PhysicsPhysics QuestionsRotational Mechanics Questions for CBSE Class 11th

Rotational Mechanics Questions for CBSE Class 11th

  1. A ballet dancer, dancing on a smooth floor is spinning about a vertical axis with her arms folded with angular velocity of 20 rad/s. When she stretches her arms fully, the spinning speed decreases to l0 rad/s. If I is the initial moment of inertia of the dancer, the new moment of inertia is
  2. A rod of mass M kg and length L m is bent in the form of an equilateral triangle as shown in Fig. The moment of inertia of the triangle about a vertical axis perpendicular to the plane of the triangle and passing through the centre O (in units of kg m 2 ) is
  3. If a circular concentric hole is made in a disc, then about its own geometrical axis
  4. Angular momentum of the particle rotating with a central force is constant due to
  5. Three identical rods each of length L and mass M are joined together to form a letter H as shown in fig. What is the moment of inertia of the system about one of the sides of H ?
  6. The flywheel of a motor has a moment of inertia of 9OO kg-m 2. If the motor produces a constant torque of 2700 N-m, then the flywheel starts from rest with an angular acceleration of
  7. A cubical block of side a is moving with velocity v on a horizontal smooth plane as shown in fig. It hits a ridge at point O. The angular speed of the block after it hits O is
  8. The moment of inertia of a uniform thin rod of length L and mass M about an axis passing through a point at a distance of L/ 3 from one its ends and perpendicular to the rod is
  9. A solid body rotates with deceleration about a stationary axis with an angular deceleration α = k ω ; where k is a constant and ω is the angular velocity of the body. If the initial angular velocity is ω 0 , then time taken by the body to come to rest—
  10. A hollow disc has inner radius r and outer radius 2r then its radius of gyration about its own geometrical axis is
  11. A solid disc of radius 0.4 m can rotate about an axis passing through O its centre and perpendicular to the plane of the disc. Initially the disc is not rotating. Now a torque τ is applied on the disc which varies with time as shown in the figure. Then angular momentum of the disc at t = 3s is
  12. A particle of mass 0.5 kg is projected horizontally with velocity 10 m/s from the top of a tower of height 40 m. The angular momentum of the particle about the base of the tower just after projection is
  13. A block of mass m is attached to the end of an inextensible string which is wound over a rough pulley p of mass m and radius R. Assuming no slipping between the string and the pulley, find the acceleration of the block when released
  14. Moment of inertia of a thin square plate about an axis passing through its diagonal is I. Its moment of inertia about an axis passing through its centre in the plane of the plate and making an angle θ with the diagonal is
  15. A solid cylinder rolls up an inclined plane of inclination θ with an initial velocity v. How far does the cylinder go up the plane?
  16. A circular disc A of radius r is made from an iron plate of thickness t and another circular disc B of radius 4r and thickness t 4 . The relation between moments of inertia I A a n d   I B and about the same axis is:
  17. Moment of inertia of a thin rod of mass M and length L about an axis passing through its centre is ML 2 12 . lts moment of inertia about a parallel axis at a distance of L 4 from this axis is given by
  18. A solid sphere is in rolling motion. In rolling motion a body possesses translational kinetic energy ( K t ) as well as rotational kinetic energy ( K r ) simultaneously. The ratio K t :   ( K t + K r ) for the sphere is
  19. A cylinder is rolling without slipping on a horizontal plane P. The friction between the plank P and the cylinder is sufficient for no slipping. The coefficient of friction between the plank and the ground surface is zero. Initially, P is attached with a string S as shown in the figure. If the string is now burned, then.
  20. Two steel ball of equal diameter are connected by a rigid bar of negligible weight as shown and are dropped in the horizontal position from height h above the heavy steel and brass base plates. If the coefficient of restitution between the ball and steel base is 0.6 and that between the other ball and the brass base is 0.4. The angular velocity of the bar immediately after rebound is. (Assume the two impacts are simultaneous).
  21. A uniform rod of mass m and length l o is rotating with a constant angular speed ω about a vertical axis passing through its point of suspension. The moment of inertia of the rod about the axis of rotation if it makes an angle θ to the vertical (axis of rotation) is
  22. Consider two objects with m 1 > m 2 connected by a light string that passes over a pulley having a moment of inertia of I about its axis of rotation as shown in figure. The string does not slip on the pulley or stretch. The pulley tums without friction. The two objects are released from rest separated by a vertical distance 2h.The translational speeds of the objects as they pass each other is
  23. Four solid spheres are made to move on a rough horizontal surface. Sphere P is given a spin and released. Sphere Q is given a forward linear velocity. Spheres R and S are given linear and rotational motions as shown in the figure. Directions of the friction force on spheres P, Q, R and S are respectively.
  24. Seven identical disc are arranged in a hexagonal, planar pattern so as to touch each neighbor, as shown in the figure. Each disc has mass m and radius r. What is the moment of inertia of the system of seven discs about an axis passing through the centre of central disk and normal to plane of all discs?
  25. Figure shows a thin metallic triangular sheet ABC. The mass of the sheet is M.The moment of inertia of the sheet about side AC is:
  26. Two spheres each of mass M and radius R 2 are connected at their centres with a mass less rod of length 2R. What will be the moment of inertia of the system about an axis passing through the centre of one of the sphere and perpendicular to the rod?
  27. A metallic sheet of mass 2 kg has the shape of an isosceles triangle OAB as shown in figure. Then its moment of inertia about an axis passing through O and perpendicular to its plane is
  28. A hollow disc of inner radius R and outer radius 2R has moment of inertia I about an axis passing through its centre and perpendicular to the plane of the disc. When the disc is melted and recast into a solid disc of same thickness, its moment of inertia about the same axis is
  29. A body spins about a fixed axis. If its angular velocity and angular momentum both are doubled, its kinetic energy
  30. Angular velocity of a body at an instant of time is ω = 2 i ^ + j ^ + 4 k ^   r a d / s . A constant torque of 2 i ^ + 2 j ^ + 2 k ^ is applied on it . Then power input to the body is
  31. A ring of radius 2m is rolling without slipping with constant velocity of 2 m/s on a plane surface. At the point of contact with the surface, O is a point on the ring. Then acceleration of O is
  32. Moment of inertia comes into play:
  33. The moment of inertia of a thin uniform circular disc about one of the diameters is I. Its moment of inertia about an axis perpendicular to the plane of the disc and passing through its centre is
  34. The moment of inertia of a ring of mass m and radius r about an axis, passing through the centre and perpendicular to the plane of the ring, is:
  35. Two rings of the same radius r and mass m are placed such that their centres are at a common point and their planes are perpendicular to each other. The moment of inertia of the system about an axis passing through the centre and perpendicular to the plane of one of the rings is:
  36. A circular disc X of radius R is made from an iron plate of thickness t and another disc f of radius 4R is made from an iron plate of thickness t 4 . Then the relation between the moment of inertia I X and I Y is
  37. A thin rod of length Land mass M is bent at its midpoint into two halves so that the angle between them is 90°. The moment of inertia of the bent rod about an axis passing through the bending point and perpendicular to the plane defined by the two halves of the rod is
  38. A cylindrical drum is pushed along by a board of length l. The drum rolls forward on the ground a distance of l 2 . There is no slipping at any instant. During the process of pushing the board, the distance moved by the man on the ground is:
  39. Find moment of inertia I of the system (axis is perpendicular to the plane passing through O)
  40. A uniform disk, a thin hoop (ring), and a uniform sphere, all with the same mass and same outer radius, are each free to rotate about a fixed axis through its centre. Assume the hoop is connected to the rotation axis by light spokes. With the objects starting from rest, identical forces are simultaneously applied to the rims, as shown. Rank the objects according to their angular momentum after a given time t, least to greatest.
  41. A thin wire of mass M and length L is turned into a ring. The moment of inertia about its will be :
  42. When a sphere rolls without slipping the ratio of its kinetic energy of translation to its total kinetic energy is
  43. Three point masses each of mass m are placed at the corners of an equilateral triangle of side b. The moment of inertia of the system about an axis coinciding with one side of the triangle is.
  44. An inclined plane makes an angle of 30 O with the horizontal. A solid sphere rolling down this inclined plane from rest without slipping has a linear acceleration equal to
  45. A uniform bar of length 6 α and mass 8 m lies on a smooth horizontal table. Two point masses m and 2 m moving in the same horizontal plane with speeds 2 v and v respectively strike the bar as shown in fig. and stick to the bar after collision. Denoting angular velocity about the centre of mass, total energy and centre of mass velocity of ω E and v c respectively, we have after collision
  46. A uniform solid sphere rolls on a horizontal surface at 20 m/s. It then rolls up an incline having an angle of inclination of 30 o with the Horizontal. If the friction losses are negligible, the value of height h above the ground where the ball stops is
  47. A ring is rolling on a surface without slipping. What is the ratio of its translational to rotational kinetic energies ?
  48. A body of M.I. of 3 kg-m 2 rotating with an angular speed of 2 rad/sec has the same KE as a mass of 12 kg moving with a speed of
  49. A ring starts from rest and acquires an angular speed of 10 rad/s in 2 s. The mass of the ring is 500 g and its radius is 20 cm. The torque on the ring is
  50. The curve for the moment of inertia of a sphere of constant mass M versus its radius of gyration K will be
  51. 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 the other along AC as shown in the figure. Negleting frictional effects, the quantities that are conserved as the beads slide down are
  52. The moment of inertia of solid cylinder about its axis is I. It is allowed to roll down an inclined plane without slipping. If its angular velocity at the bottom be ω then kinetic energy of the cylinder will be
  53. The position vector of a particle bis r = [ 2 i + 3 j + k while its linear momentum is p = (2i+3j+k). Then its angular momentum about the origin is
  54. The point of contact of an object in pure rolling will have
  55. A sphere cannot roll without applying external force on
  56. A uniform metallic rod rotates about its perpendicular bisector with constant angular speed. If it is heated uniformly to raise its temperature slightly, then [AIIMS 2015]
  57. A solid cylinder of mass 2 kg and radius 50 cm rolls up an inclined plane of angle inclination 30 0 . The centre of mass of cylinder has speed of 4 ms -1 . The distance travelled by the cylinder on the inclined surface will be (Take, g = 10 ms -2
  58. Three thin rods each of length ‘L’ and mass M are placed along X, Y and Z axis such that one end of each rod is at origin. The radius of gyration of this system about Z–axis is
  59. From a disc of radius R and mass M a circular hole of diameter R whose rim passes through the centre is cut. What is the moment of inertia of the remaining part of the disc about a perpendicular axis, passing through the centre?
  60. A wheel comprises of a ring of radius R and M and three spokes of mass m each moment of inertia of the wheel about its axis is:
  61. A rod of weight W is supported by two parallel knife edges A and B and is in equilibrium in a horizontal position. The knives are at a distance d from each other. The centre of mass of the rod is at distance x from A. The normal reaction on A is
  62. If a disc is rolling without slipping ( centre of mass is moving rightward and the body is rotating about the centre of mass clockwise), then the ratio of speeds of point C to that of B is
  63. A rope is wound around a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N ?
  64. Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities ω 1 and ω 2 . They are brought into contact face to face coinciding the axis of rotation. The expression for loss of energy during this process is
  65. A disc of mass m and radius R is rolling without slipping on a horizontal plane with angular velocity ω . P is a point in space in the plane of the disc at a height 2R above the horizontal plane. Then angular momentum of the disc about point P is
  66. A particle of given mass is moving in a circular path of radius ‘r’ with kinetic energy K. If radius of circular path is reduced to r/2 keeping the angular momentum constant, the final kinetic energy of the particle will be
  67. Two thin rings, each of mass m and radius R are rigidly attached with each other as shown in figure. Then moment of inertia of the system of rings about the diameter AB is
  68. A solid cylinder is rolling down a rough inclined surface with out slipping and it takes 3 sec to reach the bottom. Now mass of the cylinder is quadrupled and radius is doubled. If this cylinder is allowed to roll down the same inclined plane without slipping, then it will reach the bottom in time
  69. A thin rod OA of length l is pivoted about a horizontal axis through its lower end O. Initially the rod is held in vertical position. Now the rod is released. What is the angular acceleration of the rod when it makes an angle 30 o with vertical?
  70. Moment of inertia of a square sheet about one side is I. Then the moment of inertia of the same square sheet about one of its diagonals is
  71. A round object is rolling without sliding on a horizontal surface. If ratio of its translational energy to total energy is 2 : 3, then the object is a
  72. A thin solid disc of mass 1 kg and radius 25 cm is rotating about its own geometrical axis AB with angular velocity 12 rad/s. A particle of mass 1 kg is gently attached to the rim of the disc. Then new angular velocity of the disc is
  73. A force F is applied on the top of a cube as shown in figure. The coefficient of friction between the cube and the ground is μ . If F is gradually increased, the cube will topple before sliding if
  74. A solid disc is rolling without slipping on a horizontal surface. Then what fraction of its total energy is produced due to the transnational motion of the disc ?
  75. Two uniform rods of mass ‘m’ and length ‘l’ form a cross, moment of inertia of cross about an axis parallel to CD passing through A is
  76. A grinding machine whose wheel has a radius of 1/π is rotating at 2.5 rev/sec. A tool to be shar-pened is held against the wheel with a force of 40N. If the coefficient of friction between the tool and wheel is 0.2, power required is
  77. In pure rolling motion of a ring a) it rotates about instantaneous point of contact of ring and ground b) its centre of mass moves in translatory motion only c) its centre of mass will have translatory as well as rotatory motion
  78. A ballet dancer spins about a vertical axis at 60 rpm with his arms closed. Now he stretches his arms such that M.I increases by 50%. The new speed of revolution is
  79. A disc of mass ‘m’ and radius R has a concentric hole of radius ‘r’. Its M.I. about an axis through centre and normal to its plane is
  80. A sphere is performing pure rolling with constant velocity V as shown in figure. The radius of curvature of trajectory of point A as shown in the figure is (OA = R/2)
  81. A string is wrapped around a wheel of radius ‘r’. The axis of the wheel is horizontal and its M.I. about the axis is I. Weight ‘mg’ is tied to free end of the string which is released to fall down from rest position. The angular velocity of the wheel after it has fallen through distance ‘h’ will be
  82. Two point masses are attached to the two ends of a rod of negligible mass. If torque is applied they rotate with angular acceleration . If the distance between the two masses is doubled by doubling the length of the rod same torque is applied, then they move with angular acceleration
  83. In an orbital motion, the angular momentum vector is
  84. A solid sphere is rolling on a frictionless surface, shown in figure with a translational velocity vm/s. If it is to climb the inclined surface then v should be
  85. Length AB of the wedge shown in figure is rough and BC is smooth. A solid cylinder rolls without slipping from A to B. If AB=BC, then ratio of translational kinetic energy to rotational kinetic energy when the cylinder reaches point C is :
  86. A solid cylinder of mass M and radius R is rolled horizontally on a rough surface as shown in the figure. Choose the correct alternative(s)
  87. A solid sphere of radius R is rolled by a force F acting at the top of the sphere as shown in the figure. There is no slipping and initially sphere is in the rest position then : A) Work done by force F when the centre of mass moves a distance S is 2FS. B) Speed of the CM when CM moves a distance S is C) Work done by the force F when CM moves a distance S is FS D) Speed of the CM when CM moves a distance S is
  88. A ball of radius r rolls inside a hemispherical shell of radius R. It is released from rest from point A as shown in figure. The angular velocity of centre of the ball in position B about the centre of the shell is:
  89. A particle moves on a straight line with a uniform velocity. The incorrect statement about its angular momentum
  90. The ratio of the accelerations for a solid sphere (mass ‘m’ and radius ‘R’) rolling down and incline of angle ‘θ’ without slipping and slipping down the incline without rolling is
  91. A 60 cm metal rod O A is joined to another 100 cm metal rod O B to form an L shaped single piece. This piece is hung on a peg at the joint O . The two rods are observed to be equally inclined to the vertical. If the two rods are equally thick, the ratio of density of O A to that of O B is
  92. A constant torque of 1000 Nm turns a wheel of of M.I 200kg m 2 about an axis through centre. The angular velocity after 3s is
  93. A disc of radius 2m and mass 100 kg rolls on a horizontal floor. Its centre of mass has speed of 20 cm/s. How much work is needed to stop it?
  94. The following bodies have same mass and radius. The body with largest moment of inertia about geometric axis is
  95. When a hollow sphere is rolling without slipping on a rough horizontal surface then the percentage of its total K.E which is Translational is
  96. A particle is moving uniformly along a straight line as shown in the figure. During the motion of the particle from A to B, the angular momentum of the particle about ‘O’
  97. A uniform rod of length l is pivoted at point A. It is struck by a horizontal force which delivers an impulse J at a distance x from point A as shown in figure, impulse delivered by pivot is zero, if x is equal to :
  98. If 484 J of energy is spent in increasing the speed of a wheel from 60 rpm to 360 rpm, the M.I. of the wheel is
  99. Keeping the mass of earth as constant, if its radius is reduced to 1/4th of its initial value, then the period of revolution of earth about its own axis and passing through the centre, in hours, is (Assume earth to be a solid sphere and its initial period of rotation as 24 hrs).
  100. A particle of mass m moves in a circle on a smooth horizontal plane with velocity v 0 at a radius R 0 . The mass is attached to a string which passes through a smooth hole in the plane as shown. The tension in the string is increased gradually and finally it moves in a circle of radius R 0 /2. The final value of the kinetic energy is
  101. A ring of radius R is first rotated with an angular velocity ω 0 and then carefully placed on a rough horizontal surface. The coefficient of friction between the surface and the ring is µ. Time after which its angular speed is reduced to half is
  102. If the Earth shrinks such that its density becomes 8 times to the present value, then the new duration of the day in hours will be
  103. A circular platform is mounted on a frictionless vertical axle. Its radius R = 2 m and its moment of inertia about the axle is 200 kg m 2 . It is initially at rest. A 50 kg man stands on the edge of the platform and begins to walk along the edge at the speed of 1 ms – 1 relative to the ground. Time taken by the man to complete one revolution relative to the platform is
  104. An automobile moves on a road with a speed of 54 km h – 1 . The radius of its wheels is 0.45 m and the moment of inertia of the wheel about its axis of rotation is 3 kg m 2 . If the vehicle is brought to rest in 15 s, the magnitude of average torque transmitted by its brakes to the wheel is
  105. When a solid sphere rolls without slipping down an inclined plane making an angle θ with the horizontal, the acceleration of its centre of mass is a. If the same sphere slides without friction, its
  106. A thin-walled pipe rolls along the floor. What is the ratio of its translational kinetic energy to its rotational kinetic energy about the central axis parallel to its length?
  107. A copper ball of mass m = 1 kg with a radius of r = 10 cm rotates with angular velocity ω = 2 rad / s about an axis passing through its centre. The work should be performed to increase the angular velocity of rotation of the ball two fold is
  108. A wheel of moment of inertia 2 . 0 × 10 3 kgm 2 is rotating at uniform angular speed of 4 rads – 1 . What is the torque required to stop it in one second.
  109. A solid sphere is rotating freely about its symmetry axis in free space. The radius of the sphere is increased keeping its mass same. Which of the following physical quantities would remain constant for the sphere?
  110. A ball rolls without slippping. The radius of gyration of the ball about an axis passing through its centre of mass is k. If radius of the ball be R, then the fraction of total energy associated with its rotation will be
  111. A hoop of radius 2 m weighs 100 kg. It rolls along a horizontal floor so that its centre of mass has a speed of 20 cm s – 1 . How much work has to be done to stop it?
  112. Three bodies, a thin ring, a solid cylinder and a solid sphere roll down the same inclined plane without slipping. They start from rest. The radii of the bodies are same. Which of the bodies reaches the ground with maximum velocity?
  113. A rhombus ABCD is made of four identical thin wires each of mass m and length l. as shown in figure. Then moment of inertia of the rhombus about side AB is
  114. The moment of inertia of a hollow cubical box of mass M and side a about an axis passing through the centres of two opposite faces is equal to
  115. A man stands on a rotating platform with his arms stretched holding a 5 kg weight in each hand. The angular speed of the platform is 1.2 rev s – 1 . The moment of inertia of the man together with the platform may be taken to be constant and equal to 6 kg m 2 . If the man brings his arms close to his chest with the distance of each weight from the axis changing from 100 cm to 20 cm. The new angular speed of the platform is
  116. A solid disc of mass 1 kg and radius 20 cm can rotate about an axis passing through its centre and perpendicular to its plane. A constant torque of 2 N-m is applied on the disc. Then its angular momentum after 5 sec is
  117. A man spinning in free space changes the shape of his body by spreading his arms or curling up. By doing this he can change his
  118. Two identical solid cylinders roll from rest on two identical inclined planes of slant lengths s and 2s but of the same height h. Then, the velocities, v 1 and v 2 acquired by the cylinders when they reach the bottom of the incline are related as
  119. A homogeneous solid ball is placed on an inclined plane making an angle θ with the horizontal. At what values of the coefficient of friction μ will the ball roll down the plane without slipping?
  120. A solid disc rolls clockwise without slipping over a horizontal path with a constant speed v. Then the magnitude of the velocities of points A, B and C (see figure) with respect to a standing observer are, respectively,
  121. From a disc of radius R and mass M, a circular hole of diameter R, whose rim passes through the centre is cut. What is the moment of inertia of the remaining part of the disc about perpendicular axis, passing through centre?
  122. The moment of the force, F = 4 i ^ + 5 j ^ – 6 k ^ , at (2, 0,-3), about the point (2, -2, -2) is given by
  123. A light rod of length l has two masses m 1 and m 2 attached to its two ends. The moment of inertia of the system about an axis perpendicular to the rod and passing through the centre of mass is
  124. What should be the minimum coefficient of static friction between the plane and the cylinder for the cylinder not to slip on an inclined plane?
  125. A particle of mass 2kg, located at position i ^ + j ^ units has velocity 2 i ^ – j ^ – 2 k ^ units. Its angular momentum about origin is
  126. Statement I: A disc is rolling on an inclined plane without slipping. The velocity of centre of mass is V. These other points on the disc lie on a circular arc having same speed as centre of mass. Statement II: When a disc is rolling on an inclined plane. The magnitude of velocities of all the point from the contact point is same, having distance equal to radius r.
  127. Average torque on a projectile of mass m (initial speed u and angle of projection θ ) between initial and final positions P and Q as shown in figure, about the point of projection is:
  128. The rope shown in figure is wound around a cylinder of mass 4 kg and moment of inertia 0.02 kgm 2 about the cylinder axis. If the cylinder rolls without slipping, then the linear acceleration of its centre of mass is
  129. A uniform disc of mass m is fitted (pivoted smoothly) with a rod of mass m 2 . If the bottom of the rod is pulled with a velocity v, it moves without changing its angle of orientation and the disc rolls without sliding. Thd kinetic energy of the system (rod + disc) is.
  130. A thin horizontal uniform rod AB of mass m and length l can rotate freely about a vertical axis passing through its end A. At a certain moment, the end B starts experiencing a constant force F which is always perpendicular to the original position of the stationary rod and directed in a horizontal plane. The angular velocity of the rod as a function of its rotation angle θ measured relative to the initial position should be
  131. Figure shows a counterweight of mass m suspended by a cord wound around a spool of radius r, forming part of a turntable supporting the object. The turntable can rotate without friction. When the counterweight is released from rest, it descends through a distance h, acquiring a speed v. The moment of inertia I of the rotating apparatus is
  132. The top in figure has a moment of inertia of 4.00 x 10 – 4 kg . m 2 and is initially at rest. It is free to rotate about the stationary axis AA’. A string, wrapped around a peg along the axis of the top, is pulled in such a manner as to maintain a constant tension of 2.5 N. If the string does not slip while it is unwound from the peg, what is the angular speed of the top after 80.0 cm of string has been pulled off the peg?
  133. Three identical thin rods, each of mass m and length l, are joined to form an equilateral triangular frame. The moment of inertia of the frame about an axis parallel to its one side and passing through the opposite vertex is
  134. A particle of mass M may move with the velocity v, along AO, or DE or BC. Then which of the following statements is not correct about particle’s angular momentum about point O.
  135. In first figure a meter stick, half of which is wood and the other half steel is pivoted at the wooden end at A and a force is applied at the steel end at O. In second figure the stick is pivoted at the steel end at O and the same force is applied at the wooden end at A.The angular acceleration
  136. A uniform disc of mass M and radius R is mounted on an axle supported in frictionless bearings. A light cord is wrapped around the rim of the disc and a steady downward pull T is exerted on the cord. The angular acceleration of the disc is
  137. A student sits on a freely rotating stool holding two dumbbells, each of mass 5.0 kg (Fig. a). When his arms are extended horizontally (Fig. a), the dumbbells are 1.0 m from the axis of rotation and the student rotates with an angular speed of 1.0 rad/s. The moment of inertia of the student plus stool is 5.0 kg . m 2 and is assumed to be constant. The student pulls the dumbbells inward horizontally to a position 0.50 m from the rotation axis (Fig. b). The new angular speed of the student is
  138. A thin wire of length L and uniform linear mass density ρ is bent into a circular loop with centre at O as shown. The moment of inertia of the loop about an axis, XY is
  139. A uniform thin bar of mass 6 m and length 12L is bent to make a regular hexagon. Its moment of inertia about an axis passing through the centre of mass and perpendicular to the plane of the hexagon is:
  140. Three identical thin rods, each of length L and mass m, are welded perpendicular to one another as shown in figure. The assembly is rotated about an axis that passes through the end of one rod and is parallel to another. The moment of inertia of this structure about this axis is
  141. Four spheres each having mass m and radius r are placed with their centres on the four corners of a square of side a. Then the moment of inertia of the system about an axis along one of the sides of the square is
  142. Two masses each of mass M are attached to the end of a rigid massless rod of length L. The moment of inertia of the system about an axis passing centre of mass and perpendicular to its length is
  143. The moment of inertia of a uniform thin rod of mass m and length l about two axis PQ and RS passing through centre of rod C and in the plane of the rod are I PQ and I RS respectively. Then I PQ + I RS is equal to
  144. A cord is wound round the circumference of wheel of radius r. The axis of the wheel is horizontal and fixed and moment of inertia about it is I. A weight mg is attached to the free end of the cord and falls from rest. After falling through a distance h, the angular velocity of the wheel will be
  145. ABC is an equilateral triangle with O as its centre. F 1 , F 2 and F 3 represent three forces acting along the sides AB, BC and AC respectively. If the total torque about O is zero then the magnitude of F 3 is
  146. A solid cylinder of mass 50 kg and radius 0.5 m is free to rotate about the horizontal axis. A massless string is wound round the cylinder with one end attached to it and other hanging freely. Tension in the string required to produce an angular acceleration of 2 rev s – 2 is
  147. Find the acceleration of the body if a force F = 8 N pulls the string at P that passes over the body and it is o connected by another string to a rigid support at Q. (Take radius of gyration k = 2 3 m , R = 2 m, r = 1 m, and mass of the body m = 3 kg)
  148. A hollow straight tube of length l and mass m can turn freely about its centre (fixed) on a smooth horizontal table. Another smooth uniform rod of same length and mass is fitted into the tube so that their centres coincide. The system is set in motion with an initial angular velocity ω 0 . The angular velocity of the rod at an instant when the rod slips out of the tube is:
  149. A particle of mass 1 kg is moving along the line y = x + 2 (here x and y are in m) with speed 2 m/s. The magnitude of angular momentum of the particle about origin is:
  150. A uniform cube of mass M and side a is placed on a frictionless horizontal surface. A vertical force F is applied to the edge as shown in figure. Match the Column I with Column II and mark the correct choice from the given codes. Column-I Column-II (i) Mg 4 < F < Mg 2 (p) Cube will move up (ii) F >   Mg 2 (q) Cube will not exhibit motion (iii) F > Mg r) Cube will begin to rotate and slip at,4. (iv) F = Mg 4 (s) Normal reaction effectively at a 3 from A, no motion Codes
  151. A disc can rotate about an axis passing through its centre and perpendicular to the plane of the disc. Initially the disc is stationary and its moment of inertia about the axis is 4 Kg-m 2 . A torque is applied on the disc which varies with time according to the graph shown in figure. Then angular velocity of the disc at the end of 6th second is
  152. A thin rod AB of mass 1 Kg and length 2m can rotate about a spindle passing through its mid point C and perpendicular to its length. Now a 2 N-m torque is applied to the rod for 10 seconds. Then angular momentum of the rod at the end of 10 S is
  153. A body is placed on a horizontal frictionless surface. When a horizontal force F is applied on the body, it is observed that the body makes only translational motion. The which of the following points may be the positions of its centre of mass?
  154. For the adjoining diagram, a triangular lamina is shown. The correct relation between I 1 ,I 2 and I 3 , is (I-moment of inertia) :
  155. Three identical spherical shells, each of mass m and radius r are placed as shown in figure. Consider an axis XX’ which is touching to two shells and Passing through diameter of third shell. Moment of inertia of the system consisting of these three spherical shells about XX ‘axis is :
  156. A spherical ball of mass 20 kg is stationary at the top of a hill of height 100 m. It rolls down a smooth surface to the ground, then climbs up another hill of height 30 m and finally rolls down to a horizontal base at a height of 20 m above the ground. The velocity attained by the ball is :
  157. The ratio of the accelerations for a solid sphere (mass ‘m’ and radius ‘R’) rolling down an incline plane of angle ‘ θ ‘ without slipping and slipping down the incline without rolling is :
  158. A solid sphere of mass m and radius R is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation (E sphere , E cylinder , ) will be :
  159. An inclined plane makes an angle of 30° with the horizontal. A ring rolling down this inclined plane from rest without slipping has a linear acceleration equal to:
  160. The moment of inertia of a uniform circular disc about its diameter is 2 kg m 2 .What is its moment of inertia about an axis passing through its circumference and perpendicular to its circular face?
  161. The moment of inertia of a uniform flat disc about its own axis is I . The radius of the disc is a. A section ABCD of the disc (as shown in figure) is cut off and separated. The moment of inertia of the remaining part of the disc about the same axis will be :
  162. The diagram shows a uniform disc of mass M and radius ‘a’. If the moment of inertia of the disc about the axis XY is I, its moment of inertia about an axis through0and perpendicular to the plane of the disc is :
  163. The solid cylinder shown in figure is rolling without slipping between two moving planks P 1 and P 2 moving with velocities 2V and V respectively. If mass of the cylinder is m, then kinetic energy of the cylinder relative to ground frame is
  164. Between two eggs which have identical sizes, shapes and weights, one is raw and the other is half-boiled. The ratio of the moment of inertia of the raw egg to that of the half boiled egg about a central axis is
  165. Four spheres each having mass m and radius r are placed with their centres on the four corners of a square of side a. Then the moment of inertia of the system about an axis along one of the sides of the square is :
  166. A block of mass m, connected to a light string wrapped over a disc pulley of mass 2 m and radius 2r, is being pulled by a constant horizontal force F as shown. Tangential acceleration of a point distant r from the axis of pulley is (Assume that there is no slipping between the string and the pulley)
  167. A sphere rolls up the incline as shown in the diagram. The direction of friction acting on it is :
  168. Each of the three balls has a mass m and is welded to the rigid equiangular form of negligible mass. The assembly rests on a smooth horizontal surface. A force F is suddenly applied to one bar as shown in fig. (1). Which of the following statement is connect
  169. A particle of mass m is moving in a plane along a circular path of radius r. Its angular momentum about the axis of rotation is L. The centripetal force acting on the particle is
  170. The angular velocity of a body changes from ω 1 to ω 2 without applying torque but changing moment of inertia. The ratio of initial radius of gyration to the final radius of gyration is
  171. A solid cylinder of mass 50 kg and radius 0 .5 m is free to rotate about its axis which is horizontal. A string is wound around the cylinder with one end attached to it and other hanging freely as shown in fig. (O). The tension in the string required to produce an angular acceleration of 2 rev/s 2 in the cylinder is
  172. A circular disc rolls down an inclined plane without slipping. What fraction of its total energy is translational ?
  173. Two circular discs A and B of equal masses and thickness but made of metals with densities d A and d B d A > d B . If their moments of inertia about an axis through the centre and normal to the circular forces be I A and I B , then.
  174. If the moment of inertia of a disc about al axis tangentially and parallel to its surface be I, then what will be the moment of inertia about the axis tangential but Perpendicular to the surface.
  175. A thin wire AB of length l and mass m is bent in the form of a semicircle as shown in fig. Its moment of inertia about an axis joining its free ends will be
  176. Two uniform thin identical rods. AB and CD each of mass M and length I are joined so as to form a cross as shown in fig. The moment of inertia of the cross about the bisector line EF is
  177. A mass M is supported by a massless string wound round a uniform cylinder of mass M and radius E as shown in fig. On releasing the mass from rest, it wiII fall with acceleration
  178. Three point-masses m are placed at the vertices of an equilateral triangle of side L Moment of inertia of the system about an axis COD passing through a mass m at O and lying in the plane AOB and perpendicular to OA as shown in the figure is
  179. Three rings each of mass M and radius R are arranged as shown in fig. (s). The moment of inertia of the system about YY’ will be
  180. Four identical rods are joined end to end to form a square. The mass of each rod is M. The moment of inertia of the system about an axis passing through the point of intersection of diagonals and perpendicular to the plane of the square is
  181. One quarter sector is cut from a 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
  182. A wheel of bicycle is rolling without slipping on a Level road. The velocity of the centre of mass is V cm; then true statement is
  183. A round uniform body of radius R, mass M and moment of inertia I, rolls down (without slipping) an inclined plane making an angle θ with the horizontal. Then its acceleration is
  184. A disc of radius r = 20 cm is rotating about its axis with an angular speed of 20 rad/s. It is gently placed on a horizontal surface which is perfectly frictionless (see fig. 1). What is the linear speed of a point B on the disc ?
  185. If F be a force acting on a particle having the position vector r and τ be the torque of this force about the origin, then
  186. A particle of mass m is projected with a velocity v making an angle of 45 0 with the horizontal. The magnitude of angular momentum of projectile about the point of projection when the particle is at its maximum height h is
  187. A solid sphere and a solid cylinder of same mass are rolled down on two inclined planes of heights h 1 and h 2 , respectively. If at the bottom of the plane, the two objects have same linear velocities, then the ratio of h 1 : h 2 is
  188. Four rods of equal length I and each of mass m form a square as shown in figure. Moments of inertia about three axes 1, 2 and 3 are say I 1 ,I 2 and I 3 • Then, match the following columns and mark the correct option from the codes given below. Column I (A) I 1 (B) I 2 (C) I 3 Column II (p) 4 3 m l 2 (q) 2 3 m l 2 (r ) 1 2 m l 2 (s) None
  189. A disc is free to rotate about a smooth horizontal axis passing through its centre of mass. A particle is fixed at the top of the disc. A slight push is given to the disc and it starts rotating. During the process,
  190. If radius of earth is reduced to half without changing its mass, then match the following columns and mark the correct option from the codes given below. Column I (A) Angular velocity of earth (B) Time period of rotation of earth (C) Rotational kinetic energy of earth Column II (p) will become two times (q) will become four times (r) will remain constant (s) None
  191. O is the centre of an equilateral triangle ABC. F 1 , F 2 and F 3 are three forces acting along the sides AB , BC and AC respectively as shown in figure. What should be the magnitude of F 3 , so that the total torque about O is zero?
  192. A massless rod S having length 2 l has equal point masses attached to its two ends as shown in figure. The rod is rotating about an axis passing through its centre and making an angle α with the axis. The magnitude of change of angular momentum of rod i.e. d L d t equals [AIIMS 2015]
  193. Two uniform, thin identical rods each of mass M and length I are joined together to form a cross. What will be the moment of inertia of the cross about an axis passing through the point at which the two rods are joined and perpendicular to the plane of the cross?
  194. Two discs have mass ratio 1 : 2 and diameter ratio 2 : 1, then find the ratio of their moments of inertia.
  195. A disc rotates freely with rotational kinetic energy E about a normal axis through centre. A ring having the same radius but double the mass of disc is now, gently placed on the disc. The new rotational kinetic energy of the system would be
  196. A solid cylinder of mass 50 kg and radius 0.5 m is free to rotate about the horizontal axis. A massless string is wound round the cylinder with one end attached to it and other hanging freely. Tension in the string required to produce an angular acceleration of 2 revolutions s – 2 is
  197. The ratio of the accelerations for a solid sphere (mass m and radius R ) rolling down an incline of angle θ without slipping and slipping down the incline without rolling is
  198. A body initially at rest and sliding along a frictionless track from a height h (as shown in the figure) just completes a vertical circle of diameter AB=D . The height h is equal to
  199. Three identical thin rods, each of mass m and length l, are joined to form an equilateral triangular frame. The moment of inertia of the frame about an axis parallel to its one side and passing through the opposite vertex is
  200. Three objects, A : (a solid sphere), B : (a thin circular disk) and C : (a circular ring), each have the same mass M and radius R . They all spin with the same angular speed ω about their own symmetry axes. The amounts of work (W) required to bring them to rest, would satisfy the relation
  201. A uniform disc of radius ‘R” and mass ‘M’ can rotate without friction on an axis passing through its centre and perpendicular to its plane. A cord is wound at the rim of the disc and a uniform force of F is applied on the cord. The tangential acceleration of a point on the rim of the disc is
  202. The moment of the force, F = 4 i ^ + 5 j ^ – 6 k ^ at (2,0,-3), about the point (2,-2,-2) , is given by
  203. 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
  204. A solid sphere is rotating freely about its symmetry axis in free space. The radius of the sphere is increased keeping its mass same. Which of the following physical quantities would remain constant for the sphere?
  205. A disk and a sphere of same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first?
  206. A solid cylinder of mass 2 kg and radius 4 cm rotating about its axis at the rate of 3 rpm. The torque required to stop after 2 π revolutions is
  207. A solid sphere is sliding with a velocity V 0 on a rough horizontal surface. Its velocity when it starts pure rolling is
  208. Point masses m 1 and m 2 are placed at the opposite ends of a rigid rod of length L, and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position of point P on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity ω 0 is minimum, is given by
  209. An automobile moves on a road with a speed of 54 km h – 1 . The radius of its wheels is 0.45 m and the moment of inertia of the wheel about its axis of rotation is 3 k g m 2 . If the vehicle is brought to rest in 15 s, the magnitude of average torque transmitted by its brakes to the wheel is
  210. A disc of radius 2 m and mass 100 kg rolls on a horizontal floor. Its centre of mass has speed of 20 cm/s. How much work is needed to stop it?
  211. A force F = α i ^ + 3 j ^ + 6 k ^ is acting at a point r = 2 i ^ – 6 j ^ – 12 k ^ . The value of α for which angular momentum about origin is conserved is
  212. A rod PQ of mass M and length L is hinged at end P. The rod is kept horizontal by a massless string tied to point Q as shown in figure. When string is cut, the initial angular acceleration of the rod is
  213. A small object of uniform density rolls up a curved surface with an initial velocity ‘ υ ‘. It reaches upto a maximum height of 3 υ 2 4 g with respect to the initial position. The object is
  214. Two rotating bodies A and B of masses m , 2 m with moments of inertia l A and I B , I B > I B have equal kinetic energy of rotation. If L A and L B be their angular momenta respectively, then
  215. A solid sphere of mass m and radius R is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation E sphere / E cylinder will be
  216. A light rod of length l has two masses m 1 and m 2 attached to its two ends. The moment of inertia of the system about an axis perpendicular to the rod and passing through the centre of mass is
  217. Three identical spherical shells, each of mass m and radius r are placed as shown in figure. Consider an axis X X’ which is touching to two shells and passing through diameter of third shell. Moment of inertia of the system consisting of these three spherical shells about X X’ axis is
  218. The ratio of radii of gyration of a circular ring and a circular disc, of the same mass and radius, about an axis passing through their centres and perpendicular to their planes are
  219. A circular plate of radius R 2 is cut from one edge of thin circular plate of radius R. The moment of inertia of remaining portion about an axis through O perpendclicular to plane of plate is (Where M is the mass of remaining plate)
  220. Two discs are rotating about their axes, normal to the discs and passing through the centres of the discs. Disc D 1 has 2 kg mass and 0.2 m radius and initial angular velocity of 50 rad s – 1 . Disc D 2 has 4 kg mass, 0.1 m radius and initial angular velocity of 200 rad s – 1 . The two discs are brought in contact face to face, with their axes of rotation coincident. The final angular velocity (in rad s – 1 ) of the system is
  221. A particle of mass 5g is moving with a uniform speed of 3 2 cm/sec in the x-y plane along the line y=x+4. The magnitude of its angular momentum about the origin in g m c m 2 / s is
  222. The photoelectric threshold wavelength of silver is 3250 × 10 – 10 m . The velocity of the electron ejected from a silver surface by ultraviolet light of wavelength 2536 × 10 – 10 m is Given h = 4 . 14 × 10 – 15 eVs and c = 3 × 10 8 m s – 1
  223. Find the torque about the origin when a force of 3 j ^   N acts on a particle whose position vector is 2 k ^   m .
  224. A 150 kg merry go round in the shape of a uniform solid horizontal disk of radius 2 m is set in motion by wrapping a rope about the rim of the disk and pulling on the rope. What constant force must be exerted on the rope to bring the merry go round from rest to an angular speed of 0.5 rev/s in 2 sec?
  225. A cylinder of mass 4 m and a block of mass m is placed on an inclined plane as shown in the figure. The linear acceleration of cylinder is
  226. A force of F   =   2 i + 3 j − 2 k acts at a point (2, –3, 1). Then magnitude of torque about point (0,0,2) will be
  227. The kinetic energy of an object rotating about a fixed axis with angular momentum L = I ω can be written as
  228. A thin uniform rod of mass ‘M’ and length ‘L’ rotating about a perpendicular axis passing through its center with a constant angular velocity ‘ ω ‘ .Two objects each of mass M 3 are attached gently to the two ends of the rod. The rod will now rotate with an angular velocity of :
  229. The rotational kinetic energy of a solid sphere of mass 3 kg and radius 0.2 m rolling down an inclined plane of height 7 m is :
  230. The angular momentum of a rigid body of mass m about an axis is n times the linear momentum (P) of the body. Total kinetic energy of the rigid body is:
  231. A wheel of moment of inertia Ι and radius R is rotating about its axis at an angular speed ω 0 . A small fragment of mass m breaks off from its rim. Find the final angular velocity of the wheel.
  232. A flywheel, in the form of a solid disc, of mass 25 kg has a radius of 0.2 m. It is making 240 rpm. If the torque is due to a force applied tangentially on the rpm of the fly wheel, what is the magnitude of the force to bring it to rest in 20 sec ?
  233. A wheel of radius r rolls without slipping on the ground with speed v. When it is at point p, a piece of mud flies off tangentially from its highest point, and lands on the ground at a point Q. The distance PQ is
  234. Moment of inertia of the isosceles right angled triangle ABC about the side BC is I. Then its moment of inertia about side AC is
  235. A particle is moving in a circular path. If its kinetic energy is doubled, keeping its angular momentum unchanged, then new angular velocity of rotation is
  236. The angular momentum of a solid sphere executing pure rolling on a rough horizontal surface will
  237. A nearly massless rod is pivoted at one end so that it can swing freely as a pendulum. Two masses 2m and m are attached to it at distance b and 3b respectively, from the pivot. The rod is held horizontal and then released. Find its angular acceleration at the instant it is released.
  238. An imperfectly rough sphere moves from rest down a plane inclined at an angle α to the horizontal. The coefficient of friction between the inclined plane and sphere is μ .   Then
  239. A circular object is gently released on a rough inclined plane as shown in figure and the object starts rolling down the plane without slipping with acceleration of its centre 10 3    m / s 2 . Then the object must be a
  240. A solid iron sphere A rolls down an inclined plane, without slipping while an identical sphere B slides down the plane in a frictionless manner. At the bottom of the inclined plane, The total kinetic energy of sphere A is
  241. Two particles A and B of masses m and 2m are at a separation d. PQ is a line perpendicular to the line AB. Then minimum possible moment of inertia about PQ is
  242. A cubical block of mas ‘M’ and edge ‘a’ slides down a rough inclined plane of inclination ‘ θ ‘ with a uniform velocity. The torque of the normal force on the block about its center has a magnitude
  243. ABC is a triangular plate of uniform thickness. The sides are in the ratio shown in the figure. IAB, IBC, ICA are the moments of inertia of the plated about AB, BC and CA respectively. Which one of the following relation is correct?
