System of particles practice · Mixed difficulty

In uniform circular motion

A body moves with constant angular velocity on a circle. Magnitude of angular acceleration

A spherical shell has mass $M$ and radius $R$. Moment of inertia about its diameter will be

A circular turn table has a block of ice placed at its centre. The system rotates with an angular speed $\omega$ about an axis passing through the centre of the table. If the ice melts on its own without any evaporation, the speed of rotation of the system

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

A solid sphere and hollow sphere of same mass and radius rotate about their diameter with same rotational kinetic energy. If torques of equal magnitude are applied on them, ratio of their angular displacement before they are bought to rest.

The moment of inertia of a uniform cylinder of length $l$ and radius $R$ about its perpendicular bisector is $I$. What is the ratio $l/R$ such that the moment of inertia is minimum?

The ratio of angular speeds of minute hand and hour hand of a watch is

Two discs of same moment of inertia rotating about
their regular axis passing through centre and
perpendicular to the plane of disc with angular
velocity 2 rad/s and 4 rad/s. They are brought into
contact face to face coinciding the axis of rotation.
The expression for loss of energy during this procss
is:

The surface density of a circular disc of radius a depends on the distance as $\rho(r)=A+B r$. The moment of inertia about the line perpendicular to the plane of the disc is

STATEMENT-1 : Two cylinders, one hollow (metal) and the other solid (wood) with the same mass and identical dimensions are simultaneously allowed to roll without slipping down an inclined plane from the same height. The hollow cylinder will reach the bottom of the inclined plane first.

STATEMENT-2 : By the principle of conservation of energy, the total kinetic energies of both the cylinders are identical when they reach the bottom of the incline.

Two homogeneous spheres A and B of masses m and 2m having radii 2a and a respectively are placed in touch. The distance of centre of mass from first sphere is

Assertion: At the centre of earth, a body has centre of mass, but no centre of gravity.
Reason: Acceleration due to gravity is zero at the centre of earth.

A force F = 10 N is acting on a point is at 4 m distance then torque is [N-m]

A thin rod of mass 6 m and length $6l$ is bent into regular hexagon. The moment of inertia of the hexagon about an axis normal to its plane and through the centre of mass of the system

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

When a sphere rolls down an inclined plane, then identity the correct statement related to the work done by friction force

A uniform heavy disc is rotating at constant angular velocity $\omega$ about a vertical axis through its centre and perpendicular to the plane of the disc. Let L be its angular momentum. A lump of plasticine is dropped vertically on the disc and sticks to it. Which will be constant

Where will be the centre of mass on combining two masses m and (M > m)

If the equation for the displacement of a particle moving on a circular path is given by $\left( \theta \right) = 2{t^3} + 0.5$, where $\theta$ is in radians and t in seconds, then the angular velocity of the particle after 2 sec from its start is

If we hollow the ball of pendulum and fill it with sand, what would be the effect of change on the center of mass?

A solid sphere of mass $2 \mathrm{~kg}$ is rolling on a frictional horizontal surface with velocity $6 \mathrm{~m} / \mathrm{s}$. It collides on the free end of an ideal spring whose other end in fixed. The maximum compression produced in the spring will be $\sqrt{x} \mathrm{~m}$. Find the value of $x$ (Force constant of the spring is $36 \mathrm{~N} / \mathrm{m}$ ).

A solid cylinder of mass ${M}$ and radius ${R}$ rolls down an inclined plane of height ${h}$. The angular velocity of the cylinder when it reaches the bottom of the plane is :

The motion of centre of mass depends on

Two particles of masses 1 kg and 2 kg are located at ${x_1} = 0,\;{y_1} = 0\,and\,{x_2} = 1,\;{y_2} = 0$ respectively. The centre of mass of the system is at

In the absence of external torque for a body revolving about any axis, the quantity that remains constant is

${~A}$ wheel has mass of the rim $1 {~kg}$, having 50 spokes each of mass $5 {~g}$. The radius of the wheel is $40 {~cm}$. The moment of inertia

Four particles of masses m,2m,3m and 4m are arranged at the corners of a parallelogram with each inside equal to a and one of the angle between two adjacent sides is ${60^0}$. The parallelogram lies in the x-y plane with mass m at the origin and 4m on the x-axis. The centre of mass of the arrangement will be located at

What are the coordinates of centre of mass of a point mass of $2 {~kg}$ with coordinates $(2,3)$ ?

The principle of conservation of angular momentum, states that angular momentum