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 particles execute $SHM$ of same amplitude of $20 \, cm$ with same period along the same line about the same equilibrium position. The maximum distance between the two is $20 \, cm$. Their phase difference in radians is

Two particles execute simple harmonic motion $(SHM)$ along close parallel lines. Both particles have the same frequency and same amplitude. When they pass each other moving in opposite directions, their displacement is half their amplitude. Their phase difference is:

$A$ particle executes two types of $SHM$. $x_1 = A_1 \sin \omega t$ and $x_2 = A_2 \sin [\omega t + \frac{\pi}{3}]$.
$(a)$ Find the displacement at time $t = 0$.
$(b)$ Find the maximum speed of the particle.
$(c)$ Find the maximum acceleration of the particle.

Two $SHM$ are represented by equations,$y_1 = 6\cos \left( {6\pi t + \frac{\pi }{6}} \right)$ and $y_2 = 3\left( {\sqrt 3 \sin 3\pi t + \cos 3\pi t} \right)$. Which of the following statements is true?

The composition of two simple harmonic motions of equal periods at right angles to each other and with a phase difference of $\pi$ results in the displacement of the particle along