Correct Answer - Option 2 :

\(Y_d = \dfrac{Y}{1+w}\)
**Concept:**

**Water Content ( W ):**

It is defined as the ratio of the **weight of water** to the **weight of solid** in a given soil mass. It is usually expressed as a percentage.

\(w = \;\frac{{{W_W}}}{{{W_S}}} \times 100 = \;\frac{{W - {W_S}}}{{{W_S}}} \times 100\)

**Bulk density:**

It is defined as the ratio of the **total mass **( M ) of the soil to the **total volume** ( V ) of the soil. It is expressed in Kg / m^{3}

\(\rho = \;\frac{M}{V}\)

**Dry density:**

It is defined as the ratio of the **mass of Solids** to the **total Volume of the soil** ( in moist condition ).

\({\rho _d} = \;\frac{{{M_S}}}{V}\)

Relation between **dry unit weight,** bulk** unit weight**, and **water content ( w )**.

We know that,

\(w = \frac{{{W_W}}}{{{W_S}}}\)

\(w + 1 = \;\frac{{{W_W}}}{{{W_S} + 1}}\) =\(\frac{{{W_{W + {W_S}}}}}{W}\)

\(w + 1 = \;\frac{W}{{{W_S}}}\)

\({W_S} = \;\frac{W}{{1 + w}}\)

Dividing both sides by V, we get

\(\frac{W_s}{V} = \frac{W}{V(1+w)}\)

**\({\gamma _{d\;}} = \;\frac{\gamma }{{1 + w}}\;\)**