Match the following physical quantities for a particle executing Simple Harmonic Motion $(SHM)$ given by $y = A \sin(\omega t)$:
$(a)$ Velocity $(v)$
$(b)$ Potential Energy $(PE)$
$(c)$ Total Energy $(TE)$
$(d)$ Acceleration $(a)$
$(i)$ Constant
(ii) $A\omega \cos(\omega t)$
(iii) $\frac{1}{2} k A^2 \sin^2(\omega t)$
(iv) $-\omega^2 y$

Two particles,$1$ and $2$,each of mass $m$,are connected by a massless spring and are on a horizontal frictionless plane,as shown in the figure. Initially,the two particles,with their center of mass at $x_0$,are oscillating with amplitude $a$ and angular frequency $\omega$. Thus,their positions at time $t$ are given by $x_1(t) = (x_0 + d) + a \sin \omega t$ and $x_2(t) = (x_0 - d) - a \sin \omega t$,respectively,where $d > 2a$. Particle $3$ of mass $m$ moves towards this system with speed $u_0 = a \omega / 2$ and undergoes an instantaneous elastic collision with particle $2$ at time $t_0$. Finally,particles $1$ and $2$ acquire a center of mass speed $v_{cm}$ and oscillate with amplitude $b$ and the same angular frequency.
$(1)$ If the collision occurs at time $t_0 = 0$,the value of $v_{cm} / (a \omega)$ will be
$(2)$ If the collision occurs at time $t_0 = \pi / (2 \omega)$,then the value of $4b^2 / a^2$ will be

$A$ vertical spring-mass system has the same time period as a simple pendulum undergoing small oscillations. Now, both of them are placed in an elevator moving downwards with an acceleration $a = 5 \,m/s^2$. The ratio of the time period of the spring-mass system to the time period of the pendulum is (Assume, acceleration due to gravity, $g = 10 \,m/s^2$)

Two particles are in $SHM$ in a straight line. Amplitude $A$ and time period $T$ of both the particles are equal. At time $t=0$,one particle is at displacement $y_1 = +A$ and the other at $y_2 = -A/2$,and they are approaching towards each other. After what time do they cross each other?

The displacement-time graph of a particle executing $S.H.M.$ is given in the figure: (sketch is schematic and not to scale). Which of the following statements is/are true for this motion?
$(A)$ The force is zero at $t = \frac{3T}{4}$
$(B)$ The acceleration is maximum at $t = T$
$(C)$ The speed is maximum at $t = \frac{T}{4}$
$(D)$ The $P.E.$ is equal to $K.E.$ of the oscillation at $t = \frac{T}{2}$

$A$ body performs linear $S$.$H$.$M$. with amplitude $a$. When it is at a distance $\frac{a}{3}$ from the extreme position,the magnitude of velocity is $\frac{1}{3}$ times the magnitude of acceleration. The period of $S$.$H$.$M$. is: