$A$ stream-lined body falls through air from a height $h$ on the surface of a liquid. Let $d$ and $D$ denote the densities of the materials of the body and the liquid respectively. If $D$ > $d$, then the time after which the body will be instantaneously at rest is

$A$ cube of external side length $10 \ cm$ has an inner cubical cavity of side length $5 \ cm$. The density of the inner portion is twice that of the outer portion. If this cube is just floating in a liquid of density $2 \ g/cm^3$, find the density of the inner portion.

$A$ tub is filled with water and a wooden cube $10 \ \text{cm} \times 10 \ \text{cm} \times 10 \ \text{cm}$ is placed in the water. The wooden cube is found to float on the water with a part of it submerged in water. When a metal coin is placed on the wooden cube, the submerged part is increased by $3.87 \ \text{cm}$. The mass of the metal coin is . . . . . . gram. (Take water density as $1 \ \text{g/cm}^3$ and density of wood as $0.4 \ \text{g/cm}^3$)

Determine the equation for the volume of the partially immersed part of a body floating in a fluid.

$A$ metal ball of density $7800 \ kg/m^3$ is suspected to have a large number of cavities. It weighs $9.8 \ kg$ when weighed in air and $1.5 \ kg$ less when immersed in water. The fraction by volume of the cavities in the metal ball is approximately ....... $\%$

$A$ solid sphere of radius $R$ and density $\rho$ is attached to one end of a massless spring of force constant $k$. The other end of the spring is connected to another solid sphere of radius $R$ and density $3\rho$. The complete arrangement is placed in a liquid of density $2\rho$ and is allowed to reach equilibrium. The correct statement$(s)$ is (are):
$(A)$ The net elongation of the spring is $\frac{4 \pi R^3 \rho g}{3 k}$
$(B)$ The net elongation of the spring is $\frac{8 \pi R^3 \rho g}{3 k}$
$(C)$ The light sphere is partially submerged.
$(D)$ The light sphere is completely submerged.