A horizontal pipeline carries water in a streamline flow. At a point along the pipe, where the cross- sectional area is `10cm^2`, the water velocity is `1ms^-1` and the pressure is 2000 Pa. The pressure of water at another point where the cross-sectional area is `5cm^2`, is........Pa. (Density of water `=10^3kg.m^-3`)

Options

A. 4000 Pa
B. 2000 Pa
C. 1000 Pa
D. 500 Pa

Explanation

To solve the problem, we will use the principles of fluid mechanics, specifically the equation of continuity and Bernoulli's equation.

### Step-by-Step Solution

1. **Identify Given Values:**
- Cross-sectional area at point 1, \( A_1 = 10 \, \text{cm}^2 = 10 \times 10^{-4} \, \text{m}^2 = 0.001 \, \text{m}^2 \)
- Velocity at point 1, \( V_1 = 1 \, \text{m/s} \)
- Pressure at point 1, \( P_1 = 2000 \, \text{Pa} \)
- Cross-sectional area at point 2, \( A_2 = 5 \, \text{cm}^2 = 5 \times 10^{-4} \, \text{m}^2 = 0.0005 \, \text{m}^2 \)
- Density of water, \( \rho = 10^3 \, \text{kg/m}^3 \)

2. **Apply the Equation of Continuity:**
The equation of continuity states that the product of the cross-sectional area and velocity at any two points in a streamline flow is constant:
\[ A_1 V_1 = A_2 V_2 \]
Plugging in the values:
\[ 0.001 \, \text{m}^2 \times 1 \, \text{m/s} = 0.0005 \, \text{m}^2 \times V_2 \]
Solving for \( V_2 \):
\[ V_2 = \frac{0.001 \, \text{m}^2 \times 1 \, \text{m/s}}{0.0005 \, \text{m}^2} = 2 \, \text{m/s} \]

3. **Apply Bernoulli's Equation:**
Bernoulli's equation for two points in a streamline flow is given by:
\[ P_1 + \frac{1}{2} \rho V_1^2 = P_2 + \frac{1}{2} \rho V_2^2 \]
Rearranging to find \( P_2 \):
\[ P_2 = P_1 + \frac{1}{2} \rho V_1^2 - \frac{1}{2} \rho V_2^2 \]
Plugging in the known values:
\[ P_2 = 2000 \, \text{Pa} + \frac{1}{2} (1000 \, \text{kg/m}^3) (1 \, \text{m/s})^2 - \frac{1}{2} (1000 \, \text{kg/m}^3) (2 \, \text{m/s})^2 \]
Simplifying:
\[ P_2 = 2000 \, \text{Pa} + \frac{1}{2} (1000) (1) - \frac{1}{2} (1000) (4) \]
\[ P_2 = 2000 \, \text{Pa} + 500 - 2000 \]
\[ P_2 = 500 \, \text{Pa} \]

4. **Final Answer:**
The pressure of water at the point where the cross-sectional area is \( 5 \, \text{cm}^2 \) is \( \boxed{500 \, \text{Pa}} \).