When does a body sink in a liquid? Explain.

If there were no gravity, which of the following will not be there for a fluid?

$A$ 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

$A$ glass flask weighing $390 \ g$,having an internal volume of $500 \ cm^3$,just floats when half of it is filled with water. The specific gravity of the glass is:

$A$ boat carrying steel balls is floating on the surface of water in a tank. If the balls are thrown into the tank one by one,how will it affect the level of water?

$A$ solid sphere of density $\rho_s$ which is $\eta$ $( > 1)$ times lighter than water (density $\rho_w$) is suspended in a water tank by a string tied to its base as shown in the figure. If the mass of the sphere is $m$,then the tension in the string is given by: