massm and radius r are placed as shown in figure.Consider an axis XX which is touching to twoshells and pasing through diameter of thirdshell. Moment of inertia of the systemconsisting of these three spherical shellsabout XX axis is(a) 3 mr2

(b) 2165

mr

(c) 4 mr2

(d) 2115

mr

43. One quarter sector is cut from a uniformcircular disc of radius R. This sector has massM. It made to rotate about a line perpendicularto its plane and passing through the centreof the original disc. Its moment of inertiaabout the axis of rotation is

(a) 212

MR

(b) 214

MR

(c) 218

MR

(d) 22MR

44. A thin wire of length L and uniform linearmass density is bent into a circular loop withcentre at O as shown in figure. The momentof inertia of the loop about the axis XX is

(a)3

28L

(b)3

216L

(c)3

2516

L

(d)3

238

L

45. Let I be the moment of inertia of a uniformsquare plate about an axis AB that passesthrough its centre and is parallel to two of itssides. CD is a line in the plane of the platethat passes through the centre of the plateand makes an angle with AB. The moment

of inertia of the plate about the axis CD isthen equal to(a) I (b) I sin2

(c) I cos2 (d) 2cos2

I

46. Two spheres of each of mass M and radiusR/2 are connected with a massless rod oflength 2R as shown in the figure. What willbe the moment of inertia of the system aboutan axis passing through the centre of one ofthe spheres and perpendicular to the rod

(a) 2215

MR (b) 225

MR

(c) 252

MR (d) 2521

MR

47. A solid cylinder has mass ‘M’ and radius ‘R”and length ‘l’. Its moment of inertia about anaxis passing through its centre andperpendicular to its axis is

(a)2 22

3 12MR Ml (b)

2 2

3 12MR Ml

(c)2 23

4 12MR Ml (d)

2 2

4 12MR Ml

48. The moments of inertial of a non-unifromcircular disc of mass M and radius R aboutfour mutually perpendiucular tangetns AB, BC,CD, DA are I1, I2, I3 and I4 are respectively(the square ABCD circumscribes the circle).The distance of the centre of the mass of thedisc from its geometrical centre is given by

(a) 2 23 3 2 4

1 ( ) ( )4

I I I IMR

(b) 2 23 3 2 4

1 ( ) ( )12

I I I IMR

(c) 2 21 2 2 4

1 ( ) ( )3

I I I IMR

(d) 2 21 3 2 4

1 ( ) ( )2

I I I IMR

49. Moment of inertia of a sphere of mass M andradius R is I. Keeping M constant if a graphis plotted between I and R, then its form wouldbe


77

(a) (b)

(c) (d)

50. According to the theorem of parallel axesI = Icm + Mx2, the graph between I and x will be

(a)

I

x

(b)

I

x

(c)

I

x

(d)

I

x

51. Seven identical circular planar disks, eachof mass M and radius are weldedsymmetrically as shown in figure. Themoment of inertia of the arrangement aboutthe axis normal to the plane and passingthrough the point P is

(a) 2732

MR

(b) 21812

MR

(c) 2192

MR

P

O

(d) 2552

MR

52. From a uniform circular disc of radius R and

mass 9M a small disc of radius 3R is removed

as shown in the figure. The moment of inertiaof the remaining disc about an axisperpendicular to the plane of the disc andpassing through centre of the disc is

(a) 10 MR2

(b) 2379

MR

(c) 4MR22R/3

R

(d) 2409

MR

53. Point masses m1 and m2 are placed at theopposite ends of a right length L and negligiblemass. The rod is to be set rotating about anaxis perpendicular to it. The position of pointP on this rod through which the axis shouldpass so that the work required to set the rodrotating with angular velocity 0 is maximum,is given by

( - )L xx

P m2m1

0

(a) 1

2

mx Lm (b) 2

1

mx Lm

(c) 2

1 2

m Lx m m

(d) 1

1 2

m Lx m m

54. A light rod of length l has to masses m1 andm2 attached to its two ends. The moment ofinertia of the system about an axisperpendicular to the end and passing throughthe centre of mass is

(a) 21 2m m l (b) 21 2

1 2

m mlm m

(c) 21 2

1 2

m mlm m

(d) (m1 + m2)l2

3. TORQUE AND ENERGY OF ROTATION55. A cylinder of mass M and radius r is mounted

on a frictionless axle over a well. A rope ofnegligible mass is wrapped around thecylinder and a bucket of mass m is suspendedform the rope. The linear acceleration of thebucket will be

(a)mM

mg

(b)mM

mg2

(c)mM

mg2

2

(d) mgM


78

56. One end of a uniform rod of length l and massm is hinged at A. It is released from rest fromhorizontal position AB as shown in figure. Theforce exerted by the rod on the hinge whenit becomes vertical is

(a)32

mgA

B

(b)52

mg

(c) 3mg

(d) 5mg

57. A wheel is rotating about an axis through itscentre at 720 rpm. It is acted on by a constanttorque opposing its motion for 8 second tobring it to rest finally. The value of torque inNm is

(given 224I kg m

)(a) 48 (b) 72(c) 96 (d) 120

58. Two bodies of mass 1kg and 2kg are attachedto the two ends of a 3m long rod. This rod isrotating about an axis passing through centreof mass with angular velocity 10 rad/sec andperpendicular to its length. The rotational K.E.of the system will be(a) 150 J (b) 755 J(c) 300 J (d) 400 J

59. The moment of inertia of a body about a givenaxis is 21.2 kg m . Initially, the body is atrest. In order to produce a rotating kineticenergy of 1500 Joules, and angularacceleration of 225 /rad s must be appliedabout that axis for a duration of(a) 4 sec (b) 2 sec(c) 8 sec (d) 10 sec

60. A wheel of radius 20 cm has force applied toit as shown in the figure. The torque producedby the forces 4N and A, 8N at B, 6N at C, 9Nat D at angles indicated is :

4N

A

B

6N

C 9N

D20 cm

90°

30°

8N

(a) 5.40 N-m anticlockwise(b) 1.80 N-m clockwise(c) 2.0 N-m clockwise(d) 5.4 N-m clockwise

61. A ball of mass m is attached to one end of alight rod of length l, the other end of whichis hinged. What minimum velocity v shouldbe imparted to the ball downwards, so that itcan complete the circle ?

l

u

(a) gl (b) 5gl

(c) 3gl (d) 2gl

62. A uniform cube of side a and mass m rests ona rough horizontal surface. A horizontal forceF is applied normal to one face at a point thatis directly above the centre of the face at aheight /4a above the centre. The minimumvalue of F for which the cube begins to toppleabove an edge without sliding is

(a)14

mg (b) 2mg

(c)12

mg (d)23

mg

63. Let F

be the force acting on a particle havingposition vector

r and

T be the torque of this

force about the origin

(a) . 0

r T and . 0

F T

(b) . 0

r T and . 0F T

(c) . 0

r T and . 0

F T

(d) . 0

r T and . 0

F T

64. A wheel of radius 0.4 m can rotate freelyabout its axis as shown in figure. A string iswrapped over its rim and a mass of 4 kg is


79

hung. An angular acceleration of 8 rad-s2 isproduced in it due to the torque. Then,moment of inertia of the wheel is(g = 10 ms–2)(a) 2 kg–m2

(b) 1 kg-m2

4 kg

(c) 4 kg-m2

(d) 8 kg-m2

65. A force of ˆ–Fk acts on O, the origin of thecoordinate system. The torque about the point(1, –1) is(a) ˆ ˆ–F(i j) (b) ˆ ˆF(i j)(c) ˆ ˆ–F(i j) (d) ˆ ˆF(i j) ˆ ˆF(i j)

66. A uniform rod AB of length l and mass m isfree to rotate about point A. The rod isreleased from rest in horizontal position.Given that the moment of inertia of the rod

about A is 2

3ml the initial angular

acceleration of the rod will be

l

A B

(a)23gl (b) 2

lmg

(c) 32

gl (d)32gl

67. A force F = 2.0 N acts on a particle P in thexz-plane. The force F is parallel to x-axis. Theparticle P(as shown in the figure) is at adistance 3 m and the line joining P with theorigin makes an angle 30° with the axis. Themagnitude of torque on P with respect toorigin O (in N-m) is

30°

F

Fy

x

z

3m

O

(a) 2 (b) 3(c) 4 (d) 5

68. The total torque about pivot A provided by theforces shown in the figure, for L = 3.0 m s

60° 30°

60°

90°

90N 80N70N

50N

60NA B

1/2L1/2L(a) 210 Nm (b) 140 Nm(c) 95 Nm (d) 75 Nm

69. A wheel of radius R with an axle of radiusR/2 is shown in the figure and is free to rotateabout a frictionless axis through its centreand perpendicular to the edge. Three forces(F, F, 2F) are exerted tangentially to therespective rim as shown in figure. Themagnitude of the net torque acting on thesystem is nearly

2FR

R

FF

45°

(a) 3.5 FR (b) 3.2 FR(c) 3.5 FR (d) 1.5 FR

70. Consider two masses with m1 > m2 connectedby a light inextensible string that passes overa pulley of radius R and moment of inertiaI about its axis of rotation. The string doesnot slip on the pulley and the pulley turnswithout friction. Two masses are releasedfrom rest separated by a vertical distance 2H.When the two masses pass each other, thespeed of the masses is proportional to

(a)1 2

1 2 2

m mIm m

R(b)

1 2 1 2

1 2 2

( ) ( )

m m m mIm m

R

(c)1 2 2

1 2

Im mR

m m(d)

2

1 2

IR

m m

71. Two masses m1 = kg and m2 = 2 kg areconnected by a light inextensible string andsuspended by means of a weightless pulley as


80

shown in the figure. Assume that both themasses start from rest, the distance travelledby the centre of mass in two seconds is (Takeg = 10 ms–2)

m1

m2

1 kg

2 kg

(a) 209

m (b) 409

m

(c) 23

m (d) 13

m

4. ANGULAR MOMENTUM AND ITS CONSERVATION72 A body of mass m is moving with constant

velocity parallel to x-axis. The angularmomentum with respect to origin(a) Increases with time(b) Decreases with time(c) Does not change(d) None of these

73. A circular platform is free to rotate in ahorizontal plane about a vertical axis passingthrough its centre. A tortoise is sitting at theedge of the platform. Now, the platform isgiven an angular velocity 0 . When thetortoise moves along a chord of the platformwith a constant velocity (with respect to theplatform), the angular velocity of the platform

( )t will vary with time t as

(a)

(t)

0

t

(b)

(t)

0

t

(c)

(t)

0

t

(d)

(t)

0

t

74. Let g be the acceleration due to gravity atearth’s surface and K be the rotational kinetic

energy of the earth. Suppose the earth’sradius decreases by 2 %, keeping all otherquantities same, then(a) g decreases by 2% and K decreases by 4%(b) g decreases by 4% and K Increases by 2%(c) g Increases by 4% and K decreases by 4%(d) g decreases by 4% and K Increases by 4%

75. A bob of mass m attached to an inextensiblestrible string of lenght l is suspended from avertical support. The bob rotates in ahorizontal circle with an angular speed rad/s about the vertical about the point ofsuspension :(a) angular momentum is conserved.(b) angular momentum changes in magnitude

but not in direction.(c) angular momentum changes in direction

but not in magnitude.(d) angular momentum changes both in

direction but not in magnitude.76. Before jumping in water from above, a

swimmer bends his body to(a) increase moment of inertia(b) decrease moment of inertia(c) decrease the angular momentum(d) reduce the angular velocity

77. A raw egg and a hard boiled egg are made tospin on a table with the same angular speedabout the same axis. The ratio of the timetaken by the two to stop is(a) = 1 (b) < 1(c) > 1 (d) none of these

78. A particle of mass m = 5 kg is moving witha uniform speed 3 2v in the XOY planealong the line 4y x . The magnitude of theangular momentum about origin is(a) zero (b) 60 unit(c) 7.5 unit (d) 40 2

79. A uniform rod AB of length L and mass m islying on a smooth table. A small particle ofmass m strike the rod with a velocity 0v atpoint C a distance x from the centre O. Theparticle comes to rest after collision. Thevalue of x, so that point A of the rod remainsstationary just after collision is

