Oscillations

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:

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:

The equation of motion of a particle is given by $x=a\sin\left(50t+\pi/3\right)$ cm. The particle will come to rest at time $t_1$ and it will have zero acceleration at time $t_2$. The $t_1$ and $t_2$ respectively are ________.

The velocity of a particle executing simple harmonic motion along $x$-axis is described as $v^2=50-x^2$, where $x$ represents displacement. If the time period of motion is $\dfrac x7$ s, the value of $x$ is ________.

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 rigid dipole undergoes a simple harmonic motion about its centre in the presence of an electric field $\vec E_1=E_0\hat x$. If another electric field $\vec E_2=2E_0\left(\hat y+\hat z\right)$ is introduced to the system, what will be the percentage change in the frequency of the oscillation (approximate)?

Match List - I with List - II.

Choose the correct answer from the options given below :

A particle is executing simple harmonic motion. Its amplitude is $A$ and time period is 5 sec. The time required by it to move from $x=A$ to $x=\dfrac A{\sqrt2}$ is ________ sec.

A spring stretches by 2 mm when it is loaded with a mass of 200 g. From equilibrium position the mass is further pulled down by 2 mm and released. The frequency associated with the system and maxmimum energy in the spring are ________ Hz and ________ J, respectively.

(Take $g=10$ m/s²)

The frequency of oscillation of a mass $m$ suspended by a spring is $v_1$. If the length of the spring is cut to half, the same mass oscillates with frequency $v_2$. The value of $v_2/v_1$ is ________.

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):

A person sitting inside an elevator performs a weighing experiment with an object of mass 50 kg. Suppose that the variation of the height $y$ (in m) of the elevator, from the ground, with time $t$ (in s) is given by $y = 8\left[1 + \sin\left(\dfrac{2\pi t}{T}\right)\right]$, where $T = 40\pi$ s. Taking acceleration due to gravity, $g = 10$ m/s$^2$, the maximum variation of the object's weight (in N) as observed in the experiment is _____.

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:

An audio transmitter (T) and a receiver (R) are hung vertically from two identical massless strings of length 8 m with their pivots well separated along the $X$ axis. They are pulled from the equilibrium position in opposite directions along the $X$ axis by a small angular amplitude $\theta_0 = \cos^{-1}(0.9)$ and released simultaneously. If the natural frequency of the transmitter is 660 Hz and the speed of sound in air is 330 m/s, the maximum variation in the frequency (in Hz) as measured by the receiver (Take the acceleration due to gravity $g = 10\ \mathrm{m/s^2}$) is _____.

Two beads, each with charge $q$ and mass $m$, are on a horizontal, frictionless, non-conducting, circular hoop of radius $R$. One of the beads is glued to the hoop at some point, while the other one performs small oscillations about its equilibrium position along the hoop. The square of the angular frequency of the small oscillations is given by

[$\varepsilon_0$ is the permittivity of free space.]

A block of mass $5\,\mathrm{kg}$ moves along the $x$-direction subject to the force $F = (-20x + 10)\,\mathrm{N}$, with the value of $x$ in metre. At time $t = 0\,\mathrm{s}$, it is at rest at position $x = 1\,\mathrm{m}$. The position and momentum of the block at $t = (\pi/4)\,\mathrm{s}$ are

A particle of mass $m$ is moving in a circular orbit under the influence of the central force $F(r) = -kr$, corresponding to the potential energy $V(r) = kr^2/2$, where $k$ is a positive force constant and $r$ is the radial distance from the origin. According to the Bohr's quantization rule, the angular momentum of the particle is given by $L = n\hbar$, where $\hbar = h/(2\pi)$, $h$ is the Planck's constant, and $n$ a positive integer. If $v$ and $E$ are the speed and total energy of the particle, respectively, then which of the following expression(s) is(are) correct?

A small electric dipole $\vec p_0$, having a moment of inertia $I$ about its center, is kept at a distance $r$ from the center of a spherical shell of radius $R$. The surface charge density $\sigma$ is uniformly distributed on the spherical shell. The dipole is initially oriented at a small angle $\theta$ as shown in the figure. While staying at a distance $r$, the dipole is free to rotate about its center.

If released from rest, then which of the following statement(s) is(are) correct?

[$\varepsilon_0$ is the permittivity of free space.]

If the collision occurs at time $t_0 = 0$, the value of $v_{\text{cm}}/(a\omega)$ will be _____.

If the collision occurs at time $t_0 = \pi/(2\omega)$, then the value of $4b^2/a^2$ will be _____.