A thin rod of mass M and length a is free to rotate in horizontal plane about a fixed vertical axis passing through point O. A thin circular disc of mass M and of radius a /4 is pivoted on this rod with its center at a distance a/4 from the free end so that it can rotate freely about its vertical axis, as shown in the figure. Assume that both the rod and the disc have uniform density and they remain horizontal during the motion. An outside stationary observer finds the rod rotating with an angular velocity Ω and the disc rotating about its vertical axis with angular velocity 4 Ω . The total angular momentum of the system about the point O is Ma 2 Ω 48 n The value of n is .

In the figure shown mass of both, the spherical body and block is m. Moment of inertia of the spherical body about centre 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

Which of the following is/are correct?

Bodies A and B are rolling without slipping on a rough inclined plane. Match the following: Column – I Column – II (P) Velocity of A and B after travelling the same distance will be (1) Different if A and B have same size but different shape and mass (Q) Time taken by A and B to reach the bottom of the plane from the same height will be (2) Different if A and B have same size and same mass but having different shapes (R) Acceleration of centre of mass of A and B will be (3) Same if A and B have same shape and same size but of different materials (S) During rolling motion frictional force experienced by A and B will be (4) Same if A and B have same size but different shape and mass

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 through O and perpendicular to the plane of the disc is:

One of the strings at end Q is cut at t = 0 and the rod starts rotating about the other end P. Then

A uniform thin rod AB of mass M = 0.6 kg and length l = 60 cm stands at the edge of a frictionless table as shown in figure. A particle of mass m = 0.3 kg flying horizontally with velocity v 0 = 24 m/s strikes the rod at point P at a height h = 45 cm from the base and sticks to it. The rod is immediately driven off the table. If the time after collision when the rod becomes horizontal for the first time is found to be = π ∗ ∗ sec , find the value of ‘**'(g = 10 m/s 2 ).

Consider the rotation of a rod of mass m and length l from position AB to AB’. Which of the following statements are correct?

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 billiard ball (of radius R), initially at rest is given a sharp impulse by a cue. The cue is held horizontally at a distance h above the central line as shown in figure. The ball leaves the cue with a speed V o . It rolls and slides while moving forward and eventually acquires a final speed of 9/7 V 0 . If R = 5.0 cm then find the value of h in cm.

A uniform cube of side a and mass m rests on a rough horizontal table. A horizontal force F is applied normal to one of the faces at a point directly above the centre of the face, at a height 3a/4 above the base. If m = 6 kg then find the minimum value of F in N for which the cube begins to tip about an edge.

Which of the following statements are correct for instantaneous axis of rotation?

A thin uniform rod AB of mass m=1 kg moves translationally with acceleration a = 2 ms -2 due to two anti-parallel forces F 1 and F 2 . The distance between the points at which these forces are applied is equal to l = 20 cm. Besides, it is known that F 2 = 5 N. Find the length of the rod, in metre.

One end of a straight uniform 1 m long bar is pivoted on horizontal surface. It is released from rest when it makes an angle 30° 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 rod of mass ? and length L, pivoted at one of its ends, is hanging vertically. A bullet of the same mass moving at speed ? strikes the rod horizontally at a distance ? from its pivoted end and gets embedded in it. The combined system now rotates with angular speed ? about the pivot. The maximum angular speed ? ? is achieved for ? = ? ? . Then

Two discs A & B having small projections, welded on their circumference , are free to rotate about their fixed axes, which are parallel to each other as shown in figure. The disc B starts rotating with angular velocity ω . It’s projection hits the projection of other disc at rest after time T. The coefficient of restitution is 1 2 . The two discs are of same radius but their moments of inertia are 4I and I respectively.

A disc of radius r = 20 cm is rotating about its axis with an angular speed of 20 rad s -1 . It is gently placed on a horizontal surface which is perfectly frictionless (Fig). What is the linear speed of point A on the disc (in m/s)?

The height of a solid cylinder is four times its radius, lt is kept vertically at time t = 0 on a belt which is moving in the horizontal direction with a velocity v = 2.45 t 2 where v is in ms -1 and t is in second. If the cylinder does not slip, it will topple over at time t (in seconds) equal to…….[ Take g = 9 . 8 m s – 2 ]

One end of a horizontal uniform beam of weight W and length L is hinged on a vertical wall at point O and its other end is supported by a light inextensible rope. The other end of the rope is fixed at point Q, at a height L above the hinge at point O. A block of weight α W is attached at the point P of the beam, as shown in the figure (not to scale). The rope can sustain a maximum tension of (2 2 )W. Which of the following statement(s) is(are) correct?

A rod of length l = 1 m resting on a wall and the floor. Its lower end A is pulled towards right with a constant velocity v = 2 m/s. Find the angular velocity of the rod when it makes an angle θ = 60 ∘ with the vertical.

Which of the following statements are correct for instantaneous axis of rotation?

Consider the rotation of a rod of mass m and length l from position AB to AB’. Which of the following statements are correct?

A horizontal force F is applied at the top of an equilateral triangular block having mass m. The minimum coefficient of friction required to topple the block before translation will be

A ring (R), a disc (D), a solid sphere (S) and a hollow sphere with thin walls (H), all having the same mass but different radii, start together from rest at the top of an inclined plane and roll down without slipping. Then

A uniform circular disc has radius R and mass m. A particle, also of mass m, is fixed at point A on the edge of the disc as shown in figure. The disc can rotate freely about a fixed horizontal chord PQ that is at a distance R/4 from the centre C of the disc. The line AC is perpendicular to PQ. Initially, the disc is held vertical with point A at its highest position. It is then allowed to fall so that it starts rotating about PO. If R = 0.5 m then, find the linear speed of the particle (in m/s) as it reaches its lowest position. (Take g = 10m/s 2 )

A uniform cylinder of mass M and radius R is placed on a rough horizontal board, which in turn is placed on a smooth surface. The coefficient of friction between the board and the cylinder is μ . If the board starts accelerating with constant acceleration a, as shown in the figure, then

A body weighs 8 g when placed in one pan and 18 g when placed on the other pan of a false balance. If the beam is horizontal when both the pans are empty, the true weight of the body is 4xg. Find value of x.

A uniform rod is falling without rotation on a smooth horizontal plane. Assuming the collision to be perfectly elastic, the angular velocity of the rod after striking the table is maximum when the rod makes an angle cos − 1 1 ∗ with the horizontal just before striking where ∗ is not readable. Find ∗ .

Two blocks A and B each of mass m are connected by a massless spring of natural length L and spring constant k. The blocks are initially resting on a smooth horizontal floor with the spring at its natural length, as shown in figure. A third identical block C, also of mass m, moves on the floor with a speed v along the line joining A to B and collides with A. Then

A uniform disc of mass 5 kg and radius 1 m is placed vertically on a rough horizontal surface. A horizontal time varying force is applied on the center of disc and its variation with time is shown in the graph. The co–efficient of friction between disc and surface is 0.4. (Take g = 10 m / s 2 ) Then which of the following option is/are correct

A stick of mass density λ = 8 kg/m rests on a disc of radius R = 20cm as shown in the figure. The stick makes an angle θ = 37 o with the horizontal and is tangent to the disc at its upper end. Friction exists at all points of contact and assume that it is large enough to keep the system at rest. Find the friction force (in Newton) between the ground and the disc. (take g = 10 m / s 2 )

A semicircular disc of radius R and mass M is pulled by a horizontal force F so that it moves with uniform velocity. The coefficient of friction between disc and ground is μ = 4 3 π . If the angle θ = π / n radians, then find the value of n .

A uniform meter scale is balanced on fixed semicircular cylinder of radius 30 cm as shown. One end of scale is slightly depressed and released. Time period of oscillation of the meter scale is s. [ g = 10 m s − 2 , π = 3.142 ]

A solid cylinder of mass M and radius R released from rest rolls down a rough inclined plane inclined at an angle of 45 ∘ with the horizontal. The coefficient of friction between the cylinder and the inclined plane, if the heat produced is maximum , is ( 1 − n ) . Find n.