A

C

O

B

x

v0m


81

(a) 3L

(b) 6L

(c)4L

(d) 12L

80. A thin circular ring of mass M and radius Ris rotating about its axis with angular velocity . Two similar objects of mass ‘m’ each areattached gently to the opposite ends of itsdiameter. The angular velocity of the ring willnow be

(a)mM

m2ω

(b) )(ω

mMm

(c) )2()(ωmM

M (d)

mmM )2(ω

81. If the earth were to suddenly contract to th1n

of its present radius without any change inits mass, then the duration of the new daywill be nearly

(a) 24 hoursn (b) 24n hours

(c) 224 hoursn

(d) 224n hours

82. Consider a body, shown in figure, consistingof two identical balls, each of mass Mconnected by a light rigid rod. If an impulseJ = Mv is imparted to the body at one of itsends, what would be its angular velocity ?

M M

L

J Mv =

(a)vL (b)

2vL

(c)3vL (d)

4vL

83. A uniform rod AB of mass m and length l is atrest on a smooth horizontal surface. Animpulse p is applied to the end B. The timetaken by the rod to turn through right angle is

(a) 2 mlp

(b) 2 pml

(c) 12ml

p

(d)p

ml

84. A disc of mass M and radius R is rolling with

angular speed on a horizontal plane asshown. The magnitude of angular momentumof the disc about the origin O is

M

y

O x

(a)21

2MR

(b) 2MR

(c)23

2MR

(d) 22MR

85. A rigid spherical body is spinning around anaxis without any external torque. Due totemperature its volume increases by 3%. Thenpercentage change in its angular speed is(a) – 2% (b) –1%(c) – 3% (d) 1%

86. A ring of radius R is first rotated with anangular velocity 0 and then carefully placedon a rough horizontal surface. The coefficientof friction between the surface and the ringis µ. Time after which its angular speed isreduced to half is

(a)02

µRg

(b)

02

gµR

(c)02 R

µg

(d)0

2R

µg

87. A billiard ball is hit by a cue at a height habove the centre. It acquires a linear velocityv0. Mass of the ball is m and radius is r. Theangular velocity 0 acquired by the ball is

(a)02

25v hr (b)

02

52v hr

(c)2

025v r

h (d)2

052v r

h

88. A rod of length l slides down along the inclinedwall as shown in figure. At the instant shownin figure, the speed of end A is v, then thespeed of B will be

A

Bv

l

(a)sin

sinv

(b)

sinsin

v

(c)cos

cosv

(d)

coscos

v


82

89. A uniform rod AB of mass m and length l isat rest on a smooth horizontal surface. Animpulse J is applied to the end Bperpendicular to the rod in horizontaldirection. Speed of particle P at a distance

6l

from the centre towards A of the rod after time

12mltJ

is

(a) 2 Jm (b) 2

Jm

(c)Jm (d) 2 J

m

5. ROLLING MOTION

90. Portion AB of the wedge shown in figure isrough and BC is smooth. A solid cylinder rollswithout slipping from A to B. If AB BC , thenratio of translational kinetic energy torotational kinetic energy, when the cylinderreaches point C is

(a)35

(b) 5

A

C

B

D(c)75

(d)83

91. A solid cylinder of mass M and radius R rollswithout slipping down an inclined plane oflength L and height h. What is the speed ofits centre of mass when the cylinder reachesits bottom

(a) 2gh (b) 34

gh

(c) 43

gh (d) 4gh

92. A solid sphere of mass M and radius R rollson a horizontal surface without slipping. Theratio of rotational K.E. to total K.E. is(a) 1/2 (b) 3/7(c) 2/7 (d) 2/10

93. A cylinder rolls up an inclined plane, reachessome height, and then rolls down (withoutslipping throughout these motions). The

directions of the frictional force acting on thecylinder are:(a) Up the incline while ascending and down

the incline descending(b) Up the incline while ascending as well as

descending(c) Down the incline while ascending and up

the incline while descending(d) Down the incline while ascending as well

as descending

94. The speed of a homogenous solid sphereafter rolling down an inclined plane of verticalheight h from rest without sliding is

(a) 107

gh (b) gh

(c) 65

gh (d) 43

gh

95. A uniform solid cylinder of mass M and radiusR rotates about a frictionless horizontal axle.Two similar masses (m each) are suspendedwith the help of two ropes wrapped around thecylinder figure. If the system of masses isreleased from rest, what will be the tensionin each rope?

