Oscillations

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 _____.

Two massless springs with spring constants 2 k and 9 k, carry 50 g and 100 g masses at their free ends. These two masses oscillate vertically such that their maximum velocities are equal. Then, the ratio of their respective amplitudes will be :

Time period of a simple pendulum in a stationary lift is 'T'. If the lift accelerates with $\dfrac g6$ vertically upwards then the time period will be :

(Where g = acceleration due to gravity)

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 ______.

The motion of a simple pendulum executing S.H.M. is represented by the following equation.

$y=A\sin(\pi t+\phi)$, where time is measured in second.

The length of pendulum is

In figure (A), mass '2 m' is fixed on mass 'm' which is attached to two springs of spring constant k. In figure (B), mass 'm' is attached to two springs of spring constant 'k' and '2k'. If mass 'm' in (A) and in (B) are displaced by distance '$x$' horizontally and then released, then time period $T_1$ and $T_2$ corresponding to (A) and (B) respectively follow the relation.

The length of a seconds pendulum at a height $h=2R$ from earth surface will be :

(Given R = Radius of earth and acceleration due to gravity at the surface of earth, $g=\pi^2$ ms$^{-2}$)

Two waves executing simple harmonic motions travelling in the same direction with same amplitude and frequency are superimposed. The resultant amplitude is equal to the $\sqrt3$ times of amplitude of individual motions. The phase difference between the two motions is ______ (degree).

When a particle executes Simple Hormonic Motion, the nature of graph of velocity as a function of displacement will be :

As per the given figure, two blocks each of mass 250 g are connected to a spring of spring constant 2 Nm$^{-1}$. If both are given velocity $v$ in opposite directions, then maximum elongation of the spring is :

As per given figures, two springs of spring constants $k$ and $2k$ are connected to mass $m$. If the period of oscillation in figure (a) is 3 s, then the period of oscillation in figure (b) will be $\sqrt x$ s. The value of $x$ is ______.

Two identical positive charges Q each are fixed at a distance of '2a' apart from each other. Another point charge $q_0$ with mass 'm' is placed at midpoint between two fixed charges. For a small displacement along the line joining the fixed charges, the charge $q_0$ executes SHM. The time period of oscillation of charge $q_0$ will be :

A mass 0.9 kg, attached to a horizontal spring, executes SHM with an amplitude $A_1$. When this mass passes through its mean position, then a smaller mass of 124 g is placed over it and both masses move together with amplitude $A_2$. If the ratio $\dfrac{A_1}{A_2}$ is $\dfrac\alpha{\alpha-1}$, then the value of $\alpha$ will be ______.

Assume there are two identical simple pendulum clocks. Clock - 1 is placed on the earth and Clock - 2 is placed on a space station located at a height $h$ above the earth surface. Clock - 1 and Clock - 2 operate at time periods 4 s and 6 s respectively. Then the value of $h$ is -

(consider radius of earth $R_E$ = 6400 km and g on earth 10 m/s$^2$)