As per the given figures,two springs of spring constants $K$ and $2K$ are connected to a mass $m$. If the period of oscillation in figure $(a)$ is $3 \text{ s}$,then the period of oscillation in figure $(b)$ will be $\sqrt{x} \text{ s}$. The value of $x$ is $.........$

Two bodies $M$ and $N$ of equal masses are suspended from two separate massless springs of force constants $k_1$ and $k_2$ respectively. If the two bodies oscillate vertically such that their maximum velocities are equal,the ratio of the amplitude of $M$ to that of $N$ is

$A$ weightless spring of length $60\, cm$ and force constant $200\, N/m$ is kept straight and unstretched on a smooth horizontal table and its ends are rigidly fixed. $A$ mass of $0.25\, kg$ is attached at the middle of the spring and is slightly displaced along the length. The time period of the oscillation of the mass is

$A$ $3 \ kg$ block is connected as shown in the figure. The spring constants of the two springs $K_1$ and $K_2$ are $50 \ Nm^{-1}$ and $150 \ Nm^{-1}$ respectively. The block is released from rest with the springs unstretched. The acceleration of the block in its lowest position is $(g=10 \ ms^{-2})$. (in $ms^{-2}$)

One end of a long metallic wire of length $L$,area of cross-section $A$,and Young's modulus $Y$ is tied to the ceiling. The other end is tied to a massless spring of force constant $K$. $A$ mass $m$ hangs freely from the free end of the spring. It is slightly pulled down and released. Its time period is given by

$A$ plank with a small block on top of it is undergoing vertical $SHM$. Its period is $2 \ s$. The minimum amplitude at which the block will separate from the plank is: