$A$ person normally weighing $50\, kg$ stands on a massless platform which oscillates up and down harmonically at a frequency of $2.0\, s^{-1}$ and an amplitude $5.0\, cm$. $A$ weighing machine on the platform gives the person's weight against time.
$(a)$ Will there be any change in the weight of the body during the oscillation?
$(b)$ If the answer to part $(a)$ is yes,what will be the maximum and minimum reading on the machine and at which positions?

$A$ 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 $, s$)?

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 $3.$

The displacement-time graph of a particle executing $SHM$ is shown. Which of the following statements is wrong?

$A$ system is oscillating with undamped simple harmonic motion. Then the

The energy of a particle executing simple harmonic motion is given by $E = Ax^2 + Bv^2$,where $x$ is the displacement from mean position $x = 0$ and $v$ is the velocity of the particle at $x$. Choose the incorrect statement.