1. The problem statement, all variables and given/known data Consider a simple model of a free-standing dam, depicted in the diagram. Water of density ρ fills a reservoir behind the dam to a height h. Assume the width of the dam (the dimension pointing into the page) is w. (a) Determine an equation for the pressure of the water as a function of depth in the reservoir. (Note that you will need to define your own coordinate system.) (b) Consider a single horizontal layer of the water behind the dam wall. What is the magnitude dF of the force this layer of water exerts on the dam? (c) The force of the water produces a torque on the dam. What is the magnitude of the torque about point P due to the water in the reservoir. (d) Show that the torque about the base of the dam due to the force of the water can be considered to act with a lever arm equal to h/3. 2. Relevant equations P=ρgh P=F/A F=PA τ=rFsinθ Lever arm = rsinθ 3. The attempt at a solution a) Taking y to be the water depth measured from the reservoir floor, P=ρgy. Since ρwater≈1, P≈gy. b) F=PA=gyA. A=width*height of water slice = w(dy). dF=(gyw)(dy) c) I started with τ=rFsinθ and then tried to look at the torque applied by a single slice of water. I drew this picture: Then I said that the length L=sqrt(p^2+y^2). Plugging that into the equation for torque along with force, I got dτ=sqrt(p^2+y^2)*(gyw)(dy)sinθ Then integrating the above with respect to y I got τ=(1/3)gLsinθ(p^2+y^2)^(3/2) =(1/3)gLsinθ(sqrt(p^2+y^2))(p^2+y^2) d. Plugging my calculated value for L into the equation for the lever arm I got Lever arm=sqrt(p^2+y^2)sinθ=h/3 Then substituting that into the equation for torque: τ=(1/3)gp(p^2+y^2)*(h/3) = (h/9)gp(p^2+y^2) I'm not really sure where to go from here, or even if all of my solutions to the parts are actually correct. I'd really appreciate a little bit of guidance.