A metallic circular plate is rotating about its axis without friction. If the radius of plate expands by 0.1% then the % change in its moment of inertia is
If the earth were to suddenly contract to 1/nth of its present radius without any change in its mass, the duration of the new day will be nearly
A force of – F k ^ acts on O, the origin of the coordinate system. The torque about the point (1, –1) is
A uniform rod of length two meter is bent at its midpoint to make 90°. The distance of centre of mass from the centre of rod is (in cm)
A sphere can roll on a surface inclined at an angle θ if the friction coefficient is more than g 7 tan θ . If a sphere is released from rest on the incline
A Merry-go-round, made of a ring-like platform of radius R and mass M, is revolving with angular speed . 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
A system consists of two masses connected by a massless rod lies along x–axis. The distance of centre of mass from O is
Two bodies of 6 kg and 4 kg masses have their velocity 5 i ^ − 2 j ^ + 10 k ^ and 10 i ^ − 2 j ^ + 5 k ^ respectively. Then the velocity of their centre of mass is
The area of the triangle whose adjacent sides are represented by the vector ( 4 i ^ + 3 j ^ + 4 k ^ ) and 5 i ^ in sq. units is
Four particles each of mass ‘m’ are placed at the corners of a light square frame of side length ‘ l ‘. The radius of gyration of the system about an axis perpendicular to the plane of square and passing through its centre is
If a sphere of mass 2kg and diameter 10cm is rolling at speed of 5ms -1 . Its rotational kinetic energy is
Two identical hollow spheres roll down two inclined planes of same height but of different angles of inclination. Then, they reach the bottom
Four particles, each of mass 1 kg are placed at the corners of a square OABC of side 1 m. ‘O’ is at the origin of the coordinate system. OA and OC are aligned along positive X-axis and positive Y-axis respectively. The position vector of the centre of mass is (in ‘m’)
Two blocks of masses 10kg and 30 kg are placed along a vertical line if the first block is raised through a height of 7cm then the distance through the second mass should be moved to raise the centre of mass of the system by 1cm is
A boy of mass 50kg is standing at one end of a boat of length 9m and mass 400kg. He runs to the other end. The distance through which the centre of mass of the boat boy system moves is
If F = 2 i ^ − 3 j ^ N and r = 3 i ^ + 2 j ^ m then torque τ is
The radius of a solid sphere is R and its density D. When it is made to rotate about an axis passing through any diameter of sphere, expression for its moment of inertia is
Three point sized bodies each of mass M are fixed at three corners of light triangular frame of side length L. About an axis perpendicular to the plane of frame and passing through centre of frame the moment of inertia of three bodies is
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 molecule about an axis passing through centre of mass and perpendicular to the rod is :
Mass of thin long metal rod is 2 kg and its moment of inertia about an axis perpendicular to the length of rod and passing through its one end is 0.5kgm 2 . Its radius of gyration is
A solid sphere rolls down without slipping from rest on a 30° incline. Its linear acceleration is
A stationary body of mass 3 kg explodes into three equal pieces. Two of the pieces fly off at right angles to each other, one with a velocity 2 i ^ m/s and the other with a velocity 3 j ^ m/s. If the explosion takes place in 10 -5 sec, the average force acting on the third piece in Newtons is:
The diameter of a disc is 1m. It has a mass of 20kg. It is rotating about its axis with a speed of 120 rotations in one minute. Its angular momentum in kg m 2 /s is
A wheel is subjected to uniform angular acceleration about its axis. Initially its angular velocity is zero. In this first 2 sec, it rotates through an angle θ 1 ; in the next 2 sec, it rotates through an additional angle θ 2 . The ratio of θ 2 /θ 1 is
A thick straight wire of length π m is fixed at its midpoint and then bent in the form of a circle. The shift in its centre of mass is
Two bodies of different masses 2kg and 4kg are moving with velocities 2m/s and 10m/s towards each other due to mutual gravitational attraction. Then the velocity of the centre of mass is
A thin uniform rod of length “L” is bent at its mid point as shown in the figure. The distance of the centre of mass from the point “O” is
Three identical spheres each of mass ‘m’ and radius ‘R’ are placed touching each other so that their centres A, B and C lie on a straight line. The position of their centre of mass from centre of A is
A dog weighing 5kg is standing on a flat boat so that it is 10 metres from the shore. It walks 4m on the boat towards the shore and then halts. The boat weighs 20kg and one can assume that there is no friction between it and water. The dog from the shore at the end of this time is
The angle between the vectors ( i ^ + j ^ + k ^ ) and ( i ^ − j ^ − k ^ ) is
Three particles of masses 1gm, 2gm & 3gm are at 1cm, 2cm, & 3cm from the axis of rotation respectively then the moment of inertia of the system & radius of gyration of the system respectively are …….gm cm 2 and .. cm
