$A$ $1\,kg$ 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.

An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass $M$. The piston and the cylinder have equal cross-sectional area $A$. When the piston is in equilibrium, the volume of the gas is $V_0$ and its pressure is $P_0$. The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surroundings, the piston executes a simple harmonic motion with frequency

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:

$Assertion :$ In simple harmonic motion,the velocity is maximum when the acceleration is minimum.
$Reason :$ Displacement and velocity of $S.H.M.$ differ in phase by $\frac{\pi }{2}$.

As shown in the figures,a uniform rod $OO^{\prime}$ of length $l$ is hinged at the point $O$ and held in place vertically between two walls using two massless springs of same spring constant $K$. The springs are connected at the midpoint and at the top-end $(O^{\prime})$ of the rod,as shown in Fig. $1$,and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is $f_1$. On the other hand,if both the springs are connected at the midpoint of the rod,as shown in Fig. $2$,and the rod is made to oscillate by a small angular displacement,then the frequency of oscillation is $f_2$. Ignoring gravity and assuming motion only in the plane of the diagram,the value of $\frac{f_1}{f_2}$ is:

In the figure shown,there is friction between the blocks $P$ and $Q$,but the contact between the block $Q$ and the lower surface is frictionless. Initially,the block $Q$ with block $P$ over it lies at $x=0$,with the spring at its natural length. The block $Q$ is pulled to the right and then released. As the spring-block system undergoes $S.H.M.$ with amplitude $A$,the block $P$ tends to slip over $Q$. $P$ is more likely to slip at: