### Multiple Choice with ONE correct answer

1.A thin circular ring of mass M and radius r is rotating about its axis with a constant angular velocity co. Two objects each of mass m, are attached gently to the opposite ends of a diameter of the ring. The wheel now rotates with an angular velocity [1983-1 mark

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2.Two point masses of 0.3kg and 0.7kg are fixed at the ends of a rod which is of length 1.4m and of negligible mass. The rod is set rotating about an axis perpendicular to its length with a uniform angular speed. The point on the rod through which the axis should pass in order that the work required for rotation of the rod is minimum, is located at a distance of           [1995-2 marks]

(a)0.42 m from the mass of 0.3kg
(b)0.70 m from the mass of 0.7kg
(c)0.98m from the mass of 0.3kg
(d)0.98m from the mass of 0.7kg
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3.A mass m is moving with a constant velocity along a line parallel to the x axis, away from the origin. Its angular momentum with respect to the origin.[1997-lmark]

a) is zero                                        b) remains constan
c) goes on increasing                           d) goes on decreasing.
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4.A smooth sphere A is moving on a frictionless horizontal plane with angular speed co and centre of mass velocity v. It collides elastically and head on with an identical sphere B at rest. Neglect friction everywhere. After the collision, their angular speeds are (0A and 0)B, respectively. Then(1992-2Marks)

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5.A disc of mass M and radius R is rolling with angular speed co on a horizontal plane as shown.The magnitude of angular momentum of the disc about the origin O is      [1999-2 marks]

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6.A cubical block of side a is moving with velocity v on a horizontal smooth plane as shown. It hits a ridge at point O. The angular speed of the block after it hits O is  [1999-2 marks]

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7.A cubical block of side L rests on a rough horizontal surface with coefficient of friction p. A horizontal force F is applied on the block as shown. If the coefficient of friction is sufficiently high so that the block does not slide before toppling, the minimum force required to topple the block is :                  [2000]

a) infinitesimal                                             b) mg/4

c) mg/2                                                           d)mg(l-p)
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8.A thin wire of length L and uniform linear mass density p is bent into a circular loop with centre at O as shown. The moment of inertia of the loop about the axis XX is:

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9.An equilateral triangle ABC formed from a uniform wire has two small identical beads initially located at A.The triangle is set rotating about the vertical axis AO.Then the beads are released from rest simultaneously and allowed to slide down, one along AB and the other along AC as shown. Neglecting frictional effects, the quantities that are conserved as the beads slide down, are:       [2000]

(a)angular velocity and total energy (kinetic and potential)
(b)total angular momentum and total energy
(c)angular velocity and moment of inertia about the axis of rotation.
(d)total angular momentum and moment of inertia about the axis of rotation.
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10.One quarter sector is cut from a O uniform circular disc of radius R. This sector has mass M. It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc. Its moment of inertia about the axis of rotation is                                                                                                                       [2001]

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11. A cylinder rolls up an inclined plane, reaches some height, and then rolls down (without slipping throughout these motions). The directions of the frictional force acting on the cylinder are [2002]

(a).up the incline while ascending and down the incline while 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
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12.Acircular platform is free to rotate in horizontal plane about a vertical axis passing through its centre. A tortoise is sitting at the edge of the platform. Now, the platform is given an angular velocity co0. When the tortoise moves along a chord of the platform with a constant velocity (with respect to the platform), the angular velocity of the platform co(t) will vary with time t as [2002]

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13.. Consider a body, shown in figure, consisting of two identical balls, each of mass M connected by a light rigidrod. If an impulse J = Mv is imparted to the body at one of its ends, what would be its angular velocity ?             [2003]

a) v/L                                 b) 2v/L
c) v/3L                                d) V/4L
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14.A particle undergoes uniform circular motion.About which point on the plane of the circle, will the angular momentum of the particle remain conserved ?               [2003]

(a)centre of the circle
(b)on the circumference of the circle
(c)inside the circle
(d)outside the circle
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15.A horizontal circular plate is rotating about a ‘ vertical axis passing through its centre with anangular velocity co0. A man sitting at the centre having two blocks in his hands stretches out his hands so that the moment of inertia of the system doubles. If the kinetic energy of the system is K initially, its final kinetic energy will be

a) 2K                  b) K/2
c) K                    d) K/4
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16.A disc is rolling (Withoutslipping)with angular velocity to. P and Q are two points equidistant \\um\UT\m\m\ from the centre C. The order of magnitude of velocity is                                                                                     [2004]

a)VQ>Vc>Vp                                                   b) Vp > Vc > Vq

c) Vp = Vp, Vq = Vc                                       d) Vp < Vc > Vq
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17.A child is standing with folded hands at the centre of a platform rotating about its central axis. The kinetic energy of the system is K. The child now stretches his arms so that the moment of inertia of the system doubles. The kinetic energy of the  system now is

(a):2K                      (b):K/2
(c):K/4                    (d):4K
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18. From a circular disc of radius R and mass 9M, a small disc of radius R/3 is removed. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through O is

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19.A particle is confined to rotate in a circular path with decreasing linear speed. Then Which of the following is correct

(a).L(angular momentumjis conserved about the centre
(b).only direction of Lis conserved
(c).it spirals towards the centre
(d).its acceleration is towards the centre
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20.A solid sphere of mass M and radius R havingmoment of inertia / about its diameter is recast into a solid disc of radius r and thickness t. The moment of inertia of the disc about an axis passing the edge and perpendicular to the plane remains I. Then R and r

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22.A bob of mass M is suspended by a massless string of length L. The horizontal velocity V at position A is just sufficient to make it reach the point B. The angle ©at which the speed of the bob is half of that at A, satisfies

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23.A Small   object of uniform density rolls up a curved surface with an initial velocity v. It reaches upto a maximum height of3v ²/4g  With  respect to the initial position. The object

a) ring         b) solid sphere

(  c) hollow sphere            (d) disc
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24.ball of mass (m) 0.5 kg is attached to the end of a string having length (L) 0.5 m. The ball is rotated on a horizontal  circular path about vertical axis.The maximum tension that the  string can bear is 324 N. The maximum possible  value of angular velocity of ball (in radian/s) is  [2011]

a) 9 )                                              (b).18

c) 27                                                  (  d) 36
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25Aarticle of mass m is projected with a velocity v making an angle of 45 ° with the horizontal. The magnitude of the angular momentum of the projectile about the point of projection, when the particle is at its maximum height h, is  [1990-2 marks]

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26.The moment of inertia of a thin square plate ABCD, as shown in the figure, of uniform thickness about an axis passing through the centre O and perpendicular to the plane of the plate is

Where Ij, I2,13 and I4 are respectively the moments of inertia about axis 1,2,3 and 4 which are in the plane of the plate.                                                                                                   [1992-2 marks]

27.A Tube  of length L is filled completely with an incompressible liquid of mass M and closed at both the ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity oo. The force exerted by the liquid at the other end is                 [1992-2 marks]

28.Two    particles, each of mass m and charge q, are attached to the two ends of a light rigid rod of length 2R. The rod is rotated at constant angular speed about a perpendicular axis passing through its centre. The ratio of the magnitudes of the magnetic moment of the system and its angular momentum about the centre of the rod is          [1998-2 marks]

29.Let    I be the moment of inertia of a uniform square  plate about an axis AB that passes through its centre and is parallel to two of its sides. CD is a line in the plane of the plate that passes through the centre of the plate and makes an angle 0 with AB. The moment of inertia of the plate about the axis CD is then equal to                                             [1998-2 marks]

a) I                                                      b) I sin2Î¸
c) lcos2Î¸                                    d) lcos2(Î¸/2)

30.The  torque t on a body about a given point is found to be equal to AxL where A is a constant vector, and l is the angular momentum of the body about that point. From this it follows that      [1998-2 marks]

(a) dl/dt is perpendicular to L at all instants of time
(b).The component of L in the direction of A does not change with time
(c)The Magnitude of L does not change with time
(d)L  does not change with time

31.  solid cylinder is rolling down a rough inclined plane of inclination Î¸ Then [2006-5 marks]

(a) The friction force is dissipative
(b) The friction force is necessarily changing
(c)The    friction force will aid rotation but hinder translation
(d) The friction force is reduced if 0 is reduced

32. A    thin ring of mass 2 kg and radius 0.5 m is rolling without slipping on a horizontal plane  with velocity 1 m/s.    A small ball of mass 0.1 kg, moving velocity 20 m/s in the opposite direction, hits ring at a height of 0.75 m and goes vertically up with velocity 10 m/s. Immediately after the collision.

(a) The  ring has pure rotation about its stationary CM.
(b). The ring comes to a complete stop
(c) Friction between the ring and the ground is to the left.
(d) there is no friction between the ring and the ground.

### H Assertion & Reasoning type

Instructions : Each question contains statement-1 (assertion) and statement-2 (reason). Of these statements, mark correct choice if

(a). Statement-1 and 2 are true and statement-2 is a correct explanation for statement-1
(b).Statement-1 and 2 are true and statement-2 is not a correct explanation for statement-1
(c).Statement-1 is true, statement-2 is flase
(d).Statement-1 is false, statement-2 is true

33.Statement – l:Two cylinders, one hollow (metal)  and the other solid (wood) with the same mass and identical dimensions are simultaneously allowed to roll without slipping down an inclined plane from the same height. The hollow cylinder will reach the bottom of the inclined plane first;

Statement – 2:.By the principle of conservation of energy, the total kinetic energies of both the cylinders are identical when they reach the bottom of the incline.                                                     [2008]

## Comprehension based questions

### Read the passage given below and answer the questions that follow

Two discs A and B are mounted coaxially on a vertical axle. The discs have moments of inertia I and 21 respectively about the common axis. DiscA is imparted an initial angular velocity 2oa using the entire potential energy of a spring compressed by a distance xr Disc B is imparted an angular velocity to by a spring having the same spring constant and compressed by a distance x2. Both the discs rotate in the clockwise direction.

34.The ratio X1/x2 is [2007-4 marks]

(a).2                                    b) 1/2

c) âˆš2                                  d) 1âˆš2

35.When disc B is brought in contact with disc A, they acquire a common angular velocity in time t. The average frictional torque on one disc by the other during this period is         [2007-4 marks]

36.The loss of kinetic energy during the above process  is               [2008-4 marks]

### Read the passage given below and answer the questions that follow

37. A  uniform thin cylindrical disk of mass M and radius R is attached to two identical massless springs of spring constant k which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disc symmetrically on either side at a distance d from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is L. The disc is initially at its equilibrium position with its centre of mass (CM) at a distance L from the wall. The disk rolls without slipping with velocity V0 = v0i. The coefficient of friction is Î¼

37.The net external force acting on the disk when its centre of mass is at displacement x with respect to its equilibrium position is

38.The centre of mass of the disk undergoes simple harmonic motion with angular frequency u equal to

39.The maximum value of V0 for which the disk will roll without slipping is

Two discs A and B are mounted coaxially on a ‘vertical axle. The discs have moments of inertia I and 21 respectively about the common axis. Disc A is imparted an initial angular velocity 2musing the entire potential energy of a spring compressed by a distance xt. Disc B is imparted an angular velocity m by a spring having the same spring constant and compressed by a distance Xj,. Both the discs rotate in the clockwise direction.                                             [2007]

40.The ratio of xt/x2 is
(a) 2                               (b).1/2
(c)âˆš2                                 (d)1/âˆš2

41.When disc B is brought in contact with disc A, they acquire a common angular velocity in time t. The average frictional torque on one disc by the other during this period is

42.The    loss of kinetic energy during the above process

43. A    particle is projected at time t = 0 from a point P  on the ground with a speed v0, at an angle of 45 ° to the horizontal. Find the magnitude and direction of the angular momentum of the particle about P at time t = v0/g     [1984-6 marks]

44. A    small sphere rolls, down without slipping from  the top of a track in a vertical plane. The track has an elevated section and a horizontal part. The horizontal part is 1.0 metre above the ground level and the top of the track is 2.4 metre above the ground. Find the distance on the ground with respect to thepoint B (which is vertically below the end of the track as shown in fig.)where the sphere lands .During its flight as a projectile, does the sphere continue to rotate about its centre of mass? Explain. [1987-7 marks]

45.A  carpet of mass M, made of inextensible, material is rolled along its length in the from of a cylinder of radius R and is kept on rough floor. The carpet starts unrolling without sliding on the floor when a negligibly small push is given to it. Calculate the horizontal velocity of the axis of the cylindrical part of the carpet when its radius reduces to R/2.      [1990-8 marks]

46.A  homogeneous rod AB of length L*1.8m and mass M is pivoted at the centre O in such a way that it can rotate freely in the vertical plane (fig).  The rod is initially in the horizontal position. An insect S of the same mass M falls vertically with speed von the point C, midway between the points O and B. Immediately after falling, the insect moves towards the end B such that the tod rotates with a constant angular velocity ©   [1992-8 marks]

(a)Determine the angular velocity go in terms of v and L.
(b).If the insect reaches the end B when the rod has turned through an angle of 90 °, determine v.

47.  block X of mass 0.5kg is held by a long massless string on a frictionless inclined plane of inclination 30 ° to the horizontal. The string is wound on a uniform solid cylindrical drum Y of mass 2kg and of radius 0.2m as shown, in figure. The drum is given an initial angular velocity such that the block X starts moving up the plane.    [1994-6 marks]

(a).Find the tension in the string during the motion.
(b).At  a certain instant of time the magnitude of the angularvelocity of Y is 10 rads-1. Calculate the distance travelled by X from that instant of time until it comes to rest

48.Two uniform thin rods A and B of length 0.6m each and of masses 0.01kg and 0.02kg respectively are rigidly joined end to end. The combination is pivoted at object- the lighter end, P as shown is fig, such that it can , freely rotate about point P in a vertical plane. Asmall object of mass 0.05 kg, moving horizontally, hits the lower end of the combination and sticks to it.

What should be the velocity of the object so that the system could just be raised to the horizontal position?                   [1994-6 marks]

49.A    rectangular rigid fixed block has a long horizontal edge. A solid homogeneous cylinder of radius R is placed horizontally at rest Its length parallel to the edge such that the axis of the cylinder and the edge of the block are in the same vertical plane as shown in figure, there is sufficient friction present at the edge so that a very small displacement causes the cylinder to roll off the edge without slipping. Determine [1995-10 marks]

(a).The angle 0C through which the cylinder rotates  before it leaves contact with the edge,
(b).he speed of the centre of mass of the cylinder before leaving contact with the edge, and
(c).The ratio of the translational to rotational kinetic energies of die cylinder when its centre of mass is in horizontal line with the edge.

50.A    uniform disk of mass m and radius R is projected horizontally with velocity v0 on a rough horizontal floor so that it starts off with a purely sliding motion at t = 0. After t0 seconds, it acquires a purely rolling motion as shown in figure. [1997-5 marks]

(a).Calculate the velocity of the centre of mass of the disc at t0.
(b)Assuming the coefficient of friction to be p. calculate t0. Also calculate the work done by the frictional force as a function of time and the total work done by it over a time t much longer than tQ.

51.Two  thin circular discs of mass 2kg and radius 10cm each are joined by a rigid massless rod of length 20cm. The axis of the rod is along the perpendicular to the planes of the disc through their centres. This object is kept on a truck in such a way that the axis of the object is horizontal and perpendicular to the direction of motion of the truck. Its friction with the floor of the truck is large enough so that the object can roll on the truck without slipping. Take x-axis as the direction of  motion of the truck and z -axis as the vertically upwards direction. If the truck has an acceleration of 9m/s2 calculate :                       [1997-5 marks

5151515151851451515151551

(a).The force of friction on each disc

(b).The magnitude and the direction of the frictional torques acting on each disc about the centre of mass O of the object. Express the torque in the vector form in terms of unit vectors i, j and k in the x-y and z directions.

52. A    uniform circular disc has radius R and mass m. A particle also of mass m, is fixed at a point A on the edge of the disc as shown in fig.

The disc can rotate freely about a fixed horizontal chord PQ that is at a distance R/4 from the centre C of the disc. The line AC is perpendicular to PQ. Initially, the disc is held vertical with the point A at its highest position. It is then allowed to fall so that it starts rotating about PQ. Find the linear speed of the particle as it reaches its lowest position.      [1998-8 marks]

5252582502502520525252525

53.A  man pushes a cylinder of JL mass mt with the help of a plank of mass m2 as shown. There is no slipping at any contact.

The horizontal component of the force applied by the man is F. Find

(a).The acceleration of the plank and the centre of mass of cylinder, and

(b).The magnitudes and directions of frictional  forces at contact points.         [1999-10 marks]

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