$A$ horizontal platform with an object placed on it is executing $S.H.M.$ in the vertical direction. The amplitude of oscillation is $3.92 \times 10^{-3} \, m$. What must be the least period of these oscillations,so that the object is not detached from the platform (in $, s$)?

Match $List-I$ with $List-II$:
| | $List-I$ ($x-y$ graphs) | | $List-II$ (Situations) |
|---|---|---|---|
| $(a)$ | Damped oscillation graph | $(i)$ | Total mechanical energy is conserved |
| $(b)$ | Linear graph $y = -kx$ | $(ii)$ | Bob of a pendulum is oscillating under negligible air friction |
| $(c)$ | Simple harmonic motion graph | $(iii)$ | Restoring force of a spring |
| $(d)$ | Energy conservation graph ($K$.$E$. and $P$.$E$. curves) | $(iv)$ | Bob of a pendulum is oscillating along with air friction |
Choose the correct answer from the options given below:

For a particle executing $S.H.M.$,where $x$ is the displacement from the equilibrium position,$v$ is the velocity at any instant,and $a$ is the acceleration at any instant,then:

Two oscillating systems,a simple pendulum and a vertical spring-mass system,have the same time period of motion on the surface of the Earth. If both are taken to the moon,then-

$A$ $1\,kg$ mass is attached to a spring of force constant $600\,N/m$ and rests on a smooth horizontal surface with the other end of the spring tied to a wall as shown in the figure. $A$ second mass of $0.5\,kg$ slides along the surface towards the first at $3\,m/s$. If the masses make a perfectly inelastic collision,find the amplitude and time period of oscillation of the combined mass.

Due to some force $F_1$,a body oscillates with a period of $4/5 \, s$,and due to another force $F_2$,it oscillates with a period of $3/5 \, s$. If both forces act simultaneously,the new period will be .... $s$.