In a constant head permeameter, having cross-sectional area of $20 \, \text{cm}^2$, when the flow was taking place under a hydraulic gradient of $0.5$, the amount of water collected is $1200 \, \text{cm}^3$ in $60 \, \text{sec}$. The permeability of the soil is:
Step 1: State Darcy's Law.
For a constant head test, the flow rate ($Q$) is given by: \[Q = k \cdot i \cdot A \cdot t\] where: $Q =$ volume of water collected, $k =$ coefficient of permeability, $i =$ hydraulic gradient, $A =$ cross-sectional area, $t =$ time.
Step 2: Input the given values.
We are given: \[Q = 1200 \, \text{cm}^3, A = 20 \, \text{cm}^2, i = 0.5, t = 60 \, \text{s}.\]
Step 3: Rearrange the formula to solve for $k$.
Rearranging Darcy's law, we get the formula for $k$: \[k = \frac{Q}{A \cdot i \cdot t} \] Substituting the values: \[= \frac{1200}{20 \cdot 0.5 \cdot 60} = \frac{1200}{600} = 0.2 \, \text{cm/sec}.\]
Step 4: Final Result.
The permeability of the soil is determined to be $0.2 \, \text{cm/sec}$.