A particle is moving along a line y=x + a with a constant velocity v. Find the angular momentum of the particle about the origin.
Calculate the force F that is applied horizontally at the axle of the wheel which is necessary to raise the wheel over the obstacle of height 0.4 m. Radius of wheel is 1m and mass = 10 kg.F is
Three bodies, a 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 identical. Which of the body reaches the ground with maximum velocity?
Five uniform circular plates, each of diameter b and mass m, are laid out in a pattern shown. Using the origin shown, find the y coordinate of the center of mass of the five-plate system.
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:
The coordinates of centre of mass of a uniform flag shaped lamina (thin flat plate) of mass 4 kg. (The coordinates of the same are shown in figure) are :
A force F = i ^ + 2 j ^ + 3 k ^ N acts at a point 4 i ^ + 3 j ^ − k ^ m . Then the magnitude of torque about the point i ^ + 2 j ^ + k ^ m will be x N − m . The value of x is
A brick of length L is placed on the horizontal floor. The bricks of same length and size are placed on this brick, one above the other by providing a margin of L/8 from the edge of the brick placed just below, in the same direction. Find the correct option.
Given a uniform disc of mass M and radius R. A small disc of radius R/2 is cut from this disc in such a way that the distance between the centers of the two discs is R/2. Find the moment of inertia of the remaining disc about a diameter of the original disc perpendicular to the line connecting the centers of the two discs.
Three children are sitting on a see-saw in such a way that it balances. A 20 kg and a 30 kg boy are on opposite sides at a distance of 2m from the pivot. It the third boy jumps off, thereby destroying balance, then the initial angular acceleration of the board is: (Neglect weight of board)
The moment of inertia of a door of mass m, length 2l and width l about its longer side is
A girl of mass M stands on the rim of a frictionless merry go-round of radius R and rotational inertia I that is not moving. She throws a rock of mass m horizontally in a direction that is tangent to the outer edge of the merry go-round. The speed of the rock, relative to the ground, is v. Afterward, the linear speed of the girl is
A particle of mass m is attached at the end B of uniform rod of same mass m and length l . If the rod is whirled in vertical circle about end A, what is minimum speed of particle at end B, required to move it in vertical circle?
A force F acts tangentially at the highest point of a disc of mass m kept on a rough horizontal plane. If the disc rolls without slipping, the acceleration of centre of the disc is:
If a spherical ball rolls on a table without slipping, the fraction of its total energy associated with rotation is
A solid sphere of mass m is lying at rest on a rough horizontal surface. The coefficient of friction between the ground and sphere is μ . The maximum value of F, so that the sphere will not slip, is equal to
A hollow sphere of mass m starting from rest rolls down, without slipping, on an inclined plane of inclination θ . What is the total energy of the sphere after 10 -5 s if velocity after 10 -5 s is v?
A thin metal disc of radius 0.25 m and mass 2 kg starts from rest and rolls down an inclined plane. If its rotational kinetic energy is 4 J at the foot of the inclined plane, then its linear velocity at the same point is
Consider a uniform rod of mass M= 4m and length L pivoted about its center. A mass m moving with velocity ν making angle θ = π 4 to the rod’s long axis collides with one end of the rod and sticks to it. The angular speed of the rod-mass system just after the collision is:
One end of a straight uniform 1 m long bar is pivoted on horizontal table. It is released from rest when it makes an angle 30 0 from the horizontal (see figure) . Its angular speed when it hits the table is given as n s − 1 , where n is an integer. The value of n is
A uniformly thick wheel with moment of inertia I and radius R is free to rotate about its centre of mass (see fig) . A massless string is wrapped over its rim and two blocks of masses m 1 a n d m 2 m 1 > m 2 are attached to the ends of the string. The system is released from rest. The angular speed of the wheel when m 1 descents by a distance h is
A uniform cylinder of mass M and radius R is to be pulled over a step of height a (a<R) by applying a force F at its centre ‘O’ perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of F required is:
Shown in the figure is rigid and uniform one meter long rod AB held in horizontal position by two strings tied to its ends and attached to the ceiling. The rod is of mass ‘m’ and has another weight of mass 2m hung at a distance of 75cm from A. The tension in the string at A is
A rod of length 50cm is pivoted at one end. It is raised such that it makes an angle 30 0 from the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in rad/s ) will be ( g=10m/s 2 )
A sphere of radius r and mass m has a linear velocity v 0 m/s directed to the left and no angular velocity as it is placed on a horizontal platform moving to the right with a constant velocity 10 m/s. If after sliding on the platform the sphere is to have no linear velocity relative to the ground as it starts rolling on the platform without sliding. The coefficient of kinetic friction between the sphere and the platform is μ k . Determine the required value of v 0 in m/s.
Eight solid uniform cubes of edge l are stacked together to form a single cube with center O. One cube is removed from this system. Distance of the center of mass of remaining 7 cubes from O is
A boy of mass m is standing on a block of mass M kept on a rough surface. When the boy walks from left to right on the block, the center of mass (boy + block) of system:
Two discs of moment of inertia I 1 and I 2 and angular speeds ω 1 and ω 2 are rotating along collinear axes passing through their center of mass and perpendicular to their plane. If the two are made to rotate together along the same axis the rotational KE of system will be
A thick walled hollow sphere has outer radius R. It rolls down an inclined plane without slipping and its speed at the bottom is v. If the inclined plane is frictionless and the sphere slides down without rolling, its speed at the bottom is 5v/4. What is the radius of gyration of the sphere?
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 center of the sphere as shown in the figure. The value of frictional force on the sphere is
Two thin discs, each of mass M and radius r meter, are attached as shown in the figure, to form a rigid body. The rotational inertia of this body about an axis perpendicular to the plane of disc B passing through its center is
Figure shows a uniform solid block of mass M and edge lengths a, b and c. Its M.I. about an axis through one edge and perpendicular (as shown) to the large face of the block is
In rectangle ABCD, AB = 2l and BC = l. Axes xx and yy pass through the centre of the rectangle. The moment of inertia is least about
A uniform thin rod of length l and mass m is hinged at a distance l/4 from one of the end and released from horizontal position as shown in the figure. The angular velocity of the rod as it passes the vertical position is
A triangular plate of uniform thickness and density is made to rotate about an axis perpendicular to the plane of the paper (a) passing through A,(b) passing through B, by the application of the same force F, at C (mid-point of AB) as shown. Now,
Two points of a rod move with velocities 3v and v perpendicular to the rod and in the same direction separated by a distance ‘r’ . The angular velocity of the rod is
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 is
A solid sphere of mass m and radius r is released from rest from the given position. If it rolls without sliding on the circular track of radius R, its speed when it reaches its lowest position will be
In a rectangle PQRS, PQ = 5l and QR= 2l. Axes xx and yy pass through centre of the rectangle. The moment of inertia is least about
A cube of side a and mass m is to be tilted at point A by applying a force F as shown in the figure. The minimum force required is
In an experiment with a beam balance, an unknown mass m is balanced by two known masses of 16 kg and 4 kg as shown in the figure. The value of the unknown mass m is
A uniform rod of mass 15 kg and of length 5m is held stationary with the help of a light string as shown in the figure. The tension in the string is
Each pulley in the figure has radius r and moment of inertia I. The acceleration of the block is
A rectangular block of mass M and height a is resting on a smooth level surface. A force F is applied to one corner as shown in the figure. At what point should a parallel force 3F be applied in so that the block shall undergo pure translational motion? Assume normal contact force between the block and surface passes through the center of gravity of the block.
A cord is wrapped on a pulley (disc) of mass M and radius R as shown in the figure. To one end of the cord, a block of mass M is connected as shown and to other end in (a) a force of 2 Mg and in (b) a block of mass 2M. Let angular acceleration of the disc in (a) and (b) is α A and α B , respectively, then (cord is not slipping on the pulley).
A circular plate of uniform thickness has a diameter of 28 cm. A circular portion of diameter 21 cm is removed from the plate as shown . O is the center of mass of complete plate. The position of center of mass of remaining portion will shift towards left from O by (in cm)
In the figure, a cylinder A is initially rolling with velocity v on the horizontal surface of the wedge B (of same mass as A).All surfaces are smooth and B has no initial velocity. Then maximum height reached by cylinder on the wedge will be
Four spheres, each of mass M and radius r are situated at the four corners of square of side R. The moment of inertia of the system about an axis perpendicular to the plane of square and passing through its center will be
Two blocks each of the mass m are attached to the ends of a massless rod which pivots as shown in the figure. Initially, the rod is held in the horizontal position and then released. Calculate the net torque on this system above pivot.
An isosceles triangular piece is cut from a square plate of side l. The piece is one-fourth of the square and mass of the remaining plate is M. The moment of inertia of the plate about an axis passing through O and perpendicular to its plane is
Four identical rods are joined end to end to form a square. The mass of each rod is M. T he moment of inertia of the system about one of the diagonals is
Two circular discs are of same thickness. The diameter of A is twice that of B. The moment of inertia of A as compared to that of B is
A small bead of mass m moving with velocity v gets threaded on a stationary semicircular ring of mass m and radius R kept on a horizontal table. The ring can freely rotate about its center The bead comes to rest relative to the ring. What will be the final angular velocity of the system?
A thin rod of length 4l and mass 4m is bent at the points as shown in the figure. What is the moment of inertia of the rod about the axis passing through Point O and perpendicular to the plane of the paper.
