Two particles are oscillating along two close parallel straight lines side by side, with the same frequency and amplitudes. They pass each other, moving in opposite directions when their displacement is half of the amplitude. The mean positions of the two particles lie in a straight line perpendicular to the paths of the two particles. The phase difference is:
1. \(\frac{\pi}{6}\)
2. \(0\)
3. \(\frac{2\pi}{3}\)
4. \(\pi\)

A rectangular block of mass m and area of cross-section A floats in a liquid of density ρ. If it is given a small vertical displacement from equilibrium, it undergoes oscillation with a time period T. Then:
1. $T\propto \sqrt{\rho }$
2. $T\propto \frac{1}{\sqrt{A}}$
3. $T\propto \frac{1}{\rho }$
4. $T\propto \frac{1}{\sqrt{m}}$

The phase difference between the instantaneous velocity and acceleration of a particle executing simple harmonic motion is:
1. 0.5$\mathrm{\pi }$
2. $\mathrm{\pi }$
3. 0.707$\mathrm{\pi }$
4. zero

A mass of \(2.0\) kg is put on a flat pan attached to a vertical spring fixed on the ground as shown in the figure. The mass of the spring and the pan is negligible. When pressed slightly and released, the mass executes a simple harmonic motion. The spring constant is \(200\) N/m. What should be the minimum amplitude of the motion, so that the mass gets detached from the pan? 
(Take \(g=10\) m/s²) 
                

A particle executes simple harmonic oscillation with an amplitude a. The period of oscillation is T. The minimum time taken by the particle to travel half of the amplitude from the equilibrium position is:
1. $\frac{\mathrm{T}}{4}$
2. $\frac{\mathrm{T}}{8}$
3. $\frac{\mathrm{T}}{12}$
4. $\frac{\mathrm{T}}{2}$

Two points are located at a distance of \(10\) m and \(15\) m from the source of oscillation. The period of oscillation is \(0.05\) s and the velocity of the wave is \(300\) m/s. What is the phase difference between the oscillations of two points?
1. \(\frac{\pi}{3}\)
2. \(\frac{2\pi}{3}\)
3. \(\pi\)
4. \(\frac{\pi}{6}\)

A point performs simple harmonic oscillation of period \(\mathrm{T}\) and the equation of motion is given by; \(x=a \sin (\omega t+\pi / 6)\). After the elapse of what fraction of the time period, the velocity of the point will be equal to half of its maximum velocity?
1. \( \frac{T}{8} \)

2. \( \frac{T}{6} \)

3. \(\frac{T}{3} \)

4. \( \frac{T}{12}\)