Final answer:
The equation γ = (e S γ) / (1+e) relates the unit weight of soil to its void ratio, degree of saturation, and the unit weight of water, and is used in soil mechanics to describe the weight characteristics of a given soil sample.
Step-by-step explanation:
The equation γ = (e S γ) / (1+e) is a fundamental relationship in soil mechanics that relates the unit weight of the soil (γ) to its void ratio (e), degree of saturation (S), and the unit weight of water (γw). This equation is derived from the definition of bulk unit weight and the concepts of porosity and saturation within the soil matrix. The unit weight of soil is a measure of its weight per unit volume, including both solids and voids (pores filled with air or water).
To derive the equation, consider a volume of soil that consists of solids and voids. The voids can either be filled with air or water. If we assume the soil is fully saturated, all voids are filled with water. The void ratio (e) is the volume of voids divided by the volume of solids. For fully saturated soil, the unit weight of the soil can be given as the weight of the solids plus the weight of water in the voids, divided by the total volume. Here, S is the degree of saturation, which is the proportion of the void space filled with water.
By manipulating these relationships, it can be shown that the total unit weight of the soil is γ = (e S γw) / (1+e), where γw is the unit weight of water. Thus, the equation provides a way to calculate the unit weight of a particular soil when its void ratio, saturation, and the unit weight of water are known.