$A$ particle is executing $SHM$ about $y=0$ along the $y$-axis. Its position at an instant is given by $y = (7 \, m) \sin(\pi t)$. Its average velocity for the time interval $0$ to $0.5 \, s$ is ........... $m/s$.

$Assertion :$ In simple harmonic motion,the motion is to and fro and periodic.
$Reason :$ Velocity of the particle $(v) = \omega \sqrt {A^2 - x^2}$ (where $x$ is the displacement and $A$ is the amplitude).

When a particle in linear $S.H.M.$ completes two oscillations,its phase increases by

Two particles are executing Simple Harmonic Motion ($S$.$H$.$M$.). The equations of their motion are $y_1 = 10 \sin \left( \omega t + \frac{\pi}{4} \right)$ and $y_2 = 25 \sin \left( \omega t + \frac{\sqrt{3} \pi}{4} \right)$. What is the ratio of their amplitudes?

Equations $y_1 = A \sin \omega t$ and $y_2 = \frac{A}{2} \sin \omega t + \frac{A}{2} \cos \omega t$ represent $S.H.M.$ The ratio of the amplitudes of the two motions is

The equation of motion of a particle is $\frac{d^2y}{dt^2} + Ky = 0$,where $K$ is a positive constant. The time period of the motion is given by