T T

m m

(a) ( )Mmg

M m (b) ( 3 )Mmg

M m

(c) ( 4 )Mmg

M m (d) ( 2 )Mmg

M m

96. A spherical body of radius R is allowed to rolldown on an incline without slipping and itreaches with a speed 0v at the bottom. Theincline is then made smooth by waxing andthe body is allowed to slide without rolling and

now the speed attained is 054

v . The radiusof gyration of the body about an axis passingthrough its centre is

(a)43

R (b)34

R

(c)52

R (d)25

R


83

97. A thin metal disc of radius 0.25m and mass2 kg starts from rest and rolls down aninclined plane. If its rotational kinetic energyis 4J at the foot of the inclined plane, thenits linear velocity at the same point is(a) 1.2 m/s (b) 22 m/s(c) 20 m/s (d) 2 m/s

98. A body is rolling without slipping on ahorizontal surface and its rotational kineticenergy is equal to the translational kineticenergy. The body is(a) disc (b) sphere(c) cylinder (d) ring

99. A small object of uniform density rolls up acurved surface with an initial velocity v. It

reaches upto a maximum height of 23

4vg with

respect to the initial position. The object is

v

(a) ring(b) solid sphere(c) hollow sphere(d) disc

100. In the figure shown, the plank is being pulledto the right with a constant speed v. If thecylinder does not slip then

R

v

(a) the speed of the centre of mass of thecylinder is 2v

(b) the speed of the centre of mass of thecylinder is v

(c) the angular velocity of the cylinder is /v R(d) the angular velocity of the cylinder is zero

101. In the given figure, the sphere rolls withoutslipping on the plank which is moving withconstant velocity 0v . The radius and angular

velocity of the sphere is r and respectively.The velocity of centre of the sphere is

v0

Vcm

(a) 0v r (b) 0v r (c) r (d) 0v

102. The moment of inertia of a uniform rod oflength 2l and mass m about an axis xx passingthrough its centre and inclined at an angle is

x

x

A BC

(a)2

2sin3

ml (b)

22sin

12ml

(c)2

2cos6

ml (d)

22cos

2ml

103. A homogeneous cylinder of mass M and radiusR is pulled on a horizontal plane by ahorizontal force F acting through its masscentre. Assuming rolling without slipping theangular acceleration of the cylinder is

(a)3

2F

MR (b)2

3F

MR

(c) 2FMR (d)

34

FMR

104. In both the figures all other factors are same,except that in figure (i) AB is rough and BCis smooth while in figure (ii) AB is smoothand BC is rough. Kinetic energy of the ballon reaching the bottom

A

C

Bh

(i)

A

C

Bh

(ii)

(a) is same in both the cases(b) is greater in case (i)(c) is greater in case (ii)(d) information insufficient


84

105. A uniform rod of mass m and length l issuspended by means of two light inexiensiblestrings as shown in figure. Tension in onestring immediately after the other string iscut is

A B

(a) 2mg

(b) 2 mg

(c)4

mg(d) mg

106. A solid sphere, a ring and a disc all havingsame mass and radius are placed at the topof an incline and released. The frictioncoefficient between the objects and the inclineare same but not sufficient to allow purerolling. Least time will be taken in reachingthe bottom by(a) the solid sphere(b) the ring(c) the disc(d) all will take the same time

107. A ring of mass m and radius R has threeparticles attached to the ring as shown in thefigure. The centre of the ring has a speed 0v .The kinetic energy of the system is : (Slippingis absent)

m

m2m

(a) 206 mv (b) 2

012 mv(c) 2

04 mv (d) 208 mv

108. A disc of radius r rolls without slipping on arough horizontal floor. If velocity of its centreof mass is v, then velocity of point P, as shownin the figure ( /2OP r and 60QOP ), is

PO

Q

60°

(a) 0v (b) 02

v

(c) 0 72

v(d) 0 3

2v

109. A disc of radius R rolls on a horizontal groundwith linear acceleration a and angularacceleration as shown in figure. Themagnitude of acceleration of point P shownin figure at an instant when its angularvelocity is , will be

