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 masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system (see figure). Because of the torsional constant $k$,the restoring torque is $\tau = k\theta$ for an angular displacement $\theta$. If the rod is rotated by $\theta_0$ and released,the tension in it when it passes through its mean position 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, $g = 10 \,m/s^2$)

Fill in the blanks:
$1.$ If a tunnel is dug along the diameter of the Earth and a body is dropped freely in it,the motion of this body is .........,assuming there is no frictional force of the medium acting on it.
$2.$ When a body executes simple harmonic motion and makes $\frac{1}{2\pi}$ oscillations,its phase increases by ......... $rad$.
$3.$ For a body executing simple harmonic motion,the phase increases by ......... every second.
$4.$ The ratio of force constants of two springs is $1:2$ and the ratio of their mechanical energies is $2:9$. The ratio of the amplitudes of the two bodies suspended at the ends of the springs is .........

When a body is in $S.H.M.$,match the following:

List-$I$	List-$II$
$A$. Velocity is maximum	$I$. Acceleration is maximum
$B$. $K.E.$ is $\left(\frac{3}{4}\right)^{\text{th}}$ of total energy	$II$. At mean position
$C$. $P.E.$ is $\left(\frac{3}{4}\right)^{\text{th}}$ of total energy	$III$. At half of the amplitude
$D$. Acceleration is maximum	$IV$. At $\frac{\sqrt{3}}{2}$ times the amplitude

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 $3.$