$A$ mass $m$ is suspended from a spring of force constant $k$ and just touches another identical spring fixed to the floor as shown in the figure. The time period of small oscillations is

Find the ratio of time periods of two identical springs if they are first joined in series and then in parallel and a mass $m$ is suspended from them.

$A$ 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,

$A$ flat horizontal board moves up and down in $S.H.M.$ vertically with amplitude $A$. The shortest permissible time period of the vibration such that an object placed on the board may not lose contact with the board is ..........

The frequency of oscillation of a mass $m$ suspended by a spring is $v_1$. If the length of the spring is cut to half,the same mass oscillates with frequency $v_2$. The value of $v_2/v_1$ is . . . . . . .

If two similar springs each of spring constant $K$ are joined in series,the new spring constant and time period would be changed by a factor of: