System of particles practice · Mixed difficulty

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.

A body rolls down an inclined plane without slipping. The kinetic energy of rotation is 50% of its translational kinetic energy. The body is

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:

A rod of mass $5 \mathrm{~kg}$ is connected to the string at point $B$. The span of rod is along horizontal. The other end of the rod is hinged at point $A$. If the string is massless, then the reaction of hinge at the instant when string is cut, is (Take, $g=10 \mathrm{~ms}^{-2}$ )

Assertion: The centre of mass of a proton and an electron, released from their respective positions remains at rest.
Reason: The centre of mass remain at rest, if no-external force is applied.

A solid sphere of radius ${R}$ is placed on a smooth horizontal surface. A horizontal force $F$ is applied at height $h$ from the lowest point. For the maximum acceleration of centre of mass, which is correct?

Assertion: For the purpose of calculation of moment of inertia, body's mass can be assumed to be concentrated at its centre of mass.
Reason: Moment of inertia of rigid body about an axis passing through its centre of mass is zero.

A uniform rod of mass $m$ and length $l$ is struck at an end by a force $F$ perpendicular to the rod for a short time interval $t$. Calculate

(a) the speed of the centre of mass
(b) the angular speed of the rod about the centre of mass

Assume that $t$ is so small that the rod does not appreciably change its direction while the force acts.