Three-point masses 'm' each, are placed at the vertices of an equilateral triangle of side a. Moment of inertia of the system about axis COD is-
1.
2.
3.
4.
A particle is moving in a circular orbit with constant speed. Select wrong alternate
1. Its linear momentum is conserved
2. Its angular momentum is conserved
3. It is moving with variable velocity
4. It is moving with variable acceleration
One solid sphere A and another hollow sphere B are of same mass and same outer radii. Their moment of inertia about their diameters are respectively IA and IB such that
1.
2.
3.
4.
A couple produces: [NTSE 1995; CBSE PMT 1997; DCE 2004]
1. Purely linear motion
2. Purely rotational motion
3. Linear and rotational motion
4. No motion
A wheel of radius R rolls on the ground with a uniform velocity v. The velocity of topmost point relative to the bottommost point is
1. v
2. 2v
3. v/2
4. zero
Point masses and are placed at the opposite ends of a rigid of length L and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position of point P on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity is minimum is given by
Two discs are rotating about their axes, normal to the discs and passing through the centres of the discs. Disc D has 2 kg mass and 0.2 m radius and initial angular velocity of 50 rad s. Disc D has 4 kg mass, 0.1 m radius and initial angular velocity of 200 rad s. The two discs are brought in contact face to face, with their axes of rotation coincident. The final angular velocity (in rad.s) of the system is
1. 60
2. 100
3. 120
4. 40
A body rolls down an inclined plane without slipping. The fraction of total energy associated with its rotation will be
Where k is radius of gyration of the body about an axis passing through centre of mass and R is the radius of the body.
A disc and a solid sphere of the same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first?
1. Disk
2. Sphere
3. Both reach at the same time
4. Depends on their masses
From a disc of radius R and mass M, a circular hole of diameter R, whose rim passes through the centre is cut. What is the moment of inertia do the remaining part of the disc about a perpendicular axis, passing through the centre
The ratio of the radii of gyration of a circular disc to that of a circular ring, each of same mass and radius, around their respective axes is
An automobile moves on a road with a speed of 54 kmh. The radius of its wheel is 0.45 m and moment of inertia of the wheel about its axis of rotation is 3 kg m. If the vehicle is brought to rest in 15s, the magnitude of average torque transmitted by its brakes to the wheel is
1. 8.58
2. 10.86
3. 2.86
4. 6.66
A uniform rod AB of length l and mass m is free to rotate about point A. The rod is released from rest in horizontal position. Given that the moment of inerita of the rod about A is the initial angular acceleration of the rod will be
Three indentical spheres, each of mass 1 kg are kept as shown in figure, touching each other, with their centres on a straight line. If their centres are marked P, Q, R respectively, the distance of centre of mass of the system from P is
Two particles of equal mass have coordinates \((2~\text{m}, 4~\text{m}, 6~\text{m})\) and \((6~\text{m},2~\text{m},8~\text{m})\). Of these one particle has velocity \(v_1 = 2\hat i~\text{m/s}\) and another particle has velocity \(v_2 = 2\hat j ~\text{m/s}\) at time \(t=0\). The coordinate of their centre of mass at time \(t=1~\text{s}\) will be:
1. \((4~\text{m}, 4~\text{m}, 7~\text{m})\)
2. \((5~\text{m}, 4~\text{m}, 7~\text{m})\)
3. \((2~\text{m}, 4~\text{m}, 6~\text{m})\)
4. \((4~\text{m}, 5~\text{m}, 4~\text{m})\)
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
1. of the box remains constant
2. of the box plus the ball system remains constant
3. of the ball remains constant
4. of the ball relative to the box remains constant
A hollow sphere is rolling without slipping on the inclined plane. The minimum coefficient of friction is :
1.
2.
3.
4.
A rope is wound around a hollow cylinder of mass 3 kg and radius 40cm. What is the angular acceleration of the cylinder,if the rope is pulled with a force of 30 N?
(1) 25
(2) 0.25
(3) 25
(4) 5
Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities . They are brought into contact face to face coinciding the axis of rotation. The expression for loss of energy during this process is
1.
2.
3.
4.
Two rotating bodies A and B of masses m and 2m with moments of inertia and have equal kinetic energy of rotation. If and be their angular momenta respectively, then:
1.
2.
3.
4.
A solid sphere of mass m and radius R is rotating about its diameter. A soild cyclinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation will be
(a) 2:3
(b) 1:5
(c) 1:4
(d) 3:1
A light rod of length l has two masses attached to its two ends. The moment of inertia of the system about an axis perpendicular to the rod and passing through the centre of mass is
(1)
(2)
(3)
(4)
A rod of weight w is supported by two parallel knife edges A and B and is in equilibrium in a horizontal position. The knives are at a distance d from each other. The centre of mass of the rod is at distance x from A. The normal reaction on A is
(1)
(2)
(3)w(d-x)/x
(4)
Three identical spherical shells, each of mass m and radius r are placed as shown in figure. Consider an axis XX’, which is touching to two shells and passing through diameter of third shell. Moment of inertia of the system consisting of these three spherical shells about XX' axis is
1.mr2
2. 3mr2
3. mr2
4. 4mr2
A force F= is acting at a point r=--12. The value of α
for which angular momentum is conserved about the origin is:
1. -1
2. 2
3. zero
4. 1
A solid cylinder of mass 50 kg and radius 0.5 m is free to rotate about the horizontal axis.A massless string is wound round the cylinder with one end attached to it and other hanging freely.Tension in the string required to produce an angular acceleration of 2 rev/ s2 is
(1)25N
(2)50N
(3)78.5N
(4)157N
The ratio of the accelerations for a solid sphere (mass m and radius R) rolling down an incline of angle θ without slipping and slipping down the incline without rolling is
(1) 5:7
(2) 2:3
(3) 2:5
(4) 7:5
A small object of uniform density rolls up a curved surface with an initial velocity v'. It reaches up to a maximum height of 3v2/4g with respect to the initial position. The object is
(1) ring
(2) solid sphere
(3) hollow sphere
(4) disc
When a mass is rotating in a plane about a fixed point, its angular momentum is directed along
1. a line perpendicular to the plane of rotation
2. the line making an angle of to the plane of rotation
3. the radius
4. the tangent to the orbit
Two persons of mass 55 kg and 65 kg respectively, are at the opposite ends of a boat.The length of the boat is 3.0m and weighs 100 kg.The 55 kg man walks up to the 65 kg man and sits with him.If the boat is in still water the centre of mass of the system shifts by
(1)3.0m
(2)2.3m
(3)zero
(4)0.75m
A circular platform is mounted on a
frictionless vertical axle. Its radius R=2m
and its moment of inertia about the axle
is 200 kg. It is initially at rest. A 50 kg
man stands on the edge of the platform
and begins to walk along the edge at the
speed of relative to the ground.
Time taken by the man to complete one
revolution is
(1)
(2)
(3)
(4)
The moment of inertia of a uniform circular disc is maximum about an axis perpendicular to the disc and passing through:
1. B
2. C
3. D
4. A
Three masses are placed on the x-axis: \(300\) g at the origin, \(500\) g at \(x =40\) cm, and \(400\) g at \(x=70\) cm. The distance of the center of mass from the origin is:
1. | \(40\) cm | 2. | \(45\) cm |
3. | \(50\) cm | 4. | \(30\) cm |
The instantaneous angular position of a point on
a rotating wheel is given by the equation
The torque on the wheel becomes zero at
(1) t=0.5 s
(2) t=0.25 s
(3) t=2 s
(4) t=1 s
The moment of inertia of a thin uniform rod of
mass M and length L about an axis passing
through its mid-point and perpendicular to its
length is . Its moment of inertia about an
axis passing through one of its ends and
perpendicular to its length is
(1)
(2)
(3)
(4)
A circular disk of moment of inertia is rotating in a horizontal plane, about its symmetry axis, with a constant angular speed Another disk of moment of inertia is dropped coaxially onto the rotation disk. Initially the second disk has zero angular speed. Eventually both the disks rotate with a constant angular speed The energy lost by the initially rotating disc to friction is
(1)
(2)
(3)
(4)
Two particles which are initially at rest, move towards each other under the action of their mutual attraction.If their speeds are v and 2v at any instant, then the speed of centre of mass of the system will be
1. 2v 2. 0
3. 1.5v 4. v
A man of 50 kg mass is standing in a gravity free space at a height of 10m above the floor. He throws a stone of 0.5 kg mass downwards with a speed When the stone reaches the floor, the distance of the man above the floor will be
1. 9.9m
2. 10.1m
3. 10m
4. 20m
From a circular disc of radius R and mass 9M, a small disc of mass M and radius is removed concentrically. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through its centre is:
(a) (b)
(c) (d)
The solid cylinder and a hollow cylinder, both of the same mass and same external diameter are released from the same height at the same time on a inclined plane. Both roll down without slipping. Which one will reach the bottom first ?
(1) Both together only when angle of inclination of plane is
(2) Both together
(3) Hollow cylinder
(4) Solid cylinder
If is the force acting on a particle having position vector and be the torque of this force about the origin, then
(a)
(b)
(c)
(d)
A thin circular ring of mass M and radius R is rotating in a horizontal plane about an axis vertical to its plane with a constant angular velocity If two objects each of mass m be attached gently to the opposite ends of a diameter of the ring, the ring will then rotate with an angular velocity -
(1)
(2)
(3)
(4)
Two bodies of mass 1kg and 3kg have position vectors and , respectively. The centre of mass of this system has a position vector -
(1)
(2)
(3)
(4)
Four identical thin rods each of mass M and length t, form a square frame. Moment of inertia of this frame about an axis through the centre of the square and perpendicular to its plane is
(1)
(2)
(3)
(4)