System of particles practice · Mixed difficulty

(A): When couple acting on a body moment of the couple does not depends on the point about which moments are measured.
(R): When a couple acts on a body it is in translational equilibrium.

A couple produces

Assertion: When a body dropped from a height explodes in mid air, its centre of mass keeps moving in vertically downward direction.
Reason: Explosion occur under internal forces only. External force is zero.

A diver having a moment of inertia of $6.0 \mathrm{~kg}-\mathrm{m}^2$ about an axis through its centre of mass rotates at angular speed of $2 \mathrm{~rad} / \mathrm{s}$ about this axis. If he folds his hands and feet to decrease the moment of inertia to $5.0 \mathrm{~kg}-\mathrm{m}^2,$ what will be the new angular speed?

Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities ${\omega _1}\mathrm{\; and\; }{\omega _2}.$ They are brought into contact face to face coinciding the axis of rotation. The expression for loss of energy during this process is

Moment of inertia (MI) of four bodies, having same mass and radius, are reported as ${I_1} = MI$ of thin circular ring about its diameter,
${I_2} = MI$ of circular disk about an axis perpendicular to the disk and going through the centre,
${I_3} = MI$ of solid cylinder about its axis and ${I_4} = MI$ of solid sphere about its diameter. Then,

If an object starts from rest and covers angle of $60 \mathrm{~rad}$ in $10 \mathrm{~s}$ in circular motion, then magnitude of its angular acceleration will be

A star of mass twice the solar mass and radius $10^6 \mathrm{~km}$ rotates about its axis with an angular speed of $10^{-6} \mathrm{~rad} \mathrm{~s}^{-1}$. What is the angular speed of the star when it collapses (due to inward gravitational force) to a radius of $10^4 \mathrm{~km}$? Solar mass $1.99 \times 10^{30} \mathrm{~kg}$.

The angular momentum of a system of particles is not conserved

A particle is moving on a circle of radius $R$ such that at every instant the tangential and radial accelerations are equal in magnitude. If the velocity of the particle be ${v}_0$ at ${t}=0$, the time for the completion of half of first revolution will be

What is the moment of inertia of solid sphere of density $\rho$ and radius R about its diameter?

Assertion: The condition of equilibrium for a rigid body is- Translational equilibrium : $\sum \overrightarrow{\mathrm{F}}=0$, (i.e. sum of all external forces equal to zero.) Rotational equilibrium: $\sum \vec{\tau}=0$, (i.e. sum of all external forces equal to zero.)
Reason: A rigid body must be in equilibrium under the action of two equal and opposite forces.

Assertion: The centre of mass of a system of two particles of unequal masses is closer to the heavier particle.
Reason: The centre of mass of a system of two particles divides the distance between the particles in the inverse ratio of their masses.

The centre of mass of a body

Two racing cars of masses ${m_1}\,and\,{m_2}$ are moving in circles of radii ${r_1}\,and\,{r_2}$ respectively. Their speeds are such that each makes a complete circle in the same duration of time t.The ratio of the angular speed of the first to the second car is

(A): The hard boiled egg and raw egg can be distinguished on the basis of spinning of both.
(R): The moment of inertia of hard boiled egg is more as compare to raw egg.

If the torque of the rotational motion by zero,
then the constant quantity will be

A solid sphere, hollow sphere, solid cylinder, hollow cylinder and ring each of mass $M$ and radius $R$ are simultaneously released at rest from top of incline and allowed to roll down the incline. The angular acceleration is maximum for

What is the angular velocity of earth

The moment of inertia $(I)$ of a sphere of radius $R$ and mass $M$ is given by

The moment of inertia of a circular ring of radius $r$ and mass $M$ about diameter is

Two spheres, one solid and another hollow are made up of same material and are of same radius. Both the spheres are heated to same temperature $\mathrm{T}_0$. Now, both spheres are left to cool in an atmosphere having temperature $\mathrm{T_A}\left(\mathrm{T_A \ll T_0}\right)$. After time $\mathrm{t}$ the temperature of sphere becomes $\mathrm{T}_1$ and that of hollow sphere becomes $\mathrm{T}_2$. Then,

A body of mass $m$ slides down an incline and reaches the bottom with a velocity $v$. If the same mass were in the form of a ring which rolls down this incline, the velocity of the ring at bottom would have been

Assertion: For a body to be in equilibrium it must satisfy both the conditions stated simultaneously.
Reason: (a) the vector sum of all the external forces acting on the body should vanish.
(b) the vector sum of all the external torque acting on the body should vanish.

If $\omega_{\mathrm{E}}$ and $\omega_{\mathrm{H}}$ are the angular velocities of the earth rotating about its own axis and the hour hand of the clock respectively, then

Consider the following statements:

If the net external torque acting on an object is switched off, then

1. linear momentum will remain unchanged

2. angular momentum will remain unchanged

Which of the statements given above is/are correct?

A loaded spring gun of mass $M$ fires a bullet of mass $m$ with a velocity $v$ at an angle of elevation $\theta$. The gun is initially at rest on a horizontal frictionless surface. After firing, the center of mass of the gun-bullet system

Assertion: When a sphere rolls down an inclined plane without slipping, its kinetic energy is partly translational and partly rotational in nature.
Reason: When a sphere rolls down an inclined plane without slipping, it possesses both translational and rotational motions.

A constant torque acting on a uniform circular disc changes its angular momentum from L to $\frac{{4L}}{3}$ in 2 seconds. Then magnitude of the torque applied is

A diver in a swimming pool bends his head before diving, because it