A pipe has a cross sectional radius of 6.00 cm and is carrying water moving at .380 m/sec. If the pipe narrows to a radius of 3.50 cm, what will be the speed of the water in the narrow segment? What is the flow rate in each segment?
thank you!!!!!!!
lAW OF CONTINUITY:
area1*speed1=area2*speed2
speednarrow=PI(6^2)*.380/(.350^2)
volume flow rate=speed*area do that for both sides.
oops, add a PI in the denominator of the law of continuity, area=pI*radius^2
To solve this problem, we can use the principle of continuity equation in fluid mechanics, which states that the product of the cross-sectional area and the fluid velocity remains constant within an incompressible flow.
1. Speed of water in the narrow segment:
Using the continuity equation, we can write:
A1 * v1 = A2 * v2
where A1 is the cross-sectional area of the wider segment, v1 is the velocity in the wider segment, A2 is the cross-sectional area of the narrow segment, and v2 is the velocity in the narrow segment.
Given:
- Radius of the wider segment = 6.00 cm = 0.06 m
- Radius of the narrow segment = 3.50 cm = 0.035 m
- Velocity in the wider segment, v1 = 0.380 m/s
To find v2, we need to calculate A1 and A2:
- A1 = π * (r1)^2 = π * (0.06)^2
- A2 = π * (r2)^2 = π * (0.035)^2
Now we can calculate the speed of water in the narrow segment:
A1 * v1 = A2 * v2
π * (0.06)^2 * 0.380 = π * (0.035)^2 * v2
Simplifying and solving for v2:
v2 = (π * (0.06)^2 * 0.380) / (π * (0.035)^2)
v2 = (0.013608) / (0.001225)
v2 ≈ 11.12 m/s
Therefore, the speed of the water in the narrow segment is approximately 11.12 m/s.
2. Flow rate in each segment:
Flow rate is defined as the product of the cross-sectional area and the velocity of the fluid. In this case, we can calculate the flow rate in each segment as follows:
Flow rate in the wider segment:
Q1 = A1 * v1
Q1 = π * (0.06)^2 * 0.380
Flow rate in the narrow segment:
Q2 = A2 * v2
Q2 = π * (0.035)^2 * 11.12
Calculating the values:
Q1 = 0.06836 m^3/s
Q2 = 0.01399 m^3/s
Therefore, the flow rate in the wider segment is approximately 0.06836 m^3/s, and the flow rate in the narrow segment is approximately 0.01399 m^3/s.