A metallic body of material with density of $8000\  kg/m^3$ has a cavity inside. A spring balance shows its mass to be $10.0\  kg$ in air and $7.5\  kg$ when immersed in water. The ratio of the volume of the cavity to the volume of the material of the body must be

A spherical ball of radius $r$ and relative density $0.5$ is floating in equilibrium in water with half of it immersed in water. The work done in pushing the ball down so that whole of it is just immersed in water is : (where $\rho $ is the density of water)

A solid cube and a solid sphere both made of same material are completely submerged in water but to different depths. The sphere and the cube have same surface area. The buoyant force is

A fire hydrant delivers water of density $\rho $ at a volume rate $L$. The water travels vertically upward through the hydrant and then does $90^o$ turn to emerge horizontally at speed $V$. The pipe and nozzle have uniform cross-section throughout. The force exerted by the water on the corner of the hydrant is

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

Two solids $A$  and $ B$  float in water. It is observed that $A$  floats with $\frac{1}{2}$ of its body immersed in water and $ B$  floats with $\frac{1}{4}$ of its volume above the water level. The ratio of the density of $ A$  to that of $B$  is