A concrete dam has a vertical rectangular face that is wide and extends from the…
PHYSICS · Multi-part · CBSE · Physics · Mechanical Properties of Fluids · Mechanical Properties of Fluids
Problem
A concrete dam has a vertical rectangular face that is W=12m wide and extends from the lake’s surface down to a depth of H=6.0m. The lake water is at rest and has uniform density ρ. Engineers must determine the total horizontal force that the water exerts on the dam face and the depth below the surface at which this resultant force effectively acts.
(a) Derive, without using calculus, an expression for the differential force ΔF on a horizontal strip of the dam face having width W and small height Δy located at depth y below the surface.
(b) Using a geometric or summation argument, obtain an algebraic expression for the total horizontal force Ftot exerted on the entire dam face, in terms of ρ, g, W, and H.
(c) Determine the depth ycp below the surface at which the resultant force can be considered to act (the center of pressure). Express your answer as a fraction of H.
(d) Connect your results in parts (b) and (c) to the microscopic picture of pressure arising from molecular collisions, explaining why deeper regions contribute disproportionately to the total force.