Simple harmonic motion

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

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 :

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 ______.

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.

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]

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 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 :

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 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 :

Two particles are performing simple harmonic motion in a straight line about the same equilibrium point. The amplitude and time period for both particles are same and equal to A and T, respectively. At time t = 0 one particle has displacement A while the other one has displacement $\dfrac{-A}2$ and they are moving towards each other. If they cross each other at time t, then t is :

In an engine the piston undergoes vertical simple harmonic motion with amplitude 7 cm. A washer rests on top of the piston and moves with it. The motor speed is slowly increased. The frequency of the piston at which the washer no longer stays in contact with the piston, is close to :

Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies $\omega_1$ and $\omega_2$ and have total energies $E_1$ and $E_2$, respectively. The variations of their momenta $p$ with positions $x$ are shown in the figures. If $\dfrac ab=n^2$ and $\dfrac aR=n$, then the correct equation(s) is(are)

The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density $\lambda$ are kept parallel to each other. In their resulting electric field, point charges $q$ and $-q$ are kept in equilibrium between them. The point charges are confined to move in the $x$ direction only. If they are given a small displacement about their equilibrium positions, then the correct statement(s) is(are)

A particle of unit mass is moving along the $x$-axis under the influence of a force and its total energy is conserved. Four possible forms of the potential energy of the particle are given in column I ($a$ and $U_0$ are constants). Match the potential energies in column I to the corresponding statement(s) in column II.

A cylindrical block of wood (density = 650 kg m$^{-3}$), of base area 30 cm$^2$ and height 54 cm, floats in a liquid of density 900 kg m$^{-3}$. The block is depressed slightly and then released. The time period of the resulting oscillations of the block would be equal to that of a simple pendulum of length (nearly) :

A tennis ball is dropped on a horizontal smooth surface. It bounces back to its original position after hitting the surface. The force on the ball during the collision is proportional to the length of compression of the ball. Which one of the following sketches describes the variation of its kinetic energy $K$ with time $t$ most appropriately? The figures are only illustrative and not to the scale.

A particle moves with simple harmonic motion in a straight line. In first $\tau$ s, after starting from rest it travels a distance $a$, and in next $\tau$ s it travels $2a$, in same direction, then :

Two bodies of masses 1 kg and 4 kg are connected to a vertical spring, as shown in the figure. The smaller mass executes simple harmonic motion of angular frequency 25 rad/s, and amplitude 1.6 cm while the bigger mass remains stationary on the ground. The maximum force exerted by the system on the floor is (take $g=10$ ms$^{-2}$).

Which of the following expressions corresponds to simple harmonic motion along a straight line, where $x$ is the displacement and a, b, c are positive constants ?

A body is in simple harmonic motion with time period half second (T = 0.5 s) and amplitude one cm (A = 1 cm). Find the average velocity in the interval in which it moves from equilibrium position to half of its amplitude.