Water flowing through a horizontal pipe line having a constriction. Then,

Options

A. pressure will be the same througghlut the length of the pipe.
B. pressure will be greater at the constiction.
C. pressuce will be smaller in the wider portion.
D. pressure will smaller at the constriction.

Answer

### Step-by-Step Solution:

1. **Understanding the Problem**: We have water flowing through a horizontal pipe that has a constriction. We need to analyze how the pressure and velocity of the water change as it flows through the constriction.

2. **Applying Bernoulli's Equation**: Bernoulli's equation states that for an incompressible, non-viscous fluid flowing in a streamline, the total mechanical energy along a streamline is constant. The equation is given by:
\[ P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant} \]
Since the pipe is horizontal, the height (h) remains constant, and we can ignore the gravitational potential energy term.

3. **Identifying Areas and Velocities**: Let:
- \( A_1 \) = cross-sectional area of the pipe before the constriction
- \( A_2 \) = cross-sectional area at the constriction
- \( v_1 \) = velocity of water before the constriction
- \( v_2 \) = velocity of water at the constriction

According to the continuity equation for incompressible fluids:
\[ A_1 v_1 = A_2 v_2 \]
This implies that if \( A_1 > A_2 \), then \( v_2 > v_1 \).

4. **Relating Pressures and Velocities**: Using Bernoulli's equation between two points (before and at the constriction), we can write:
\[ P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2 \]

5. **Solving for Pressure**: Rearranging the equation gives:
\[ P_2 = P_1 + \frac{1}{2} \rho v_1^2 - \frac{1}{2} \rho v_2^2 \]
Since \( v_2 > v_1 \), it follows that \( \frac{1}{2} \rho v_2^2 > \frac{1}{2} \rho v_1^2 \). Thus,
\[ P_2 < P_1 \]
This means that the pressure at the constriction (P2) is less than the pressure before the constriction (P1).

6. **Conclusion**: The pressure decreases as the water flows through the constriction in the pipe.