$A$ container contains mercury $(\rho = 13.6 \text{ g/cm}^3)$ and oil $(\rho = 0.8 \text{ g/cm}^3)$. $A$ sphere floats such that half of its volume is in mercury and half of its volume is in oil. What is the density of the material of the sphere in $\text{g/cm}^3$?

$A$ concrete sphere of radius $R$ has a cavity of radius $r$ which is packed with sawdust. The specific gravities of concrete and sawdust are $2.4$ and $0.3$ respectively. For this sphere to float with its entire volume submerged under water,the ratio of the mass of concrete to the mass of sawdust will be:

An object suspended by a wire stretches it by $10 \,mm$. When the object is immersed in a liquid,the elongation in the wire reduces by $\frac{10}{3} \,mm$. The ratio of the relative densities of the object and the liquid is ............

$A$ 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

$A$ soft plastic bottle,filled with water of density $1 \text{ g/cc}$,contains an inverted glass test-tube with some air (ideal gas) trapped inside,as shown in the figure. The test-tube has a mass of $5 \text{ g}$,and it is made of thick glass with a density of $2.5 \text{ g/cc}$. Initially,the bottle is sealed at atmospheric pressure $P_0 = 10^5 \text{ Pa}$,such that the volume of the trapped air is $V_0 = 3.3 \text{ cc}$. When the bottle is squeezed from the outside at a constant temperature,the pressure inside increases and the volume of the trapped air decreases. It is observed that the test-tube begins to sink at a pressure $P_0 + \Delta P$ without changing its orientation. At this pressure,the volume of the trapped air is $V_0 - \Delta V$.
Let $\Delta V = X \text{ cc}$ and $\Delta P = Y \times 10^3 \text{ Pa}$.
$(1)$ The value of $X$ is
$(2)$ The value of $Y$ is

An object of mass $10 \ kg$ is released from rest in a liquid. If the object moves a distance of $2 \ m$ while sinking in a time duration of $1 \ s$,then the mass of the liquid displaced by the submerged object is $(g = 10 \ m/s^2)$. (in $kg$)