An ideal fluid of density 800 kg m⁻³ flows smoothly through a bent pipe that tapers in cross-sectional area from a to a/2. As shown in the figure, the wide section of area a runs horizontally, and the pipe then steps down and narrows, so that the narrow section of area a/2 runs horizontally at a de…

Question

An ideal fluid of density 800 kg m⁻³ flows smoothly through a bent pipe that tapers in cross-sectional area from a to a/2. As shown in the figure, the wide section of area a runs horizontally, and the pipe then steps down and narrows, so that the narrow section of area a/2 runs horizontally at a depth h=1 m below the wide one. The pressure difference between the wide and narrow sections of the pipe is 4100 Pa. At the wider section the velocity of the fluid is (√x)/6 m s⁻¹, for x= . (Given g=10 m s⁻²)

Accepted Answer

363. Given: ρ=800 kg m⁻³, areas a and a/2, p₁-p₂=4100 Pa, the narrow section h=1 m lower, g=10 m s⁻². Continuity: av₁=a/2v₂, so v₂=2v₁. Bernoulli between the two sections, with the drop in height adding to the pressure difference: p₁-p₂+ρ gh=1/2ρ(v₂²-v₁²). 4100+800(10)(1)=12100, and 1/2(800)(3v₁²)=1200v₁². v₁²=(12100)/(1200), so v₁=(√12100)/(√1200)=(110)/(34.64)=3.175 m s⁻¹, and x=(6v₁)²=363.

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