A cubical block of side 30cm is moving with velocity 2m/s on a smooth horizontal surface. The surface has a bump at a point O as shown in figure. The angular velocity (in rad/sec) of the block immediately after it hits the bump, is:
A ball is released from the top (A) of a semi cylindrical surface of which part AB is sufficiently rough and BC is smooth. Then:
A solid sphere and circular disc of same mass and radius are allowed to roll down an inclined plane from the same height without slipping. Find the ratio of times taken by these two to come to the bottom of incline:
A sphere has to purely roll upwards. At an instant when the velocity of sphere is v, frictional force acting on it is
A homogenous rod of length l = η x and mass M is lying on a smooth horizontal floor. A bullet of mass m hits the rod at a distance x from the middle of the rod at a velocity v 0 perpendicular to the rod and comes to rest after collision. If the velocity of the farther end of the rod just after the impact is in the opposite direction of v 0 , then
A rigid body of radius R, either hollow or solid, lies on a smooth horizontal surface. The body is pulled by a horizontal force acting tangentially from the highest point. The distance travelled by the body in the time in which it makes one full rotation is the same that it will make in one fulI rotation during pure rolling. The rigid body will be
A uniform solid sphere rolls down a vertical surface without sliding. If the vertical surface moves with an acceleration a = g 2 , the minimum coefficient of friction between the sphere and vertical surfaces so as to prevent relative sliding is
A solid sphere is set into rotation at an angular velocity and it is then placed on a rough horizontal surface. The ratio of distances covered by rotational and translational motions up to the start of the pure rolling is (Assume uniformly accelerated motion up to start of pure rolling):
In the figure, the velocities are in ground frame and the cylinder is performing pure rolling on the plank, velocity of point ‘A’ would be
A small solid sphere of radius r rolls down an incline without slipping which ends into a vertical loop of radius R. Find the height above the base so that it just loops the loop
A solid cylinder of mass 3 kg is placed on a rough inclined plane of inclination 30 o . If g = 10 m s – 2 , then the minimum frictional force required for it to roll without slipping down the plane is
A yo-yo is placed on a rough horizontal surface and a constant force F, which is less than its weight, pulls it vertically. Due to this
A sphere is rolling down an inclined plane without slipping. The ratio of rotational kinetic energy to total kinetic energy is
A rolling object rolls without slipping down an inclined plane (angle of inclination θ ), then the minimum acceleration it can have is
A solid sphere rests on a horizontal surface. A horizontal impulse is applied at height h from centre. The sphere starts rolling just after the application of impulse. The ratio h/r will be
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 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)
A carpet of mass M made of inextensible material is rolled along its length in the form of a cylinder of radius R and is kept on a rough floor. The carpet starts unrolling without sliding on the floor when a negligibly small push is given to it. If R = 1.05 m then calculate the horizontal velocity of the axis (in m/s) of the cylindrical part of the carpet when its radius reduces to R/2. (Take g = 10 m/s 2 )
From a circular disc of radius R and mass 9M, a small disc of radius R 3 is removed as shown in figure. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through O is
The centre of mass of a body
A circular portion of diameter R is cut out from a uniform circular disc of mass M and radius R as shown in Fig. The moment of inertia of the remaining (shaded) portion of the disc about an axis passing through the centre O of the disc and perpendicular to its plane is
A 198 cm tall girl lies on a light (massless) board which is supported by two scales one under the top of her head and one beneath the bottom of her feet (Figure). The two scales read respectively 36 and 30 kg. What distance is the centre of gravity of this girl from the bottom of her feet?
The moment of inertia of a body about a given axis is 1.2 kg-m 2 . Initially, given axis the body is at rest. In order to produce a rotational kinetic energy of 1500 J and angular acceleration of 25 rads -2 a torque that must be applied about that axis for a duration of
Three points particles of masses 1.0kg, 1.5kg and 2.5kg are placed at three corners of right angle triangle of sides 4.0cm, 3.0cm and 5.0cm as shown in the figure. The centre of mass of the system is at a point.
