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:

Column $I$ describes some situations in which a small object moves. Column $II$ describes some characteristics of these motions. Match the situation in Column $I$ with the characteristics in Column $II$.

Column $I$	Column $II$
$(A)$ The object moves on the $x$-axis under a conservative force such that its speed $v = c_1 \sqrt{c_2 - x^2}$,where $c_1, c_2 > 0$.	$(p)$ The object executes simple harmonic motion.
$(B)$ The object moves on the $x$-axis such that its velocity $v = -kx$,where $k > 0$.	$(q)$ The object does not change its direction.
$(C)$ An object is attached to a spring in an elevator accelerating upwards with constant acceleration $a$. The motion is observed from the elevator.	$(r)$ The kinetic energy of the object keeps on decreasing.
$(D)$ The object is projected vertically upwards with speed $2 \sqrt{GM_e / R_e}$.	$(s)$ The object can change its direction only once.

$A$ body describes simple harmonic motion with an amplitude of $5\; cm$ and a period of $0.2\; s$. Find the acceleration and velocity of the body when the displacement is $(a)\; 5\; cm$,$(b)\; 3\; cm$,and $(c)\; 0\; cm$.

Determine whether the following statements are True or False:
$1.$ The acceleration of $SHO$ at the mean position is maximum.
$2.$ The mechanical energy of $SHO$ depends on the maximum displacement.
$3.$ The periodic time for a seconds pendulum is $1 \, s$.
$4.$ If the frequency of $SHM$ is $v$,then the frequency of kinetic energy is also $v$.

$Assertion :$ 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 $\frac{\pi }{2}$.

Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along the horizontal axis and momentum is plotted along the vertical axis. The phase space diagram is the $x(t)$ vs. $p(t)$ curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum upwards (or to the right) is positive and downwards (or to the left) is negative.
$1.$ The phase space diagram for a ball thrown vertically up from the ground is:
$2.$ The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and $E_1$ and $E_2$ are the total mechanical energies respectively. Then:
$(A) E_1 = \sqrt{2} E_2$
$(B) E_1 = 2 E_2$
$(C) E_1 = 4 E_2$
$(D) E_1 = 16 E_2$
$3.$ Consider the spring-mass system, with the mass submerged in water, as shown in the figure. The phase space diagram for one cycle of this system is:
Give the answer for questions $1, 2,$ and $3.$