A uniform rod of length 2m and mass M = 20 k g is hinged at point O such that it is free to rotate in a vertical plane about a horizontal axis passing through point O and perpendicular to the plane of the paper. The rod is kept in equilibrium with the help of an ideal string. Then

A uniform disc of mass 5kg and radius 1m is placed vertically on a rough horizontal surface. A horizontal time varying force is applied on the centre of disc and its variation with time is shown in the graph. The co-efficient of friction between disc and surface is 0.4 ( T a k e g = 10 m / s 2 ) Then which of the following option is/are correct

A uniform rod of mass m = 5 kg and length L = 90cm rests on a smooth horizontal surface. One of the ends of the rod is struck with the impulse J = 3 N.s in a horizontal direction perpendicular to the rod. As a result, the rod attains the momentum p = 3 N.s . Find the force (in newtons) with which one half of the rod will apply on the other half in the process of motion.

A rod of mass ‘M’ and length 6R is pivoted at its centre (point P) to rotate in vertical plane as shown. Two discs of same radius ‘R’ are joined at the ends O 1 and O 2 of the rod. Mass of the discs A and B are ‘M’ and ‘2M’ respectively. Initial angular velocities of discs are shown in figure. The system is released from rest from horizontal position. The angular velocity of the rod when it is vertical is g λ R . Find the value of λ .

A ring of mass M = 90 gram and radius R = 3 meter is kept on a frictionless horizontal surface such that its plane is parallel to horizontal plane. A particle of mass m = 10 gram is placed in contact with the inner surface of ring as shown figure. An initial velocity v = 2 m/s is given to the particle along the tangent of the ring. The magnitude of the force of interaction in milli-newton between them after 1 sec from the start is ………..

In the List-I below, four different paths of a particle are given as functions of time. In these functions, α and β are positive constants of appropriate dimensions and α ≠ β . In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned: p is the linear momentum, L is the angular momentum about the origin, K is the kinetic energy, U is the potential energy and E is the total energy. Match each path in List-I with those quantities in List-II, which are conserved for that path. LIST-I LIST-II P. r t = α t i ^ + β t j ^ 1. p Q. r t = α cos ω t i ^ + β sin ω t j ^ 2. L R. r t = α cos ω t i ^ + sin ω t j ^ 3. K S. r t = α t i ^ + β 2 t 2 j ^ 4. U 5. E

One twirls a circular ring (of mass M and radius R) near the tip of one’s finger as shown in Figure 1. In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the path traced out the point where the ring and the finger is in contact is r. The finger rotates with an angular velocity ω 0 . The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger is in contact (Figure 2). The coefficient of friction between the ring and the finger is μ and the acceleration due to gravity is g. Take r ≪ R

A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity ω is an example of a non-inertial frame of reference. The relationship between the force F r o t experienced by the particle of mass m moving on the rotating disc and the force F i n experienced by the particle in an inertial frame of reference is F r o t = F i n + 2 m v r o t × ω + m ω × r × ω , where v r o t is the velocity of the particle in the rotating frame of reference and r is the position vector of the particle with respect to the center of the disc Now consider a smooth slot along a diameter of a disc of radius R rotating counter-clockwise with a constant angular speed ω about its vertical axis through its center. We assign a coordinate system with the origin at the center of the disc, the x-axis along the slot, the y-axis perpendicular to the slot and the z-axis along the rotation axis ω = ω k ^ . A small block of mass m is gently placed in the slot at r = R / 2 i ^ at t = 0 and is constrained to move only along the slot

Rods AB,BC,CD, & DA are joined to form a square loop having current ‘i’. Mass and length of each rod is m & / respectively. This loop is situated in uniform magnetic field B as shown, and can rotate freely about axis p 1 p 2 Match the following, if initially plane of the loop is parallel to magnetic field Column – I Column – II A Angular acceleration of the loop at t = 0 P i B l 2 2 B Angular velocity when loop has rotated by 30 0 Q Zero C Torque on the loop when it has rotated by 60 0 R 3 B i 2 m D Angular acceleration of the loop when it has rotated by 90 0 S 3 B i 2 m