As shown in the figure, a bob of mass m is tied by a massless string whose other end portion is wound on a fly wheel (disc) of radius r and mass m. when released from rest the bob starts falling vertically. When it has covered a distance of h, the angular speed of the wheel will be
The radius of gyration of a uniform rod of length l , about an axis passing through a point L 4 away from the center of the rod, and perpendicular to it is
Mass per unit area of a circular disc of radius a depends on the distance r from its centre as σ r = A + B r . The moment of inertia of the disc about the axis, perpendicular to the plane and passing through its center is:
Consider a uniform cubical box of side a on a rough floor that is to be moved by applying minimum possible force F at a point b above its center of mass (see figure). If the coefficient of friction is μ = 0.4 , the maximum possible value of 100 × b a for box not to topple before moving is
A uniform sphere of mass 500 g rolls without slipping on a plane horizontal surface with its centre moving at a speed of 5.00 cm/s. Its kinetic energy is :
As shown in fig. when a spherical cavity (centered at O) of radius 1 is cut out of a uniform sphere of radius R (centered at C), the center of mass of remaining (shaded) part of sphere is at G, i.e. on the surface of the cavity. R can be determined by the equation :
Three solid spheres each of mass m and diameter d are stuck together such that the lines connecting the centre form an equilateral triangle of side of length d. The ratio I 0 / I A of moment of inertia I 0 of the system about an axis passing the centroid and about centre of any of the spheres I A and perpendicular to the plane of the triangle is :
A rod of length L has non-uniform linear mass density given by ρ x = a + b x L 2 , where a and b are constants and 0 ≤ x ≤ L . The value of x for the centre of mass of the rod is at
Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are 0.1 kg – m 2 and 10 rad s – 1 respectively while those for the second one are 0.2 kg – m 2 and 5 rad s – 1 respectively. At some instant they get stuck together and start rotating as a single system about their common axis with some angular speed. The kinetic energy of the combined system is:
A square shaped hole of side l = a 2 is carved out at a distance d = a 2 from the centre ‘O’ of a uniform circular disk of radius a. If the distance of the centre of mass of the remaining portion from O is − a X , value of X (to the nearest integer) is
A block of mass m = 1 kg slides with velocity υ = 6 m / s on a frictionless horizontal surface and collides with a uniform vertical rod and sticks to it as shown. The rod is pivoted about O and swings as a result of the collision making angle θ before momentarily coming to rest. If the rod has mass M = 2 kg, and length l = 1 m, the value of θ is approximately: t a k e g = 10 m / s 2
Moment of inertia of a cylinder of mass M, length L and radius R about an axis passing through its centre and perpendicular to the axis of the cylinder is I = M R 2 4 + L 2 12 . If such a cylinder is to be made for a given mass of material, the ratio L/R for it to have minimum possible I is :
A person of 80 kg mass is standing on the rim of a circular platform of mass 200 kg rotating about its axis at 5 revolutions per minute (rpm). The person now starts moving towards the centre of the platform. What will be the rotational speed (in rpm) of the platform when the person reaches its centre .
A uniform rod of length ‘ l ’ is pivoted at one of its ends on a vertical shaft of negligible radius. When the shaft rotates at angular speed ω the rod makes an angle θ with it (see figure). To find θ equate the rate of change of angular momentum (direction going into the paper) m l 2 12 ω 2 sin θ cos θ about the centre of mass (CM) to the torque provided by the horizontal and vertical forces FH and FV about the CM. The value of θ is then such that :
A massless equilateral triangle EFG of side ‘a’ (as shown in figure) has three particles of mass m situated at its vertices. The moment of inertia of the system about the line EX perpendicular to EG in the plane of EFG is N 20 ma 2 where N is an integer. The value of N is
A circular disc of mass M and radius R is rotating about its axis with angular speed ω 1 . If another stationary disc having radius R 2 and same mass M is dropped co-axially on to the rotating disc. Gradually both discs attain constant angular speed ω 2 . The energy lost in the process is p% of the initial energy. Value of p is .
ABC is a plane lamina of the shape of an equilateral triangle. D, E are mid points of AB, AC and G is the centroid of the lamina. Moment of inertia of the lamina about an axis passing through G and perpendicular to the plane ABC is I 0 . If part ADE is removed, the moment of inertia of the remaining part about the same axis is N I 0 16 where N is an integer. Value of N is .
For a uniform rectangular sheet shown in the figure, the ratio of moments of inertia about the axes perpendicular to the sheet and passing through O (the centre of mass) and O’ (corner point) is:
Consider two uniform discs of the same thickness and different radii R 1 = R and R 2 = α R made of the same material. If the ratio of their moments of inertia I 1 and I 2 , respectively, about their axes is I 1 : I 2 = 1 : 16 then the value of α is:
A wheel is rotating freely with an angular speed ω on a shaft. The moment of inertia of the wheel is I and the moment of inertia of the shaft is negligible. Another wheel of moment of inertia 3 I initially at rest is suddenly coupled to the same shaft. The resultant fractional loss in the kinetic energy of the system is:
A thin rod of mass 0.9 kg and length 1 m is suspended, at rest, from one end so that it can freely oscillate in the vertical plane. A particle of mass 0.1 kg moving in a straight line with velocity 80 m/s hits the rod at its bottom most point and sticks to it (see figure). The angular speed (in rad/s) of the rod immediately after the collision will be
A particle of mass m is kept on a smooth cube of mass M and side L as shown in figure. Cube starts moving with a constant velocity v . Displacement of the center of mass along the horizontal direction when particle hits the ground is
The moment of inertia of a rod of length L about an axis passing through its center of mass and perpendicular to rod is I. The moment of inertia of hexagonal shape formed by six such rods, about an axis passing through its center of mass and perpendicular to its plane will be
A particle of mass m is projected with a velocity v making an angle of 45 0 with the horizontal. The magnitude of the angular momentum of the projectile about the point of projection when the particle is at maximum height h is m v 3 ( x g ) . Find the value of x.
A wire of uniform cross-section is bent in the shape shown in the figure. The coordinates of the center of mass of each side are shown in the figure. The coordinates of the center of mass of the system are
The ‘y’ coordinate of the center of mass of the system of three rods of length ‘2a’ and two rods of length ‘a’ as shown in the figure is: (Assume all rods to be of uniform density)
Two semicircular rings of linear mass densities λ and 2 λ and of radius ‘R’ each are joined to form a complete ring. The distance of the center of the mass of complete ring from its center is
In the figure, the L-shaped shaded piece is cut from a metal plate of uniform thickness. The point that corresponds to the center of mass of the L-shaped piece is
A system of particles is free from any external force. If v and a be the velocity and acceleration of the centre of mass, then it necessarily follows that
The velocity of the CM of a system changes from v 1 = 4 i ^ m / s to v 2 = 3 j ^ m / s during time Δ t = 2 s . If the mass of the system is 10 kg, the constant force acting on the system is
A child is sitting at one end of a long trolley moving with a uniform speed v on a smooth horizontal track. If the child starts running towards the other end of the trolley with a speed u (w.r.t trolley), the speed of the center of mass of the system will
A mass m is at rest on an smooth inclined plane of mass M which is further resting on a smooth horizontal plane. Now if the mass starts moving the position of C.M. of mass of system will
Two particles are shown in the figure. At t=0, a constant force F=6 N starts acting on the 3 kg man. Find the velocity of the centre of mass of these particles at t=5 s
A ball kept in a closed box moves in the box making collisions with the walls. The box is kept on a smooth surface. The velocity of the centre of mass
A machinist starts with three identical square plates but cuts one corner from one of them, two corners from the second and three corners from the third. Rank the three according to the x-coordinate of their cetre of mass, from smallest to largest.
Three particles of equal masses are placed at the corners of an equilateral triangle as shown in the figure. Now particle A starts with a velocity v 1 towards line AB, particle. B starts with a velocity v 2 towards line BC and particle C starts with velocity v 3 towards line CA. The displacement of CM of three particle A, B and C after time t will be (given if v 1 = v 2 = v 3 )
Three particles of masses 1 kg, 2kg and 3 kg are situated at the corners of an equilateral triangle move at speed 6 ms − 1 , 3 ms − 1 and 2 ms − 1 respectively. Each particle maintains a direction towards the particle at the next corner symmetrically. Find velocity of CM of the system at this instant.
A circular disc X of radius R is made from an iron plate of thickness t, and another disc Y of radius 4R is made from an iron plate of thickness t/4. Then the relation between the moment of inertia l X and I Y is
Four thin rods of same mass M and same length l, form a square as shown in the figure. Moment of inertia of this system about an axis through centre O and perpendicular to its plane is
Three rings each of mass M and radius R are arranged as shown in the figure. The moment of inertia of the system about YY’ will be
Let l be the moment of inertia of an uniform square plate about an axis AB that passes through its center and is parallel to two of its sides. CD is a line in the plane of the plate that passes through the center of the plate and makes an angle θ with AB. The moment of inertia of the plate about the axis CD is then equal to
The moment of inertia of a rod of length L about an axis passing through its center of mass and perpendicular to rod is I . The moment of inertia of hexagonal shape formed by six such rods, about an axis passing through its center of mass and perpendicular to its plane will be n I . Find the value of n.
A circular platform is free to rotate in a horizontal plane about a vertical axis passing through its center. 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 the platform with a constant velocity (with respect to the platform), the angular velocity of the platform ω (t) will vary with time t as
A horizontal heavy uniform bar of weight W is supported at its ends by two men. At the instant, one of the men lets go off his end of the rod, the other feels the force on his hand changed to
In the following figure, a body of mass m is tied at one end of a light string and this string is wrapped around the solid cylinder of mass M and radius R. At the moment t=0,the system starts moving. If the friction is negligible, angular velocity at time t would be
Two circular discs A and B are of equal masses and thicknesses but made of metal with densities d A and d B (d A > d B ).If their moments of inertia about an axis passing through their centers and perpendicular to circular faces are I A and I B , then
A solid sphere of radius R starts rotating on rough horizontal surface with translational velocity v 0 and initial angular velocity ω 0 = 2 v 0 / 3 R . The sphere starts pure rolling after some time t. Find the angle by which sphere rotates upto the instant at which pure rolling starts, if v is the translational velocity at pure rolling. Assume uniformly accelerated motion up to start of pure rolling.
In the figure mass of both spherical body and block is m. Moment of inertia of the spherical body about center of mass is 2mR 2 . The spherical body rolls on the horizontal surface. There is no slipping at any surfaces in contact. The ratio of kinetic energy of the spherical body to that of block is
Four identical rods are joined end to end to form a square. The mass of each rod is M. T he moment of inertia of the system about one of the diagonals is
Four identical rods are joined end to end to form a square. The mass of each rod is M. T he moment of inertia of the system about one of the diagonals is
The moment of inertia of a system of four rods, each of length I and mass m, about the axis perpendicular to plane of square as shown is
About which axis, the moment of inertia in the given triangular lamina is maximum?
Three identical rods, each of mass m and length l, form an equilateral triangle. Moment of inertia about one of the sides is
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
A cube of side a is placed on an inclined plane of inclination θ . What is the maximum value of θ for which the cube will not topple?
