Rotational Motion

Consider a large disk of radius $R$ and two smaller disks, each of radius $r=R/50$, lying on its circumference, as shown in the figure. The smaller disks are initially in contact with each other, with an angular separation $\Delta\theta$ between their centers. They are made to roll without slipping in opposite directions, with constant angular velocities $\omega$ and $2\omega$ while the large disk is held stationary. The time $\tau$ at which the smaller disks are again in contact is:
[Use $\sin(\Delta\theta)=\Delta\theta$ and ignore gravity.]

A solid cylinder of radius $R$ rolls without slipping with a center of mass speed $v_0=\sqrt{\dfrac{gR}3}$ on a horizontal surface with a vertical edge, as shown in the figure. Here, $g$ is the acceleration due to the 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:

A tank contains two immiscible liquids of densities $6\rho$ and $2\rho$. The higher density liquid is filled up to a height $L/2$ from the bottom. A thin rod of density $\rho$ and length $L$ is fully immersed and hinged at the bottom so that it can oscillate freely, as shown in the figure. If the rod is slightly disturbed from its equilibrium, the time period of small oscillations is $\dfrac{2\pi}n\sqrt{\dfrac Lg}$, where $g$ is the acceleration due to gravity. The value of $n$ 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.

A particle of mass $m$, and angular momentum $\ell$ is moving in a circular orbit of radius $r_0$ under the influence of an attractive force $\vec F(r)=-\dfrac{k}{r^2}\hat r$. Keeping its angular momentum unchanged, the particle is displaced radially by a small distance $\delta r\ll r_0$, due to which its radial distance varies periodically. The corresponding time period is:

Consider an electric dipole comprising two charges $+q$ and $-q$ each with mass $m$, separated by a fixed distance $d$ and initially at rest with its dipole moment pointing along $\hat\imath$. A uniform electric field $E\hat\jmath$ is turned on at time $t=0$ and it is turned off at $t=t_f$, when the dipole moment makes an angle $\theta_f$ with $\hat\imath$. Neglecting any sources of energy loss, correct option(s) is/are:

After the collision the disk starts to rotate around point $C$ in the $XY$ plane. The maximum change in the height (in m) of its center $O$ is:

Amount of energy loss (in J) in the collision is:

The position of an object having mass 0.1 kg as a function of time $t$ is given as $\vec r=\left(10t^2\hat i+5t^3\hat j\right)$ m. At $t=1$ s, which of the following statements are correct ?

A. The linear momentum $\vec p=\left(2\hat i+1.5\hat j\right)$ kg·m/s.

B. The force acting on the object $\vec F=\left(2\hat i+3\hat j\right)$ N.

C. The angular momentum of the object about its origin $\vec L=15\hat k$ J s.

D. The torque acting on the object about its origin $\vec\tau=20\hat k$ N m.

Choose the correct answer from the options given below :

A particle is rotating in a circular path and at any instant its motion can be described as $\theta=\dfrac{5t^4}{40}-\dfrac{t^3}3$. The angular acceleration of the particle after 10 seconds is ________ rad/s².

Moment of inertia about an axis $AB$ for a rod of mass 40 kg and length 3 m is same as that of a solid sphere of mass of 10 kg and radius $R$ about an axis parallel to $AB$ axis with separation of 3 m as shown in figure below. The value of $R$ is given as $\sqrt{\dfrac\alpha2}$. The value of $\alpha$ is ________.

A solid sphere of mass $M$ and radius $R$ is divided into two unequal parts. The smaller part having mass $M/8$ is converted into a sphere of radius $r$ and the larger part is converted into a circular disc of thickness $t$ and radius $2R$. If $I_1$ is moment of inertia of a sphere having radius $r$ about an axis through its centre and $I_2$ is the moment of inertia of a disc about its diameter, the ratio of their moment of inertia $I_2/I_1$ = ________.

A uniform disc of radius $R$ and mass $M$ is free to oscillate about the axis $A$ as shown in the figure. For small oscillations the time period is ________.
($g$ is acceleration due to gravity)

A wheel initially at rest is subjected to a uniform angular acceleration about its axis. In the first 2 s it rotates through an angle $\theta_1$ and in the next 2 s it rotates through an angle $\theta_2$. The ratio $\dfrac{\theta_2}{\theta_1}$ is ________.

An object of uniform density rolls up the curved path with the initial velocity $v_o$ as shown in the figure. If the maximum height attained by an object is $\dfrac{7v_o^2}{10\,g}$ ($g$ = acceleration due to gravity), the object is a ________.

A solid sphere of radius 4 cm and mass 5 kg is rotating (rotation axis is passing through the centre of the sphere) with an angular velocity of 1200 rpm. It is brought to rest in 10 s by applying a constant torque. The torque applied and the number of rotations it made before it comes to rest are ________ and ________ respectively.

A solid sphere ($A$) of mass $5m$ and a spherical shell ($B$) of mass $m$, both having same radius, are placed on a rough surface. When a force of same magnitude is applied tangentially at the highest points of $A$ and $B$, they start rolling without slipping with an acceleration of $a_A$ and $a_B$, respectively. The ratio of $a_A$ and $a_B$ is ________.

A solid cylinder having radius $R$ and length $L$ is slipping on a rough horizontal plane. At time $t=0$ the cylinder has a translational velocity $v_o=49$ m/s, perpendicular to its axis and a rotational velocity $v_o/4R$ about the centre. The time taken by the cylinder to start rolling is ________ seconds. (coefficient of kinetic friction $\mu_K=0.25$ and $g=9.8$ m/s²)

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 $T = \dfrac{2\pi}{\omega}$. The correct expression for $\omega$ is ($g$ is the acceleration due to gravity):

As shown in the figures, a uniform rod $OO'$ of length $l$ is hinged at the point $O$ and held in place vertically between two walls using two massless springs of same spring constant. The springs are connected at the midpoint and at the top-end ($O'$) of the rod, as shown in Fig. 1 and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is $f_1$. On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2 and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is $f_2$. Ignoring gravity and assuming motion only in the plane of the diagram, the value of $\dfrac{f_1}{f_2}$ is: