$A$ body floats in water with $40 \%$ of its volume outside water. When the same body floats in an oil,$60 \%$ of its volume remains outside oil. The relative density of oil is

$A$ solid sphere of specific gravity $27$ has a concentric spherical cavity and it just sinks in water. The ratio of the cavity radius to the outer radius of the sphere is

$A$ piece of metal weighs $45 \ g$ in air and $25 \ g$ in a liquid of density $1.5 \times 10^3 \ kg \ m^{-3}$ kept at $30^{\circ} C$. When the temperature of the liquid is raised to $40^{\circ} C$,the metal piece weighs $27 \ g$. The density of the liquid at $40^{\circ} C$ is $1.25 \times 10^3 \ kg \ m^{-3}$. The coefficient of linear expansion of the metal is:

$A$ cube of ice has an iron piece frozen inside it. The cube floats in a beaker filled with water. When the ice melts, the level of water in the beaker:

$A$ cubical block of wood having a mass of $160 \,g$ has a metal piece fastened underneath as shown in the figure. Find the maximum mass of the metal piece which will allow the block to float in water. The specific gravity of wood is $0.8$, the specific gravity of the metal is $10$, and the density of water is $1 \,g/cm^3$. (in $\,g$)

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