Match the List I with the List-II and select the correct answer using the code given below the lists. List – I List – II (P) Three identical uniform rods each of mass ‘m’ and length ‘L’ are joined as shown in figure. The system is hinged about one end in a vertical plane. At equilibrium position, the value of ‘ tan θ ‘ is (1) 1 3 (Q) Two identical uniform rods each of mass ‘m’ and length ‘L’ are joined as shown in figure. The system is hinged about one end in a vertical plane. At equilibrium position, the value of ‘tan θ ’ is (2) 3 4 (R) A ball collides with a smooth and fixed inclined plane (of inclination θ and coefficient of restitution e = 1 4 ) after falling vertically through a distance h. If it moves horizontally just after the impact, the value of tan θ is (3) 1 (S) A ball is projected with velocity u at an angle ‘ θ ’ with the horizontal. The value of “tan θ ” for which radius at the highest point is half the range is (4) 1 2

Two identical uniform discs roll without slipping on two different surfaces AB and CD (see figure) starting at A and C with linear speed v 1 and v 2 , respectively, and always remain in contact with the surfaces, If they reach B and D with the same linear speed and v 1 = 3 m/s, v 2 in m/s is ( g = 10 m / s 2 )

Match the List I with the List-II and select the correct answer using the code given below the lists. List – I List – II (P) Three identical uniform rods each of mass ‘m’ and length ‘L’ are joined as shown in figure. The system is hinged about one end in a vertical plane. At equilibrium position, the value of ‘ tan θ ‘ is (1) 1 3 (Q) Two identical uniform rods each of mass ‘m’ and length ‘L’ are joined as shown in figure. The system is hinged about one end in a vertical plane. At equilibrium position, the value of ‘tan θ ’ is (2) 3 4 (R) A ball collides with a smooth and fixed inclined plane (of inclination θ and coefficient of restitution e = 1 4 ) after falling vertically through a distance h. If it moves horizontally just after the impact, the value of tan θ is (3) 1 (S) A ball is projected with velocity u at an angle ‘ θ ’ with the horizontal. The value of “tan θ ” for which radius at the highest point is half the range is (4) 1 2

A 45 kg girl and her 60 kg brother are at rest at the center of a frozen pond ( μ = 0 ) almost touching each other. He pushes her so that she slides away relatively at 5.6 ms -1 . Assume that he extends his hands by 60 cm to complete the push and that he applies a constant horizontal force F on his sister.

A solid cylinder of mass M and radius R released from rest rolls down a rough inclined plane inclined at an angle of 45 ∘ with the horizontal. The coefficient of friction between the cylinder and the inclined plane if the heat produced is maximum is 1 − n . Find n.

Imagine a hole drilled along the radius of earth. A uniform rod of length equal to radius (R) of earth is inserted into the hole. The distance of centre of gravity of the rod from the centre of earth is m n R , where m and n are positive integers and co-primes. Find the value of m + n.

A uniform rod of mass M and length L is hinged at lower end on a fixed horizontal table. The rod can rotate freely in vertical plane and there is no friction at the hinge. A ball of mass M and radius R = L 3 is placed in contact with the vertical rod and a horizontal force F is applied at the upper end of the rod. There is no friction between rod and the ball and there is sufficient friction between ball and ground for pure rolling. Mark the correct statement(s). (assume the ball to be a uniform solid sphere)

On a horizontal smooth plane there is a motionless vertical cylinder and disc A. A is connected to cylinder by thread AB. The disc is set in motion with initial velocity V. as shown. Then the angular momentum of the body is

A system consists of two point masses, A and B of masses 1 kg and 2 kg respectively. At an instant the kinetic energy of A with respect to the centre of mass is 2 joules and the velocity of centre of mass is 2 m/s. The kinetic energy of the system at this instant is:

A football of radius R is kept on a hole of radius r (? < ?) made on a plank kept horizontally. One end of the plank is now lifted so that it gets tilted making an angle ? from the horizontal as shown in the figure below. The maximum value of ? so that the football does not start rolling down the plank satisfies (figure is schematic and not drawn to scale) –

A small roller of diameter 20 cm has an axle of diameter 10 cm (see figure below on the left). It is on a horizontal floor and a meter scale is positioned horizontally on its axle with one edge of the scale on top of the axle (see figure on the right). The scale is now pushed slowly on the axle so that it moves without slipping on the axle, and the roller starts rolling without slipping. After the roller has moved 50 cm, the position of the scale will look like (figures are schematic and not drawn to scale)-

Put a uniform meter scale horizontally on your extended index fingers with the left one at 0.00 cm and the right one at 90.00 cm. When you attempt to move both the fingers slowly towards the center, initially only the left finger slips with respect to the scale and the right finger does not. After some distance, the left finger stops and the right one starts slipping. Then the right finger stops at a distance ? ? from the center (50.00 cm) of the scale and the left one starts slipping again. This happens because of the difference in the frictional forces on the two fingers. If the coefficients of static and dynamic friction between the fingers and the scale are 0.40 and 0.32, respectively, the value of ? ? (in cm) is

A horizontal disc rotates freely about a vertical axis through it centre. A ring, having the same mass and radius as the disc, is now gently placed on the disc. After some time, the two rotate with a common angular velocity. Select the correct statements from the following.

The densities of two solid spheres A and B of the same radii R vary with radial distance r as ρ A ( r ) = k r R and ρ B ( r ) = k r R 5 , respectively, where k is a constant. The moments of inertia of the individual spheres about axes passing through their centres are I A and I B , respectively. If I B I A = n 10 , the value of n is

A spool has the shape shown in figure. Radii of inner and outer cylinders are R and 2 R respectively. Mass of the spool is 3 m and its moment of inertia about the shown axis is 2mR 2 . Light threads are tightly wrapped on both the cylindrical parts. The spool is placed on a rough surface with two masses m 1 = m and m 2 = 2m connected to the strings as shown. The string segment between spool and the pulleys P 1 and P 2 are horizontal. The centre of mass of the spool is at its geometrical centre. System is released from rest. What is minimum value of coefficient of friction between the spool and the table so that it does not slip?

Two disc are suspended with a ribbon between them as shown in figure. At t = 0 the ribbon has 100 complete anti-clock twists in it, i.e., the ribbon is twisted by keeping the upper disc fixed and rotating the lower disc anti-clock when viewed from above. At t = 0 the upper disc starts to rotate at ω u = 2 rpm anticlockwise, and the lower disc, starting from rest, begins to rotate with a clockwise angular acceleration of a magnitude of 1 rpm/s 2 . How many relative revolutions does lower disc make between t = 0 and t = t f with respect to upper disc?

Two disc are suspended with a ribbon between them as shown in figure. At t = 0 the ribbon has 100 complete anti-clock twists in it, i.e., the ribbon is twisted by keeping the upper disc fixed and rotating the lower disc anti-clock when viewed from above. At t = 0 the upper disc starts to rotate at ω u = 2 rpm anticlockwise, and the lower disc, starting from rest, begins to rotate with a clockwise angular acceleration of a magnitude of 1 rpm/s 2 . How many relative revolutions does lower disc make between t = 0 and t = t f with respect to upper disc?

A mass m of radius r is rolling horizontally without any slip with a linear speed v. It then rolls up to a height given by 3 4 v 2 g

A uniform solid disc is rolling on a horizontal surface. At a certain instant B is the point of contact and A is at height 2R from ground, where R is radius of disc:

A system of uniform cylinders and plates is shown. All the cylinders are identical and there is no slipping at any contact. Velocity of lower and upper plate is V and 2V respectively as shown. Then the ratio of angular speed of the upper cylinders to angular speed of lower cylinders is:

Assume body B to be uniform: Column-A Column-B (Torque of normal reaction on body B about centre of mass) (a) (F is directed towards| centre of mass) (p) Clockwise (b) (q) Anticlockwise (c) (r) Zero (s) Conclusion cannot be drawn with the information given

Two particles P 1 and P 2 of equal masses situated at (0 , 0) and (10, 0) respectively at t = 0 moving with constant velocities collided head on at point (4 ,0) after time t 0 . If the coefficient o f restitution is 1 then what is the x-co-ordinate of centre of mass of the two particles at t = 2t 0 .

ACB is a smooth quater circular path of radius R. Four forces are acting at a particle placed at A. F 1 is always horizontal, F 2 is always vertical, F 3 is always tangential to the path, F 4 is always directed from particle ‘s position to point B. Magnitude of all forces are equal to F. Column I Column II A) work done by F 1 is P) FR B) work done by F 2 is Q) F ⋅ R 2 C) work done by F 3 is R) πFR 2 D) work done by F 4 is S) F.R 2

A horizontal force F is applied at the centre of mass of a cylindrical object of mass m and radius R, perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is μ . The centre of mass of the object has an acceleration a. The acceleration due to gravity is g. Given that the object rolls without slipping, which of the following statement(s) is(are) correct ?

A uniform thin flat isolated disc is floating in space. It has radius R and mass m. A force F is applied to it at a distance d = (R/2) from the centre in the y-direction. Treat this problem as two-dimensional. Just after the force is applied :

Illustrated below is a uniform cubical block of mass M and side a. Mark the correct statement(s).

A rod of mass m and length l is connected with a light rod of length l. The composite rod is made to rotate with angular velocity ω as shown in the figure. Find the ratio of the translational and rotational kinetic energy of the rod

A disc of radius R = 1 2 m is rolling on a plank. If the disc is moving on the plank with angular velocity ω = 2 rad / s and speed v C = 2 m / s . Find the speed of the plank (in m/s) for pure rolling.

A block is placed on a rough inclined plane with inclination θ = 45 ∘ . The coefficient of static friction between the block and the incline is μ = 0 .25 . Determine the maximum ratio h/b for which the homogenous block will slide without tipping under the action of force F.

A uniform circular disc has radius R and mass m. A particle, also of mass m, is fixed at point A on the edge of the disc as shown in figure. The disc can rotate freely about a fixed horizontal chord PQ that is at a distance R/4 from the centre C of the disc. The line AC is perpendicular to PQ. Initially, the disc is held vertical with point A at its highest position. It is then allowed to fall so that it starts rotating about PQ. If R = 0.5 m then, find the linear speed of the particle (in m/s) as it reaches its lowest position. (Take g = 10 m/s 2 )

A bobbin has inner radius r and outer radius R is placed on a rough horizontal surface. A light is string wrapped over inner core, connects a block with bobbin as shown in the figure. Now system is released from rest and bobbin moves on the horizontal surface without sliding, and the string does not slide from bobbin, If r R = 0 .25 , find the ratio of the acceleration of the block and bobbin a A .

An inextensible thread is wound round a cylinder on mass M, the cylinder is placed over a rough horizontal surface and the thread is passed over massless and frictionless pulley such that the part of the thread between cylinder and pulley is horizontal as shown in figure. When a block of mass m is attached with free end of the thread and system is released, it is observed that friction just prevents slipping of the cylinder over the horizontal surface. If M m = 4 , then find minimum coefficient of friction between horizontal surface and cylinder.

A clockwise torque of 6 N-m is applied to the circular cylinder as shown in the figure. There is no friction between the cylinder and the block.

A solid cylinder of mass m is kept in balance on a fixed incline of angle α = 37 ∘ with the help of a thread fastened to its jacket. The cylinder does not slip.

A T shaped object with dimensions shown in the figure is lying on a smooth floor. A force is applied at the point P parallel to AB such that the object has only the translational motion without rotation. Find the location of P with respect to C.

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 bucket of water of mass 21 kg is suspended by a rope wrapped around a solid cylinder 0.2 m in diameter. The mass of the solid cylinder is 21 kg. The bucket is released from rest. Which of the following statements are correct?

Illustrated below is a uniform cubical block of mass M and side a. Mark the correct statement (s).

Two blocks of masses m and 2m are connected by an ideal spring. The system is placed on smooth horizontal surface. The blocks are given impulse so that the blocks start moving as shown in the figure. In this spring-blocks system, find the ratio of the initial kinetic energy of centre of mass to the initial kinetic energy of system w.r.t. centre of mass .