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
A string is wrapped around a cylinder of mass M and radius R. The string is pulled vertically upwards to prevent the centre of mass from falling as the cylinder unwinds the string. The tension in the string is
The mass of a sphere placed on a smooth horizontal surface is distributed non-uniformly. A horizontal force equal to weight of sphere is applied at a distance ‘a’ from centre of the sphere for a short time. The sphere is rotated by an angle θ but it behaves as if in stable equilibrium. Find the distance between centre and CG of the sphere:
Two forces F and 2F are applied on a rod of length l and mass m, as shown. The angular acceleration of the rod is
A smooth tube of certain mass is rotated in a gravity-free space and released. The two balls shown in the figure move towards the ends of the tube. For the whole system, which of the following quantities is not conserved.
Two bodies with moments of inertia I 1 and I 2 I 1 > I 2 have equal angular momenta. If their kinetic energies of rotation are E 1 and E 2 , respectively, then
A disc of mass M and radius R rolls without slipping on a horizontal surface. If the velocity of its center is v 0 , then the total angular momentum of the disc about a fixed point P at a height (3/2)R above the center C
A child is standing with folded hands at the center of a platform rotating about its central axis. The kinetic energy of the system is K. The child now stretches his arms so that the moment of inertia of the system doubles. The kinetic energy of the system now is
A block of mass m is attached to a pulley disc of equal mass m and radius r by means of a slack string as shown. The pulley is hinged about its centre on a horizontal table and the block is projected with an initial velocity of 5 m/s. Its velocity when the string becomes taut will be
A conical pendulum consists of a simple pendulum moving in a horizontal circle as shown in the figure. C is the pivot, O is the center of the circle in which the pendulum bob moves and ω is the constant angular velocity of the bob . If L is the angular momentum about point C, then
Two discs, each having moment of inertia 5 kg m 2 about its central axis, rotating with speeds 10 rad s -1 and 20 rad s -1 are brought in contact face to face with their axes of rotation coincided. The loss of kinetic energy in the process is
A boy stands over the center of a horizontal platform which is rotating freely with a speed of 2 revolutions per second about a vertical axis through the center of the platform and straight up through the boy. He holds 2 kg masses in each of his hands close to his body. The combined moment of inertia of the system is 1 kg x metre 2 . The boy now stretches his arms so as to hold the masses far from his body. In this situation, the moment of inertia of the system increases to 2 kg x metre 2 . The kinetic energy of the system in the latter case as compared with that in the previous case will
A sphere is released on a smooth inclined plane from the top. when it moves down, its angular momentum is
A solid uniform disk of mass m and radius R is pivoted about a horizontal axis through its center and a small body of mass m is attached to the rim of the disk. If the disk is released from rest with the small body, initially at the same level as the centre, the angular velocity when the small body reaches the lowest point of the disk is
A disk with moment of inertia I 1 rotates about a frictionless, vertical a:de with angular speed ω 1 . A second disk, this one having moment of inertia l 2 and initially not rotating, drops onto the first disk (figure). Because of friction between the surfaces, the two eventually reach the same angular speed ω f The value of ω f is
Two men of equal masses stand at opposite ends of the diameter of a turntable disc of a certain mass, moving with constant angular velocity. The two men make their way to the middle of the turntable at equal rates. In doing so
An insulated particle of mass m is moving in a horizontal plane (x-y) along the X-axis. At a certain height above the ground, it suddenly explodes into two fragments of masses m/4 and 3m/4.An instant later, the smaller fragment is at Y= +15 cm. The larger fragment at this instant is at:
A man stands at one end of a boat which is stationary in water. Neglect water resistance. The man now moves to the other end of the boat and again becomes stationary. The center of mass of the ‘man plus boat’ system will remain stationary with respect to water
A thin circular ring of mass M and radius R is rotating about its axis with a constant angular velocity ω . Four objects each of mass m, are kept gently to the opposite ends of two perpendicular diameters of the ring. The angular velocity of the ring will be
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 center of the rod and the system is rotating with angular velocity ω 0 about an axis perpendicular to the rod and passing through the mid point of the rod (see figure). There are no external forces. When the beads reach the ends of the rod, the angular velocity of the system is
ABC is a triangular plate of uniform thickness. The sides are in the ratio shown in the figure. I AB , I BC , I CA are the moments of inertia of the plated about AB, BC and CA respectively. Which one of the following relation is correct?
