$A$ rectangular block of mass $M$ and cross-sectional area $A$ floats on a liquid of density $\rho$. It is given a small vertical displacement from equilibrium; it starts oscillating with frequency $n$. Then:

$A$ wooden block of volume $1000\, cm^3$ is suspended from a spring balance. It weighs $12\, N$ in air. It is suspended in water such that half of the block is below the surface of water. The reading of the spring balance is ...... $N$.

$A$ 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

The weight indicated on a balance is $X$ when a beaker of water is placed on it. $A$ solid object has weight $Y$ in air and displaces weight $Z$ of water when completely immersed. The given diagram shows the object suspended from a light string and completely immersed in the beaker of water. What is the balance reading in the given arrangement?

$A$ 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

$A$ cube of ice has an iron piece frozen inside it. The cube floats in a beaker filled with water. When the ice melts, the level of water in the beaker: