Problem *3.122[Difficulty: 3]Given:Rectangular container of base dimensions 0.4 m x 0.2 m and a height of 0.4 m is filled with water to a depth of d =0.2 m. Mass of empty container is Mc= 10 kg. The container slides down an incline of θ= 30 deg with respect tothe horizontal without friction.Find:(a) The angle of the water surface relative to the horizontal.(b) The slope of the free surface for the same acceleration up the plane.θ yxα Solution:We will apply the hydrostatics equations to this system.agpGGρ=+∇−Governing Equations:(Hydrostatic equation)aMFGG=(Newton's Second Law)Assumptions:(1) Incompressible fluid(2) Rigid body motionxaxp=∂∂−xaxp−=∂∂yagyp=−∂∂−()yagyp+−=∂∂Writing the component relations:dyypdxxpdp∂∂+∂∂=We write the total differential of pressure as:Now along the free surface of the water dp = 0. Thus:yxagaypxpdxdy+−=∂∂∂∂−=and αatandydx−⎛⎝⎞⎠=To determine the acceleration components we analyze a free-body diagram:
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