Example 14.4 The Force on a Dam Water is lled to a heightH = 108 m behind a dam of width w = 125 m (see the gure). Determine the resultant force exerted by the water on the darn. SOLVE IT Conoeptualize Because pressure varies with depth, we cannot calculate the force simply by multiplying the area by the pressure. categorize Because of the variation of pressure with depth, we must use integration to solve this example, so we categorize it as an analysis problem. Analyze Let's imagine a vertical y axis, with y = 0 at the bottom of the dam. We divide the face of the dam into narrow horizontal strips at a distance y above the bottom, such as the red strip in the gure. The pressure on each such strip is due only to the water; atmospheric pressure acts on both sides of the dam. 0 Water exerts a force on a dam. Use the equation to calculate the pressure P = pgh = pg(H - y) due to the water at the depth h: Use the equation to nd the force exerted db" = PM = gU-I - y) wdy on the shaded strip of area dA = wdy: Integrate to nd the total force on the darn (Use the following as necessary: p, w, H, and g): Substitute the known values to determine F [j N the total force acting on the dam. Finalize Notice that the thickness of the darn shown in the gure increases with depth. This design accounts for the greater pressure the water exerts on the dam at greater depths. Analyze Let's imagine a vertical y axis, with y = 0 at the bottom of the dam. We divide the face of the dam into narrow horizontal strips at a distance y above the bottom, such as the red strip in the gure. The pressure Water xer\" 3 f0\" 0\" a dam- on each such strip is due only to the water; atmospheric pressure acts on both sides of the dam. Use the equation to calculate the pressure P = pgh = pg(H - y) due to the water at the depth h: Use the equation to nd the force exerted dF = PM = pg(H - y) w dy on the shaded strip of area M = wdy: Integrate to find the total force on the dam H (Use the following as necessary: p, w, H, F _ /P{]A _ A pg (H _ y) WI\" and g): Substitute the known values to determine the total force acting on the dam. Finalize Notice that the thickness of the darn shown in the gure increases with depth. This design accounts for the greater pressure the water exerts on the dam at greater depths. MASTE T A glass with a square cross section when viewed directly from above is filled with water. The glass is 8.0 cm wide on each side and is 16.4 cm high. (3) Find the net horizontal force exerted by the water on each side. i N (b) Find the net horizontal force exerted by the water on all four walls of the glass. E" (c) Find the net force exerted by the water on the bottom of the glass