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 . . . . . . .

$A$ stiff spring having spring constant $k = 400 \text{ N/m}$ is attached to the floor vertically. $A$ mass $m = 10 \text{ kg}$ is placed on top of the spring. The block oscillates if it is pressed downward and released. Find the extension in the spring at which the block loses contact with the spring. (Take $g = 10 \text{ m/s}^2$) (in $\text{ cm}$)

$A$ spring has a certain mass suspended from it and its period for vertical oscillation is $T$. The spring is now cut into two equal halves and the same mass is suspended from one of the halves. The period of vertical oscillation is now

$A$ $5\; kg$ collar is attached to a spring of spring constant $500\; N m^{-1}$. It slides without friction over a horizontal rod. The collar is displaced from its equilibrium position by $10.0\; cm$ and released. Calculate
$(a)$ the period of oscillation.
$(b)$ the maximum speed and
$(c)$ maximum acceleration of the collar.

$A$ particle of mass $m$ is attached to three identical massless springs of spring constant $k$ as shown in the figure. The time period of vertical oscillation of the particle is

$A$ block is fastened to a horizontal spring. The block is pulled to a distance $x = 10 \, cm$ from its equilibrium position (at $x = 0$) on a frictionless surface from rest. The total energy of the block at $x = 5 \, cm$ is $0.25 \, J$. The spring constant of the spring is $......... \, N \, m^{-1}$.