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Problem 4.172 - Problem 4.172[Difficulty 4 y x CS(moves at...

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Problem 4.172 [Difficulty: 4] CS (moves at speed U) c d F f R y y x Given: Water jet striking moving vane Find: Plot of terminal speed versus turning angle; angle to overcome static friction Solution: Basic equations: Momentum flux in x and y directions Assumptions: 1) Incompressible flow 2) Atmospheric pressure in jet 3) Uniform flow 4) Jet relative velocity is constant Then F f M a rfx u 1 ρ V 1 A 1 ( ) u 2 ρ V 2 A 2 ( ) + = V U ( ) ρ V U ( ) A [ ] V U ( ) cos θ ( ) ρ V U ( ) A [ ] + = a rfx ρ V U ( ) 2 A 1 cos θ ( ) ( ) F f M = (1) Also R y M g v 1 ρ V 1 A 1 ( ) v 2 ρ V 2 A 2 + = 0 V U ( ) sin θ ( ) ρ V U ( ) A [ ] + = R y M g ρ V U ( ) 2 A sin θ ( ) + = At terminal speed a rfx = 0 and F f = μ k R y . Hence in Eq 1 0 ρ V U t ( ) 2 A 1 cos θ ( ) ( ) μ k M g ρ V U t ( ) 2 A sin θ ( ) + M = ρ V U t ( ) 2 A 1 cos θ ( ) μ k sin θ ( ) ( ) M μ k g = or V U t μ k M g ρ A 1 cos θ ( ) μ k sin θ ( ) ( ) = U t V μ k M g ρ A 1 cos θ ( ) μ k sin θ ( ) ( ) = The terminal speed as a function of angle is plotted below; it can be generated in
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