The ratio of moments of inertia of two solid spheres of same mass but densities in the ratio 1:8 is
A hoop of mass 500gm & radius 10cm is placed on a nail. then the moment of inertia of the hoop, when it is rotated about the nail will be– kgm 2
Four point size bodies each of mass M are fixed at four corners of a light squre frame of side length L. The moment of inertia of the four bodies about an axis perpendicular to the plane of frame and passing through its centre is
The diameter of a fly wheel is R. Its coefficient of linear expansion is α . If its temperature is increased by ∆ T the percentage increase in its moment of inertia is
The radius of gyration of a body is 18 cm when it is rotating about an axis passing through centre of mass of body. If radius of gyration of same body is 30 cm about a parallel axis to first axis then, perpendicular distance between two parallel axes is
I is moment of inertia of a thin square plate about an axis passing through opposite corners of plate. The moment of inertia of same plate about an axis perpendicular to the plane of plate and passing through its centre is
A particle performs uniform circular motion with an angular momentum L. If the angular frequency ω of the particle is doubled, and kinetic energy is halved, its angular momentum becomes
A ballet dancer is rotating about his own vertical axis at an angular velocity 100 rpm on smooth horizontal floor. The ballet dancer folds himself close to his axis of rotation by which is moment of inertia decreases to half of initial moment of inertia then his final angular velocity is
A particle of mass m is moving along a circle of radius r with a time period T. Its angular momentum is
If the radius of earth shrinks by 0.2% without change in its mass, the % change in its angular velocity is
A mass is whirled in a circular path with an angular momentum L. If the length of string and angular velocity, both are doubled, the new angular momentum is
An automobile engine develops 100 KW when rotating at a speed of 1800 rev/min. The torque it delivers ( in N-m )
An electric motor exerts a constant torque 5Nm on a fly wheel by which it is rotated at the rate of 420rpm The power of motor is
For a body rolling along a level surface, the translational and rotational K.E. are equal. The body is
A shaft rotating at 3000rpm is transmitting a power of 3.14KW. The magnitude of the driving torque is
A hollow sphere rolls down a 30° incline of length 6m without slipping. The speed of centre of mass at the bottom of plane is
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
A ring and a disc of same mass roll without slipping along a horizontal surface with same velocity. If the K.E. of ring is 8J, then that of disc is
The distance of centre of mass from ‘O’ is
Four bodies of masses 1,2,3,4 kg respectively are placed at the comers of a square of side ‘a’. Coordinates of centre of mass are (take 1kg at the origion, 2kg on X-axis and 4kg on Y-axis)
Particles of masses 1kg and 3kg are at ( 2 i + 5 j + 13 k ) m and ( − 6 i + 4 j − 2 k ) m then instantaneous position of their centre of mass is
Two bodies of masses 5kg and 3kg are moving towards each other with 2ms -1 and 4ms -1 respectively. Then velocity of centre of mass is
A boat of mass 50kg is at rest. A dog of mass 5kg moves in the boat with a velocity of 20m/s. What is the velocity of boat?
A circular disc of radius 20cm is cut from one edge of a larger circular disc of radius 50cm. The shift of centre of mass is
Two particles of masses 4kg and 6kg are separated by a distance of 20m and are moving towards each other under mutual force of attraction, the position of the point where they meet is
A uniform metre rod is bent into L shape with the bent arms at 90° to each other. The distance of the center of mass from the bent point is
The position of a particle is given by r = i ^ + 2 j ^ − k ^ and its momentum is p = 3 i ^ + 4 j ^ − 2 k ^ . The angular momentum is perpendicular to
Two objects of masses 200g and 500g have velocities of 10i m/s and (3i + 5 j)m/ s respectively. The velocity of their centre of mass is
The area of the parallelogram whose adjacent sides are P = 3 i ^ + 4 j ^ ; Q = − 5 i ^ + 7 j ^ is (in sq.units)
If A = 3 i + j + 2 k and B = 2 i − 2 j + 4 k is the angle between the two vectors, then sin θ is equal to
A uniform sphere has radius R. A sphere of diameter R is cut from its edge as shown. Then the distance of centre of mass of remaining portion from the centre of mass of the original sphere is
The handle of a door is at a distance 40cm from axis of rotation. If a force 5N is applied on the handle in a direction 30° with plane of door, then the torque is
A door can just be opened with 10N force on the handle of the door. The handle is at a distance of 50cm from the hinges. Then, the torque applied on the door (in Nm) is
A particle of mass m is projected with an initial velocity u at an angle θ to horizontal. The torque of gravity on projectile at maximum height about the point of projection is
A uniform rod is 4m long and weights 10kg. If it is supported on a knife edge at one meter from the end, what weight placed at that end keeps the rod horizontal.
The ratio of moments of inertia of a solid sphere about axes passing through its centre and tangent respectively is
If I is moment of inertia of a thin circular plate about an axis passing through tangent of plate in its plane. The moment of inertia of same circular plate about an axis perpendicular to its plane and passing through its centre is
The moment of inertia of a solid sphere about an axis passing through its centre is 0.8kgm 2 The moment of inertia of another solid sphere whose mass is same as mass of first sphere, but the density is 8 times density of first sphere, about an axis passing through its centre is
Moment of inertia of a hoop suspended from a peg about the peg is
Four particles each of mass 1kg are at the four corners of square of side 1m. The M.I.of the system about a normal axis through centre of square is
Three identical masses, each of mass 1kg, are placed at the corners of an equilateral triangle of side l. Then the moment of inertia of this system about an axis along one side of the triangle is
A wire of mass m and length l is bent in the form of circular ring. The moment of inertia of the ring about its axis is
Four point size bodies each of mass m are fixed at four corners of light square frame of side length 1m. The radius of gyration of these four bodies about an axis perpendicular to the plane of frame passing through its centre is
The moment of inertia of a thin uniform rod of mass M and length L about an axis perpendicular to the rod, through its centre is I. The moment of inertia of the rod about an axis perpendicular to rod through its end point is
Uniform square plate of mass 240 gram is made to rotate about an axis passing through any diagonal of plate. If its moment of inertia is 2 x 10 -4 kgm 2 then its side length is
Two objects of masses 1kg and 2kg separated by a distance of 1.2m are rotating about their centre of mass. Find the moment of inertia of the system
The M.I. of a thin rod about a normal axis through its centre is I. It is bent at the centre such that the two parts are perpendicular to each other and perpendicular to the axis. The M.I. of the system about the same axis will be
The moment of inertia of two spheres of equal masses about their diameters are the same. One is hollow, then ratio of their diameters
If the mass of earth and radius suddenly become 2 times and 1/4th of the present value, the length of the day becomes
A circular disc of mass 4kg and of radius 10cm is rotating about its natural axis at the rate of 5 rad/sec. its angular momentum is
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
The radius of gyration of a body about an axis at a distance of 4cm from its centre of mass is 5cm. The radius of gyration about a parallel axis through centre of mass is
If radius of earth shrinks by 0.1% without change in its mass, the percentage change in the duration of one day
A ballet dancer spins about a vertical axis at 60rpm with his arms closed. Now he stretches his arms such that M.I. increases by 50%. The new speed of revolution is
A metallic circular wheel is rotating about its own axis without friction. If the radius of wheel expands by 0.2%, percentage change in its angular velocity
F = a i ^ + 3 j ^ + 6 k ^ and r = 2 i ^ − 6 j ^ − 12 k ^ The value of ‘a’ for which the angular momentum is conserved is
A mass is whirled in a circular path with a constant angular velocity and its angular momentum is L. If the length of string is now halved keeping the angular velocity same, the new angular momentum is
A uniform circular disc of radius R is rotating about its own axis with moment of inertia I at an angular velocity ω If a denser particle of mass m is gently attached to the rim of disc than its angular velocity is
A disc rotates with angular velocity w and kinetic energy E. Then its angular momentum
If earth shrinks to 1/64 of its volume with mass remaining same, duration of the day will be
A wheel at rest has M.I. 2 π 2 kgm 2 It is rotated by a 60W motor for one minute. The number of rotations made by the wheel in one minute is
The shaft of a motor is making 1260rpm. The torque supplied by the motor is 100Nm. the power of motor is ( in KW)
An electric motor rotates a wheel at a constant angular velocity 10rps while opposing torque is 10 Nm . The power of that electric motor is
The work done in increasing the angular frequency of a circular ring of mass 2kg and radius 25cm from 10 rpm to 20rpm about is axis
A ring is allowed to roll down on an incline of 1 in 10 without slipping. The acceleration of its center of mass is
A cylinder is released from rest from the top of an incline of inclination θ and length ‘l’. If the cylinder roles without slipping, its speed at the bottom
For a body rolling along a level surface, without slipping the translational and rotational kinetic energies are in the ratio 2:1.The body is
A solid sphere and a spherical shell roll down an incline from rest from same height. The ratio of times taken by them is
When a solid sphere is rolling along level surface the percentage of its total kinetic energy that is translational is
A thin ring of mass 1kg and radius 1m is rolling at a speed of 1ms -1 . Its kinetic energy is
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
O is the center of an equilateral triangle A B C, F 1 , F 2 and F 3 are three forces acting along the, sides A, B, B C and A C as shown in figure. what should be the magnitude of F 3 so that the total torque about O is zero
Relation between torque and angular momentum is similar to the relation between
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. How will his angular speed change?
When running boy jumps on to a rotating table, the quantity that is conserved is
A shaft rotating at 3000 rpm is transmitting a power of 3.14 KW. The magnitude of the driving torque is ………. Nm
A thin uniform rod of length l and mass m is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is ω . Its centre of mass rises to a maximum height
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. 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
