System of particles practice · Mixed difficulty

The total linear momentum of a system of particle about the centre of mass is

S-I: In rolling, all points of a rigid body have the same linear speed.
S-II: The rotational motion does not affect the linear velocity of rigid body.

A body falling vertically downwards under gravity breaks in two parts of unequal masses. The centre of mass of the two parts taken together shifts horizontally towards

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

A mass is revolving in a circle which is in the plane of paper. The direction of angular acceleration is

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

A solid cylinder starts rolling from a height ${h}$ on an inclined plane. At some instant $t$, the ratio of its rotational K.E. and the total K.E. would be:

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 sphere of mass 10kg and radius 1m rotates about a tangent. The moment of inertia of the solid sphere is

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.

What happens to the centripetal acceleration of a particle, when its speed is doubled and angular velocity is halved?

A point mass $(m)$ is located at $(0, a)$. Where should the $2 m$ mass be kept, so that centre of mass lies at origin?

A boat of mass 20 kg is at rest. A dog of mass 1 kg in the boat with a velocity of $5\,m{s^{ - 1}}$. What is the velocity of boat (nearly)?

A particle is under uniform circular motion with angular momentum L. If its kinetic energy is tripled and frequency of motion is halved then new angular momentum becomes

Centre of mass for a uniform semi-circular ring will be at (from centre)

From a circular disc of radius R and mass 4m, a small disc of radius $\frac{R}{2}$ is removed concentrically. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through its centre is

A uniform rod of length $\mathrm{L}=60 \mathrm{~cm}$ is suspended through an end and is set into oscillation with small amplitude under gravity. Find the time period of oscillation. $\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\right)$

Moment of inertia of an object does not depend upon

If linear density of a rod of length 3m varies as $\lambda = 2 + x$, then the position of the centre of gravity of the rod is

A body of mass ‘M’ is moving on a circular track of radius ‘r’ in such a way that its kinetic energy ‘k’ depends on the distance travelled by the body ‘s’ according to relation $k = \beta s$, where $\beta$ is a constant. The angular acceleration of the particle is

A wheel of mass 16 kg has a moment of inertia of 160 kg-m² about its own axis, the radius of gyration will be

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

Position vector of centre of mass of two particles system is given by:

A thin uniform disc of mass $M$ and radius $R$ is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity $\omega$. Another disc of same dimension but mass $\frac{M}{2}$ is placed gently on the first disc co-axially. The angular velocity of the system now is

In the equation of angular displacement of a particle moving on a circular path is given as $\theta = 2{t^3} + 0.5,\theta$ is in radian and t in second. The angular velocity of the particle at t=2s is

A metre stick is placed vertically at the origin on a frictionless surface. A gentle push in $+x$-direction is given to the top most point of the rod. When it has fallen completely, the $x$-coordinate of centre of rod is at

A uniform disc of mass 2 kg and radius 15 cm is revolving around the central axis by 4 $\mathrm{rad}{\mathrm{s}^{ - 1}}$. The linear momentum of disc will be

A large tray of mass ${M}$ has in it a cubical block of ice of mass ${m}$ and edge length ${L}$. The ice starts melting. The shift in the height of centre of mass of system (ice + tray) is

The correct relation between moment of inertial $I$, radius of gyration $K$ and mass $M$ of the body is:

What is moment of inertia in terms of angular momentum (L) and kinetic energy (K)?