  244. A thin uniform circular ring of mass M and radius R is in XY plane with its center coinciding with the origin. If it’s moment of inertia about Z-axis is same as its moment of inertia about the line y = x + c , the value of c is
  245. P( a , a , a ) is a point in space. Three forces each of magnitude F are acting at P in the positive X-direction, positive Y-direction and positive Z-direction. The magnitude of the torque due to the system of forces about the origin is
  246. From a given sample of uniform wire, two circular loops P and Q are made, P of radius r and Q of radius nr. If the M.I. of Q about its axis is four times that of P about its axis (assuming the wire to be of diameter much smaller than either radius), the value of n is
  247. A uniform rod of mass m, length l rests on a smooth horizontal surface. Rod is given a sharp horizontal impulse p perpendicular to the rod at a distance l/4 from the centre. The angular velocity of the rod will be
  248. A disc of mass M and radius R rolls without slipping on a horizontal surface. If the velocity of its centre is v, then the total angular momentum of the disc about a fixed point P at a height 3R/2 above the centre C
  249. A closed tube partly filled with water lies in a horizontal plane. If the tube is rotated about perpendicular bisector, the moment of inertia of the system
  250. A pot-maker rotates pot-making wheel of radius 3 m by applying a force of 200 N tangentially; because of this if wheel completes exactly 1½ revolution, the work done by him is
  251. A circular disc of mass m and radius R rests flat on a horizontal frictionless surface. A bullet, also of mass m and moving with a velocity v, strikes the disc and gets embedded in it. If the disc is free to rotate about a fixed axis passing through its center then the angular velocity with which the system rotates after the bullet strikes the hoop is
  252. A 13 m ladder is placed against a smooth vertical wall with its lower end 5m from the wall. What should be the minimum coefficient of friction between ladder and floor so that it remains in equilibrium.
  253. A thin hollow sphere of mass ‘m’ is completely filled with a liquid of mass ‘m’. When the sphere rolls with a velocity ‘V’ at center of mass , then kinetic energy of the system is [neglect friction]
  254. A sphere is rolling down an inclined plane without slipping. The ratio of rotational kinetic energy to total kinetic energy is
  255. The wheels of an airplane are set into rotation just before landing so that the wheels do not slip on the ground. If the airplane is travelling in the east direction, what should be the direction of angular velocity vector of the wheels?
  256. An impulse I is applied at the end of a uniform rod if mass m. then :
  257. The moment of inertia of a thin rod of mass M and length L, about an axis perpendicular to the rod at a distance ¼ th from one end
  258. A thin rod AB of mass 1 kg and length 1m is bent as its mid point c at an angle 60 o as shown in figure. PQ is the bisector of the angle ACB. Then moment of inertia of the bent rod about the line PQ is
  259. A number of forces are acting on a system. If net external force on the system is zero, Then
  260. The position of a particle is given by r   =   i ^ + 2 j ^ − k ^   , linear momentum by   P   =   ( 3 i ^ + 4 j ^ − 2 k ^ ) . The angular momentum is perpendicular to
  261. Two wheels A and B of same mass are rotating with same angular velocity about their respective geometrical axis. Wheel A has radius R and wheel B has radius 2R. Same breaking torque is applied on the wheels. If wheel A stops in 1 minute, wheel B will stop in
  262. A then ring of radius 0.5 m is rolling without slipping on horizontal ground with angular velocity 5 rad/sec. Then speed of top most point on the ring is
  263. An equivalent triangle is formed by three identical rods each of mass m and length l. Then moment of inertia of the triangle about any of its sides is
  264. A solid Copper sphere is spinning about a diameter with angular velocity 10 rad/s. Due to rise in temperature, the radius of sphere increase by 1%. What will be the change in its angular velocity?
  265. A square is made by joining four rods each of mass M and length L. Its moment of inertia about an axis PQ, in its plane and passing through one of its corner is
  266. Moment of inertia of a solid disc about a diameter is I. Then moment of inertia of the disc about an axis which passes through a point on its circumference and perpendicular to the plane of the disc is
  267. A cubical block of side a is moving with a velocity v on a horizontal smooth plane as shown in the figure, it hits a bridge at O. The angular speed of the block after it hits the ridge at O, is
  268. If a particle of mass m is moving with constant velocity v parallel to x-axis in x-y plane as shown in fig. Its angular momentum with respect to origin at any time t will be
  269. A cylinder rolls up an inclined plane, reaches some height and then rolls down (without slipping through out these motions). The directions of frictional force acting on the cylinder are
  270. A railway track is banked for a speed v, by making the height of the outer rail h higher than that of the inner rail. The distance between the rails is d. The radius of curvature of the track is r
  271. Moment of inertia of a body is independent of
  272. Rotational analogue of inertia in translatory motion is
  273. A particle of mass 0.2kg is moving with linear velocity (i-j+2k). If the radius vector r = 4i+j-k, the angular momentum of the particle is
  274. Three identical spherical shells, each of mass m and radius r are placed as shown in figure. Consider an axis XX’ which is touching the lower shells and passing through diameter of third shell. Moment of inertia of the system consisting of these three spherical shells about XX’ axis is
  275. Two small spheres of masses 10kg and 30 kg are joined by a rod of length 0.5 m and of negligible mass. The M.I of the system about a normal axis through centre of mass of the system is
  276. A ring has a mass of 0.5 kg and radius 1 m. The moment of inertia of the ring about its diameter is
  277. The moment of inertia of a solid cylinder about its own geometrical axis is equal to its moment of inertia about an axis passing through its centre and normal to its length. The ratio of length to radius is
  278. A straight thin uniform rod of length 4L and mass 4M is bent into a square. Its M.I about one side is
  279. Two circular rings of equal mass M, and radius R are placed touching each other. The moment of inertia of the system about tangential axis in the plane of system passing through point of contact of rings is
  280. Two identical rods are joined to form an ‘X’. The smaller angle between the rods is ‘ θ ’. The moment of inertia of the system about an axis passing through the point of intersection of the rods and perpendicular to their plane is
  281. If the mass of earth and radius suddenly becomes 2 times and 1/4 th of the present value respectively, the length of the day becomes
  282. A Particle of mass M is moving in a straight line with uniform speed v parallel to X-axis in X- Y plane at a height ‘h’ from the X-axis. Its angular momentum about the origin is
  283. Relation between torque and angular momentum is similar to the relation between
  284. A boy standing on a rotating table with heavy spheres in his hands, suddenly brings his hands close to his body. The angular velocity of the table
  285. The principle used by a gymnast to increase the number of somersaults, is law of conservation of
  286. If polar ice caps melt, the length of the day
  287. A man turns on rotating table with an angular speed ω. He is holding two equal masses at arms length. Without moving his arms, he just drops the two masses on to rotating table. How will his angular speed change?
  288. When a stopper is pulled from a filled wash basin, the water drains out while circulating like a small whirl pool. The angular velocity of a fluid element about a vetical axis through the orifice appears to be greater near the orifice. The angular velocity of fluid element varies
  289. When running boy jumps on to a rotating table, the quantity that is conserved is
  290. A solid sphere, solid cylinder and a disc are allowed to roll down from the top of an incline plane from the same height. Then
  291. A circular disc A of radius r is made from an iron plate of thickness t and another circular disc B of radius 4r is made from an iron plate of thickness t/4. The relation between the moments of inertia I A and I B is
  292. Let I 1 and I 2 be the moments of inertia of two bodies of identical geometrical shape, the first made of aluminium and the second of iron.
  293. A closed cylindrical tube containing some water (not filling the entire tube) is rotated about a perpendicular bisector, the moment of inertia of water about the axis
  294. Consider the following two equations A) L = I ω B) dL / dt = In noninertial frames
  295. A hollow sphere and a solid sphere having same mass and same radii are rolled down a rough inclined plane without slipping,
  296. Two uniform solid spheres having unequal masses and unequal radii are released from rest from the same height on a rough incline. If the spheres roll without slipping,
  297. Three identical solid spheres moves down three incline planes A, B and C – all of the same dimensions. A is without friction, the friction between B and sphere is sufficient to cause rolling without slipping, the friction between C and sphere causes rolling with slipping. The kinetic energies on A, B and C at the bottom of the inclines are E A , E B and E C
  298. An external device, e.g., an electric motor, supplies constant power to a rotating system, e.g., a flywheel, through a torque τ . The angular velocity of the system is ω. Both τ and ω are variable
  299. A solid sphere is rotating freely about its symmetry axis in free space. The radius of the sphere is increased keeping its mass same. Which of the following physical quantities would remain constant for the sphere ?
  300. A man is spinning in the gravity free-space changes the shape of the body by spreading his arms. By doing this he can change his (a) moment of inertia (b) angular momentum (c) angular velocity (d) rotational kinetic energy Which one of the following are correct?
  301. Moment of inertia of a rigid body depends on A) Mass of body B) Position of axis of rotation C) Angular velocity of the body D) Time period of its rotation
  302. Four objects with the same mass and radius are spinning freely about a diameter with the same angular speed. Arrange the work required to stop them in the decreasing order a) Solid sphere b) Hollow sphere c) Disc d) Hoop
  303. Identify the correct order in which the values of M.I. decreases for the following i) M.I. of solid sphere of mass ‘M’ and radius ‘R’ about its diameter of rotation ii) M.I. of uniform ring of mass ‘M’ and radius ‘R’ about its tangent perpendicular to its plane iii) M.I. of uniform disc of mass ‘M’ and radius ‘R’ about its diameter iv) M.I of a uniform solid cylinder of mass M about its own axis of rotation
  304. A thin disc rotates about an axis passing through its centre and perpendicular to its plane with a constant angular velocity ‘ ω’ I is the moment of inertia of that disc and ‘L’ is its angular momentum about the given axis. Then rotational kinetic energy of the disc ‘E’ is A) E L 2 B) E L -2 C) E l D)E l -1
  305. A child is standing with folded hands at the centre of a platform rotating about its central axis. The K.E of the system is K. The child now stretches his arms so that the M.I of the system doubles. The K.E of the system now is
  306. Two wheels of M.I. 3kg m 2 and 5kg m 2 are rotating at the rate of 600 rpm and 800 rpm respectively in the same direction. If the two are coupled so as to rotate with the same axis of rotation, the resultant speed of rotation will be (in rpm)
  307. A disc of mass 100g and radius 10cm has a projection on its circumference. The mass of projection is negligible. A 20g bit of putty moving tangential to the disc with a velocity of 5ms -1 strikes the projection and sticks to it. The angular velocity of disc is
  308. A Merry-go-round, made of a ring-like platform of radius R and mass M, is revolving with angular speed ω . A person of mass M is standing on it. At one instant, the person jumps off the round, radially away from the centre of the round (as seen from the round). The speed of the round afterwards is
  309. A solid cylinder of mass 50kg and radius 0.5m is free to rotate about the horizontal axis. A massless string is wound round the cylinder with one end attached to it and other hanging freely. Tension in the string required to produce an angular acceleration of 2 rad/s 2 is
  310. A solid sphere is in rolling motion. In rolling motion, the body possesses translational kinetic energy (K t ) as well as rotational kinetic energy (K r ) simultaneously. The ratio K t : (K t +K r ) for the sphere is
  311. If 27 small spheres each of M.I. ‘ I ’ about their diametrical axis are combined to form a big sphere, then the M.I. of the big sphere about its diameter is
  312. From a disc of radius R and mass M, a circular hole of diameter R, whose rim passes through the centre is cut. What is the moment of inertia of the remaining part of the disc about a perpendicular axis, passing through the centre?
  313. A particle of mass 2 kg is on a smooth horizontal table and moves in a circular path of radius 0.6m. The height of the table from the ground is 0.8 m. If the angular speed of the particle is 12 rad s –1 , the magnitude of its angular momentum about a point on the ground right under the centre of the circle is
  314. A heavy wheel of radius 20 cm and weight 10 kg is to be dragged over a step of height 10 cm, by a horizontal force F applied at the centre of the wheel. The minimum value of F is
  315. A uniform rod of length l and mass m is suspended by two vertical inextensible string as shown in the fig. The tension in the left string when right string snaps, is
  316. 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 other along AC as shown. Neglecting frictional effects, the quantities that are conserved as the beads slide down, are
  317. A force f acts tangentially at the highest point of a solid cylinder of mass ‘m’ kept on a rough horizontal plane. If the cylinder rolls without slipping, find the acceleration of the center of cylinder
  318. A sphere rolls without slipping on an incline of inclination θ. The minimum coefficient of static friction to support pure rolling is to be
  319. Consider a cylinder of mass M resting on a rough horizontal rug that is pulled out from under it with acceleration ‘a’ perpendicular to the axis of the cylinder. What is F friction at point P? It is assumed that the cylinder does not slip
  320. A thin wire of length l and mass m is bent in the form of a semicircle as shown in the figure. Its moment of inertia about and axis joining its free ends will be
  321. A uniform disc of mass ‘m’ and radius ‘r’ rotates along an axis passing through its centre of mass and normal to its plane. An unstretchable light rope is wound on the disc. Tangential acceleration of a point P on peripheral of the disc when a uniform force F is applied on the rope is
  322. A rope of negligible mass is wound round a hollow cylinder of mass 3 kg and radius 40cm. If the rope is pulled with a force of 30N, then the angular acceleration produced in the cylinder is
  323. The graph between log L and log P where L and P are angular momentum and momentum of a body undergoing rotational motion. (Perpendicular distance r is constant)
  324. The graph between angular momentum (L) and angular velocity ω if its rotational kinetic energy is to remain constant is
  325. A circular turn table rotates about its normal axis with uniform angular speed . A circular thick layer of ice and of radius much smaller than the table – top rotates along with the table. The new angular speed of the table when ice starts melting is
  326. A homogeneous cylinder of mass M and radius R is pulled on a horizontal plane by a horizontal force F acting through its centre of mass. Assuming rolling without slipping the angular acceleration of the cylinder is
  327. A ball of radius r rolls inside a hemispherical shell of radius R. It is released from rest from point A as shown in figure and the ball starts rolling without slipping. The angular velocity of centre of the ball in position B about the centre of the shell is:
  328. A disc of mass M and radius R is rolling with angular speed ω on a horizontal plane as shown in figure. The magnitude of angular momentum of the disc about the origin O is
  329. A solid cylinder rolls up along a long inclined plane of inclination θ with an initial velocity v. How far does the cylinder go up the plane? Assume pure rolling.
  330. A uniform ring of mass m and radius R is released from top of an inclined plane. The plane makes an angle θ with horizontal. The coefficient of friction between the ring and the plane is µ. Initially, the point of contact of ring and plane is P. Angular momentum of the ring about point P as a funtion of time t is
  331. A solid sphere of radius R is resting on a smooth horizontal surface. A constant force F is applied at a height h from the bottom. Choose the correct alternative :
  332. If the velocity of a rotating body of mass 50 kg is given by and radius vector , then its angular momentum is.
  333. A solid sphere rolls down without slipping from rest on a 30 0 incline. Its linear acceleration is
  334. A small object of uniform density roll up a curve surface with an initial velocity v. It reaches up to a maximum height of 3 v 2 4 g with respect to the initial position. The object is
  335. Four small spheres each of radius ‘r’ and mass ‘m’ are placed with their centres at the four corners of a square of side ‘L’. The M.I. of the system about any side of square is
  336. Consider a rod of mass M and length L pivoted at its centre is free to rotate in a vertical plane. The rod is at rest in the vertical position. A bullet of mass M moving horizontally at a speed v strikes and embedded in one end of the rod. The angular velocity of the rod just after the collision will be
  337. The speed of a homogeneous solid sphere after rolling down an inclined plane of vertical height h from rest without slipping will be
  338. A solid cylinder of mass M and radius R rolls down an inclined plane of height h without slipping. The speed of its centre when it reaches the bottom is
  339. A particle is whirled in a circular path by a string with a constant angular velocity and its angular momentum is L. If the length of the string is now halved keeping the angular velocity same, the angular momentum is:
  340. An ice skater starts a spin with her arms stretched out to the sides. She balances on the tip of one skate to turn without friction. She then pulls her arms in so that her moment of inertia decreases by a factor of 2. In the process of her doing so, what happens to her kinetic energy?
  341. In a bicycle the radius of rear wheel is twice the radius of front wheel. If v F and v r are the speeds of top most points of front and rear wheels respectively, then
  342. A hollow cylinder and a solid cylinder of the same mass and radius are released simultaneously from rest at the top of the same inclined plane, which will reach the ground first?
  343. A particle of mass m is moving in a plane along a circular path of radius r. Its angular momentum about the axis of rotation is L. The centripetal force acting on the particle is
  344. From a given sample of uniform wire, two circular loops P and Q are made, P of radius r and Q of radius nr. If the M.I. of Q about its axis is 4 times that of P about its axis (assuming wire diameter much smaller than either radius of the loops), the value of n is:
  345. A solid sphere and a disc of same radii are rolling down an inclined plane without slip. One reaches earlier than the other due to
  346. A round object is rolling without slipping on a horizontal surface. If its kinetic energy of translation is equal to 250% of kinetic energy due to rotational motion, then the object must be a
  347. A thin rod OA of length 2R and mass m is attached to a thin ring of mass m and radius R. as shown in the figure. Then moment of inertia of the composite body about an axis passing through O and perpendicular to the plane of the ring is
  348. A body of mass m slides down an incline (without friction) and reaches the bottom with a velocity v. If the same mass were in the form of a ring which rolls down(without slipping) this incline, the velocity of the ring at bottom would have been
  349. If F is the force acting on a particle having position vector r and τ be the torque of this force about the origin, then
  350. A sphere of outer radius R having some cavity inside is allowed to roll down on an incline without slipping and it reaches a speed v 0 at the bottom of the incline. The incline is then made smooth by waxing and the sphere is allowed to slide without rolling and now the speed attained is ( 5 4 ) v 0 . What is the radius of gyration of the sphere about an axis passing through its centre?
  351. Four identical thin rods each of mass M and length l form a square frame. Moment of inertia of this frame about an axis through the centre of the square and perpendicular to its plane is
  352. Two discs of same moment of inertia rotating about their respective geometrical axis passing through centre and perpendicular to the plane of disc with angular velocities ω 1 and ω 2 . They are brought into contact face to face coinciding the axis of rotation. The expression for loss of energy during this process is
  353. With reference to figure of a cube of edge a and mass m, which of the following is the correct statement? (O is the centre of the cube)
  354. A solid sphere of mass m and radius R is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation ( E sphere E cylinder ) will be
  355. A disc and a sphere of same radius but different masses are rolling down two inclined planes of the same altitude and length.Which one of the two objects gets to the bottom of the plane first?
  356. A force F = α i ^ + 3 j ^ + 6 k ^ is acting at a point r = 2 i ^ – 6 j ^ – 12 k ^ . The value of α for which angular momentum about origin is conserved is
  357. The moment of inertia of a thin uniform rod of mass M and length L about an axis passing through its midpoint and perpendicular to its length is I o Its moment of inertia about an axis passing through one of its ends and perpendicular to its length is
  358. A body of radius R and mass m is rolling smoothly with speed v on a horizontal surface. It then rolls up a hill to a maximum height h. If h = 3 v 2 4 g , what might the body be?
  359. A force F = ( 5 i ^ + 6 j ^ ) N is acting at point A ( 2 m , 4 m ) . Then the moment of the force about the point B ( 1 m ,   1 m ) in N-m is
  360. A solid disc of mass 2 kg and radius 50 cm is rotating about a spindle passing through its centre and perpendicular to the plane of the disc with constant angular velocity of 5 rad/s. A constant torque of 2 N-m is applied on the disc. Then angular velocity of the disc after 10 sec will be
  361. A solid sphere rolls down two different inclined planes of the same heights but different angles of inclination. In both cases
  362. A solid iron sphere A rolls down an inclined plane, while an identical hollow sphere B of same mass slides down the plane in a frictionless manner. At the bottom of the inclined plane, the total kinetic energy of sphere A is
  363. A solid cylinder of mass 2 kg and radius 4 cm rotating about its axis at the rate of 3 rpm. The torque required to stop after 2 π revolution is
  364. The ratio of the acceleration for a solid sphere (mass m and radius R) rolling down an incline of angle ‘ θ ’ without slipping and slipping down the incline without rolling is
  365. The instantaneous angular position of a point on a rotating wheel is given by the equation θ ( t ) = 2 t 3 – 6 t 2 The torque on the wheel becomes zero at
  366. A circular disc of moment of inertia I i is rotating in a horizontal plane, about its geometrical axis, with a constant angular speed ω i . Another disc of moment of inertia I f is dropped coaxially onto the rotating disc. Initially the second disc has zero angular speed. Eventually both the discs rotate with a constant angular speed ω f . The energy lost by the initially rotating disc due to friction is:
  367. Two wheels A and B of same radii starts rolling with same angular velocity up a rough inclined and smooth inclined plane respectively. The angle of inclination of both the planes is same. Column-I Column-II i. the maximum height gained depends upon p. angular velocity ii. The retarding force for translational velocity is more for q. translational velocity iii. The wheel which will go upto to a greater height rolls up on a r. rough inclined plane iv. The maximum height gained does not depend s. smooth inclined plane Now match the given columns and select the correct option from the codes given below.
  368. A ring of radius R is rotating with an angular speed ω 0 about a horizontal axis. It is placed on a rough horizontal table. The coefficient of kinetic friction is μ k . The time after which it starts rolling is
  369. A body is rolling down an inclined plane. If kinetic energy of rotation is 40% of kinetic energy in translatory state, then the body is a
  370. A wheel of mass 5 kg and radius 0.40 m is rolling on a road without sliding with angular velocity of 10 rad s – 1 . The moment of inertia of the wheel about the axis of rotation is 0.65 kg m 2 . The percentage of kinetic energy of rotation in the total kinetic energy of the wheel is
  371. Two bodies with moment of inertia I 1 and I 2 ( I 1 > I 2 ) have equal angular momenta. If their kinetic energy of rotation are E 1 and E 2 respectively , then
  372. A thin ring and a disc have same mass and roll without slipping with the same linear velocity v. If the total kinetic energy of the loop is 8 J, the kinetic energy of the disc must be
  373. A body rolls without slipping. The radius of gyration of the body about an axis passing through its centre of mass is K. The radius of the body is R. The ratio of rotational kinetic energy to translational kinetic energy is
  374. When a sphere rolls without slipping the ratio of its kinetic energy of translation to its total kinetic energy is
  375. A hoop rolls down on an inclined plane. The fraction of its kinetic energy that is associated with only the rotational motion is
  376. A uniform disc of mass M and radius R is mounted on an axle supported in frictionless bearings. A light cord is wrapped around the rim of the disc and a steady downward pull T is exerted on the cord. If we hang a body of mass m from the cord, the tangential acceleration of a point on rim of the disc is
  377. A uniform disc of mass M and radius R is mounted on an axle supported in frictionless bearings. A light cord is wrapped around the rim of the disc and a steady downward pull T is exerted on the cord. If we hang a body of mass m from the cord, the tangential acceleration of a point on rim of the disc is
  378. A wheel of moment of inertia 2 . 0 × 10 3 kgm 2 is rotating at uniform angular speed of 4 rads – 1 . What is the torque required to stop it in one second .
  379. A uniform disc of mass M and radius R is mounted on an axle supported in frictionless bearings. A light cord is wrapped around the rim of the disc and a steady downward pull T is exerted on the cord. The tangential acceleration of a point on the rim is
  380. A ring starts from rest and acquires an angular speed of 10 rad s – 1 in 2 s. The mass of the ring is 500 gm and its radius is 20 cm. The torque on the ring is
  381. A solid cylinder of mass M and radius R rolls without slipping down an inclined plane making an angle θ with the horizontal. Then its acceleration is
  382. A horizontal 90 kg merry-go-round is a solid disk of radius 1.50 m and is started from rest by a constant horizontal force of 50.0 N applied tangentially to the edge of the disk. The kinetic energy of the disk after 3.00 s is
  383. A solid sphere of radius R is rotating about an axis passing through its centre with a time period 0.5 seconds. During the rotation, if the sphere suddenly changes into a solid cylinder of height R 3 such that axis of cylinder coincides with axis of rotation, the time period of rotation of cylinder is
  384. A square ABCD is fabricated by four indentical thin rods each of mass m and length l. Then moment of inertia of the square about an axis passing through its centre O and perpendicular to its plane is
  385. The wheel on a grinder is a uniform 2 kg disc of 25 cm radius. It comes uniformly to rest from 1440 rpm in a time of 31.4 s. How large a friction torque slows its motion?
  386. An equilateral triangle ABC is fabricated by three identical rods each of mass m and length l. Then the moment of inertia of the triangular frame about an axis passing through vertex A and perpendicular to the plane of the triangular frame is
  387. A disc of mass 4 kg and radius 50 cm is rotating with angular velocity 30 rad/s about a spindle passing through its centre and perpendicular to the plane of the disc. A constant braking torque is applied on the disc and the disc stops spinning in 10 second. Then the magnitude of the braking torque is
  388. A solid disc of mass 2 kg and radius 0.5 m is rotating about its own geometrical axis at 2 rps. A constant torque π   N − m is applied on the disc. Then angular velocity of the disc after 10 sec will be
  389. A solid sphere is rolling without sliding on a horizontal surface. Then ratio of its translational kinetic energy to rotational kinetic energy is
  390. Which of the following has the highest moment of inertia when each of them has the same mass and the same radius?
  391. A solid aluminium sphere of radius R has moment of inertia l about an axis through its centre. The moment of inertia about a central axis of a solid aluminium sphere of radius 2R is
  392. A man standing at the centre of a turn table which is rotating at an angular velocity of 10 rad/s, with its arms out stretched. He suddenly folds his arms. If the moment of inertia of the turn table is 2.5 k g − m 2 , moment of inertia of the man with stretched arms is 0.3 k g − m 2 and moment of inertia of the man with folded arms is 0.1 k g − m 2 , then final angular velocity of the turn table is
  393. Four particles, each of mass m, are placed at the corners of a square having side l. Then moment of inertia of the particles about an axis passing through the centre of the square and perpendicular to its plane is
  394. A sphere of radius R is executing pure rolling on a rough horizontal surface. The acceleration of the point of contact is
  395. A uniform solid sphere is executing pure rolling without sliding on a rough horizontal surface. The ratio of its rotational kinetic energy to the total kinetic energy due to rolling is
  396. A disc of radius R and mass M is rolling horizontally without slipping with speed v. It then moves up an incline as shown in figure. The maximum height upto which it can reach is
  397. A ball of mass m and radius r rolls inside a hemispherical shell of radius R. It is released from rest from point A as shown in figure. The angular velocity of centre of the ball in position B about the centre of the shell is
  398. A rope is wound around a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N?
  399. Point masses m 1 and m 2 are placed at the opposite ends of a rigid rod of length L, and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position of point P on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity ω 0 is minimum is given by
  400. Three identical spherical shells, each of mass m and radius r are placed as shown in the figure. Consider an axis XX ‘ which is touching to two shells and passing through diameter to third shell. Moment of inertia of the system consisting of these three spherical shells about XX ‘ axis is
  401. A mass m moves in a circle on a smooth horizontal plane with velocity v 0 at a radius R 0 . The mass is attached to a string which passes through a smooth hole in the plane as shown. The tension in the string is increased gradually and finally m moves in a circle of radius R 0 2 The final value of the kinetic energy is
  402. A rod PQ of mass M and length L is hinged at end P. The rod is kept horizontal by a massless string tied to point Q as shown in the figure. When string is cut, the initial angular acceleration of the rod is
  403. A ball of mass m and radius r is rolling without slipping inside a hemispherical shell of radius R. It is released from rest from point A as shown in figure. The angular velocity of centre of the ball in position B about the centre of the shell is
  404. The moment of inertia of a uniform circular disc is maximum about an axis perpendicular to the disc and passing through
  405. A uniform cylinder has a radius R and length L. If the moment of inertia of this cylinder about an axis passing through its centre and normal to its circular face is equal to the moment of inertia of the same cylinder about an axis passing through its centre and normal to its length; then:
  406. Statement I: A Moment of inertial of uniform disc and solid cylinder of equal mass and equal radius about an axis passing through centre and perpendicular to plane will be same. Statement II: Moment of inertia depends upon distribution of mass from the axis of rotation i.e., perpendicular distance from the axis.
  407. A cubical block of side a is moving with velocity v on a horizontal smooth plane as shown in figure. It hits a ridge at point O. The angular speed of the block after it hits O is
  408. Figure shows a rough track, a portion of which is in the form of a cylinder of radius R. With what minimum velocity a sphere of radius r be set rolling on the horizontal part so that it completely goes around the circle on the cylindrical part?
  409. A small sphere I of radius r and mass m rolls without slipping inside a large hemispherical bowl of radius R as shown in figure. The sphere I starts from rest at the top point on the rim of the hemisphere. Find the normal force exerted by small sphere on the hemisphere when it touches the bottom of hemisphere.
  410. A cylindrical rod of mass M, length L and radius R has two cords wound around it whose ends are attached to the ceiling. The rod is held horizontally with the two cords vertical. When the rod is released, the cords unwind and the rod rotates the linear acceleration of the cylinder as it falls, is:
  411. A track is mounted on a large wheel that is free to tum with negligible friction about a vertical axis (Fig). A toy train of mass Mis placed on the track and, with the system initially at rest, the train’s electrical power is turned on. The train reaches speed v with respect to the track. What is the wheel’s angular speed if its mass is m and its radius is r? (Treat it as a hoop, and neglect the mass of the spokes and hub.)
  412. A solid sphere of mass M and radius r slips on a rough horizontal plane. At some instant it has translational velocity v 0 and rotational velocity about the centre v 0 2 r . The translational velocity after the sphere starts pure rolling is
  413. The two uniform discs rotate separately on parallel axles. The upper disc (radius a and momentum of inertia I 1 ) is given an angular velocity ω 0 and the lower disc of (radius b and momentum of inertia I 2 ) is at rest. Now the two discs are moved together so that their rims touch. Final angular velocity of the upper disc is
  414. When a body rolls without sliding up an inclined plane, the frictional force is:
  415. A bobbin is pushed along on a rough stationary horizontal surface as shown in the figure. The board is kept horizontal and there is no slipping at any contact points. The distance moved by the board when distance moved by the axis of the bobbin is l is
  416. A solid sphere of mass 10 kg is placed on a rough surface having coefficient of friction μ = 0 . 1 . A constant force F = 7 N is applied along a line passing through the centre of the sphere as shown in the figure. The value of frictional force on the sphere is
  417. A cotton reel rolls without sliding such that the point P of the string has velocity v = 6 m/s. If r = 10 cm and R = 20 cm then the velocity of its centre C is
  418. A disc is pulled by a force F acting at a point above the centre of mass of the disc. The direction of frictional force ( f r ) acting on disc pushed on a rough surface will be represented by
  419. A cylindrical drum, pushed along by a board rolls forward on the ground. There is no slipping at any contact. The distance moved by the man who is pushing the board, when axis of the cylinder covers a distance L will be
  420. A uniform solid disk of radius R and mass M is free to rotate on a frictionless pivot through a point on its rim. If the disk is released from rest in the position shown in figure. The speed of the lowest point on the disk in the dashed position is
  421. The reel shown in figure has radius R and moment of inertia I. One end of the block of mass m is connected to a spring of force constant k, and the other end is fastened to a cord wrapped around the reel. The reel axle and the incline are frictionless. The reel is wound counterclockwise so that the spring stretches a distance d from its un-stretched position and the reel is then released from rest. The angular speed of the reel when the spring is again un-stretched is
  422. Three identical thin rods, each of length L and mass m, are welded perpendicular to one another as shown in figure. The assembly is rotated about an axis that passes through the end of one rod and is parallel to another. The moment of inertia of this structure about this axis is
  423. A conical pendulum consists of a bob of mass m in motion in a circular path in a horizontal plane as shown in figure. During the motion, the supporting wire of length l, maintains a constant angle θ with the vertical. The magnitude of the angular mornentum of the bob about the vertical dashed line is
  424. A disk with moment of inertia I 1 rotates about a frictionless, vertical axle with angular speed ω i . A second disk, this one having moment of inertia I 2 and initially not rotating, drops onto the first disk (Fig.). Because of friction between the surfaces, the two eventually reach the same angular speed ω f The value of ω f is
  425. A projectile of mass m is launched with an initial velocity v i making an angle θ with the horizontal as shown in figure. The projectile moves in the gravitational field of the Earth. The angular momentum of the projectile about the origin when it is at the highest point of its trajectory is
  426. The moment of inertia of an angular wheel shown in figure is 3200 kgm 2 . If the inner radius is 5 cm and the outer radius is 20 cm, and the wheel is acted upon by the forces shown, then the angular acceleration of the wheel is
  427. A wheel of radius 20 cm has four forces applied to it as shown in fig. Then, the torque produced by these forces about O is
  428. Figure shows a lamina in x-.y plane. Two axes z and z’ pass perpendicular to its plane. A force F acts in the plane of lamina at point P as shown. Which of the following statements is incorrect? (The point P is closer to z’-axis than the z-axis).
  429. A hoop rolls on a horizontal ground without slipping with linear speed v. Speed of a particle P on the circumference of the hoop at angle θ is:
  430. A ring of mass M and radius R lies in x-y plane with its centre at origin as shown. The mass distribution of ring is non uniform such that, at any point P on the ring, the mass per unit length is given by λ = λ 0 cos 2 θ ( where λ 0 is a positive constant ). Then the moment of inertia of the ring about z-axis is:
  431. A triangular plate of uniform thickness and density is made to rotate about an axis perpendicular to the plane of the paper and (i) passing through A, (ii) passing through B, by the application of some force F at C (mid-point of AB) as shown in the figure. In which case angular acceleration is more?
  432. Two thin discs each of mass M and radius r are attached as shown in figure, to form a rigid body. The rotational inertia of this body about an axis perpendicular to the plane of disc B and passing through its centre is:
  433. In a rectangle ABCD, AB = 2l and BC = l. Axes xx and yy pass through centre of the rectangle. The moment of inertia is least about:
  434. The moment of inertia of a door of mass m, length 2l and width l about its longer side is
  435. A rectangular loop has mass M and sides a and b. An axis OO ‘ passes tlrrough the centre C of the loop and is parallel to side a (lie in the plane of the loop). Then the radius of gyration of the loop, for the axis OO ‘ is
  436. A uniform square plate has a small piece Q of an irregular shape removed and glued to the centre of the plate leaving a hole behind in figure. The moment of inertia about the z-axis is then,
  437. Moment of inertia of uniform triangular plate about axis passing through sides AB, AC, BC are I p , I B and I H respectively and about an axis perpendicular to the plane and passing through point C is I C . Then:
  438. Three identical thin rods each of length l and mass M are joined together to form a letter H. What is the moment of inertia of the system about one of the sides of H?
  439. A right circular solid cylinder of radius 10 cm and height 20 3   c m is placed on a rough inclined surface having angle of inclination θ with horizontal. Upto what maximum value of θ , will the cylinder not topple? Assume that there is sufficient friction between the cylinder and the inclined surface to prevent slipping.
  440. Two uniform, thin identical rods each of mass M and length L are joined at middle so as to form a cross as shown. The moment of inertia of the cross about a bisector line EF (in the plane of cross) is
  441. Three identical thin rods, each of mass m and length L, are joined to form an equilateral triangular frame. The moment of inertia of the frame about an axis parallel to its one side and passing through the opposite vertex is
  442. Four spheres each having mass m and radius r are placed with their centres on the four corners of a square of side a. Then the moment of inertia of the system about an axis along one of the sides of the square is
  443. A ring is rolling without slipping on a plane surface as shown in figure. If speed of A is 4 m/s, speed of B is
  444. A solid disc is rolling without slipping on a plane surface. If its kinetic energy of translation is 2 Joule, total kinetic energy of the disc
  445. A cylindrical container contains some ice and the container is rotating with constant angular velocity ω 0 about its own geometrical axis. Now heat is supplied to the cylinder and the ice is completely melted. Then angular velocity of the container becomes ω . Then
  446. A sphere is spinning about a diameter and its angular momentum is varying with time according to the graph shown in figure. If moment of inertia of the sphere is 2 Kg-m 2 , its maximum angular acceleration is
  447. The end A of a thin rod AB of mass m and length l is hinged and the end B is attached to X string BC hanging from a ceiling. Just after the string is burnt
  448. A rod AB of length l is moving in a plane and at an instant of time velocities of its ends A and B are 4V and V as shown in figure. Then angular velocity of the rod is
  449. A solid disc of mass 2 kg and radius 50 cm can rotate about an axis through its centre O and perpendicular to the plane of the disc. Forces are applied at points A, B and C as shown in figure. Then angular acceleration of the disc is
  450. A metallic sheet, in the form of a square ABCD, has non-uniform distribution of mass. Two forces 10 N and 20 N are applied on the sheet as shown in figure. It is observed that the sheet makes only translational motion. Then distance of centre of mass from the corner A may be
  451. A pulley one meter in diameter rotating at 600 revolutions a minute is brought to rest in 80 sec by a constant force of friction on its shaft. How many revolutions does it make before coming to rest?
  452. A point P consider at contact point of a wheel on ground which rolls on ground without slipping then value of displacement of point P when wheel completes half of rotation (If radius of wheel is 1 m) :
  453. Moment of inertia plays the same role in rotatory motion as in translatory motion is played by:
  454. The dimensions of moment of inertia are:
  455. Which of the following has the smallest moment of inertia about the central axis if all have equal masses and radii?
  456. For the same total mass which of the following will have the largest moment of inertia about an axis passing through the centre of mass and perpendicular to the plane of the body?
  457. If the moment of inertia of a rod of mass M and length L about an axis passing through its centre of mass and perpendicular to its length is (ML 2 /12), then moment of inertia of the same rod about an axis passing through one of its ends and parallel to the given axis will be:
  458. Three uniform rods, each of mass ‘M’ and length ‘L’ form the sides of an equilateral triangle. Moment of inertia of this system about an axis passing through a corner and perpendicular to the plane of triangle is:
  459. The momentum of inertia of a rod about an axis through its centre and perpendicular to it is 1 12 ML 2 (where M is the mass and L is the length of the rod). The rod is bent in the middle so that the two halfs make an angle of 60 o . The moment of inertia of the bent rod about the same axis would be :
  460. Three particles, each of mass m gram, are situated at the vertices of an equilateral triangle ABC of side l cm (as shown in the figure). The moment of inertia of the system about a line AX perpendicular to AB and in the plane of ABC, in gram cm 2 units will be :
  461. The ratio of the radii of gyration of a circular disc about a tangential axis in the plane of the disc and of a circular ring of the same radius about a tangential axis in the plane of the ring is :
  462. Four identical thin rods each of mass M and length l form a square frame. Moment of inertia of this frame about an axis through the centre of the square and perpendicular to its plane is :
  463. From a circular disc of radius R and mass 9M, a small disc of mass M and radius R 3 is removed concentrically. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through its centre is :
  464. Point masses m 1 and m 2 , are placed at the opposite ends of a rigid rod of length L, and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position of point P on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity ω o it minimum, is given by :
  465. Find moment of inertia of a uniform annular disc of mass 100 g, having inner radius l0 cm and outer radius 20 cm, about an axis passing through its centre and perpendicular to its plane :
  466. A cylinder having mass = 200 g, length = 6 cm, inner radius = 5cm and outer radius = 9 cm. Find moment of inertia about axis of symmetry.
  467. A body of mass m slides down an incline and reaches the bottom with a velocity v . If the same mass was in the form of a ring which rolls down this incline, the velocity of the ring at the bottom would have been:
  468. A ring is rolling on a rough horizontal surface without slipping with a linear speed v. Referring to Figure linear speed of point X is:
  469. A ring is rolling on a rough horizontal surface without slipping with a linear speed v. Referring to Figure linear speed of point X is:
  470. A cylinder of radius R is spinned and then placed on an incline having coefficient of friction μ = tanθ ( θ is the angle of incline). The cylinder continues to spin without falling for time :
  471. When a stick is released (as shown in figure), its free end velocity when its strikes the ground is :
  472. A solid sphere of radius R is placed on smooth horizontal surface. A horizontal force ‘F’ is applied at height ‘h’ from the lowest point. For the maximum acceleration of centre of mass, which is correct
  473. A uniform rod of length l, hinged at the lower end is free to rotate in the vertical plane. If the rod is held vertically in the beginning and then released, the angular acceleration of the rod when it makes an angle of 45° with the horizontal is:
  474. Figure shows a flywheel of radius 10 cm. Its moment of inertia about the rotation axis is 0.4 kg-m 2 . A massless string passes over the flywheel and a mass 2kg is attached at its lower end. Angular acceleration of the pulley is nearly
  475. A uniform disc of radius 20 cm and mass 2kg is fixed at its centre and can rotate about an axis through the centre and perpendicular to its plane. A massless cord is round along the rim of the disc. If a uniform force 2 newton is applied on the cord, tangential acceleration of a point on the rim of the disc will be:
  476. A door 1.6 m wide requires a force of 1 N to be applied at the free end to open or close it. The force that is required at a point 0.4 m distant from the hinges for opening or closing the door is:
  477. In Figure, a cubical block is held stationary against a rough wall by applying a force F, then the incorrect statement among the following is:
  478. 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 slip before toppling, the minimum force required to topple the block is:
  479. A torque of 2 N-m produces an angular acceleration of 2 rad/sec 2 in a body. If its radius of gyration is 2m, its mass will be :
  480. A weightless rod is acted on by upward parallel forces of 2 N and 4 N at ends A and B respectively. The total length of the rod AB = 3 m. To keep the rod in equilibrium a force of 6 N should act in the following manner:
  481. Four equal and parallel forces are acting on a rod (as shown in Figure) at distances of 20 cm, 40 cm, 60 cm and 80 cm respectively from one end of the rod. Under the influence of these forces the rod:
  482. Let F be the force acting on particle having position vector r and τ be. the torque of this force about the origin. Then :
  483. If a street light of mass M is suspended from the end of a uniform rod of length L in the different possible patterns as shown in figure, then :
  484. O is the centre of an equilateral triangle ABC. F 1 , F 2 and F 3 are three forces acting along the sides AB, BC and AC as shown in figure. What should be the magnitude of F 3 so that total torque about O is zero
  485. A wheel having moment of inertia 2 kg-m 2 about its vertical axis, rotates at the rate of 60 rpm about an axis. The torque which can stop the wheel’s rotation in the minute would be :
  486. When there is no external torque acting on a system the quantity which has a necessarily zero magnitude is
  487. ABC is an equilateral triangle with O as its centre. F 1 , F 2 and F 3 represent three forces acting along the sides AB, BC and AC respectively. If the total torque about O is zero then the magnitude of F 3 is :
  488. A rod PQ of mass M and length L is hinged at end P. The rod is kept horizontal by a massless string tied to point Q as shown in figure. When string is cut, the initial angular acceleration of the rod is :
  489. A force ( 5 i ^ + 7 j ^ − 3 k ^ ) N acts on a particle at position ( i ^ + j ^ − k ^ ) m . Find torque of this force on the particle about origin
  490. A solid cylinder of mass 50 kg and radius 0.5 m is free to rotate about the horizontal axis. A massless string is wound round the cylinder with one end attached to it and other hanging freely. Tension in the string required to produce an angular acceleration of 2 revolutions s -2 is
  491. A wheel and an axle is made to rotate about a horizontal axis with the help of a body of mass 5 kg attached to a string wound around the axle. The radius of the axle is l0 cm. The body falls vertically through 5 m in 10 s starting from rest. The moment of inertia of the wheel and axle is :
  492. What will be the value of maximum acceleration of the truck in the forward direction so that the block kept on the back does not topple
  493. A wheel of moment of inertia I and radius R is rotating about its axis at an angular speed ω 0 . It picks up a stationary particle of mass mat its edge. Find the new angular speed of the wheel.
  494. A uniform disc of mass M and radius R is supported vertically by a pivot at its periphery as shown. A particle of mass M is fixed to the rim and raised to the highest point above the centre. The system is released from rest and it can rotate about pivot freely. The angular speed of the system when it attached object is directly beneath the pivot, is
  495. The angular momentum of a rotating body changes from A 0 to 4A 0 s in 4 sec. The torque acting on the body is :
  496. An object of mass ‘m’ is projected with a velocity ‘u’ at an angle 45° with the horizontal. When the object is at maximum height, its angular momentum about the point of projection is
  497. A uniform rod of length 2 m and mass 10 kg is placed on a table. It is made to rotate about an axis that passes through one of its ends and is perpendicular to the plane of the table with an angular velocity 10 π rad/s. Angular momentum of the rod about the given axis is:
  498. A solid uniform sphere rotating about its axis (with rotational kinetic energy ω 0 ) is placed on a rough horizontal plane without any translational push. Friction coefficient p is not same everywhere on the plane and it may differ even at each point. After sometime the sphere begins pure rolling with total kinetic energy E. Then:
  499. A particle of mass m is released from rest at point A falling parallel to vertical y-axis. Its angular momentum after falling through distance h about O is:
  500. A disc of mass M and radius R is rolling with angular speed o on a horizontal plane as shown. The magnitude of angular momentum of the disc about the origin O is:
  501. A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same which one of the following will not be affected?
  502. A small particle of mass m is projected at an angle θ with the x-axis with an initial velocity us in the x-y plane as shown in the figure. At a time t < v 0 sin ⁡ θ g , angular momentum of the particle is :
  503. A merry-go-round, made of a ring-like platform of radius R and mass M, is revolving with angular speed ω . A person of mass M is standing on it. At one instant, the person jumps off the round radially away from the centre of the round as seen from the round. The speed of the round afterwards is :
  504. A particle of mass m moves along line PC with velocity v as shown. What is the angular momentum of the particle about O ?
  505. A plank P is moving with constant velocity V on a horizontal surface. A disc of radius R is rolling without slipping on the plank with a velocity V relative to the plank. Then radius of curvature of the trajectory of a point on the circumference of the ring when it is at position A, as viewed by an observer standing on the ground is
  506. The moment of inertia of a body does not depend on:
  507. Four masses are fixed on a massless rod as shown in Figure . The moment of inertia about the axis P is about:
  508. The moment of inertia of a thin square plate ABCD of uniform thickness about an axis passing through the centre O and perpendicular to the plate is: (i) I 1 + I 2 (ii) I 3 + I 4 (iii) I 1 + I 3 (iv) I 1 + I 2 + I 3 + I 4 where I 1 ,I 2 , I 3 and l 4 are respectively the moments of inertia about axes 1,2,3 and 4 which are in the plane of the plate.
  509. A spool is rolling without slipping on a strip with velocity v as shown in figure. Then acceleration of point A relative to ground is
  510. The moment of inertia of a thin uniform rod of mass M and length I about an axis passing through its midpoint and perpendicular to its length is I 0 . Its moment of inertia about an axis passing through one of its ends and perpendicular to its length is
  511. A disc is rotating about its own axis with angular frequency 100 rpm. When a mass of 100 g is placed at l0 cm from its centre then its angular frequency becomes 50 rpm. The moment of inertia of the disc is :
  512. Lower end B of a thin rod AB of length 1 m is made to move with a constant velocity of 1 m/s as shown in figure. Initial angle of inclination the rod with horizontal is 90 0 . Then magnitude of angular acceleration of the rod at t = 0.5 second is
  513. End B of a thin rod of length ‘ l ’ is hinged so that it can swing in a vertical plane. ‘S’ is a stopper at a distance ‘x’ below the hinge B. Initially the rod is held in horizontal position by an external agent as shown in figure. After the rod is released it swings in a vertical plane and when it becomes vertical it strikes the stopper S and stops. If during the course of impact the horizontal component of reaction at the hinge B is zero, find x.
  514. A solid uniform cylinder is gently released on an inclined surface AB whose angle of inclination with horizontal can be varied. If coefficient of static friction between the cylinder and the inclined surface AB is 1 3 3 , The maximum possible linear acceleration of the centre of mass of the cylinder for pure rolling, is (Take g = 10 m/s 2 )
  515. A thin ring of radius is rolling without slipping on the horizontal surface AB with velocity. Then the ring strikes the inclined surface BC and starts rolling up the plane. What fraction of initial energy is lost in the process ?
  516. A thin rod AB of mass m and Length ‘ l’ is lying on a frictionless horizontal surface. Two forces, each of magnitude F are applied to the rod at points A and M. Then select the correct option.
  517. A solid sphere of mass M and radius R is placed on a rough horizontal surface. It is pulled by a horizontal force F acting through its centre of mass as a result of which it begins to roll without slipping. Angular acceleration of the sphere can be expressed as:
  518. Figure shows a hemisphere of radius 4R. A ball of radius R is released from position P. It rolls without slipping along the inner surface of the hemisphere. Linear speed of its centre of mass when the ball is at position Q, is:
  519. When a mass is rotating in a plane about a fixed point its angular momentum is directed along :
  520. A small object of uniform density rolls up a curved surface with an initial velocity v. It reaches up to a maximum height of 3 v 2 4 g with respect to the initial position. The object is:
  521. A small mass attached to a string rotates on a frictionless table top as shown. If the tension in the string is increased by pulling the string causing the radius of the circular motion to decrease by a factor of 2,the kinetic energy of the mass will :
  522. A solid sphere is rolling on a frictionless surface, shown in figure with a translational velocity v m/s. If it is to climb the inclined surface then u should be :
  523. A force F = α i ^ + 3 j ^ + 6 k ^ is acting at a point r = 2 i ^ − 6 j ^ − 12 k ^ The value of α for which angular momentum about origin is conserved is :
  524. The radius of a rotating sphere decreases by 2% keeping the mass constant, the change in its rotational kinetic energy?
  525. A solid cylinder of mass 3 kg rolling on a horizontal surface with velocity 4 m/s. It collides with a horizontal spring of force constant 200 N/m. The maximum compression produced in the spring will be :
  526. A disc is rolling (without slipping) on a rough surface. C is its centre and Q and P are two points equidistant from C. Let V p ,V Q andV, be the magnitudes of velocities of points P,Q and C respectively, then :
  527. Find out angular velocity of disk. If disk is confined to roll without slipping at points of contact. Radius of disk is 5 m :
  528. A force of – F k ^ acts on O, the origin of the co-ordinate system. The torque about the point (1, -1) is :
  529. Two identical rings each of mass m with their planes mutually perpendicular, radius R are welded at their point of contact O. If the system is free to rotate about an axis passing through the point P perpendicular to the plane of the paper the moment of inertia of the system about this axis is equal to :
  530. A uniform beam of weight W is attached to a vertical wall by a hinge H. The beam is held horizontal by a rope as shown below : Which one of the following best shows the direction of the reaction force R at the hinge?
  531. A vertical rectangular door with its centre of gravity at O (see figure) is fixed on two hinges A and B along one vertical length side of the door. The entire weights of the door is supported by the hinge A. Then the free body force diagram for the door (the arrows indicate the direction of the forces) is :
  532. A solid cylinder S and a hollow cylinder T of same mass and same outer radius have a thin thread wound over them. One end of each thread is fixed to the same ceiling. They are initially held at rest with their axis horizontal and parallel to each other, at same height above ground. tf the cylinders are released simultaneously with strings taut then the correct statement is :
  533. A solid sphere of mass M and radius r slips on a rough horizontal plane. At some instant it has translational velocity v 0 and rotational velocity about the centre v 0 /2r . The translational velocity after the sphere starts pure rolling
  534. The following figure shows two situations in which a uniform round rigid body is released from rest from the positions shown, such that it is just able to loop the loop without leaving contact with the track. Assuming that radius of the track is large in comparison to the radius of round body, the ratio h 1 /h 2
  535. A particle is projected with some velocity at an acute angle to the horizontal. The graph between its angular momentum about the point of projection and time during flight is a :
  536. A solid cylinder of mass 3 kg rests on a rough horizontal plane. A horizontal force F is applied to the top most point of the cylinder as shown in the figure. Coefficient of friction between the cylinder and the surface is 0.6. Then the maximum value of F for which the cylinder will roll without slipping is
  537. In the arrangement shown, the end B of the rod of length ‘ l ’ is made to move with constant velocity V. Then velocity of its centre of mass when θ = 30 0 is
  538. A constant torque acting on a uniform circular wheel changes its angular momentum from A to 4A in 4sec. The torque acted on it is
  539. A particle of mass m moves with a constant velocity. Which of the following statements is not correct about its angular momentum about origin?
  540. A hollow sphere of mass ‘M’ and radius ‘R’ is rolling on horizontal surface with speed ‘V’. Find the angular momentum of sphere about point of contact on surface.
  541. A particle of mass m is projected from point P on the ground with initial speed v at an angle θ with horizontal. Find the angular momentum of particle w.r.t. point P when it is at highest point of its trajectory.
  542. If the moment of inertia of a disc about an axis tangentially and parallel to its surface be I, what will be the moment of inertia about the axis tangential but perpendicular to the surface-
  543. The moment of inertia of sphere is 20 kg- m 2 about the diameter. The moment of inertia about any tangent will be-
  544. A constant torque acting on a uniform circular wheel changes its angular momentum from A 0 to 4 A 0 in 4 seconds. The magnitude of this torque is-
  545. The angular momentum of a particle performing uniform circular motion is L. If the kinetic energy of particle is doubled and frequency is halved, then angular momentum becomes
  546. A uniform cylindrical disc of radius R and mass M is pulled over a horizontal friction less surface by a constant force. The force is applied by means of a string wound around the disc as shown in fig. (2). If it starts from rest at t = O, the linear and angular displacements respectively at time , are
  547. A body of mass m is projected with vertical making an angle 45 o with the horizontal from the point P. Its highest point is H and it returns to the ground at Q. The change in angular momentum in going from P to Q is
  548. A uniform rod PQ of length I is hinged at one end p and is kept in horizontal position by a massless string tied to point Q as shown in fig. ( ). If the string is cut, the initial angular acceleration of the rod will be
  549. A particle of mass m is projected with a velocity u making an angle of 45 o 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
  550. If L, m denote the angular momentum and mass of the particle a:rd p is linear momentum, which of the following can represent the kinetic energy of the particle moving in a circle of radius .R ?
  551. A particle of mass m moves along the path PC with velocity V as shown in fig. (9) . The angular momentum of the particle about O is
  552. A thin circular ring of mass M and radius r is rotating about an axis passing through its centre and perpendicular to its plane with a constant angular velocity ω . Two objects, each of mass m are attached gently to the opposite ends of a diameter of the ring’ The ring now rotates with angular velocity ω
  553. A thin uniform circular disc of mass M and radius R is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity ω : Another disc of same dimensions but of mass M/4 is placed gently on the first disc co-axially. The angular velocity ω ‘of the system is
  554. What will be the moment of inertia of thin rod of mass M and length L about an axis passing through its end and perpendicular to the length ?
  555. Two spheres each of mass M and radius R/2 are connected with a massless rod of length 2 R as shown in fig. (12). What will be the moment of inertia of the system about an axis passing through the centre of one of the sphere and perpendicular to the rod ?
  556. A thin, uniform, circular ring is rolling down an inclined plane of inclination 30 o without slipping. Its linear acceleration along the inclined plane will be
  557. Four sphere of diameter 2a and mass M are placed with their centres of the four comers of a square of side b. Then the moment of inertia of the system about an axis along one of the sides of the square is
  558. Two discs of moments of inertia I 1 and I 2 about their respective axes, rotating with angular frequencies ω 1 and ω 2 respectively, are brought into contact face to face with their axes of rotation coincident. The angular frequency of the composite disc will be
  559. Three thin rods each of length I and mass Mare placed along X, Y and Z axes in such a way that one end of each of the rods is at the origin as shown in fig The moment of inertia of this system about Z-axis
  560. Four identical rods (each of length I) are joined end to end to form a square as shown in fig. The mass of each rod is M. The moment of inertia of the square about the median line is
  561. Fig. shows a thin metallic triangular sheet,{BC. The sides AB and BC are equal of length L. The mass of the sheet is M. The moment of inertia of the sheet about AC is
  562. Fig. shows a uniform solid block of mass M and edge length a, b and c. The moment of inertia of the block about an axis through one edge and perpendicular (as shown) to the large face is
  563. The curve between angular moment L and angular velocity is:
  564. For the given uniform square lamina ABCD, whose centre is O find the relation between I A C and I E F
  565. The moment of inertia of a body about a given axis is 1.2kgm 2 ,Initially, the body is at rest. In order to produce a rotational kinetic energy of 1500J,an angular acceleration of 25 rad s – 2 must be applied about that axis for a duration of
  566. A particle performs uniform circular motion with an angular momentum L. If the angular frequency of the particle is doubled and kinetic energy is halved, its angular momentum becomes
  567. If a constant torque of 5OO N-m turns a wheel of moment of inertia 1OO kg-m 2 about an axis passing though its centre, then the gain in angular velocity in 2 sec. is
  568. A wheel of moment of inertia 2 x 10 3 kg-m 2 is rotating at uniform angular speed of 4 rad/s. What is the torque required to stop it in one second
  569. The moment of inertia of a cylinder about its own axis is equal to its moment of inertia about an axis passing through its centre and perpendicular to its length. The ratio of length to radius is
  570. A thin wire of length L and uniform linear mass density p is bent into a circular loop with centre O as shown in fig. The moment of inertia of the loop about the axis XX’ is
  571. A circular platform is free to rotate in a 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 ω 0 When the tortoise moves along a chord of platform with a constant velocity (with respect to the platform), the angular velocity of the platform ω (t) will vary with time t as
  572. A cylinder of mass M, radius R is resting on a horizontal platform (which is parallel to x-y plane) with its axis fixed along the y-axis and free to rotate about its axis. The Platform is given a motion in x-direction given by x = acos ⁡ ωt There is no slipping between the cylinder and platform. The maximum torque acting on the cylinder during its motion is
  573. A smooth uniform rod of length L and mass M has two identical beads of negligible size, each of mass m, which can slide freely along the rod. Initially, the two beads are at the centre of the rod and the system is rotating with angular velocity ω 0 , about an axis perpendicular to rod and passing through the mid-point of rod. There are no external forces. When the beads reach the ends of the rod, the angular velocity of the system is
  574. A circular disc of mass m is revolving with angular velocity ω along a line passing perpendicular to its centre. When a mass m is gently attached to the revolving disc, then what will be new angular velocity
  575. A one kg stone attached to the end of a 60 cm chain is revolving at the rate of 3 revolutions/second. If after 30 second, it is making only one revolution per second, find the mean torque
  576. A flywheel is a uniform disc of mass 72 kg and 50 cm. When it is rotating at the rate of 70 rpm its kinetic energy is
  577. A solid cylinder of mass M and radius R without slipping down an inclined plane of length L and height h. What is the speed of centre of mass when the cylinder reaches bottom ?
  578. A circular disc of radius R and thickness (R/6) has moment of inertia I about an axis passing through its centre and perpendicular to its plane. It is melted and recast into a solid sphere. The moment of inertia of the sphere about its diameter on axis of rotation is
  579. Circular disc X of radius R is made from an iron plate of thickness t, and another disc Y of radius 4 R is made from an iron plate of thickness. t 4 .Then, the relation between the moment of inertia I X , and Iy is
  580. A diatomic molecule is formed by two atoms which may be treated as mass points m 1 and m 2 ,joined by a massless rod of length r. Then the moment of inertia of the molecule about an axis passing through the centre of mass and perpendicular to rod is
  581. The radius of gyration of a body about an axis at a distance 6 cm from its centre of mass is 10 cm. Then its radius of gyration about a parallel axis through its centre of mass will be
  582. A wheel having moment of inertia of 2 kg-m 2 about its vertical axis, rotates at the rate of 60 rpm about this axis. The torque which can stop the wheel’s rotation in one minute would be
  583. A round disc of moment of inertia I 2 , about its axis perpendicular to its plane and passing through its centre is placed over another disc of moment of inertia I 1 , rotating with an angular velocity ω about the same axis. The final angular velocity of the combination of discs is
  584. A disc is rolling (without slipping) on a horizontal surface. C is its centre and Q and P are two points equidistant from C. Let v p , v Q . and v C be the magnitudes of velocities of points, P, Q and C respectively, then
  585. A solid sphere is rotating about a diameter at an angular velocity ω If it cools so that its radius reduces to 1 n of its original value, its angular velocity becomes
  586. Two bodies have their moments of inertia I and 2 I respectively about their axis of rotation’ If their kinetic energies of rotation are equal, their angular momenta will be in the ratio
  587. The moment of inertia of a uniform circular disc of radius R and mass M about an axis passing from the edge of the disc and normal to the disc is
  588. A drum of radius R and mass M rolls down without slipping along an incline plane of angle θ . The frictional force
  589. The moment of inertia of a uniform semicircular disc of mass M and radius r about a line perpendicular to the plane of the disc through the centre is
  590. Four point masses, each of value m, are placed at the corners of a square ABCD of side l. The moment of inertia of the system about an axis passing through A and parallel to BD is
  591. A heavy uniform bar is carried by two men on their shoulders. The weight of the bar is W If one man lets it fall from the end carried try him, the weight experienced by the other at the instant the rod is falling is
  592. A particle of mass m is describing a circular path of radius r with uniform speed. If L is the angular momentum of the particle about the axis of the circle, the kinetic energy of the particle is given by
  593. A particle is making uniform circular motion with angular momentum L. If its kinetic energy is made half and angular frequency be doubled, its new angular momentum m will be
  594. A particle of mass m = 5 units is moving with a uniform speed v = 3 2 , units in the XOY plane along the line y = x + 4. The magnitude of the angular momentum of the Particle about the origin is
  595. Find the torque of a force F = -3 i ^ + j ^ + 5 k ^ acting at the point r = 7 i ^ + 3 j ^ + k ^
  596. The moment of force F= i ^ + j ^ + k ^ acting at points (- 2,3, 4) about point (1, 2, 3) is
  597. A particle of mass m is moving with constant velocity v along X-axis in X-Y plane as shown in figure. Its angular momentum with respect to origin at any time t, if position vector is r is
  598. A force of – F k ^ acts on O, the origin of the coordinate system. The torque about the point (1, – 1) is
  599. A diver in a swimming pool bends his head before diving, because it
  600. If a person standing on a rotating disc stretches out his hands, the angular speed will
  601. A wheel is rotating at 900 rpm about its axis. When power is cut-off, it comes to rest in 1 min. The angular retardation (in rad s -2 ) is
  602. A particle performs uniform circular motion with an angular momentum L . If the frequency of the particle motion is doubled, the angular momentum becomes
  603. A flywheel having a radius of gyration of 2 m and mass 10 kg rotates at an angular speed of 5 rad s -1 about an axis perpendicular to it through its centre. The kinetic energy of rotation is
  604. A ring of diameter 0.4 m and mass 10 kg is rotating about its axis at the rate of 1200 rpm. The angular momentum of the ring is
  605. A particle of mass 2 kg located at the position ( i ^ + j ^ ) m has a velocity 2 ( + i ^ – j ^ + k ^ ) ms – 1 Its angular momentum about Z-axis (in kg – m 2 s – 1 ) is
  606. Five particles of masses 2 kg each are attached to the rim of a circular disc of radius 0.1 m and negligible mass. Moment of inertia of the system about the axis passing through the centre of the disc and perpendicular to its plane is
  607. A constant torque of 1000 N-m turns a wheel of moment of inertia 200 kg-m 2 about an axis through its centre. Its angular velocity after 3 s is
  608. A rod is placed along the line y = 2 x with its centre at origin. The moment of inertia of the rod is maximum about
  609. Work done by friction in case of pure rolling
  610. A disc of mass m and radius R is rolling on horizontal ground with linear velocity v . What is the angular momentum of the disc about an axis passing through bottom most point and perpendicular to the plane of motion?
  611. A solid sphere of mass 2 kg rolls up a 30 0 incline with an initial speed of 10 ms -1 . The maximum height reached by the sphere is (Take, g = 10 ms -2 )
  612. Let I A and I B be moments of inertia of a body about two axes A and B , respectively. The axis A passes through the centre of mass of the body but B does not. Choose the correct option.
  613. A uniform square plate has a small piece Q of an irregular shape removed and glued to the centre of the plate leaving a hole behind as shown in figure. The moment of inertia about the Z-axis is
  614. Moment of a force of magnitude 10 N acting along positive y -direction at point (2m, 0, 0) about the point (0, 1m, 0) (in N-m) is
  615. The radius of gyration of a uniform rod of length L about an axis passing through its centre of mass is
  616. A flywheel is in the form of a uniform circular disc of radius 1 m and mass 2 kg. The work which must be done on it to increase its frequency of rotation from 5 rps to 10 rps is approximately
  617. The ratio of the radii of gyration of a hollow sphere and a solid sphere of the same radii about a tangential axis is
  618. A uniform disc of radius a and mass m is rotating freely with angular speed ω in a horizontal plane about a smooth fixed vertical axis through its centre. A particle of mass m is suddenly attached to the rim of the disc and rotates with it. The new angular speed is
  619. A particle of mass 5 g is moving with a uniform speed of 3 2 cm s – 1 in the XY-plane along the line y = 2 5 cm The magnitude of its angular momentum about the origin (in g-cm 2 s -1 ) is
  620. A square lamina is as shown in figure. The moment of inertia of the frame about the three axes as shown in figure are I 1 , I 2 and I 3 , respectively. Select the correct alternative
  621. A thin uniform circular disc of mass M and radius R is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with an angular velocity ω . Another disc of same dimensions but of mass 1 4 M is placed gently on the first disc co-axially. The angular velocity of the system is
  622. A ball rolls without slipping. The radius of gyration of the ball about an axis passing through its centre of mass is K . If radius of the ball be R, then the fraction of total energy associated with its rotational energy will be
  623. A merry-go-round made of a ring-like platform of radius R and mass M , is revolving with angular speed ω . A person of mass M is standing on it. At one instant, the person jumps off the round, radially away from the centre of the round (as seen from the round). The speed of the round afterwards is
  624. A disc is rolling without slipping on a horizontal surface with C as its centre and Q and P the two points equidistant from C . Let v P , v Q and v C be the magnitudes of velocities of points P, Q and C respectively, then
  625. If a disc of mass m and radius r is reshaped into a ring of radius 2 r , the mass remaining the same, the radius of gyration about centroidal axis perpendicular to plane goes up by a factor of
  626. A rope is wound around a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder, if the rope is pulled with a force of 30 N? [NEET 2017]
  627. What is the moment of inertia of solid sphere of density ρ and radius R about its diameter? [UP CPMT 2012]
  628. A particle moving in a circular path has an angular momentum of L. If the frequency of rotation is halved, then its angular momentum becomes [Kerala CEE 2013]
  629. A rod PQ of mass M and length L is hinged at end P. The rod is kept horizontal by a massless string tied to point Q as shown in figure. When string is cut, the initial angular acceleration of the rod is [NEET 2013)
  630. A rod of mass 5 kg is connected to the string at point B. The span of rod is along horizontal. The other end of the rod is hinged at point A If the string is massless, then the reaction of hinge at the instant when string is cut, is (Take, 9 = 10 ms -2 ) [UP CPMT 2015]
  631. The moment of inertia of a thin uniform rod of mass M and length L about an axis passing through its mid-point and perpendicular to its length is I 0 , Its moment of inertia about an axis passing through one of its ends and perpendicular to its length is
  632. Which of the following statements are correct? [NEET 2017] I. Centre of mass of a body usually coincides with the centre of gravity of the body. II. Centre of mass of a body is the point at which the total gravitational torque on the body is zero. III. A couple on a body produces both translational and rotational motion in a body. IV. Mechanical advantage greater than one means that small effort can be used to lift a large load.
  633. The moment of the force F = 4 i ^ + 5 j ^ − 6 k ^ acting at (2 0, – 3), about the point (2 – 2 – 2) is given by
  634. Three identical spherical shells, each of mass m and radius r are placed as shown in figure. Consider an axis XX’ , which is touching the two shells and passing through diameter of third shell. Moment of inertia of the system consisting of these three spherical shells about XX’ axis is [CBSE AIPMT 2015]
  635. A wheel of radius 10 cm can rotate freely about its centre as shown in figure. A string wrapped over its rim is pulled by a force of 5N. It i” observed that the torque produces angular acceleration of 2 rad s -2 in the wheel. What is the moment of inertia of the wheel? [NEET 2016]
  636. If two circular discs A and B are of same mass but of radii rand 2 r respectively, then the moment of inertia of A is [Kerala CEE 2015]
  637. A solid cylinder of mass 2 kg and radius 4 cm is rotating about its axis at the rate of 3 rpm. The torque required to stop after 2 π revolutions is [NEET 2019]
  638. Moment of inertia of ring about its diameter is I . The moment of inertia of the same ring about that axis perpendicular to its plane and passing through centre is [KCET 2014]
  639. The moment of inertia ( I ) of a sphere of radius R and mass M is given by IJ&K CET 2013]
  640. A mass m moves in a circle on a smooth horizontal plane with velocity v 0 at a radius R 0 . The mass is attached to a string which passes through a smooth hole in the plane as shown in the figure. The tension in the string is increased gradually and finally m moves in a circle of radius R 0 /2. The final value of the kinetic energy is [CBSE AIPMT 2015]
  641. Moment of inertia of a disc of radius R about a diametric axis is 25 kg m -2 . The moment of inertia of the disc about a parallel axes at a distance R/2 from the centre is [MPPMT 2013]
  642. One solid sphere A and another hollow sphere B are of same mass and same outer radius. Their moments of inertia about their diameters are respectively, I A and I B such that
  643. If earth suddenly shrinks to 1 8 th of its original volume while mass remaining the same, then duration of day (in hour) will be
  644. Two discs have same mass and thickness. Their materials are made of densities d 1 and d 2 . The ratio of their moments of inertia about an axis passing through the centre and perpendicular to the plane is
  645. Figure 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 ?
  646. When a body is projected at an angle with the horizontal in a uniform gravitational field of the earth, the angular momentum of the body about the point of projection, as it proceeds along its path
  647. A particle of mass m = 5 units is moving with a uniform speed ν = 3 2 units in the XY -plane along the line y = x + 4. The magnitude of the angular momentum about origin is
  648. A particle of mass m is moving in YZ -plane with a uniform velocity v with its trajectory running parallel to + ve Y -axis and intersecting Z-axis at z = a as shown in figure. The change in its angular momentum about the origin as it bounces elastically from a wall at y = constant is [NCERT Exemplar)
  649. A wheel comprises a ring of radius R and mass M and three spokes each of mass m . The moment of inertia of the wheel about its axis is
  650. A portion of a ring of radius R has been removed as shown in figure. Mass of the remaining portion is m . Centre of the ring is at origin O . Let I A and I O be the moments of inertia passing through points A and O are perpendicular to the plane of the ring. Then,
  651. A square is made by joining four rods each of mass M and length L . Its moment of inertia about an axis PQ , in its plane and passing through one of its corner is
  652. Forces are applied on a wheel of radius 20 cm as shown in the figure. The torque produced by the forces 8 N at A, 8 N at E, 6 N at C and 9 N at D at angles indicated is
  653. The speed of a homogeneous solid sphere after rolling down an inclined plane of vertical height h , from rest without sliding is
  654. For the uniform T shaped structure with mass 3 M , moment of inertia about an axis normal to the plane and passing through O would be
  655. A ring is kept on a rough inclined surface. But the coefficient of friction is less than the minimum value required for pure rolling. At any instant of time, let K T and K R be the translational and rotational kinetic energies of the ring respectively, then
  656. The figure shows a uniform rod lying along the X -axis. The locus of all the points lying on the XY -plane, about which the moment of inertia of the rod is same as that about O is
  657. I. Speed of any point on rigid body executing rolling motion can be calculated by the expression ν = r ω where r is the distance of point from instantaneous centre of rotation. II. Rolling motion of rigid body can be considered as a pure rotation about instantaneous centre of rotation. Which of the above statement{s} is/are correct?
  658. I. Angular momentum of a particle moving in a straight line is always constant with respect to fixed point. II. Moment of inertia of a body remains same irrespective of position of axis of rotation. Which of the statement{s} is/are correct?
  659. Directions : These questions consists of two statements each printed as Assertion and Reason. While answering these questions, you are required to choose any one of the following four responses. Assertion : A body is moving along a circle with a constant speed. Its angular momentum about the centre of the circle remains constant. Reason : In this situation, a constant non-zero torque acts on the body.
  660. Find the torque about the origin when a force of 3 j ^ N acts on the particle whose position vector is 2 k ^ m .
  661. A solid sphere is rotating about an axis as shown in figure. An insect follows the dotted path on the circumference of sphere as shown in the figure. Then, match the following columns and mark the correct option from the codes given below. Column I (A) Moment of inertia (B) Angular velocity (C) Angular momentum (D) Ratational kinetic energy Column II (p) will remain constant (q) will first increase, then decrease (r) will first decrease, then increase (s) will continuously decrease (t) will continuously increase
  662. Directions : These questions consists of two statements each printed as Assertion and Reason. While answering these questions, you are required to choose any one of the following four responses. Assertion : If a particle moves with a constant velocity, then angular momentum of this particle about any point remains constant. Reason : Angular momentum has the units of Planck’s constant.
  663. A particle of mass m is attached to a thin uniform rod of length a at a distance of a 4 from the mid-point C as shown in the figure. The mass of the rod is 4 m . The moment of inertia of the combined system about an axis passing through 0 and perpendicular to the rod is [EAMCET 2013]
  664. The moment of inertia of a circular loop of of mass M and radius radius R , about an axis at a distance of R/ 2 parallel to horizontal diameter of loop is [AIIMS 2012]
  665. Two particles A and B are rnoving as shown in the figure. Their total angular momentum about the point O is [WB JEE 2015]
  666. The instantaneous angular position of a point on a rotating wheel is given by the equation θ (t) = 2 t 3 – 6 t 2 . The torque on the wheel becomes zero at [CBSE AIPMT 2011]
  667. Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities ω 1 and ω 2 . They are brought into contact face to face coinciding the axis of rotation. The expression for loss of energy during this process is [NEET 2017]
  668. ABC is right angled triangular plane of uniform thickness. The sides are such that AB > BC as shown in figure. I 1, I 2 , I 3 are moments of inertia about AB, BC and AC , respectively. Then, which of the following relations is correct? UIPMER 2017]
  669. From a disc of radius R and mass M , a circular hole of diameter R , whose rim passes through the centre is cut. What is the moment of inertia of the remaining part of the disc about a perpendicular axis, passing through the centre? [NEET 2016]
  670. If two discs have same mass and thickness but their densities are ρ 1 and ρ 2 then the ratio of their moments of inertia about central axis will be [BCECE (Mains) 2012]
  671. A force F = 2.0 N acts on a particle P in the XZ-plane. The force F is parallel to X -axis. The particle P (as shown in the figure) is at a distance 3 m and the line joining P with the origin makes angle 30° with the X -axis. The magnitude of torque on P w.r.t. origin O (in N-m) is [AMU 2011)
  672. A particle moves with a constant velocity parallel to the X-axis. Its angular momentum with respect to the origin [UP CPMT 2015]
  673. The torque of a force F = 2 i ^ − 3 j ^ + 5 k ^ acting at a point whose position vector r = 3 i ^ − 3 j ^ + 5 k ^ about the origin is [Kerala CEE 2013]
  674. The moment of inertia (I) and the angular momentum ( L ) are related by the expression [J&K CET 2013]
  675. A sphere pure rolls on a rough inclined plane with initial velocity 2.8 ms -1 . Find the maximum distance on the inclined plane.

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