$A$ boat containing some iron pieces is floating in a pond. If the iron pieces are thrown into the water,the level of the water will:

$A$ pendulum has a time period $T$ in air. When it is made to oscillate in water,it acquires a time period $T' = \sqrt{2} T$. The density of the pendulum bob is (density of water $= 1$)

$A$ 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

State and prove Archimedes' principle.

$A$ 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:

$A$ hollow spherical shell of outer radius $R$ floats just submerged under the water surface. The inner radius of the shell is $r$. If the specific gravity of the shell material is $\frac{27}{8}$ with respect to water,the value of $r$ is $......R$.