System of particles practice · Mixed difficulty

A point mass of $10 \mathrm{~kg}$ is placed at the centre of earth. The weight of the point mass is

Assefion: Radius of gyration of body is a constant quantity.

Reason: The radius of gyration of a body about an axis of rotation may be defined as the root mean square distance of the particle from the axis of rotation.

In pure rolling motion of a ring
a) It rotates about Instantaneous point of contact of ring and ground
b) Its centre of mass moves in translatory motion only
c) Its centre of mass will have translatory as well as rotatory motion

Assertion: A judo fighter in order to throw his opponent on to the mat tries to initially bend his opponent and then rotate him around his hip.
Reason: As the mass of the opponent is brought closer to the fighter's hip, the force required to throw the opponent is reduced.

There are some passengers inside a stationary railway compartment. The centre of mass of the compartment itself (without the passengers) is $C_1,$ while the centre of mass of the 'compartment plus passengers' system is $C_2.$ If the passengers move about inside the compartment then

Two particles move towards each other from rest under a mutual force of attraction. At the instant, when the speed of A is V, the speed of B is V/2. The speed of centre of mass of the system is

Assertion (A): A disc is rolling on a rough horizontal surface without slipping. The velocity of centre of mass is $v .$ Then all the other points on the disc lying on a circular arc with point of contact as the center and this arc passing through center of mass of disc will have the same speed $v.$
Reason (R): When a disc is rotating without sliding on a rough horizontal surface the magnitude of velocities of all the points at a distance $r$ from point of contact is same.

If the earth is a point mass of $6 \times {10^{24}}$kg revolving around the sun at distance of $1.5 \times {10^8}$km and in time T = $3.14 \times {10^7}$ seconds, then the angular momentum of the earth around the sun is [in $kg - {m^2}$ / s]

Assertion: The centre of mass of a body may lie where there is no mass.
Reason: The centre of mass does not depend upon the mass of the body.

(A): The spin angular velcoity of a star is greatly enhanced when it collapse under gravitational pull and become a neutron star.
(R): According to law of conservation of angular momentum there is increase in angular velocity of collapsing star.

Mass is non-uniformly distributed on the circumference of a ring of radius $a$ and centre at origin. Let $b$ be the distance of the centre of mass of the ring from origin. Then

A disc rotates about its axis with a constant angular acceleration of $4 \mathrm{~rad} / \mathrm{s}^2.$ Find the radial and tangential accelerations of a particle at a distance of $1 \mathrm{~cm}$ from the axis at the end of the first second after the disc starts rotating.

A ring of mass 10 kg and diameter 0.4 m is rotated about its axis. If it makes 2100 revolutions per minute, then its angular momentum will be

If torque is zero, then

The angular displacement of a particle is given by $\theta = 4{t^3} + 3{t^2} + 2t + 1$ where ‘t’ is time in seconds. Its angular velocity at $t = 2\,\sec$ is [in rad/sec]

One solid sphere A and another hallow sphere B are of same mass and same outer radii. Their moment of inertia about their diameters are respectively ${I_A}$ and ${I_B}$ such that

A solid sphere is rotating freely about its symmetry axis in free space, The radius of the sphere is increased keeping
its mass same Which of the following physical quantities would remain constant for the sphere ?

The ratio of moments of inertia of a solid sphere about axes passing through its centre and tangent respectively is

The moment of inertia of semicircular ring about an axis which is perpendicular to the plane of the ring and passes through the centre

A square plate is kept in yz-plane. Then according to perpendicular axis theorem

Assertion (A): A body rolling without slipping has only rotational kinetic energy.
Reason (R): The centre of mass of a rolling body docs not move forward.

The radius of gyration of a uniform rod of length L about on axis passing through its centre of mass and perpendicular to its length is

Two particles of masses ${m_1}\,and\,{m_2}$ initially at rest start moving towards each other under their mutual force of attraction. The speed of the centre of mass at any time $t$, when they are at a distance $r$ apart, is

A circular table rotates about a vertical axis with a constant angular speed $\omega$. A circular pan rests on the turn table (with the centre coinciding with centre of table) and rotates with the table. The bottom of the pan is covered with a uniform small thick layer of ice placed at centre of pan. The ice starts melting. The angular speed of the turn table

Assertion (A): A ball connected to a string is in circular motion on a frictionless horizontal table and is in equilibrium.

Reason (R) : Magnitude of the centripetal force is equal to the magnitude of the tension in the string.

The correct answer is

Statement I: Total torque on a system is independent of
the origin if the total external force is zero.
Statement II: Torque due to a couple is independent of
the origin.

Identify the correct statement for the rotational motion of a rigid body.

Assertion: If the ice on the polar caps of the earth melts, then length of day will increase.

Reason: Moment of inertia of the earth increases as ice on polar caps melts.

A solid sphere of mass 2 kg and radius 5 cm rolls without shipping along a horizontal plane. The velocity of its centre of mass is $10\,\,cm\,\,\,{s^{ - 1}}.$ The kinetic energy of the sphere is

A solid sphere rolls down without slipping on an inclined plane at angle ${60^0}$ over a distance of 10 m. The acceleration (in $\mathrm{m}{\mathrm{s}^{ - 2}}$) is