System of particles practice · Medium

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

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

Assertion (A): If two different axes are at same distance from the centre of mass of a rigid body then moment of inertia of the given rigid body about both the axes will always be equal.
Reason (R): According to perpendicular axis theorem $\mathrm{I}=\mathrm{I}_{\mathrm{cm}}+\mathrm{Md}^2$ where symbols have their usual meaning.

A sphere of mass m and radius r rolls on a horizontal plane without slipping with the speed u. Now, if it rolls up vertically, the maximum height it would attain will be

Assertion: A shell at rest, explodes. The centre of mass of fragments moves along a straight path.
Reason: In explosion the linear momentum of the system is not conserved.

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.

Very thin ring of radius R is rotated about its centre. It’s radius will

A T shape with dimensions shown in Figure is lying on a smooth floor. A force $\vec F$ is applied at the point $P$ parallel to $AB$, such that the object has only the translation motion without rotation. Find the location of $P$ with respect to $C$

If the torque is zero, what will be the value of angular momentum?

Assertion: The centre of mass of an electron and proton, when released moves faster towards proton.
Reason: Proton is lighter than electron.

Which relation is not correct of the following?

Two balls are thrown simultaneously in air. The acceleration of the centre of mass of the two balls in air

When a motor cyclist takes a $U$-turn in 4 s what is the average angular velocity of the motor cyclist.

Three identical metal balls, each of radius $r$ are placed touching each other on a horizontal surface, such that an equilateral triangle is formed, when centres of three balls are joined. The centre of mass of the system is located at:

The coordinates of the centre of mass of a system of three particles of masses $1 \mathrm{~mg}, 2 \mathrm{~mg}$ and $3 \mathrm{~mg}$ are $(2,2,2)$. The coordinates of the fourth particle of mass $4 \mathrm{~mg}$ to be positioned so that the centre of mass of the four particles system is at the origin of the three-dimensional rectangular coordinate system are

The moment of inertia of a body does not depend on

A fan rotating with $\omega=100 \mathrm{rad} / \mathrm{s}$, is switched off. After $2 \mathrm{n}$ rotation its angular velocity reduces to $50 \mathrm{rad} / \mathrm{s}$. Find out the angular velocity of the fan after $n$ rotations.

A thin hollow cylinder is free to rotate about its geometrical axis. It has a mass of $8 \mathrm{~kg}$ and a radius of 20 $\mathrm{cm}$. A rope is wrapped around the cylinder. What force must be exerted along the rope to produce an angular acceleration of $3\, \mathrm{rad} / \mathrm{s}^2$ ?

If the external forces acting on a system have zero resultant, the center of mass

A flywheel of mass $2 \mathrm{~kg}$ has radius of gyration $0.5 \mathrm{~m}$. If it makes 10 r.p.s. then its rotational kinetic energy will be

A particle of mass m moving with a velocity $\left( {3\hat i + 2\hat j} \right)m{s^{ - 1}}$ collides with a stationary body mass M and finally moves with a velocity $\left( { - 2\hat i + \hat j} \right)\mathrm{m}{\mathrm{s}^{ - 1}}.\,If\,\frac{m}{M} = \frac{1}{{13}},$ then

M.I of a solid sphere about its diameter is $64\,kg\,{m^2}$. If that sphere is recast into 8 identical small spheres then M.I. of such small sphere about its diameter is

Find the angular velocity of a body rotating with an acceleration of $2 \mathrm{~rev} / \mathrm{s}^2$ as it completes the 5th revolution after the start.

A flywheel rotating about an axis experiences an angular retardation proportional to the angle through which it rotates. If itsrotational kinetic energy gets reduced by $\Delta {E}$ while it rotates through an angle $\theta$ then

A wheel rotating with uniform angular acceleration
covers 50 revolutions in the first five seconds after the
start. Find the angular acceleration and the angular
velocity at the end of five seconds.

A rigid system consists of two point masses, $A$ and $B$ of masses $1 \mathrm{~kg}$ and $2 \mathrm{~kg}$ respectively. At an instant the kinetic energy of $A$ and $B$ with respect to the centre of mass is 2 Joule and 1 Joule respectively. The velocity of centre of mass is $2 \mathrm{~m} / \mathrm{s}$. The kinetic energy of the system with respect to ground at this instant is: