An orifice of diameter 8 mm is made on one side of a tank in which water level is 10 mm above the orifice. What is the rate of discharge of water through the orifice?

Explanation

To solve the problem of determining the rate of discharge of water through an orifice, we will follow these steps:

### Step 1: Calculate the Area of the Orifice
The formula for the area \( A \) of a circular orifice is given by:
\[ A = \frac{\pi D^2}{4} \]
where \( D \) is the diameter of the orifice.

Given:
- Diameter \( D = 8 \text{ mm} = 8 \times 10^{-3} \text{ m} \)

Substituting the values:
\[ A = \frac{\pi (8 \times 10^{-3})^2}{4} \]
Calculating \( A \):
\[ A = \frac{3.14 \times (64 \times 10^{-6})}{4} = \frac{3.14 \times 64}{4} \times 10^{-6} \]
\[ A = \frac{200.96}{4} \times 10^{-6} = 50.24 \times 10^{-6} \text{ m}^2 \]

### Step 2: Calculate the Velocity of Water Flowing Through the Orifice
The velocity \( V \) of water flowing through the orifice can be calculated using Torricelli's theorem:
\[ V = \sqrt{2gh} \]
where:
- \( g = 9.8 \text{ m/s}^2 \) (acceleration due to gravity)
- \( h = 10 \text{ mm} = 10 \times 10^{-3} \text{ m} \)

Substituting the values:
\[ V = \sqrt{2 \times 9.8 \times (10 \times 10^{-3})} \]
Calculating \( V \):
\[ V = \sqrt{2 \times 9.8 \times 0.01} = \sqrt{0.196} \approx 0.443 \text{ m/s} \]

### Step 3: Calculate the Rate of Discharge
The rate of discharge \( Q \) is given by the formula:
\[ Q = A \times V \]
Substituting the values of \( A \) and \( V \):
\[ Q = (50.24 \times 10^{-6}) \times (0.443) \]
Calculating \( Q \):
\[ Q \approx 22.26 \times 10^{-6} \text{ m}^3/\text{s} \]

### Final Answer
The rate of discharge of water through the orifice is approximately:
\[ Q \approx 22.26 \times 10^{-6} \text{ m}^3/\text{s} \]

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