Question

A rectangular gate of width 2 m and height 3 m lies in a vertical plane, the top of which is at the surface of water. If the unit weight of water is 10 kN/m$^3$, then the force exerted by the water in the gate is

• A. 20 kN
• B. 30 kN
• C. 60 kN
• D. 90 kN

Hint:

The formula for hydrostatic force is "Pressure at the centroid times the Area".
$F = P_{centroid} \times A = (w \bar{h}) \times A$.
Remember that $\bar{h}$ is the vertical depth to the centroid, not the slant distance.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks to calculate the total hydrostatic force (or thrust) on a vertical rectangular gate submerged in water.

Step 2: Key Formula or Approach:
The total hydrostatic force ($F$) exerted on a submerged plane surface is given by the formula:
\[ F = \rho g A \bar{h} = w A \bar{h} \] where:
$w = \rho g$ = unit weight of the fluid
$A$ = area of the submerged surface
$\bar{h}$ = vertical depth from the free surface to the centroid of the submerged area.

Step 3: Detailed Explanation:
First, let's find the values for each term in the formula.

• Unit weight of water ($w$) = 10 kN/m$^3$.

• The gate is rectangular with width = 2 m and height = 3 m.

• Area of the gate ($A$) = width $\times$ height = $2 \text{ m} \times 3 \text{ m} = 6 \text{ m}^2$.

• The gate is vertical, and its top edge is at the water surface. The centroid of a rectangle is at its geometric center, which is at half its height.

• The depth to the centroid ($\bar{h}$) = height / 2 = $3 \text{ m} / 2 = 1.5$ m.


Now, substitute these values into the force formula:
\[ F = w A \bar{h} \] \[ F = (10 \text{ kN/m}^3) \times (6 \text{ m}^2) \times (1.5 \text{ m}) \] \[ F = 60 \times 1.5 \text{ kN} \] \[ F = 90 \text{ kN} \]

Step 4: Final Answer:
The force exerted by the water on the gate is 90 kN.