In the pulley system shown, if radii of the bigger and smaller pulley are 2 m and 1 m, respectively, and the acceleration of block A is 5 ms -2 in the downward direction, the acceleration of block B will be
Two rigid bodies A and B rotate with angular momenta L A and L B , respectively. The moments of inertia of A and B about the axes of rotation are I A and I B respectively. If I A = I B / 4 and L A = 5 L B ,then the ratio of rotational kinetic energy K A of A to the rotational kinetic energy K B of B is given by
A particle of mass m moves in the XY-plane with a constant velocity ‘V’ along the straight line X ‘ Y ‘ . If the angular momentum of the particle with respect to origin O is L A when it is at A, and L B when it is at B, then
An L -shaped object, made of thin rods of uniform mass density, is suspended with a string as shown in figure(the object is in equilibrium). If AB = BC, and the angle made by AB with downward vertical is θ , then :
The drawing shows a bicycle wheel of radius r resting against a small step whose height is h = r/5. A clockwise torque is applied to the axle of the wheel. As the magnitude of torque increases, there comes a time when the wheel just begins to rise up and loses contact with the ground. Let this torque be T. What is the magnitude of the horizontal component of the acceleration of the centre of the wheel when a torque of 2T is applied? Assume that the wheel doesn’t slip at the edge of the step when this torque is applied (ignore the mass of the spokes)
A disc of radius r is rotating about its center with an angular speed ω 0 . It is gently placed on a rough horizontal surface. After what time it will start pure rolling?
A square plate of mass m and side length ‘a’ is kept as shown in figure. In equilibrium two of its side are horizontal and two are vertical. What is its angular acceleration just after lower spring is cut.
An arrangement of rods each of mass m and length l are welded (where ever required) to form a shape as shown. The moment of inertia about an axis passing through point C and perpendicular to the plane of figure is:
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
In the figure, a sphere of radius 2 m rolls on a plank. The accelerations of the sphere and the plank are indicated. The value of α is
A body is rolling without slipping on a horizontal plane. The rotational energy of the body is 40% of the total kinetic energy. Identify the body.
A solid sphere rolls down two different inclined planes of the same height but of different inclinations:
A body of mass m slides down an smooth incline and reaches the bottom with a velocity v, Now smooth incline surface is made rough and 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:
A ring, cylinder and solid sphere are placed on the top of a rough incline on which the sphere can just roll without slipping. When all of them are released at the same instant from the same position, then
In the figure shown, a ball without sliding on a horizontal surface. It ascends a curved track up to height h and returns. The value of h is h 1 for sufficiently rough curved track to avoid sliding and is h 2 for smooth curved track, then
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 w.r.t. the initial position. The object is
In both the figures all other factors are same, except that in Figure (i) AB is rough and BC is smooth while in Figure (ii) AB is smooth and BC is rough. In Figure (i), if a sphere is released from rest it starts rolling. Now consider the Figure (ii), if same sphere is released from top of the inclined plane, what will be the kinetic energy of the sphere on reaching the bottom:
A solid cube ABCD of side a and mass M is placed on a rough horizontal surface. A bullet of mass m and speed v is shot at the top of the cube so that cube rotates about point D. The bullet sticks to the cube, m << M. Find the angular velocity of the cube.
The angular momentum of the disc which spins with ω = 3 v R k ^ and its CM moves with a velocity v = v i ^ about O will be
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
One fourth length of a uniform rod of length 2l and mass m is placed on a horizontal table and the rod is held horizontally. The rod is released from rest. The normal reaction on the rod as soon as the rod is released will be
Two heavy right circular rollers of radii R and r respectively rest on a rough horizontal plane as shown in figure. The larger roller has a string wound around it to which a horizontal force P can be applied as shown. Assuming that the coefficient of friction μ has the same value for all surfaces of contact and the smaller cylinder should neither roll nor slide. The minimum coefficient of friction so that the larger roller can be pulled over the smaller one is
A spherical ball of radius r initially at rest on a rough horizontal surface is hit horizontally at a point at a distance x above the central line. Due to this sharp impulse the centre of the ball acquires a velocity v c . After some time, the ball will start pure rolling with a velocity equal to
A uniform rod of length L (in between the supports) and mass m is placed on two supports A and B. The rod breaks suddenly at length L/10 from the support B. Find the reaction at support A immediately after the rod breaks.
A solid sphere, a hollow sphere and a disc, all having same mass and radius, are placed at the top of an incline and released. The friction coefficients between the objects and the incline are the same and not sufficient to allow pure rolling. Least time will be taken in reaching the bottom by
The string of a step rolling wheel is pulled by applying force F with different lines of action in two situations as shown. The wheel starts rolling without slipping due to application of the force.
If a spherical ball rolls on a table without slipping, the fraction of its total energy associated with rotation is:
A solid cylinder of mass M and radius R rolls without slipping down an inclined plane of length L and height h. What is the speed of its centre of mass when the cylinder reaches its bottom ?
