RECITATION #4 1.060 ENGINEERING MECHANICS II THERE'S A HOLE IN THE BUCKET The Problem [Diagram of a large container (bucket) with a small hole at the bottom. Water flows out of the hole, forming a jet with a vena contracta. Labels are present for the height of the water (h), the hole area (A_h), the vena contracta area (A_o), and the flow rate (Q_out).] Large container (bucket): Surface Area A_s Small hole in bucket: Hole Area = A_h = C_v A_o Vena contracta in jet: Area = A_o How fast does water exit the bucket, or what is Q_i as the bucket is emptied? Conservation of Volume (Continuity) CV₁ has top surface just below free surface in bucket. CV₂ has top surface just above free surface in bucket. Otherwise, all CV₁ = CV₂ CV₁ is filled with water at all times. Conservation of volume (of water) within CV₁ (it is constant) gives In - Out = Q_in - Q_out = V₁A₁ - V₂A₂ = change = 0 (1.1) CV₂ does not have an inflow (of water), but volume of water inside CV₂ changes with time. Conservation of volume gives In - Out = Q_out = -V₂A₂ = d/dt (hA_s) - V₂A₂ = 0 Since (-dh/dt) = velocity at which free surface is moving down, this must equal V₂, and (1.1) and (1.2) are one and the same! Problem No: 2 [Diagram: A plate of length l is shown rotating about a pivot point (O) towards a stationary surface. The gap between the plate and the surface is filled with fluid.] A plate of length l (very long into paper, so problem may be treated as 2-D) rotates around a pivot point, O, at an angular velocity ω [rad/s], forcing the plate down towards a horizontal surface. The angle between the rotating plate and the surface, θ, is decreasing with time and water filling the gap is being squeezed out. Treat fluid as incompressible and inviscid. Determine the velocity in the gap as a function of radial distance r from O, (since cosθ ≈ 1 you call it x or r, up to you), θ and ω. Assuming steady flow and neglecting elevation differences (take z = 0 everywhere) determine the pressure variation, p(x), along the plate.

Bernoulli Experiment

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