$A$ flat horizontal board moves up and down in $SHM$ with an amplitude $\alpha$. What is the shortest permissible time period of the vibration such that an object placed on the board does not lose contact with the board?

Figure $(a)$ shows a spring of force constant $k$ clamped rigidly at one end and a mass $m$ attached to its free end. $A$ force $F$ applied at the free end stretches the spring. Figure $(b)$ shows the same spring with both ends free and attached to a mass $m$ at either end. Each end of the spring in Figure $(b)$ is stretched by the same force $F$.
$(a)$ What is the maximum extension of the spring in the two cases?
$(b)$ If the mass in Figure $(a)$ and the two masses in Figure $(b)$ are released,what is the period of oscillation in each case?

$A$ uniform cylinder of length $L$ and mass $M$ having cross-sectional area $A$ is suspended,with its length vertical,from a fixed point by a massless spring,such that it is half submerged in a liquid of density $\sigma$ at equilibrium position. When the cylinder is given a downward push and released,it starts oscillating vertically with a small amplitude. The time period $T$ of the oscillations of the cylinder will be

In the adjacent figure,if the inclined plane is smooth and the springs are identical,then the period of oscillation of this body is

If a watch with a wound spring is taken on to the moon,it

The total spring constant of the system as shown in the figure will be