$A$ horizontal board is performing simple harmonic oscillations horizontally with an amplitude of $0.3 \ m$ and a period of $4 \ s$. What is the minimum coefficient of friction between a heavy body placed on the board and the board itself if the body is not to slip?

The amplitude of a particle executing $SHM$ about $O$ is $10 \, cm.$ Then:

Two pendulums with identical bobs and lengths are suspended from a common support such that in the rest position the two bobs are in contact. After being displaced by $5^o$,the bob $A$ is released from rest at $t = 0$. Subsequently,it collides elastically head-on with the other bob $B$. Identify the graph showing the variation in energy of pendulum $A$ with time for $0 \leqslant t \leqslant T$ (where $T$ is the period of either pendulum).

$A$ system of two identical rods ($L$-shaped) of mass $m$ and length $l$ are resting on a peg $P$ as shown in the figure. If the system is displaced in its plane by a small angle $\theta$,find the period of oscillations:

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:

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