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

Which of the following examples represent periodic motion?
$(a)$ $A$ swimmer completing one (return) trip from one bank of a river to the other and back.
$(b)$ $A$ freely suspended bar magnet displaced from its $N-S$ direction and released.
$(c)$ $A$ hydrogen molecule rotating about its centre of mass.
$(d)$ An arrow released from a bow.

If the displacement $(x)$ and velocity $(v)$ of a particle executing simple harmonic motion are related through the expression $4v^2 = 25 - x^2$, then the time period is

Show that the motion of a particle represented by $y = \sin \omega t - \cos \omega t$ is simple harmonic with a period of $\frac{2\pi}{\omega}$.

The figure shows the circular motion of a particle. The radius of the circle is $B$. The particle starts at $t=0$ from the positive $y$-axis and moves clockwise. The simple harmonic motion of the $x$-projection of the radius vector of the rotating particle is given by:

$Assertion :$ 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).