The amplitude of the vibrating particle due to the superposition of two $SHMs$,$y_1 = \sin \left( \omega t + \frac{\pi}{3} \right)$ and $y_2 = \sin \omega t$ is:

Two simple harmonic motions are represented by the equations $y_{1} = 10 \sin(3 \pi t + \frac{\pi}{3})$ and $y_{2} = 5(\sin 3 \pi t + \sqrt{3} \cos 3 \pi t)$. The ratio of the amplitude of $y_{1}$ to $y_{2}$ is $x : 1$. The value of $x$ is ...... .

Two particles $P$ and $Q$ perform $SHM$ with the same amplitude $a$ and frequency $v$ along the same straight line. The maximum distance between the two particles is $a \sqrt{2}$. The initial phase difference between the particles is:

The resultant of two rectangular simple harmonic motions of the same frequency and equal amplitudes but differing in phase by $\frac{\pi}{2}$ is

$A$ particle which is simultaneously subjected to two perpendicular simple harmonic motions represented by $x = a_1 \cos \omega t$ and $y = a_2 \cos 2 \omega t$ traces a curve given by:

Two particles are executing simple harmonic motion of the same amplitude $A$ and frequency $\omega$ along the $x$-axis. Their mean positions are separated by a distance $X_0$ $(X_0 > A)$. If the maximum separation between them is $(X_0 + A)$,the phase difference between their motions is: