Relation Between Dry unit Weight, Bulk unit Weight & Water Content

In this article,we will make the formula/relation between the dry unit weight( $\gamma_{d}$ ), bulk unit weight( $\gamma$ ) and water content(w).

[Note: Bulk unit weight means bulk density, and dry unit weight means dry density. So, be careful if the question is like that; derive the relation between bulk density, dry density and water content, therefore, the answer will be same.]

Let,

W = Total weight of given soil mass.
V = Total volume of the given soil mass.
$W_{d}$ = Weight of soil solid in a given soil mass.
$W_{w}$ = Weight of water present in the given soil mass.
w = Moisture content or Water content.

From the definition of water content, we can write,

$w = \frac{W_{w}}{W_{d}}$
[Add 1 in both sides]
So, $1 + w = 1 + \frac{W_{w}}{W_{d}}$

Or, $1 + w = \frac{W_{w} + W_{d}}{W_{d}}$
[ We know, Total weight = Weight of water + Weight of soil solid. So, $W = W_{w} + W_{d}$ ]

Or, $1 + w = \frac{W}{W_{d}}$
Or, $W_{d} = W/(1+w)$

Again from the definition of dry unit weight, We have,

$\gamma_{d} = \frac{W_{d}}{V}$
[Now, put the value of $W_{d}$ , which we have got from the above equation]
Hence, $\gamma_{d} = \frac{W}{V(1+w)}$
[We know, $\gamma = \frac{W}{V}$ ]
Or, $\gamma_{d} = \frac{\gamma }{(1+w)}$