System of particles practice · Mixed difficulty

Assertion: The velocity of a body at the bottom of an inclined plane of given height is more than when it slides down the plane, compared to, when it rolls down the same plane.
Reason: In rolling down, a body acquires, both kinetic energies of translational and rotational motions.

Which of the following statements are correct:

(a) Centre of mass of a body always coincides with the centre of gravity of the body.

(b) A couple on a body produce pure rotational motion in a body.

(c) A couple on a body produce both translational and rotational motion in a body.

(d) Mechanical advantage greater than one means that small effort can be used to lift a large load.

A solid sphere of mass $m$ rolls without slipping on a horizontal surface with a linear velocity $v.$ Then its total kinetic energy is

Choose correct statement:

A. The position of centre of mass of a system is dependent on the choice of coordinate system.

B. Newton's second law of motion is applicable to the centre of mass of the system.

C. Internal forces cannot change the state of centre of mass.

D. Internal forces can change the state of centre of mass.

If a hollow cylinder and a solid cylinder are allowed to roll down an inclined plane, which will take more time to reach the bottom

The times taken by a solid sphere, a solid cylinder a thin-walled hollow sphere and a thin-walled hollow cylinder all having the same mass to roll down an inclined plane when released at the top are denoted as $t_{s s}, t_{s c}, t_{h s}$, and ${t}_{{hc}}$ respectively. The following is true with regard to the roll down times.

An angular ring with inner and outer radii ${R_1}\,and\,{R_2}$ is roiling without slipping with a uniform angular speed. The ratio of the forces experienced by the two particles situated on the inner and outer parts of the ring. ${F_1}/{F_2}$ is

For a rigid body made of total $\mathrm{N}$ particles each of mass $\mathrm{m}$, the radius of gyration about a given axis equals to

A man standing on a platform holds weights in his outstretched arms. The system is rotated about a central vertical axis. If the man now pulls the weights inwards close to his body, then

When a body is projected at an angle with the horizontal in the uniform gravitational field of the earth, the angular momentum of the body about the point of projection, as it proceeds along the path

For the same total mass, which of the following will have the largest moment of inertia about an axis passing through the centre of mass and perpendicular to the plane of the body

Consider a uniform wire of mass $M$ and length $L$. It is bent into a semicircle. Its moment of inertia about a line perpendicular to the plane of the wire passing through the centre is

The shaft of a motor is making 1000 rpm. The torque supplied by the motor is 120 N m. The power supplied by the motor is

If all of a sudden the radius of the earth decreases, then

A parallel undergoes uniform circular motion. About which point on the plane of the circle, will the angular momentum of the particle remain conserved

A disc is rotating with angular speed $\omega$. If a child sits on it, what is conserved?

Statements:
a) In translatory motion each point of the body experiences the same displacement as any other point as time goes on, so that the motion of any particle represents the motion of the whole body.
b) In rotatory motion, all the particles in the rigid body moves in concentric circles about the axis of rotation.

There are two identical balls of same material, one being solid and the other being hollow. How will you distinguish them without weighing?

If applied torque on a system is zero, i.e., $\tau=0$, then for that system

Assertion (A) : $I_{\mathrm{s}}$ and $I_{\mathrm{H}}$ are the moments of inertia about the diameters of a solid and thin walled hollow sphere respectively. If the radii and the masses of the above spheres are equal, $l_{\mathrm{H}} > l_{\mathrm{s}}$.

Reason (R) : In solid sphere, the mass is continuously and regularly distributed about the centre whereas the mass, surface of hollow sphere.

A solid sphere of mass $m$ rolls down an inclined plane a height $h$. Find rotational kinetic energy of the sphere.

One circular ring and one circular disc, both are having the same mass and radius. The ratio of their moments of inertia about the axes passing through their centres and perpendicular to their planes, will be

Assertion: A rigid disc rolls without slipping on a
fixed rough horizontal surface with uniform
angular velocity. Then the acceleration of lowest
point on the disc is zero.

Reason: For a rigid disc rolling without slipping
on a fixed rough horizontal surface, the velocity of
the lowest point on the disc is always zero.

The centre of mass of system of particles does not depend on

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

A stone of mass $2 {~kg}$ is attached to one end of string of length $5 {~m}$, is revolved around a stick so that the string winds upon the stick and gets shortened. Angular velocity of stone was $5 \,{rad} / {s}$ initially. Find angular velocity of the stone when length of the string become $2 {~m}$.

Assertion: The position of centre of mass of a body does not depend upon shape and size of the body.
Reason: Centre of mass of a body lies always at the centre of the body

(A): Polar ice melts, day will be longer.
(R): Moment of inertia increases and thus angular velocity decreases.

The angular momentum of a system of particles is conserved

A 2 kg body and a 3 kg body are moving along the x-axis. At a particular instant the 2 kg body has a velocity of $3\;m{s^{ - 1}}$ and the 3 kg body has the velocity of $2m{s^{ - 1}}$. The velocity of the centre of mass at that instant is