Or

Pa

,

(a) 2 2 2( ) ( )a r r (b)a rR

(c) r (d) 2 2 2 4r r

6. INTEGER TYPE QUESTIONS1. A thin wire of length L and uniform densities

is bent into a circular loop with centre at O asshown. The moment of inertia of the loop about

the axis xx is 23 L

. What is the value of .

x x'90°

O

2. A small disc of uniform density rolls up a curvedsurface an initial velocity v. If reaches up to a

maximum height 2v

g

with respect to the

initial position. Then the value of + is_______.

v

3. A particle of mass 1 kg moving with constantvelocity 2 m/s along the line y = x + 22. Findthe angular momentum of the particle aboutorigin.

4. A man of mass m standing at distance R/2 fromthe centre of the disc of mass 2 m and radiusR. Initially both man and disc are at rest anddisc is free to rotate about the vertical axispassing through the centre. The man startswalking on the circular path of radius R/2 aboutthe axis of rotation of the disc with angularspeed 5 rad/s with respect to disc. Find the


85

angular speed of the disc in rad/s.

5. A stone of mass m, tied to the end of a string, iswhirted around in a horizontal circle(neglecting the force due to gravity). The lengthof the strip is reduced gradually keeping theangular momentum of the stone about thecentre of the circle constant. Then, the tension

in the string is given by nATr

where A is aconstant, r is the instantaneous radius of thecircle, what is the value of n.

6. In the pulley system shown. If radii of the biggerand smaller pulley are 2m and 1m respectivelyand the acceleration of block A is 5 m/s2 inthe downward direction, then the accelerationof block B is a m/s2. What is the value of(2 × a)?

2m1m

A

B

7. 50 uniform rods each of mass 117g are kept atx = 20 cm, 40 cm, 60 cm ..... parallel to the y-axis. The moment of inertia of the systemapproximately is 25x kg-m2 about y-axis. Findthe value of x.

8. Two identical semi-circular rings each of massm and radius r are placed as shown in the

figure. The moment of inertia of the systemabout x-axis is n × mr2. Find the value of n.

r

m

r

my

x

9. Two wheels A and C are connected by a belt Bas shown in figure. The radius of C is threetimes the radius of A. What would be the ratio

of the rotational inertias C

A

II

if both the wheels

have the same rotational kinetic energy?

P QCA

B

rArc

10. A solid cylinder C and a hollow pipe of samediameter are in contact when they are releasedfrom rest as shown the figure on a long inclineplane. Cylinder C and pipe P roll withoutslipping. Determine the clear gap (in m)between them after 2 3 seconds.

CP

30°


86

1. CIRCULAR MOTION1. b 2. a 3. c 4. b 5. b6. d 7. d 8. a 9. c 10. b11. c 12. a 13. a 14. a 15. b

2. MOMENT OF INERTIA16. d 17. a 18. a 19. a 20. c21. b 22. b 23. b 24. c 25. c26. a 27. c 28. d 29. d 30. a31. a 32. a 33. a 34. d 35. d36. b 37. c 38. b 39 a 40. c41. a 42. b 43. a 44. d 45. a46. a 47 d 48. c 49. d 50. c51. b 52. c 53. c 54. b

3. TORQUE AND ENERGY OF ROTATION55. c 56. a 57. b 58. c 59. b60. b 61. c 62. d 63. a 64. a65. b 66. d 67. b 68. d 69. a70. a 71. a

4. ANGULAR MOMENTUM AND ITS CONSERVATION72. c 73. b 74. c 75. c 76. b77. c 78. b 79. b 80. c 81. c82. a 83. c 84. c 85. a 86. d87. b 88. c 89. d

5. ROLLING MOTION90. b 91. c 92. c 93. b 94. a95. c 96. b 97. b 98. d 99. d100. c 101. c 102. b 103. b 104. b105. c 106. d 107. a 108. b 109. a

6. INTEGER TYPE QUESTIONS1. (8) 2. (2) 3. (4) 4. (1) 5. (3)6. (5) 7. (8) 8. (1) 9. (9) 10. (5)

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Topper’s Package Physics - XI Circular Motion & Rotational