Oscillations

The potential energy of a particle of mass 4 kg in motion along the $x$-axis is given by $U=4(1-\cos4x)$ J. The time period of the particle for small oscillation $(\sin\theta\simeq\theta)$ is $\left(\dfrac\pi K\right)$ s. The value of K is ______.

The time period of oscillation of a simple pendulum of length L suspended from the roof of a vehicle, which moves without friction down an inclined plane of inclination α, is given by :

The metallic bob of simple pendulum has the relative density 5. The time period of this pendulum is 10 s. If the metallic bob is immersed in water, then the new time period becomes $5\sqrt x$ s. The value of $x$ will be _______.

An oscillating simple pendulum of time period T$_o$ is placed in a lift which is accelerating upwards with 2.5 m/s$^2$. The time period of pendulum in the lift will be (if g = 10 m/s$^2$)

A particle executing SHM has its velocity 20 cm s$^{-1}$ at mean position and acceleration of 25 cm s$^{-2}$ at one of its extreme position. The amplitude of the particle will be _______ cm.

In the given figure, a mass M is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is k. The mass oscillates on a frictionless surface with time period T and amplitude A. When the mass is in equilibrium position, as shown in the figure, another mass m is gently fixed upon it. The new amplitude of oscillation will be :

One end of a spring of negligible unstretched length and spring constant $k$ is fixed at the origin (0,0). A point particle of mass $m$ carrying a positive charge $q$ is attached at its other end. The entire system is kept on a smooth horizontal surface. When a point dipole $\vec p$ pointing towards the charge $q$ is fixed at the origin, the spring gets stretched to a length $l$ and attains a new equilibrium position (see figure below). If the point mass is now displaced slightly by $\Delta l \ll l$ from its equilibrium position and released, it is found to oscillate at frequency $\dfrac1\delta\sqrt{\dfrac km}$. The value of $\delta$ is ______.

A block of mass $2M$ is attached to a massless spring with spring-constant $k$. This block is connected to two other blocks of masses $M$ and $2M$ using two massless pulleys and strings. The accelerations of the blocks are $a_1$, $a_2$ and $a_3$ as shown in the figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is $x_0$. Which of the following option(s) is/are correct?

[$g$ is the acceleration due to gravity. Neglect friction]

A perfectly reflecting mirror of mass $M$ mounted on a spring constitutes a spring-mass system of angular frequency $\Omega$ such that $\dfrac{4\pi M\Omega}h = 10^{24}\ \mathrm{m^{-2}}$ with $h$ as Planck's constant. $N$ photons of wavelength $\lambda = 8\pi\times10^{-6}$ m strike the mirror simultaneously at normal incidence such that the mirror gets displaced by 1 μm. If the value of $N$ is $x\times10^{12}$, then the value of $x$ is ______.

[Consider the spring as massless]

A spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is $2.0\ \mathrm{N\,m^{-1}}$ and the mass of the block is 2.0 kg. Ignore the mass of the spring. Initially the spring is in an unstretched condition. Another block of mass 1.0 kg moving with a speed of $2.0\ \mathrm{m\,s^{-1}}$ collides elastically with the first block. The collision is such that the 2.0 kg block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is ______.

In the List-I below, four different paths of a particle are given as functions of time. In these functions, $\alpha$ and $\beta$ are positive constants of appropriate dimensions and $\alpha \ne \beta$. In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned: $\vec{p}$ is the linear momentum, $\vec{L}$ is the angular momentum about the origin, $K$ is the kinetic energy, $U$ is the potential energy and $E$ is the total energy. Match each path in List-I with those quantities in List-II, which are conserved for that path.

A particle is executing simple harmonic motion with a time period T. At time t = 0, it is at its position of equilibrium. The kinetic energy - time graph of the particle will look like :

A magnetic needle of magnetic moment $6.7\times10^{-2}$ Am$^2$ and moment of inertia $7.5\times10^{-6}$ kg m$^2$ is performing simple harmonic oscillations in a magnetic field of 0.01 T. Time taken for 10 complete oscillations is :

A 1 kg block attached to a spring vibrates with a frequency of 1 Hz on a frictionless horizontal table. Two springs identical to the original spring are attached in parallel to an 8 kg block placed on the same table. So, the frequency of vibration of the 8 kg block is :

The ratio of maximum acceleration to maximum velocity in a simple harmonic motion is 10 s$^{-1}$. At, t = 0 the displacement is 5 m. What is the maximum acceleration ? The initial phase is $\dfrac\pi4$.

A block of mass 0.1 kg is connected to an elastic spring of spring constant 640 Nm$^{-1}$ and oscillates in a damping medium of damping constant $10^{-2}$ kg s$^{-1}$. The system dissipates its energy gradually. The time taken for its mechanical energy of vibration to drop to half of its initial value, is closest to :

In an experiment to determine the period of a simple pendulum of length 1 m, it is attached to different spherical bobs of radii $r_1$ and $r_2$. The two spherical bobs have uniform mass distribution. If the relative difference in the periods, is found to be $5\times10^{-4}$ s, the difference in radii, $|r_1-r_2|$ is best given by :

A block with mass $M$ is connected by a massless spring with stiffness constant $k$ to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude $A$ about an equilibrium position $x_0$. Consider two cases: (i) when the block is at $x_0$; and (ii) when the block is at $x=x_0+A$. In both the cases, a particle with mass $m\,(<M)$ is softly placed on the block after which they stick to each other. Which of the following statement(s) is(are) true about the motion after the mass $m$ is placed on the mass $M$?

A pendulum clock loses 12 s a day if the temperature is 40°C and gains 4 s a day if the temperature is 20°C. The temperature at which the clock will show correct time, and the co-efficient of linear expansion (α) of the metal of the pendulum shaft are respectively :

A particle performs simple harmonic motion with amplitude A. Its speed is trebled at the instant that it is at a distance $\dfrac{2A}3$ from equilibrium position. The new amplitude of the motion is :