An object of volume $V_0$ and density $d_0$ is placed in a liquid of density $d$. What fraction of its volume remains outside the liquid?

To determine the composition of a bimetallic alloy, a sample is first weighed in air and then in water. These weights are found to be $w_1$ and $w_2$ respectively. If the densities of the two constituent metals are $\rho_1$ and $\rho_2$ respectively, then the weight of the first metal in the sample is (where $\rho_w$ is the density of water):

$A$ solid floats such that its $(1/3)$rd part is above the water surface. Then,the density of the solid is

$A$ sphere of solid material of relative density $9$ has a concentric spherical cavity and floats just submerged in water. If the radius of the sphere is $R$,then the radius of the cavity $(r)$ is related to $R$ as:

$Assertion :$ The buoyant force on a submerged rigid object can be considered to be acting at the centre of mass of the object.
$Reason :$ For a rigid body,a force field distributed uniformly through its volume can be considered to be acting at the centre of mass of the body.

$A$ large ship can float but a steel needle sinks because of