$A$ metallic body of material with density of $8000\ kg/m^3$ has a cavity inside. $A$ spring balance shows its mass to be $10.0\ kg$ in air and $7.5\ kg$ when immersed in water. The ratio of the volume of the cavity to the volume of the material of the body must be

An ice cube contains a large air bubble. The cube is floating on the horizontal surface of water contained in a trough. What will happen to the water level when the cube melts?

$A$ wax candle floats vertically in a liquid of density twice that of wax. The candle burns at the rate of $4\ cm/hr$. Then, with respect to the surface of the liquid, the upper end of the candle will:

Why do bubbles rise in a soda water bottle?

$A$ 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

$A$ dumbbell is placed in water of density $\rho$. It is observed that by attaching a mass $m$ to the rod,the dumbbell floats with the rod horizontal on the surface of water and each sphere exactly half submerged as shown in the figure. The volume of the mass $m$ is negligible. The value of length $l$ is