The maximum velocity and maximum acceleration of a particle performing a linear $S.H.M.$ are $\alpha$ and $\beta$ respectively. Then the path length of the particle is

$A$ 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?

Two particles $P$ and $Q$ describe simple harmonic motions of same period,same amplitude,along the same line about the same equilibrium position $O.$ When $P$ and $Q$ are on opposite sides of $O$ at the same distance from $O,$ they have the same speed of $1.2 \, m/s$ in the same direction. When their displacements are the same,they have the same speed of $1.6 \, m/s$ in opposite directions. The maximum velocity in $m/s$ of either particle is

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

The center of a disk of radius $r$ and mass $m$ is attached to a spring of spring constant $k$,inside a ring of radius $R > r$ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring,without slipping. The spring can only be stretched or compressed along the periphery of the ring,following Hooke's law. In equilibrium,the disk is at the bottom of the ring. Assuming small displacement of the disc,the time period of oscillation of the center of mass of the disk is written as $T = \frac{2 \pi}{\omega}$. The correct expression for $\omega$ is ($g$ is the acceleration due to gravity):

For a particle executing simple harmonic motion, match the following statements (conditions) from Column-$I$ to statements (shapes of graph) in Column-$II$.

Column-$I$	Column-$II$
$(A)$ Velocity-displacement graph $(\omega \neq 1)$	$(i)$ Straight line
$(B)$ Acceleration-displacement graph	$(ii)$ Sinusoidal
$(C)$ Acceleration-time graph	$(iii)$ Circle
$(D)$ Acceleration-velocity graph $(\omega \neq 1)$	$(iv)$ Ellipse