Q10. Figure 2 below shows a stream of water in steady flow from a kitchen faucet. At the faucet, the diameter of the stream is 0.960 cm. The stream fills a 125-cm3 container in 16.3 s. Find the diameter of the stream 13.0 cm below the opening of the faucet.See answer
Q10 Figure 2 below shows a stream of water in steady flow from a kitchen faucet At the faucet the diameter of the stream is 0960 cm The stream fills a 125 cm3 container in 163 s Find the diameter of…
Question
Basic Answer
Step 1: Find the flow rate
The flow rate (Q) is the volume of water divided by the time it takes to fill the container.
Q = Volume / Time = 125 cm³ / 16.3 s ≈ 7.66 cm³/s
Step 2: Calculate the cross-sectional area at the faucet
The cross-sectional area (A1) at the faucet is given by the formula for the area of a circle:
A1 = π * (d1/2)²
where d1 is the diameter at the faucet (0.960 cm).
A1 = π * (0.960 cm / 2)² ≈ 0.724 cm²
Step 3: Calculate the speed of water at the faucet
The speed (v1) of the water at the faucet can be calculated using the flow rate and the cross-sectional area:
v1 = Q / A1 = 7.66 cm³/s / 0.724 cm² ≈ 10.6 cm/s
Step 4: Apply the equation of continuity
The equation of continuity states that the flow rate remains constant in a steady flow:
A1 * v1 = A2 * v2
where A2 and v2 are the cross-sectional area and speed at a point 13.0 cm below the faucet.
Step 5: Apply Bernoulli’s equation
Bernoulli’s equation relates the pressure, speed, and height of a fluid in steady flow:
P1 + (1/2)ρv1² + ρgh1 = P2 + (1/2)ρv2² + ρgh2
where:
- P1 and P2 are the pressures at the faucet and 13.0 cm below.
- ρ is the density of water.
- g is the acceleration due to gravity (approximately 980 cm/s²).
- h1 and h2 are the heights at the faucet and 13.0 cm below (h2 = h1 – 13.0 cm).
Assuming atmospheric pressure at both points (P1 = P2), we can simplify Bernoulli’s equation:
(1/2)ρv1² + ρgh1 = (1/2)ρv2² + ρgh2
Step 6: Solve for v2
Rearrange the simplified Bernoulli’s equation to solve for v2:
v2² = v1² + 2g(h1 – h2) = v1² + 2g(13.0 cm)
v2 = √(v1² + 2g(13.0 cm)) ≈ √(10.6² + 2 * 980 * 13) ≈ 51.2 cm/s
Step 7: Calculate A2
Using the equation of continuity (A1 * v1 = A2 * v2), solve for A2:
A2 = (A1 * v1) / v2 = (0.724 cm² * 10.6 cm/s) / 51.2 cm/s ≈ 0.15 cm²
Step 8: Calculate d2
Since A2 = π * (d2/2)², solve for d2:
d2 = 2 * √(A2 / π) = 2 * √(0.15 cm² / π) ≈ 0.437 cm
Final Answer
The diameter of the stream 13.0 cm below the faucet is approximately 0.437 cm.