Three particles each of mass m, are placed at the corners of an equilateral triangle of side a, as shown in Fig. The position vector of the centre of mass is
A light rod of length L is suspended from a support horizontally by means of two vertical wires A and B of equal length as shown in Fig. The cross sectional area of A is half that of B and the Young’s modulus of A is twice that of B. A weight W is hung as shown. The value of x so that W produces equal stress in wires A and B is—– L 3
Four forces are applied to a wheel of radius 20 cm as shown in Fig. The net torque produced by the forces about O is Nm
A cylinder, released from the top of an inclined plane, rolls without sliding and reaches the bottom with speed v r . Another identical cylinder, released from the top of the same inclined plane, slides without rolling and reaches the bottom with speed v s . Then
A comet is moving in a highly elliptical orbit round the sun. When it is closest to the sun, its distance from the sun is r and its speed is v. When it is farthest from the sun, its distance from the sun is R and its speed will be
Two blocks of masses 10 kg and 4 kg are connected by a spring of negligible mass and placed on a frictionless horizontal surface. An impulse gives a velocity of 14 m/s to the heavier block in the direction of the lighter block. The velocity of the center of mass is m/s
A solid cylinder of mass M and radius R rolls down an inclined plane of height h. The angular velocity of the cylinder when it reaches the bottom of the plane will be
A cord is wound around the circumference of a bicycle wheel (without tyre) of diameter 1 m. A mass of 2 kg is tied to the end of the cord and it is allowed to fall from rest. The weight falls 2m in 4 s. The axle of the wheel is horizontal and the wheel rotates with its plane vertical. (Take g = 10 ms − 2 ) then the angular acceleration of the wheel is (rad/s 2 )
In an experiment with a beam balance, an unknown mass m is balanced by two known masses of 16 kg and 4 kg as shown in figure. The value of the unknown mass m is
In the figure shown, the instantaneous speed of end I of the rod is v to the left. The angular velocity of the rod of length L must be
Each pulley in figure has radius r and moment of inertia I. The acceleration of the block is
An equilateral prism of mass m rests on a rough horizontal surface with coefficient of friction μ . A horizontal force F is applied on the prism as shown in figure. If the coefficient of friction is sufficiently high so that the prism does not slide before toppling, the minimum force required to topple the prism is
A cord is wrapped on a pulley (disk) of mass M and radius R as shown in figure. To one end of the cord, a block of mass M is connected as shown and to other end in (a) a force of 2 Mg and in (b) a block of mass 2M. Let angular acceleration of the disk in A and B is α A and α B , respectively, then (cord is not slipping on the pulley).
Two beads each of mass m are welded at the ends of two light rigid rods each of length l. If the pivots are smooth then find the ratio of translational and rotational kinetic energy of system.
A uniform rod of mass m and length l which can rotate freely in vertical plane without friction, is hinged at its lower end on a table. If a sphere of mass m and radius R = L 3 is placed in contact with the vertical rod and a horizontal force F = 80 N is applied at the upper end of the rod. Find the horizontal component of hinge reaction (in N) acting on the rod just after force F starts acting.
A plank of mass M is placed on a smooth surface over which a cylinder of mass m (= M) and radius R = 1 m is placed as shown in figure. Now the plank is pulled towards the right with an external force F (= 2Mg). If the cylinder does not slip over the surface of the plank, find the linear acceleration of the cylinder (in m/s 2 ). (Take g = 10 m/s 2 ).
Two wheels A and C are connected by a belt B as shown in figure. The radius of C is three times the radius of A. What would be the ratio of the rotational inertias I C / I A if both the wheels have the same rotational kinetic energy?
Equilibrium of two Planks on a Rough Plane : Figure shows two planks inclined to horizontal rough surface and connected by a string. Find the minimum angle θ (in degrees) for which this system remain in equilibrium if friction coefficient is μ = 0.5.
Figure shows a plank of mass m moving at speed u on a smooth surface. A spinning ball (sphere) of mass m and radius r normally strikes it at speed v. The friction coefficient between ball and upper surface of plank is μ = 0.5 and coefficient of restitution is e = 0.5. Find the angular velocity of sphere just after collision.
The moment of inertia of a solid sphere of mass M and radius R about a tangent to the sphere is
A solid sphere of mass 3 kg is kept on a horizontal surface. The coefficient of static friction between the surfaces in contact is 2/7. What maximum force can be applied at the highest point in the horizontal direction so that the sphere does not slip on the surface? (in N)
A thin rod of length 4l and mass 4m is bent at the points as shown in figure. What is the moment of inertia of the rod about the axis passing through point O and perpendicular to the plane of the paper?
An equilateral prism of mass m rests on a rough horizontal surface with coefficient of friction m. A horizontal force F is applied on the prism as shown in figure. If the coefficient of friction is sufficiently high so that the prism does not slide before toppling, the minimum force required to topple the prism is
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
A flywheel rotating about a fixed axis has a kinetic energy of 360 J when its angular speed is 30 rad s -1 . The moment of inertia of the wheel about the axis of rotation is
A ring and a disc of different masses are rotating with the same kinetic energy. If we apply a retarding torque T on the ring it stops after making n revolutions, then in how many revolutions will the disc stop under the same retarding torque?
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 on the rod through which the axis should pass in order that the work required to set the rod rotating with angular velocity ω 0 be minimum is
Velocity of the centre of a small cylinder is v. There is no slipping anywhere. The velocity of the centre of the larger cylinder is
A yo-yo, arranged as shown, rests on a frictionless surface. When a force F is applied to the string, the yo-yo
A particle of mass m is projected with velocity v at an angle q with the horizontal. Find its angular momentum about the point of projection when it is at the highest point of its trajectory.
