Question

The critical depth in a 2 m wide rectangular channel carrying a discharge of 10 m³/s and taking $g = 9.81$ m/s² is __________ (in m, rounded off to two decimal places).

Hint:

Critical depth depends only on unit discharge and gravity, not on channel slope or roughness. It represents the unique depth where specific energy is minimum.

Answer 1

Correct Answer: 1.35

Step 1: Formula for critical depth in a rectangular channel.
For a rectangular channel, \[ y_c = \left(\frac{q^2}{g}\right)^{1/3} \] where $q$ is the discharge per unit width.
Step 2: Find the discharge per unit width.
\[ q = \frac{Q}{b}, Q = 10 \, m^3/s, \, b = 2 \, m \] \[ q = \frac{10}{2} = 5 \, m^2/s \]
Step 3: Substitute values into formula.
\[ y_c = \left(\frac{q^2}{g}\right)^{1/3} = \left(\frac{5^2}{9.81}\right)^{1/3} \] \[ = \left(\frac{25}{9.81}\right)^{1/3} = (2.548)^{1/3} \]
Step 4: Cube root calculation.
\[ (2.548)^{1/3} \approx 1.366 \, m \] Rounded to two decimal places: $1.37 \, m$. Final Answer: \[ \boxed{1.37 \, m} \]