HW05sols - Problem*3.114 Given[Difficulty 3 Cylindrical...

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Problem *3.114 [Difficulty: 3] Given: Cylindrical container rotating as in Example 3.10 R 0.25 m = h o 0.3 m = f 2 Hz = Find: (a) height of free surface at the entrance (b) if solution depends on ρ Solution: We will apply the hydrostatics equations to this system. Governing Equations: (Hydrostatic equation) Assumptions: (1) Incompressible fluid (2) Atmospheric pressure acts everywhere In order to obtain the solution we need an expression for the shape of the free surface in terms of ω , r, and h o . The required expression was derived in Example 3.10. The equation is: z h o ω R ( ) 2 2 g 1 2 r R 2 = The angular velocity ω is related to the frequency of rotation through: ω 2 π f = ω 2 π 2 × rad s 12.57 rad s = = Now since h 1 is the z value which corresponds to r = 0: h 1 h o ω R ( ) 2 4 g = Substituting known values: h 1 0.3 m 1 4 12.57 rad s 0.25 × m 2 × s 2 9.81 m × = h 1 0.05 m = The solution is independent of ρ
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