A homogeneous solid cylinder of length $L$$(L < H/2)$. Cross-sectional area $A/5$ is immersed such that it floats with its axis vertical at the liquid-liquid interface with length $L/4$ in the denser liquid as shown in the fig. The lower density liquid is open to atmosphere having pressure ${P_0}$. Then density $D$  of solid is given by

A hydraulic automobile lift is designed to lift cars with a maximum mass of $3000\, kg$. The area of cross section a of piston carrying the load is $425\, cm ^{2}$. What is the maximum pressure () would smaller piston have to bear ?

A wooden cube first floats inside water when a $200\,g$ mass is placed on it. When the mass is removed the cube is $2\,cm$ above water level. The side of cube is ........ $cm$

A thin vertical uniform wooden rod is pivoted at the top and immersed in water as shown. The container is slowly raised. At a certain moment, the equilibrium becomes unstable. If density of water is $9/5$ times the density of wood, then ratio of total length of rod to the submerged length of rod, at that moment is

A fluid container is containing a liquid of density $\rho $ is accelerating upward with acceleration a along the inclined place of inclination $\alpha$  as shown. Then the angle of inclination $ \theta $ of free surface is :

A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If $\rho_0$ is the relative density of the material of the shell with respect to water, then the correct statement is that the shell is