An ideal fluid flows through a horizontal pipe made of two sections with diameters of 1.0 and 2.0 cm, respectively. The speed of the fluid flow through the 2.0-cm section will be what factor times that through the 1.0-cm section?

Question

BlackTom AI · Accepted Answer

To analyze this question, we start from the principle of conservation of mass for an ideal (incompressible) fluid in a horizontal pipe: A1 v1 = A2 v2, where A is the cross-sectional area and v is the flow speed.

Option 1: 1/2. If v2 were 1/2 of v1, then v2/v1 = 1/2. This would require A1/A2 = 1/2. However, since A ∝ diameter^2, and d1 = 1.0 cm while d2 = 2.0 cm, we have A1/A2 = (d1^2)/(d2^2) = (1^2)/(2^2) = 1/4, not 1/2. Therefore this option is incorrect.

Option 2: 1/4. This would mean v2/v1 = 1/4, which matches the continuity relation since A1/A2 = 1/4 for the given diameters. Specifically, A1 = π(0.5 cm)^2 and A2 = π(1.0 cm)^2, giving A1/A2 = (0.5^2)/(1.0^2) = 1/4. Hence v2 is indeed 1/4 of v1, so this option is consistent with the physics.

Option 3: 4. This would imply v2 is four times v1, which would require A1/A2 = 4. But with the actual diameters, A1/A2 = 1/4, not 4, so this contradicts the continuity equation. Therefore this option is incorrect.

Option 4: 2. Similar to the previous incorrect options, v2/v1 = 2 would require A1/A2 = 2, which does not hold for the given diameters. This option is incorrect.

Option 5: None of these. Since option 2 correctly reflects the v2/v1 ratio derived from A1/A2 = 1/4, the statement 'None of these' is false. Therefore this option is incorrect.

In summary, applying A1 v1 = A2 v2 and the fact that A ∝ d^2 leads to v2/v1 = (d1/d2)^2 = (1.0/2.0)^2 = 1/4, which identifies option 1/4 as the correct choice.

2025 Fall PHYS 2205 TR 11:00 - 12:15 _ 89122 2205 Final Exam

An ideal fluid flows through a horizontal pipe made of two sections with diameters of 1.0 and 2.0 cm, respectively. The speed of the fluid flow through the 2.0-cm section will be what factor times that through the 1.0-cm section?

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