$A$ cylinder of mass $m$ and material density $\rho$ hanging from a string is lowered into a vessel of cross-sectional area $A$ containing a liquid of density $\sigma$ $(< \rho)$ until it is fully immersed. The increase in pressure at the bottom of the vessel is

The fraction of a floating object of volume $V_0$ and density $d_0$ above the surface of a liquid of density $d$ will be

$A$ cubical block is floating in a liquid with one-fourth of its volume immersed in the liquid. If the whole system accelerates upward with an acceleration of $g / 4$,what will be the fraction of the volume immersed in the liquid?

$A$ hot air balloon is carrying some passengers and a few sandbags of mass $1 kg$ each,so that its total mass is $480 kg$. Its effective volume providing buoyancy is $V$. The balloon is floating at an equilibrium height of $100 m$. When $N$ number of sandbags are thrown out,the balloon rises to a new equilibrium height of $150 m$ with its volume $V$ remaining unchanged. If the variation of the density of air with height $h$ from the ground is $\rho(h) = \rho_0 e^{-\frac{h}{h_0}}$,where $\rho_0 = 1.25 kg m^{-3}$ and $h_0 = 6000 m$,the value of $N$ is:

$A$ boat carrying a number of large stones is floating in a water tank. What would happen to the water level if a few stones are unloaded into the water?

$A$ solid cube is floating in a liquid. The coefficient of linear expansion of the cube is $\alpha$ and the coefficient of volume expansion of the liquid is $\gamma$. On increasing the temperature of the (liquid + cube) system,the cube will sink if