A small block is connected to one end of a massless spring of un-stretched length 4.9 m. The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by 0.2 m and released from rest at t = 0. It then executes simple harmonic motion with angular frequency $\omega$ = ($\pi$/3) rad/s. Simultaneously, at t = 0, a small pebble is projected with speed v from point P at an angle of 45$^\circ$ as shown in the figure. Point O is at a horizontal distance of 10 m from O. If the pebble hits the block at t = 1 s, the value of v is (take g = 10 m/s²)

The phase space diagram for a ball thrown vertically up from ground is

The phase space diagram for simple harmonic motion is a circle centred at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and E₁ and E₂ are the total mechanical energies respectively. Then

Consider the spring-mass system, with the mass submerged in water, as shown in the figure. The phase space diagram for one cycle of this system is

If the total energy of the particle is E, it will perform periodic motion only if

For periodic motion of small amplitude A, the time period T of this particle is proportional to

The acceleration of this particle for $|x| > {X_0}$ is

The mass M shown in the figure below oscillates in simple harmonic motion with amplitude A. The amplitude of the point P is

A uniform rod of length L and mass M is pivoted at the centre. Its two ends are attached to two springs of equal spring constants $k$. The springs are fixed to rigid supports as shown in the figure, and the rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle $\theta$ in one direction and released. The frequency of oscillation is

The $x$-$t$ graph of a particle undergoing simple harmonic motion is shown in the figure. The acceleration of the particle at $t=4/3$ s is

A student performed the experiment to measure the speed of sound in air using resonance air-column method. Two resonances in the air-column were obtained by lowering the water level. The resonance with the shorter air-column is the first resonance and that with the longer air-column is the second resonance. Then,

Column I gives a list of possible set of parameters measured in some experiments. The variations of the parameters in the form of graphs are shown in Column II. Match the set of parameters given in Column I with the graphs given in Column II. Indicate your answer by darkening the appropriate bubbles of the 4 $\times$ 4 matrix given in the ORS.

	Column I		Column II
(A)	Potential energy of a simple pendulum (y-axis) as a function of displacement (x) axis	(P)
(B)	Displacement (y-axis) as a function of time (x-axis) for a one dimensional motion at zero or constant acceleration when the body is moving along the positive x-direction	(Q)
(C)	Range of a projectile (y-axis) as a function of its velocity (x-axis) when projected at a fixed angle	(R)
(D)	The square of the time period (y-axis) of a simple pendulum as a function of its length (x-axis)	(S)

Function $x=\mathrm{A} \sin ^2 \omega t+\mathrm{B} \cos ^2 \omega t+\mathrm{C} \sin \omega t \cos \omega t$ represents SHM

A small body attached to one end of a vertically hanging spring is performing SHM about its mean position with angular frequency $\omega$ and amplitude $a$. If at a height $y^{\prime}$ from the mean position, the body gets detached from the spring, calculate the value of $y^{\prime}$ so that the height $\mathrm{H}$ attained by the mass is maximum. The body does not interact with the spring during its subsequent motion after detachment $\left(a \omega^{2}>g\right)$