System of particles practice · Mixed difficulty

Assertion (A): Angular velocity of the seconds hand of a watch is $\frac{\pi}{30} \mathrm{rad} / \mathrm{s}.$
Reason (R): Angular velocity is equal to $\frac{2 \pi}{T}$ where $T$ is the time period.

A constant torque is acting on a wheel. If starting from rest, the wheel makes $n$ rotations in $t$ second, the angular acceleration $\alpha=$

Assertion: Angular velocity of moon revolving about earth is more than angular velocity of earth revolving around sun.
Reason: Time taken by moon to revolve around earth is less than time taken by earth to revolve around sun.

A sphere of mass $2 \mathrm{~kg}$ and radius $5 \mathrm{~cm}$ rolls without slipping along a horizontal plane. The velocity of its centre of mass is $10 \mathrm{~cm} / \mathrm{s}$. The kinetic energy of the sphere is

Two rigid bodies A and B rotate with rotational kinetic energies ${E_A}\,and\,{E_B}$ respectively. The moments of inertia of A and B about the axis of rotation are ${I_A}\,and\,{I_B}$ respectively. If ${I_A} = {I_B}/4\,and\,{E_A} = 100\;{E_B}$ the ratio of angular momentum $\left( {{L_A}} \right)$ of A to the angular momentum $\left( {{L_B}} \right)$ of B is

The velocities of three particles of masses $20 \mathrm{~g}$, $30 \mathrm{~g}$ and $50 \mathrm{~g}$ are $10 \hat{{i}}, 10 \hat{{j}}$ and $10 \hat{{k}}$ respectively. The velocity of the centre of mass of the three particles is

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

S-I: Motion of centre of mass depends on external forces applied on it
S-II: The motion of the centre of mass of the body is called the translational motion of the body

The centre of mass of a system cannot change its state of motion, unless there is external force acting on it. Yet the internal force of the brakes can bring a car to rest. Then

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

I. Angular momentum of a particle moving in a straight line is always constant with respect to fixed point.

II. Moment of inertia of a body remains same irrespective of position of axis of rotation.

Which of the statement(s) is/are correct?

The angular velocity of a particle is given by $\omega = {t^4} + 2{t^3}$ then its angular acceleration of the particle at $t = 1\,sec$ is $\left( {rad\,{s^{ - 2}}} \right)$

If the rotational kinetic energy of a body is increased by $300 \%$ then the percentage increase in its angular momentum :-

The angular momentum of planet of mass ${M}$ moving around the sun in as elliptical orbit is $\vec{L}$. Then torque acting on the planet is

Two particles of masses 4 kg and 6 kg are separated by a distance of 20 m and are moving towards each other under mutual force of attraction, the position of the point where they meet is

A uniform $\operatorname{rod} A B$ of length $2 l$ and mass $m$ is rotating in a horizontal plane about a vertical axis through $A$, with angular velocity $\omega$, when the mid-point of the rod strikes a fixed nail and is brought immediately to rest. Find the impulse exerted by the nail.

A body rotating at $20 \mathrm{~rad} / \mathrm{s}$ is acted upon by a constant torque providing it a deceleration of $2 \mathrm{~rad} / \mathrm{s}^2$. At what time will the body have kinetic energy same as the initial value if the torque continues to act?

A uniform circular disc of mass $200 \mathrm{~g}$ and radius $4.0 \mathrm{~cm}$ is rotated about one of its diameter at an angular speed of $20 \mathrm{rad} / \mathrm{s}$. Find its angular momentum ( in $\mathrm{kg}-\mathrm{m} / \mathrm{s}$ ) about the axis of rotation.

At any instant, a rolling body may be considered to be in pure rotation about an axis through the point of contact. This axis is translating forward with speed

A transistor has an $\alpha = 0.95,\,$ then $\beta$ is equal to

A wheel has angular acceleration of $6\,rad/{\sec ^2}$ and an initial angular speed of 4 rad/sec. In a time of 4 sec it has rotated through an angle (in radian) of

A child is standing at one end of a long trolley moving with a speed $v$ on a smooth horizontal
track. If the child starts running towards the other end of the trolley with a speed $u,$ the centre of
mass of the system (trolley + child) will move with a speed

Moment of inertia of an object does not depend upon

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

For a hollow cylinder and a solid cylinder rolling without slipping on an inclined plane, then which of these reaches earlier

Statements:
a) Position of C.m depends on acceleration due to gravity where as position of C.g independent on acceleration due to gravity.
b) Centre of gravity of a body is defined to know the amount of stability of the body and the center of mass of the body is defined to describe the nature of motion of the body.

A ballet dancer spins about a vertical axis at 120 rpm with arms out stretched with her arms folded, moment of inertial about the axis of rotation decreases by $40\%.$ Calculate the new rate of rotation is [in rpm]

A horizontal disc rotates freely with angular velocity $\omega$ about a vertical axes through its centre. A ring, having the same mass and radius as the disc, is now gently placed coaxially on the disc. After some time, the two rotate with a common angular velocity. Then

If the body is moving in a circle of radius r with a constant speed v, its angular velocity is

For a particle of a purely rotating body, $v=r \omega$, so correct relation will be