A solid cylinder of radius R rolls without slipping with a center of mass speed v₀ = $\sqrt{\frac{gR}{3}}$ on a horizontal surface with a vertical edge, as shown in the figure. Here, g is the acceleration due to gravity. At the moment when the cylinder loses contact with the surface due to rotation around the corner, the speed of its center of mass is:

List-I shows four planar structures made of uniform solid rods each of mass $m$ and length $l$. In the List-II the possible moment of inertia of these structures about an axis $OCO'$, which lies in the plane of the structures, are given.

Choose the option that describes the correct match between the entries in List-I to those in List-II.

List-I	List-II
(P)	(1) $\frac{5}{4}ml^2$
(Q)	(2) $\frac{1}{6}ml^2$
(R)	(3) $\frac{1}{12}ml^2$
(S)	(4) $\frac{2}{3}ml^2$
	(5) $\frac{1}{3}ml^2$

The center of a disk of radius $r$ and mass $m$ is attached to a spring of spring constant $k$, inside a ring of radius $R>r$ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following the Hooke's law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $T=\frac{2 \pi}{\omega}$. The correct expression for $\omega$ is ( $g$ is the acceleration due to gravity):

A flat surface of a thin uniform disk $A$ of radius $R$ is glued to a horizontal table. Another thin uniform disk $B$ of mass $M$ and with the same radius $R$ rolls without slipping on the circumference of $A$, as shown in the figure. A flat surface of $B$ also lies on the plane of the table. The center of mass of $B$ has fixed angular speed $\omega$ about the vertical axis passing through the center of $A$. The angular momentum of $B$ is $n M \omega R^{2}$ with respect to the center of $A$. Which of the following is the value of $n$ ?

List I describes four systems, each with two particles $A$ and $B$ in relative motion as shown in figures. List II gives possible magnitudes of their relative velocities (in $m s^{-1}$ ) at time $t=\frac{\pi}{3} s$.

List-I	List-II
(I) $A$ and $B$ are moving on a horizontal circle of radius $1 \mathrm{~m}$ with uniform angular speed $\omega=1 \mathrm{rad} \mathrm{s}^{-1}$. The initial angular positions of $A$ and $B$ at time $t=0$ are $\theta=0$ and $\theta=\frac{\pi}{2}$, respectively.	(P) $\frac{\sqrt{3}+1}{2}$
(II) Projectiles $A$ and $B$ are fired (in the same vertical plane) at $t=0$ and $t=0.1 \mathrm{~s}$ respectively, with the same speed $v=\frac{5 \pi}{\sqrt{2}} \mathrm{~m} \mathrm{~s}^{-1}$ and at $45^{\circ}$ from the horizontal plane. The initial separation between $A$ and $B$ is large enough so that they do not collide. $\left(g=10 \mathrm{~ms}^{-2}\right)$.	(Q) $\frac{\sqrt{3}-1}{\sqrt{2}}$
(III) Two harmonic oscillators $A$ and $B$ moving in the $x$ direction according to $x_{A}=x_{0} \sin \frac{t}{t_{0}}$ and $x_{B}=x_{0} \sin \left(\frac{t}{t_{0}}+\frac{\pi}{2}\right)$ respectively, starting from $t=0$. Take $x_{0}=1 \mathrm{~m}, t_{0}=1 \mathrm{~s}$.	(R) $\sqrt{10}$
(IV) Particle $A$ is rotating in a horizontal circular path of radius $1 \mathrm{~m}$ on the $x y$ plane, with constant angular speed $\omega=1 \mathrm{rad} \mathrm{s}^{-1}$. Particle $B$ is moving up at a constant speed $3 \mathrm{~m} \mathrm{~s}^{-1}$ in the vertical direction as shown in the figure. (Ignore gravity.)	(S) $\sqrt{2}$
	(T) $\sqrt{25\pi^{2}+1}$

Which one of the following options is correct?

Which of the following statements about the instantaneous axis (passing through the centre of mass) is correct?

Which of the following statements regarding the angular speed about the instantaneous axis (passing through the centre of mass) is correct?

A small mass m is attached to a massless string whose other end is fixed at P as shown in the figure. The mass is undergoing circular motion in the xy-plane with centre at O and constant angular speed $\omega$. If the angular momentum of the system, calculated about O and P are denoted by ${\overrightarrow L _O}$ and ${\overrightarrow L _P}$, respectively, then

A thin uniform rod, pivoted at O, is rotating in the horizontal plane with constant angular speed $\omega$, as shown in the figure. At time t = 0, a small insect starts from O and moves with constant speed v, with respect to the rod towards the other end. It reaches the end of the rod at t = T and stops. The angular speed of the system remains $\omega$ throughout. The magnitude of the torque (|$\tau$|) about O, as a function of time is best represented by which plot ?

A metal rod of length L and mass m is pivoted at one end. A thin disk of mass M and radius R ( < L) is attached at its centre to the free end of the rod. Consider two ways the disc is attached : (case A). The disc is not free to rotate about its centre and (case B) the disc is free to rotate about its centre. The rod-disc system performs SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is(are) true?

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

If the resultant of all the external forces acting on a system of particles is zero, then from an inertial frame, one can surely say that

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:

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.

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

STATEMENT 1

If there is no external torque on a body about its center of mass, then the velocity of the center of mass remains constant.

Because

STATEMENT 2

The linear momentum of an isolated system remains constant.

The ratio ${{{x_1}} \over {{x_2}}}$ 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 solid sphere of radius $R$ has moment of inertia $I$ about its geometrical axis. If it is melted into a disc of radius $r$ and thickness $t$. If its moment of inertia about the tangential axis (which is perpendicular to plane of the disc), is also equal to $I$, then the value of $r$ is equal to

A solid cylinder of mass m and radius $r$ is rolling on rough inclined plane of inclination $\theta$. The coefficient of friction between the cylinder and incline is $\mu$. then

A wooden log of mass M and length L is hinged by a frictionless nail at O; a bullet of mass m strikes with velocity $v$ and sticks to it. Find angular velocity of the system immediately after the collision about O.

A cylinder of mass m and radius R rolls down on an inclined plane of inclination $\theta$. Calculate the linear acceleration of axis of cylinder.

Two identical ladders, each of mass M and length L are resting on the rough horizontal surface as shown in the figure. A block of mass $m$ hangs from P. If the system is in equilibrium, find the magnitude and the direction of frictional force at A and B.

One quarter section is cut from a uniform circular disc of radius $R$. This section has a mass $M$. It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc. Its moment of inertia about the axis of rotation is

A thin wire of length $L$ and uniform linear mass density $\rho$ is bent into a circular loop with centre at $O$ as shown. The moment of inertia of the loop about the axis $XX'$ is: