System of particles practice · Medium

A rod of mass $5 \mathrm{~kg}$ is connected to the string at point $B$. The span of rod is along horizontal. The other end of the rod is hinged at point $A$. If the string is massless, then the reaction of hinge at the instant when string is cut, is (Take, $g=10 \mathrm{~ms}^{-2}$ )

Assertion: The centre of mass of a proton and an electron, released from their respective positions remains at rest.
Reason: The centre of mass remain at rest, if no-external force is applied.

A solid sphere of radius ${R}$ is placed on a smooth horizontal surface. A horizontal force $F$ is applied at height $h$ from the lowest point. For the maximum acceleration of centre of mass, which is correct?

Assertion: For the purpose of calculation of moment of inertia, body's mass can be assumed to be concentrated at its centre of mass.
Reason: Moment of inertia of rigid body about an axis passing through its centre of mass is zero.

A uniform rod of mass $m$ and length $l$ is struck at an end by a force $F$ perpendicular to the rod for a short time interval $t$. Calculate

(a) the speed of the centre of mass
(b) the angular speed of the rod about the centre of mass

Assume that $t$ is so small that the rod does not appreciably change its direction while the force acts.

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.

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

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

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.

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.

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

A thin hollow cylinder open at both ends.

(i) Sliding without rolling

(ii) Rolls without slipping, with the same speed.

The ratio of kinetic energy in the two cases is

A wheel of moment of inertia $5 \times {10^{ - 3}}kg - {m^2}$ is making 20 revolutions/sec. The torque required to stop it in 10 sec is