How does the velocity and pressure of water in a pipe change when the cross-sectional area is widened?
a. What will be the velocity of the water through this section of the pipe?
b. What is the pressure through this section of the pipe?
Explanation
a. Velocity Calculation
The continuity equation tells us that the product of velocity and cross-sectional area must remain constant for an incompressible fluid like water. So, as the cross-sectional area increases, the velocity must decrease to maintain this product:
Initial velocity * Initial area = Final velocity * Final area
(8 m/s) * (0.05 m²) = Final velocity * (0.08 m²)
Final velocity ≈ 5 m/s
b. Pressure Calculation
Bernoulli's principle helps us understand the pressure changes. As the pipe widens, the fluid's velocity decreases, leading to an increase in pressure due to Bernoulli's principle:
Initial pressure + 0.5 * density * initial velocity² = Final pressure + 0.5 * density * final velocity²
150,000 Pa + 0.5 * density * (8 m/s)² = Final pressure + 0.5 * density * (5 m/s)²
Final pressure ≈ 112,500 Pa
In summary, when the pipe widens, the velocity of the water decreases to around 5 m/s while the pressure increases to approximately 112,500 Pa. These changes are explained by the principles of fluid dynamics, specifically the continuity equation and Bernoulli's principle.