Rotational Motion

A small mass m is attached to a massless string whose other end is fixed at P as shown in the figure. The mass is undergoing circular motion in the xy-plane with centre at O and constant angular speed \(\omega\). If the angular momentum of ...

Four solid spheres each of diameter \(\sqrt 5\) cm and mass 0.5 kg are placed with their centers at the corners of
a square of side 4 cm. The moment of inertia of the system about the diagonal of the square is N \(\times\) 10−4
kg-m2, t...

A boy is pushing a ring of mass 2 kg and radius 0.5 m with a stick as shown in the figure. The stick applies a force of 2 N on the ring and rolls it without slipping with an acceleration of 0.2 m/s2. The coefficient of friction between the ...

A metal rod of length L and mass m is pivoted at one end. A thin disk of mass M and radius R ( < L) is attached at its centre to the free end of the rod. Consider two ways the disc is attached : (case A). The disc is not free to rotate abou...

A sphere is rolling without slipping on a fixed horizontal plane surface. In the figure below, A is the point of contact, B is the centre of the sphere and C is its topmost point. Then,

If the resultant of all the external forces acting on a system of particles is zero, then from an inertial frame, one can surely say that

The maximum value of V\(_0\) for which the disk will roll without slipping is:

The centre of mass of the disk undergoes simple harmonic motion with angular frequency \(\omega\) equal to:

The net external force acting on the disk when its centre of mass is at displacement x with respect to its equilibrium position is :

STATEMENT - 1 :
Two cylinders, one hollow (metal) and the other solid (wood) with the same mass and identical dimensions are simultaneously allowed to roll without slipping down an inclined plane from the same height. The hollow cylinder wi...

STATEMENT 1
If there is no external torque on a body about its center of mass, then the velocity of the center of mass remains constant.
Because
STATEMENT 2
The linear momentum of an isolated system remains constant.

A small object of uniform density rolls up a curved surface with an initial velocity \(v\). It reaches up to a maximum height of \(\frac{3 v^{2}}{4 g}\) with respect to the initial position. The object is

The loss of kinetic energy during the above process is :

When disc B is brought in contact with disc A, they acquire a common angular velocity in time t. The average frictional torque on one disc by the other during this period is

The ratio \({{{x_1}} \over {{x_2}}}\) is

A solid cylinder of mass m and radius $r$ is rolling on rough inclined plane of inclination $\theta$. The coefficient of friction between the cylinder and incline is $\mu$. then

A solid sphere of radius $R$ has moment of inertia $I$ about its geometrical axis. If it is melted into a disc of radius $r$ and thickness $t$. If its moment of inertia about the tangential axis (which is perpendicular to plane of the disc)...

Two identical ladders, each of mass M and
length L are resting on the rough horizontal
surface as shown in the figure. A block of
mass \(m\) hangs from P. If the system is in
equilibrium, find the magnitude and the
direction of frictio...

A cylinder of mass m and radius R rolls
down on an inclined plane of inclination \(\theta\).
Calculate the linear acceleration of axis of
cylinder.

A wooden log of mass M and length L is
hinged by a frictionless nail at O; a bullet
of mass m strikes with velocity \(v\) and sticks
to it. Find angular velocity of the system
immediately after the collision about O.

One quarter section is cut from a uniform circular disc of radius $R$. This section has a mass $M$. It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc. Its moment of inertia abou...

A thin wire of length $L$ and uniform linear mass density $\rho$ is bent into a circular loop with centre at $O$ as shown. The moment of inertia of the loop about the axis $XX'$ is: