$A$ body performs linear $S$.$H$.$M$. with amplitude $a$. When it is at a distance $\frac{a}{3}$ from the extreme position,the magnitude of velocity is $\frac{1}{3}$ times the magnitude of acceleration. The period of $S$.$H$.$M$. is:

In the figure shown,there is friction between the blocks $P$ and $Q$,but the contact between the block $Q$ and the lower surface is frictionless. Initially,the block $Q$ with block $P$ over it lies at $x=0$,with the spring at its natural length. The block $Q$ is pulled to the right and then released. As the spring-block system undergoes $S.H.M.$ with amplitude $A$,the block $P$ tends to slip over $Q$. $P$ is more likely to slip at:

$A$ 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 oscillating systems,a simple pendulum and a vertical spring-mass system,have the same time period of motion on the surface of the Earth. If both are taken to the moon,then-

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

Consider two identical cylinders (each of mass $m$,density $\rho_0$,horizontal cross-section area $s$) in equilibrium,partially submerged in two containers filled with liquids of densities $\rho_1$ and $\rho_2$ as shown in the figure. Find the period of small oscillations of this system about its equilibrium. Neglect the changes in the level of liquids in the containers. Neglect the mass of the strings. The acceleration due to gravity is $g$. ($v$ is the volume of each block).