$A$ small mass $m$ is suspended at the end of a wire having negligible mass,length $L$,and cross-sectional area $A$. The frequency of oscillation for the $S.H.M.$ along the vertical line is ($Y =$ Young's modulus of the wire).

If the period of oscillation of mass $m$ suspended from a spring is $2 \ s$,then the period of oscillation of suspended mass $4m$ with the same spring will be: (in $s$)

$A$ body of mass $1 \,kg$ is suspended from a spring of negligible mass. Another body of mass $500 \,g$ moving vertically upwards hits the suspended body with a velocity of $3 \,ms^{-1}$ and gets embedded in it. If the frequency of oscillation of the system of the two bodies after collision is $\frac{10}{\pi} \,Hz$,the amplitude of the motion and the spring constant are respectively,

$A$ spring of length $l$ and force constant $k$ is attached to a mass $m$ and oscillates with frequency $f_1$. If the spring is cut into two equal parts and one part is attached to the same mass $m$ to oscillate,its frequency becomes $f_2$. Find the relation between $f_1$ and $f_2$.

Two identical particles each of mass $m$ are interconnected by a light spring of stiffness $k$. The time period for small oscillations is equal to:

$A$ block of mass $m$ is attached to two springs of spring constants $k_1$ and $k_2$ as shown in the figure. The block is displaced by $x$ towards the right and released. The velocity of the block when it is at $x/2$ will be