A cylindrical block of density $d$ stays fully immersed in a beaker filled with two immiscible liquids of different densities $d_1$ and $d_2.$ The block is in equilibrium with half of it in liquid $1$ and the other half in liquid $2$ as shown in the given figure. If the block is given a displacement downwards and released, then neglecting friction study the following statements.

As shown in the figure a horizontal platform with a mass $m$ placed on it is executing SHM along $y$-axis. If the amplitude of oscillation is $2.5 \mathrm{cm}$, the minimum period of the motion for the mass not to be detached from the platform is: $(g= 10 \mathrm{m} {\sec }^{-2} = {\mathrm{\pi }}^2)$ 

A small body of mass $m$ is connected to two horizontal springs of elastic constant $k$, natural length $3 d / 4$. In the equilibrium position both springs are stretched to length $d$, as shown in the given figure. What wil be the ratio of period of the motion $\left(T_b/T_0\right)$ if the body is displaced horizontally by a small distance where $T_a$ is the time period when the particle oscillates along the line of springs and $T_b$ is time period when the particle oscillates perpendicular to the plane of the figure? Neglect effects of gravity.

A uniform disc of mass $m$ and radius $R$ is pivoted smoothly at its centre of mass. A light spring of stiffness $k$ is attached with the disc tangentially as shown in the following figure. Find the angular frequency in $\mathrm{rad}/\mathrm{s}$ of torsional oscillations of the disc. (Take $m=5 \mathrm{kg}$ and $K=10 \mathrm{N}/\mathrm{m}$.)

A uniform disc of mass $m$ and radius $R=\frac{80}{23{\pi }^2}m$ is pivoted smoothly at $P$ If a uniform disc of mass $m$ and radius $R$ is welded at the lowest point of the disc, find the period of SHM of the system (disc + ring). (in seconds)