$A$ 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

Two cubical blocks identical in dimensions float in water in such a way that the $1$st block floats with half part immersed in water and the $2$nd block floats with $3/4$ of its volume inside the water. The ratio of densities of the blocks is ..........

$A$ wooden block floats in a liquid with $40\%$ of its volume inside the liquid. When the vessel containing the liquid starts rising upwards with an acceleration $a = g/2$, the percentage of volume inside the liquid is ......... $\%$

$A$ wide-bottomed cylindrical massless plastic container of height $9 \, cm$ contains $40$ identical coins and is floating on water with $3 \, cm$ of its height submerged. If we start placing additional identical coins on its lid,it is observed that after $N$ coins are added,the equilibrium changes from stable to unstable. Equilibrium in floating is stable if the geometric centre of the submerged portion is above the centre of mass of the object. The value of $N$ is closest to:

$A$ cube of side $40 \ cm$ is floating with $\frac{1}{4}$th of its volume immersed in water. When a circular disc is placed on the cube, it floats with $\frac{2}{5}$th of its volume immersed in water. The mass of the disc is (in $kg$)

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