$A$ solid sphere of density $\rho_s$ which is $\eta$ 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:

$A$ 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?

$A$ $0.5\ kg$ mass of lead is submerged in a container filled to the brim with water and a block of wood floats on top. The lead mass is slowly lifted from the container by a thin wire and as it emerges into air the level of the water in the container drops a bit. The lead mass is now placed on the block of wood. As the lead is placed on the wood,what happens to the water level in the container?

$A$ metallic sphere weighing $3 \,kg$ in air is held by a string so as to be completely immersed in a liquid of relative density $0.8$. The relative density of the metal is $10$. The tension in the string is ........ $N$.

Two bodies placed on a balance are in equilibrium when immersed in water. If the mass of the first body is $36 \, g$ and its density is $9 \, g/cm^{3}$,and the mass of the second body is $48 \, g$,what is the density of the second body in $g/cm^{3}$?

$A$ 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