Two particles $P$ and $Q$ describe $S.H.M.$ of same amplitude $a$,same frequency $f$ along the same straight line. The maximum distance between the two particles is $a\sqrt{2}$. The initial phase difference between the particles is:

Two simple harmonic motions are represented by the equations $x_{1}=5 \sin \left(2 \pi t+\frac{\pi}{4}\right)$ and $x_{2}=5 \sqrt{2}(\sin 2 \pi t+\cos 2 \pi t)$. The ratio of the amplitude of $x_{1}$ and $x_{2}$ is

Two simple harmonic motions are represented as $y_1 = 10 \sin \omega t$ and $y_2 = 10 \sin \omega t + 5 \cos \omega t$. The ratio of the amplitudes of $y_1$ and $y_2$ is

Two simple harmonic motions are represented by the equations ${y_1} = 0.1 \sin(100\pi t + \frac{\pi}{3})$ and ${y_2} = 0.1 \cos(\pi t)$. The phase difference of the velocity of particle $1$ with respect to the velocity of particle $2$ is

Two particles are executing $SHM$ in a straight line. The amplitude $A$ and time period $T$ of both particles are equal. At time $t = 0$,one particle is at displacement $x_1 = +A$ and the other is at $x_2 = -A/2$,and they are approaching each other. The time after which they will cross each other is:

$A$ body of mass '$m$' performs linear $S$.$H$.$M$. given by the equation $x = P \sin \omega t + Q \sin \left(\omega t + \frac{\pi}{2}\right)$. The total energy of the particle at any instant is