Pressure difference between two cross sections of a venturimeter, of area `40 cm^(2)` & `20 cm^(2)` is 480 Pascals.Find the rate of flow of water in the ventrurimeter. ( Density of water `= 1000 Kgm^(-3) ` )

Options

A. `1200cm^(3)//s`
B. `1600sqrt(2)cm^(3)//s`
C. `1600cm^(3)//s`
D. `1200sqrt(2)cm^(3)//s`

Explanation

To find the rate of flow of water in the venturimeter, we will use the principle of Bernoulli's theorem and the equation for a venturimeter.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Area of larger cross-section, \( A_1 = 40 \, \text{cm}^2 = 40 \times 10^{-4} \, \text{m}^2 \)
- Area of smaller cross-section, \( A_2 = 20 \, \text{cm}^2 = 20 \times 10^{-4} \, \text{m}^2 \)
- Pressure difference, \( \Delta P = P_1 - P_2 = 480 \, \text{Pa} \)
- Density of water, \( \rho = 1000 \, \text{kg/m}^3 \)

2. **Use the Venturimeter Equation:**
The equation relating pressure difference to flow velocity is given by:
\[ \frac{P_1 - P_2}{\rho} = \frac{v_1^2}{2} \left( \frac{A_1^2}{A_2^2} - 1 \right) \]
Rearranging gives:
\[ v_1^2 = \frac{2(P_1 - P_2)}{\rho} \left( \frac{A_2^2}{A_1^2} \right) \]

3. **Calculate the Area Ratios:**
\[ \frac{A_1^2}{A_2^2} = \left(\frac{40 \times 10^{-4}}{20 \times 10^{-4}}\right)^2 = \left(2\right)^2 = 4 \]
Therefore,
\[ \frac{A_2^2}{A_1^2} = \frac{1}{4} \]

4. **Substitute Values into the Equation:**
\[ v_1^2 = \frac{2 \times 480}{1000} \left(4 - 1\right) \]
\[ v_1^2 = \frac{960}{1000} \times 3 = 0.96 \times 3 = 2.88 \]

5. **Calculate \( v_1 \):**
\[ v_1 = \sqrt{2.88} = \frac{2\sqrt{2}}{5} \, \text{m/s} \]

6. **Calculate the Rate of Flow \( Q \):**
The rate of flow \( Q \) is given by:
\[ Q = A_1 \times v_1 \]
Substituting the values:
\[ Q = 40 \times 10^{-4} \times \frac{2\sqrt{2}}{5} = \frac{80\sqrt{2}}{5} \times 10^{-4} = 16\sqrt{2} \times 10^{-4} \, \text{m}^3/\text{s} \]

7. **Convert to \( \text{cm}^3/\text{s} \):**
\[ Q = 16\sqrt{2} \times 10^{-4} \times 10^6 = 1600\sqrt{2} \, \text{cm}^3/\text{s} \]

### Final Answer:
The rate of flow of water in the venturimeter is \( 1600\sqrt{2} \, \text{cm}^3/\text{s} \).