A bobbin with inner radius r and outer radius R is arranged with light strings and a block as shown in the figure. The strings are not sliding over the cylinder. The system is released from rest. The block and bobbin move down with accelerations a and A respectively. If R r = 4 , then find the ration a A .

A uniform rod is resting freely over a smooth horizontal plane. A particle moving horizontally strikes at one end of the rod normally and gets stuck. Then

A semicircular disc of radius R and mass M is pulled by a horizontal force F so that it moves with uniform velocity. The coefficient of friction between disc and ground is μ = 4 3 π . Find the value of θ in degree.

A solid cylinder of mass m = 4 kg is kept in balance on a fixed incline of angle θ = 37 ∘ with the help of a thread fastened to its jacket. The cylinder does not slip. What force F is required to keep the cylinder in balance when the thread is held vertically? (Use g = 10 m/s 2 )

A uniform rod of length L and mass M is lying on a frictionless horizontal plane and is pivoted at one of its ends as shown in figure. There is no friction at the pivot. An inelastic ball of mass m is fixed with the rod at a distance L/3 from O. A horizontal impulse J is given to the rod at a distance 2L/3 from O in a direction perpendicular to the rod. Assume that the ball remains in contact with the rod after the collision and impulse J acts for a small time interval Δt Now answer the following questions:

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 m s -2 in the downward direction, the acceleration of block B will be

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 clockwise torque of 6 N-m is applied to the circular cylinder as shown in the figure. There is no friction between the cylinder and the block.

A uniform rod AB, 12 m long weighing 24 kg, is supported at end B by a flexible light string and a lead weight (of very small size) of 12 kg attached at end A. The rod floats in water with one-half of its length submerged. Find the volume of the rod in × 10 − 3 m 3 Take g = 10 m / s 2 , density of water = 1000 kg / m 3

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 figure. At what point should a parallel force 3F be applied in order that the block shall undergo pure translational motion? Assume normal contact force between the block and surface passes through the centre of gravity of the block.

Two blocks A and B each of equal masses m are released from the top of a smooth fixed wedge as shown in the figure. Find the magnitude of the acceleration of the centre of mass of the two blocks (in m/s 2 ). Take g = 10 m/s 2

The spool shown in the figure is placed on a rough horizontal surface and has inner radius r and outer radius R. The angle θ between the applied force and the horizontal can be varied. The critical angle ( θ ) for which the spool does not roll and remains stationary is given by

A thin uniform rod of mass m and length l is hinged at the upper end and free to rotate about its upper end in vertical plane . When it is at rest, it receives an impulse J at its lowest point, normal to its length. Immediately after impact,

A conical pendulum consists of a simple pendulum moving in a horizontal circle as shown in the figure. C is the pivot, O the centre of the circle in which the pendulum bob moves and ω the constant angular velocity of the bob. If L is the angular momentum about point C, then

A block is placed on a rough inclined plane with inclination θ = 45 ∘ . The coefficient of static friction between the block and the incline is μ = 0 . 75 . Determine the maximum ratio h/b for which the homogenous block will slide without tipping under the action of force F.

Three spools A, B and C each having moment of inertia I= MR 2 /4 are placed on rough ground and equal force F is applied at positions as shown in the figures (a), (b) and (c). Then

Three spools A, B and C each having moment of inertia I = MR 2 /4 are placed on rough ground and equal force F is applied at positions as shown in the figures (a), (b) and (c). Then

A ring of mass 3 kg is rolling without slipping with linear velocity 1 ms -1 on a smooth horizontal surface. A rod of same mass is fitted along its one diameter. Find total kinetic energy of the system (in J).

A loop and a disc roll without slipping with same linear velocity v. The mass of the loop and the disc is same. If the total kinetic energy of the loop is 8 J, find the kinetic energy of the disc (in J)

A solid sphere rolls without slipping on a rough horizontal floor, moving with a speed v. It makes an elastic collision with a smooth vertical wall. After impact

A block of mass m moving with a velocity v 0 collides with a stationary block of mass M at the back of which a spring of spring constant k is attached, as shown in the figure. Select the correct alternative(s)