Rotational Kinetic Energy, Work & Power

A body is rotating about its own axis. Its rotational kinetic energy is '$x$' and its angular momentum is '$y$'. Hence its moment of inertia about its own axis is

The power (P) is supplied to rotating body having moment of inertia 'I' and angular acceleration '$\alpha$'. Its instantaneous angular velocity is

A flywheel rotating about a fixed axis has a kinetic energy of $500$ joule, if its angular frequency is $5$ Hz, then calculate the moment of inertia of the wheel about the axis of rotation. ($\pi^2 = 10$)

The moment of inertia of two rotating bodies 'X' and 'Y' are $I_1$ and $I_2$ respectively where $I_2 > I_1$. If $K_x$ and $K_y$ are their rotational kinetic energies and their angular momenta are equal, then

Two bodies rotate with kinetic energies $E_1$ and $E_2$, moment of inertia about their axes of rotation is $I_1$ and $I_2$. If $3I_1 = I_2$ and $E_1 = 27E_2$ then the ratio of the angular momenta $L_1$ to $L_2$ is

The angular momentum of a rotating body is 'L'. When the frequency of rotating body is tripled and its kinetic energy is made one-third, the new angular momentum becomes

Two bodies A and B have moment of inertia $I_A$ and $I_B$, and angular momenta $L_A$ and $L_B$ respectively. Both of them have same kinetic energy of rotation. So the ratio $L_A$ to $L_B$ is

If the earth suddenly contracts to $\left(\frac{1}{3}\right)^{\text{rd}}$ of its present size without change in its mass, the ratio kinetic energy of the earth after and before contraction will be (Earth is assumed to be a rotating sphere about itself)

A uniform rod of length $l$ is free to rotate in a vertical plane about a fixed horizontal axis through point B. The rod begins rotating from rest, from its unstable equilibrium position. The angular velocity $\omega$, when it has turned through an angle $\theta$, is

A thin uniform rod of length $L$ and mass $m$ is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is $\omega$. Its centre of mass rises to a maximum height of ($g$ = acceleration due to gravity)

A thin uniform rod AB of mass '$m$' and length '$l$' is hinged at one end A to the ground level. Initially the rod stands vertically and is allowed to fall freely to the ground in the vertical plane. The angular velocity of the rod when its B end strikes the ground is

($g$ = acceleration due to gravity)

A thin uniform rod $(X)$ of mass $M$ and length $L$ is pivoted at a height $\left(\frac{L}{3}\right)$ as shown in the figure. The rod is allowed to fall from a vertical position and lie horizontally on the table. The angular velocity of this rod when it hits the table top, is ______.

( $g =$ gravitational acceleration)

A body is rotating about its own axis. Its rotational kinetic energy is ' x ' and its angular momentum is ' $y$ '. Hence its moment of inertia about its own axis is

The angular momentum of a rotating body is ' $L$ '. When the frequency of rotating body is tripled and its kinetic energy is made one-third, the new angular momentum becomes

Using Bohr's quantisation condition, what is the rotational energy in the second orbit for a diatomic molecule?

( $I =$ moment of inertia of diatomic molecule and $h =$ Planck's constant)

A solid cylinder of mass ' $M$ ' and radius ' $R$ ' is rotating about its geometrical axis. A solid sphere of same mass and same radius is also rotating about its diameter with an angular speed half that of the cylinder. The ratio of the kinetic energy of rotation of the sphere to that of the cylinder will be

A solid sphere of mass ' $m$ ' and radius ' $R$ ' is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with angular speed twice that of sphere. The ratio of kinetic energy of sphere to kinetic energy of cylinder will be

A thin uniform rod of length ' $L$ ' and mass ' $M$ ' is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is ' $\omega$ '. Its centre of mass rises to a maximum height of ( $g =$ acceleration due gravity)

A thin uniform rod of mass ' $m$ ' and length ' $L$ ' is pivoted at one end so that it can rotate in a vertical plane. The free end is held vertically above pivot and then released. The angular acceleration of the rod when it makes an angle ' $\theta$ ' with the vertical is [consider negligible friction at the pivot] ( $g =$ acceleration due to gravity)

If the angular velocity of a body rotating about the given axis increases by $20\%$, then its kinetic energy of rotation will increase by

Two bodies A and B have their moments of inertia $I_1$ and $I_2$ respectively about their axis of rotation. If their kinetic energies of rotation are equal and their angular momenta $L_1$ and $L_2$ respectively are in the ratio $1 : \sqrt{3}$, then $I_2$ will be

A rotating body has angular momentum ' $L$ '. If its frequency is doubled and kinetic energy is halved, its angular momentum will be

A solid cylinder of mass M and radius R is rotating about its geometrical axis. A solid sphere of the same mass and same radius is also rotating about its diameter with an angular speed half that of the cylinder. The ratio of the kinetic energy of rotation of the sphere to that of the cylinder will be

A string is wrapped around the rim of a wheel of moment of inertia $0.40\text{ kg m}^{2}$ and radius $10\text{ cm}$. The wheel is free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of $40\text{ N}$. The angular velocity of the wheel after $10\text{ s}$ is $x\text{ rad/s}$, where $x$ is __________.

A thin uniform metal rod of mass ' $M$ ' and length ' $L$ ' is swinging about a horizontal axis passing through its end. Its maximum angular velocity is ' $\omega$ '. Its centre of mass rises to a maximum height of ( $g =$ Acceleration due to gravity)

A thin uniform rod of length ' $L$ ' and mass ' $M$ ' is swinging freely along a horizontal axis passing through its centre. Its maximum angular speed is ' $\omega$ '. Its centre of mass rises to a maximum height of [ $g =$ gravitational acceleration]

A rigid body rotates with an angular momentum L. If its rotational kinetic energy is made four times, its angular momentum will become

Two bodies have their moments of inertia I and 2I respectively about their axis of rotation. If their kinetic energies of rotation are equal, their angular momenta will be in the ratio

A solid sphere of mass $500\text{ g}$ and radius $5\text{ cm}$ is rotated about one of its diameters with angular speed of $10\text{ rad s}^{-1}$. If the moment of inertia of the sphere about its tangent is $x \times 10^{-2}$ times its angular momentum about the diameter. Then the value of $x$ will be ___________.

A thin uniform rod of mass '$m$' and length '$l$' is suspended from one end which can oscillate in a vertical plane about the point of suspension. It is pulled to one side and then released. It passes through the equilibrium position with angular speed '$\omega$'. The kinetic energy while passing through mean position is

A thin uniform $\text{rod } AB$ of mass $m$ and length $l$ is hinged at one end $A$ to the ground level. Initially the rod stands vertically and is allowed to fall freely to the ground in the vertical plane. The angular velocity of the rod when its end $B$ strikes the ground is ( $g =$ acceleration due to gravity)

An energy of 484 J is spent in increasing the speed of a flywheel from 60 rpm to 360 rpm. The moment of inertia of the flywheel is:

The moment of inertia of a body about the given axis, rotating with angular velocity 1 rad/s is numerically equal to 'P' times its rotational kinetic energy. The value of 'P' is

A molecule consists of two atoms each of mass '$m$' and separated by a distance '$d$'. At room temperature, if the average rotational kinetic energy is '$E$' then the angular frequency is

Two rotating bodies $P$ and $Q$ of masses '$m$' and '$2m$' with moment of inertia $I_P$ and $I_Q \, (I_Q > I_P)$ have equal Kinetic energy of rotation. If $L_P$ and $L_Q$ be their angular momenta respectively then

A solid sphere of mass '$M$' and radius '$R$' is rotating about its diameter. A solid cylinder of same mass and same radius is also rotating about its geometrical axis with an angular speet twice that of the sphere. The ratio of the kinetic energy of rotation of the sphere to that of the cylinder is

A particle performs rotational motion with an angular momentum 'L'. If frequency of rotation is doubled and its kinetic energy becomes one fourth, the angular momentum becomes.

Two bodies rotate with kinetic energies 'E$1$' and 'E$2$'. Moments of inertia about their axis of rotation are 'I $1$' and 'I$2$'. If $I_1 = \frac{I_2}{3}$ and E $1$ = 27 E$2$, then the ratio of angular momenta 'L $1$' to 'L$2$' is

The moment of inertia of a body about a given axis is $1.2\text{ kg m}^{2}$. Initially the body is at rest. In order to produce rotational kinetic energy of $1500\text{ J}$, an angular acceleration of $25\text{ rad/s}^{2}$ must be applied about an axis for a time duration of

In the given figure, two wheels P and Q are connected by a belt B. The radius of P is three times as that of Q. In case of same rotational kinetic energy, the ratio of rotational inertias $\left(\frac{I_1}{I_2}\right)$ will be x: 1. The value of x will be _____________.

A 2 kg steel rod of length 0.6 m is clamped on a table vertically at its lower end and is free to rotate in vertical plane. The upper end is pushed so that the rod falls under gravity, ignoring the friction due to clamping at its lower end, the speed of the free end of rod when it passes through its lowest position is ____________ ms______1. (Take g = 10 ms______2)

Four point masses, each of mass m, are fixed at the corners of a square of side $l$. The square is rotating with angular frequency $\omega$, about an axis passing through one of the corners of the square and parallel to its diagonal, as shown in the figure. The angular momentum of the square about this axis is:

One end of a straight uniform 1m long bar is pivoted on horizontal table. It is released from rest when it makes an angle 30° from the horizontal (see figure). Its angular speed when it hits the table is given as $\sqrt{n}$ s-1, where n is an integer. The value of n is _________.

When a 12000 joule of work is done on a flywheel, its frequency of rotation increases from 10 Hz to 20 Hz. The moment of inertia of flywheel about its axis of rotation is $(\pi^2 = 10)$

Moment of inertia of a body about a given axis is 1.5 kg m2. Initially the body is at rest. In order to produce a rotational kinetic energy of 1200 J, the angular accleration of 20 rad/s2 must be applied about the axis for a duration of :-

Three objects, A: (a solid sphere), B: (a thin circular disk) and C: (a circular ring), each have the same mass M and radius R. They all spin with the same angular speed $\omega$ about their own symmetry axes. The amounts of work (W) required to bring them to rest, would satisfy the relation

A thin rod MN, free to rotate in the vertical plane about the fixed end N, is held horizontal. When the end M is released the speed of this end, when the rod makes an angle $\alpha$ with the horizontal, will be proportional to: (see figure)

A slender uniform rod of mass M and length $l$ is pivoted at one end so that it can rotate in a vertical plane (see figure). There is negligible friction at the pivot. The free end is held vertically above the pivot and then released. The angular acceleration of the rod when it makes an angle $\theta$ with the vertical is

Two rotating bodies A and B of masses m and 2m with moments of inertia $I_A$ and $I_B$ ($I_B$ > $I_A$) have equal kinetic energy of rotation. If LA and LB be their angular momenta respectively, then

A solid sphere of mass m and radius R is rotating about its diameter. A solid cylinder of same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation (Esphere / Ecylinder) will be

A thin uniform rod of length $l$ and mass $m$ is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is $\omega$. Its center of mass rises to a maximum height of:

Two bodies have their moments of inertia I and 2I respectively about their axis of rotation. If their kinetic energies of rotation are equal, their angular velocity will be in the ratio

A particle performing uniform circular motion has angular momentum $L$. If its angular frequency is doubled & its kinetic energy halved, then the new angular momentum is

A fly wheel rotating about fixed axis has a kinetic energy of 360 joule when its angular speed is 30 radian/sec. The moment of inertia of the wheel about the axis of rotation is

The moment of inertia of a body about a given axis is $1.2\text{ kg m}^{2}$. Initially, the body is at rest. In order to produce a rotational kinetic energy of 1500 joule, an angular acceleration of 25 radian/sec2 must be applied about that axis for a duration of

A ring of mass $m$ and radius $r$ rotates about an axis passing through its centre and perpendicular to its plane with angular velocity $\omega$. Its kinetic energy is