System of particles practice · Mixed difficulty

A flywheel having moment of inertia $10 \mathrm{~kg} . \mathrm{m}^2$ is rotating at $50 \mathrm{~rad.s}^{-1}$. What amount of work needs to be done in order to bring this flywheel to rest in 10 seconds?

In free space, a shell moving with velocity $60\,m/s$ along the positive x-axis. When it passes through origin, it explodes into two pieces of mass ratio 1: 2. After the explosion, the velocity of the centre of mass is:-

A particle with position vector $r$ has a linear momentum $p$. Which of the following statements is true in respect of its angular momentum L about the origin

A rigid body of mass $m$ rotates about an axis passing through its centre. If the angular velocity of the body is unity, the moment of inertia of the body will be equal to

A ball kept in a closed box moves in the box making collisions with the walls. The box is kept
on a smooth surface. The center of mass

Find the position of centre of mass from the base of a uniform solid cone of height $20 \mathrm{~cm}$.

Statement-I: A coin is placed on phonogram turn table. The motor is started, coin moves along the moving table.
Statement-II: The rotating table is providing necessary centripetal force to the coin.

Assertion: As star collapse its angular velocity increases.
Reason: The mass of star decreases.

An astronaut is rotated in a horizontal circle at a radius of $5 \mathrm{~m}$. If the astronaut can withstand a maximum acceleration of $7 \mathrm{~g}$, then maximum numbers of permissible revolution per second is $\left(1 \mathrm{~g}=10 \mathrm{~m} / \mathrm{s}^2\right)$.

A shaft rotating at 6000 rpm is transmitting a power of $2\pi$ kilowatt. Then magnitude of the driving torque is

When an external force is applied at the centre of mass of a rigid body at rest, then it undergoes

Assertion: Location of centre of mass is independent of the reference frame.
Reason: Centre of mass is same as centre of gravity.

A horizontal platform is rotating with uniform angular velocity around the vertical axis passing through its centre. At some instant of time a viscous fluid of mass ‘m’ is dropped at the centre and is allowed to spread out and finally fall. The angular velocity during this period

A solid cylinder of radius $4 \mathrm{~cm}$ and mass $250 \mathrm{~g}$ rolls down an inclined plane (1 in 10). Calculate the acceleration and the total energy of the cylinder after $5 \mathrm{~s}$.

The radius of gyration of disc of mass and radius $R$ about an axis which is at distance $2 R$ from its centre and in the plane of the disc will be

A particle of mass 1 kg is projected with an initial velocity 10 $\mathrm{m}{\mathrm{s}^{ - 1}}$  at an angle of projection ${45^0}$ with the horizontal. The average torque acting on the projectile, between the time at which it is projected and the time at which it strikes the ground, about the point of projection in newton-metre is

Two equal and opposite forces act on a rigid body at a certain distance. Then

A projectile of mass $1 \mathrm{~kg}$ is projected with a speed of $10 \mathrm{~m} / \mathrm{s}$ at an angle of $60^{\circ}$ from the horizontal when projectile is at its highest point, its magnitude of angular momentum (about point of projection)

A car wheel is rotated to uniform angular acceleration about its axis. Initially its angular velocity is zero. It rotates through an angle ${\mathrm{\theta }_1}$ in the first 2 s , in the next 2 s, it rotates through an additional angle ${\mathrm{\theta }_2}$, the ratio of $\frac{{{\mathrm{\theta }_2}}}{{{\mathrm{\theta }_1}}}$ is

Assume the earth's orbit around the sun as circular and the distance between their centre as $\mathrm{D}$. Mass of the earth is $\mathrm{M}$ and its radius is $\mathrm{R}$. If earth has an angular velocity $\omega_0$ with respect to its centre and $\omega$ with respect to the centre of the sun, the total kinetic energy of earth is

STATEMENT-1: If there is no external torque on a body about its center of mass, then the velocity of the center of mass remains constant.

STATEMENT-2: The linear momentum of an isolated system remains constant.

Two uniform rods $A$ and $B$ of lengths $5 {~m}$ and $3{~m}$ are placed end to end. If their linear densities are $3 {~kg} / {m}$ and $2 {~kg} / {m}$, the position of their centre of mass from their interface is

A flywheel rotates with uniform angular acceleration of $2 \pi {rad} / {s}^2$. Its initial angular velocity is $20 \pi {rad} / {s}$. What are the total number of rotations made in $20 {~s}$ ?

Statement-1: If the centre of mass of a system is the origin then the total mass to the right of origin is same as total mass to left of origin.
Statement-2: $x_{\mathrm{cm}}=\frac{\Sigma_{m_i x_i}}{\Sigma_{m_i}}$

(A): If body or a system has neither translational nor rotational motion then it is in complete mechanical equilibrium.
(R): A body or system is said to be in mechanical equilibrium when net force and net torque acting on the body are zero.

Given that force $(5 \hat{i}+7 \hat{j}-3 \hat{k}) \mathrm{N}$ acts on a particle at position $(\hat{i}+\hat{j}-\hat{k}) \mathrm{m}$. Find torque of this force on the particle about origin.

Radius of gyration of a thin rod of mass 1 kg about an axis perpendicular to its length and passing through its centre is $\sqrt 3 \,m$ . Then length of the rod is

A non-zero external force acts on a systems of particles. The velocity and acceleration of the centre of mass are found to be ${v_0}\,and\,{a_0}$ at an instant t. It is possible that

Two particles of equal mass have velocities ${v_1} = 4\hat i\,and\,{v_2} = 4\hat j\,m{s^{ - 1}}$. First particle has an acceleration ${a_1} = \left( {5\hat i + 5\hat j} \right)m{s^{ - 2}}$ , while the acceleration of the other particle is zero. The centre of mass of the two particles moves in a path of

When no external torque acts on a rotating system,