A vertical concrete dam wall 40 m wide holds back a freshwater reservoir to a maximum depth of 30 m. The upstream water surface is open to atmospheric pressure, and the air on the downstream side is at the same atmospheric pressure. Gravity is 9.8ms−2 and freshwater density is 1.00×103kgm−3. Neglect any effects due to the curvature of Earth or water motion.
(a) Express the gauge pressure P(y) exerted by the water on the dam wall as a function of the depth y measured downward from the surface (y=0 at the surface).
(b) Describe qualitatively the shape of a graph of pressure versus depth from y=0 to y=30m. Identify the key features that justify your description.
(c) Using your expression from part (a) and appropriate geometric reasoning, determine the magnitude of the total horizontal force that the water exerts on the dam wall.
(d) At what depth below the surface does the line of action of this resultant force (the center of pressure) act? Provide a clear explanation of your method.