Simple harmonic motion

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 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.

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 _____.

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

Two point-like objects of masses 20 gm and 30 gm are fixed at the two ends of a rigid massless rod of length 10 cm. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass, as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is $1.2\times10^{-8}\ \mathrm{N\,m\,rad^{-1}}$. The angular frequency of the oscillations in $n\times10^{-3}\ \mathrm{rad\,s^{-1}}$. The value of $n$ is _____.

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 $\mathrm{m\,s^{-1}}$) at time $t = \dfrac\pi3$ s.

Which one of the following options is correct?

A particle of mass 1 kg is subjected to a force which depends on the position as $\vec F = -k(x\hat i + y\hat j)\ \mathrm{kg\,m\,s^{-2}}$ with $k = 1\ \mathrm{kg\,s^{-2}}$. At time $t = 0$, the particle's position $\vec r = \left(\frac{1}{\sqrt2}\hat i + \sqrt2\hat j\right)$ m and its velocity $\vec v = \left(-\sqrt2\hat i + \sqrt2\hat j + \frac2\pi\hat k\right)\ \mathrm{m\,s^{-1}}$. Let $v_x$ and $v_y$ denote the $x$ and the $y$ components of the particle's velocity, respectively. Ignore gravity. When $z = 0.5$ m, the value of $(x v_y - y v_x)$ is _____ $\mathrm{m^2s^{-1}}$.

On a frictionless horizontal plane, a bob of mass $m = 0.1$ kg is attached to a spring with natural length $l_0 = 0.1$ m. The spring constant is $k_1 = 0.009\ \mathrm{N\,m^{-1}}$ when the length of the spring $l > l_0$ and is $k_2 = 0.016\ \mathrm{N\,m^{-1}}$ when $l < l_0$. Initially the bob is released from $l = 0.15$ m. Assume that Hooke's law remains valid throughout the motion. If the time period of the full oscillation is $T = (n\pi)$ s, then the integer closest to $n$ is _____.

The displacement of simple harmonic oscillator after 3 seconds starting from its mean position is equal to half of its amplitude. The time period of harmonic motion is :

The equation of a particle executing simple harmonic motion is given by $x=\sin\pi\left(t+\dfrac13\right)$ m. At $t=1$ s, the speed of particle will be

(Given: $\pi=3.14$) -

A particle executes simple harmonic motion. Its amplitude is 8 cm and time period is 6 s. The time it will take to travel from its position of maximum displacement to the point corresponding to half of its amplitude, is ______ s.

A body is performing simple harmonic with an amplitude of 10 cm. The velocity of the body was tripled by air Jet when it is at 5 cm from its mean position. The new amplitude of vibration is $\sqrt x$ cm. The value of $x$ is ______.