A pipe of circular cross-section of inner radius r and outer radius 2r is bent into a semi circle as shown in the diagram. A fluid of density ρ = 1 0 3 kg / m 3 is flowing through it. The breaking str

Question

A pipe of circular cross-section of inner radius r and outer radius 2r is bent into a semi circle as shown in the diagram. A fluid of density ρ = 1 0 3 kg / m 3 is flowing through it. The breaking stress of the material of the pipe is 3 × 1 0 7 N / m 2 . The maximum velocity with which the fluid can flow in the pipe is k × 100 m / s . Find value of k.

TestHub · Accepted Answer

Force needed to turn the fluid through the semicircle: $2 ρ A v^{2} = 2 F$

Load-bearing wall area: $π (2 r)^{2} - π r^{2} = 3 π r^{2}$ , with $A = π r^{2}$

$σ = \frac{F}{3 π r ^{2}} = \frac{ρ v ^{2}}{3}$

$v^{2} = \frac{3 σ}{ρ} = \frac{3 \times 3 \times 1 0 ^{7}}{1 0 ^{3}} \Rightarrow v = 300$ m/s

$v = k \times 100 \Rightarrow k = 3$

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