A thin uniform vertical rod of mass $m$ and length $l$ pivoted at point $O$ is shown in the given figure. The system is placed on smooth horizontal table. The combined stiffness of the springs is equal to $k.$ The mass of the spring is negligible. The frequency of small oscillation is

A block $A$ is connected to spring and performs simple harmonic motion with a time period of $2 \mathrm{s}.$ Another block $B$ rests on$A.$ The coefficient of static friction between $A$ and $B$ is ${\mu }_s=0.6.$ the maximum amplitude of oscillation which the system can have so that there is no relative motion between $A$ and $B$ is (take ${\pi }^2=g=10$)

Two springs, each of unstretched length $20 \mathrm{cm}$ but having different spring constants $k_1=1000 {\mathrm{Nm}}^{-1}$ and $k_2=3000 {\mathrm{Nm}}^{-1},$ are attached to two opposite faces of a small block of mass $m=100 \mathrm{g}$ kept on a smooth horizontal surface as shown in the given figure.

The outer ends of the two springs are now attached to two pins $P_1$ and $P_2$ whose locations are shown in the figure. As a result of this, the block acquires a new equilibrium position. The block has been displaced by small amount from its equilibrium position and released to perform simple harmonic motion; then

The coefficient of friction between block of mass $m$ and $2\mathrm{m}$ is $\mu =2\tan \left(\theta \right)$. There is no friction between a block of mass $2\mathrm{m}$ and inclined plane. The maximum amplitude of two block system for which there is no relative motion between both the blocks-

One end of a spring of force constant $K$ is fixed to a vertical wall and the other to a body of mass $m$ resting on a smooth horizontal surface. There is another wall at a distance $x_0$ from the body. The spring is then compressed by $3x_0$ and released. The time taken to strike the wall from the instant of release is (given ${\sin }^{-1}(1/3)=(\pi /9)$

A uniform stick of mass $M$ and length $L$ is pivoted at its centre. Its ends are tied to two springs each of force constant $K.$ In the position shown in figure, the springs are in their natural length. When the string is displaced through a small angle $\theta$ and released, the stick