System of particles practice · Mixed difficulty

A cylinder of mass $m$ and radius $R$ rolls purely over an inclined surface while moving up on it. Correct statement regarding friction acting on it is

The motion of a rigid body which is not pivoted or fixed in some way is either a pure ___A___ or a combination of translation and rotation. The motion of a rigid body which is pivoted or fixed in some way is ___B___ Here, A and B refer to:

A solid cylinder of mass 4 kg and radius 8 cm is rotating about its axis at the rate of 6 rpm. The torque required to stop after $4 \pi$ revolutions is:

A circular plate is rotating about its own axis at an angular velocity 100 revolution per minute. The linear velocity of a particle ‘P’ of plate at a distance 4.2 cm from axis of rotation (in m/s) is

If a rigid body is rotating about an axis with a
constant velocity, then

Moment of inertia of a circular loop of radius R about the axis of rotation parallel to horizontal diameter at a distance R/2 from it is

Two uniform solid spheres having unequal masses and unequal radii are released from rest from the same height on a rough incline. If the spheres roll without slipping,

A man is standing at the edge of a circular plate which is rotating with a constant angular speed about a perpendicular axis passing through the centre. If the man walks towards the axis along the radius, its angular velocity

A disc and a hoop (ring) of the same mass and size roll down an inclined plane simultaneously. The object which reaches the bottom of the incline first is

If solid sphere and solid cylinder of same radius and density about their own axis, the moment of inertia will be greater for (L = R)

S-I: Radius of gyration of body is a variable quantity.
S-II: The radius of gyration of a body about an axis of rotation may be defined as the root mean square distance of the particles from the axis of rotation.

A gymnast takes turns with her arms & legs stretched. When she pulls her arms and legs in

Which of the following relation is correct? (Where symbol have their usual meanings)

S-I: In rolling, all points of a rigid body have the same linear speed.
S-II: The rotational motion does not affect the linear velocity of rigid body.

Consider the following two statements

a. Among a given set of axis for a rigid body, the moment of inertia is always minimum about the one passing through the center of mass.

b. Perpendicular axis theorem for moment of inertia would be applicable for a solid sphere.

The correct statement(s) is/are

A body is rotating with angular velocity 20 rad/s. If
its kinetic energy is 400 J, then its moment of
inertia is

If a solid sphere is rolling the ratio of its rotational energy to the total kinetic energy is given by :-

The ratio of rotational and translatory kinetic energies of a sphere is

The torque of a force $\mathbf{F}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ acting at a point whose position vector $\mathbf{r}=3 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ about the origin is

The angular speed of a body changes from $\omega_1$ to $\omega_2$ without applying a torque but due to change in its moment of inertia. The ratio of radii of gyration in the two cases is

Statement-I: Gravitational torque due to weight of a body about its centre of gravity is zero.

Statement-II: In the absence of external torque the angular momentum of a body changes.

Two masses of 6 and 2 units are at positions $(6 \hat{{i}}-7 \hat{{j}})$ and $(2 \hat{{i}}+{5} \hat{{j}}-{8} \hat{{k}})$ respectively. The co-ordinates of the centre-of-mass (c.m.) are

Assertion: When a body rolls down an inclined plane it has both kinetic and potential energy.
Reason: The rolling body follow the conservation law of mechanical energy.

A solid body rotates about a stationary axis with an angular retardation $\alpha=k \sqrt{\omega}$ where $\omega$ is the angular velocity of body. Find time after which body will come to rest if at $t=0$ angular velocity of body was $\omega_0$.

A thick straight wire of length $4\pi \,m\,$ is fixed at its midpoint and the bent in the form of a circle. The shift in its centre of mass is

Out of the following bodies of same mass, which one will have maximum moment of inertia about an axis passing through its center of gravity and perpendicular to its plane?

The moment of inertia of a uniform circular disc of radius $R$ and mass $M$ about an axis passing from the edge of the disc and normal to the disc is:

A uniform rod is placed vertically on a smooth surface and then released then,

A rigid body rotates about a fixed axis such that its angular velocity $(\omega)$ depends on $\theta$ as $\omega=\frac{k}{\theta}$, where $k$ is a positive constant. At $t=0, \theta=\pi$, then the $\theta$ is related to time $(t)$ as

(A): The total kinetic energy of rolling solid sphere is the sum of translational and rotational kinetic energies.
(R): For all solid bodies total kinetic cnergy is always twice translational kinetic energy.