Rotational Motion

A diatomic molecule has moment of inertia $I$. By Bohr's quantization condition its rotational energy in the $n^{\text{th}}$ level ($n=0$ is not allowed) is

It is found that the excitation frequency from ground to the first excited state of rotation for the CO molecule is close to $\dfrac4\pi\times10^{11}$ Hz. Then the moment of inertia of CO molecule about its center of mass is close to (Take $h=2\pi\times10^{-34}$ J s)

In a CO molecule, the distance between C (mass = 12 a.m.u.) and O (mass = 16 a.m.u.), where 1 a.m.u. $=\dfrac53\times10^{-27}$ kg, is close to

A block of base 10 cm × 10 cm and height 15 cm is kept on an inclined plane. The coefficient of friction between them is $\sqrt3$. The inclination $\theta$ of this inclined plane from the horizontal plane is gradually increased from 0°. Then

Column II shows five systems in which two objects are labelled as X and Y. Also in each case a point P is shown. Column I gives some statements about X and/or Y. Match these statements to the appropriate system(s) from Column II.

A uniform rod of length $L$ and mass $M$ is pivoted at the centre. Its two ends are attached to two springs of equal spring constants $k$. The springs are fixed to rigid supports as shown in the figure, and the rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle $\theta$ in one direction and released. The frequency of oscillation is

Under the influence of the Coulomb field of charge $+Q$, a charge $-q$ is moving around it in an elliptical orbit. Find out the correct statement(s).

A sphere is rolling without slipping on a fixed horizontal plane surface. In the figure, A is the point of contact, B is the centre of the sphere and C is its topmost point. Then,

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.

and

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.

The net external force acting on the disk when its centre of mass is at displacement $x$ with respect to its equilibrium position is

The centre of mass of the disk undergoes simple harmonic motion with angular frequency $\omega$ equal to

The maximum value of $V_0$ for which the disk will roll without slipping is

The ratio $x_1/x_2$ is

When disc B is brought in contact with disc A, they acquire a common angular velocity in time $t$. The average frictional torque on one disc by the other during this period is

The loss of kinetic energy during the above process is

A small object of uniform density rolls up a curved surface with an initial velocity $v$. It reaches up to a maximum height of $\dfrac{3v^2}{4g}$ with respect to the initial position. The object is