Question

As the reservoir behind a dam is filled with water, the pressurethat the water exerts on the dam increases. Eventually, the forceon the dam becomes substantial, and itcould cause the dam tocollapse. There are two significant issues to be considered: First,the base of the dam should be able to withstand the pressure, where is the density of the water behind thedam, is its depth, and is the magnitude of the acceleration due togravity. This means that the material of which the dam is madeneeds to be strong enoughso that it doesn't crack (compressivestrength).

The second issue has to do with the strength of the foundationof the dam. The water pressure exerts a clockwise torque on thedam, as shown in the figure. Thefoundation of the dam should bestrong enough so that the dam does not topple. The material has tobe strong enough that the dam does not snap (shear strength).

To study this phenomenon, consider the simple model of a damdepicted in the diagram. A reservoir of water (density ) behind the dam is filled to a height. Assume that the width of the dam (thedimension pointing into the screen) is.

QUESTION:Consider a horizontal layer of the dam wall ofthickness located a distance above the reservoir floor. What isthemagnitude of the force on this layer due to the water inthereservoir?Express your answer in terms of, , the magnitude of the acceleration due togravity , and any quantities from theproblem introduction.



So I found that the pressure at a height x above the bottom is P =ρg(h-x).This was confirmed by answering the hint problemassociated with thisquestion.I then set this equal toForce/Area, solved for F, and differentiated.

P = ρg(h-x)
F = ρg(h-x) * A
F = ρg(h-x) * L*(h-x)
dF = ρLg(2x-2h).....which is unfortunately incorrect, and themastering physics website says that there is an incorrect numericalmultiplier.

Any suggestions?

Answer 1

I guessed after seeing your answer up above, and somehow got it right. the answer is: (ρ⋅−gL)(x−h)dx