CP CALC The upper edge of a gate in a dam runs along the water surface. The gate is 2.00 m high and 4.00 m wide and is hinged along a horizontal line through its center (Fig. P12.55). Calculate the torque about the hinge arising from the force due to the water. (Hint: Use a procedure similar to that used in 12.53; calculate the torque on a thin, horizontal strip at a depth h and integrate this over the gate.) 12.53 ... CALC A swimming pool is 5.0 m long, 4.0 m wide, and 3.0 m deep. Compute the force exerted by the water against (a) the bottom and (b) either end. (Hint: Calculate the force on a thin, horizontal strip at a depth h , and integrate this over the end of the pool.) Do not include the force due to air pressure.

Solution 54P Step 1: Pressure of the water column, P = gh Where, - Density of water g - Acceleration due to gravity h - Height of the water column Equation for force, F = PA Where, P - Pressure acting on the surface A - Area of the surface Then, for an elemental area dA, the force acting, dF = PdA Step 2: The torque acting on the elemental area due to the force on the elemental area due to the water, d = dF . y Where, dy - perpendicular distance That is, d = PdA y We know that, dA = L dy Or, d = gh Ly dy Since, the vertical height is varying from the top, we can write, h = 1-y That is, at the center of the strip, consider, y = 0, so, h = 1m. At the bottom, h = 2m and at the top of the strip, h = 0 m So, the values of y will be from -1 m to 1 m Therefore, d = g(1-y) Ly dy Or, d = g L (y - y ) dy