Explain the concepts of density and relative density, and describe how a U-tube can be used to determine the relative density of a liquid.

a) Density is a measure of how much mass is contained in a given volume. It tells you how compact a substance is.
The formula for density ($\rho$) is:

$$\rho = \frac{\text{mass}}{\text{volume}}$$

The SI unit for density is kilograms per cubic meter ($\text{kg/m}^3$).

b) Relative density (also known as specific gravity) is the ratio of the density of a substance to the density of a reference substance. For liquids and solids, the reference substance is typically water at $4^\circ\text{C}$. For gases, the reference is usually air.
Since it is a ratio of two densities, relative density has no units.

$$\text{Relative density} = \frac{\text{Density of substance}}{\text{Density of reference substance}}$$

c) A U-tube is a simple glass tube bent into a 'U' shape. To use it for a relative density experiment:

1. Two immiscible liquids (liquids that do not mix, like oil and water) are poured into the U-tube.
2. The denser liquid will settle at the bottom, forming an interface with the less dense liquid in both arms of the U-tube.
3. The heights of the liquid columns are measured from the common interface to the free surface of each liquid. Let these heights be $h_1$ and $h_2$, and their respective densities be $\rho_1$ and $\rho_2$.
4. At equilibrium, the pressure exerted by both columns at the interface is equal. This means: $$\rho_1 h_1 g = \rho_2 h_2 g$$ Where $g$ is the acceleration due to gravity.
5. The relative density of liquid 1 with respect to liquid 2 can then be found by: $$\frac{\rho_1}{\rho_2} = \frac{h_2}{h_1}$$ So, the relative density is the inverse ratio of their heights.