Po  A hinged gate is placed at the bottom of a tank containing

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Presentation transcript:

Po  A hinged gate is placed at the bottom of a tank containing water. Find the resultant force on the gate due to hydrostatic pressure, the torque on the gate about hinge B, and the point of application of the resultant force. D=10 ft, L= 6 ft and W = 1 ft D Water x B L y ---------------------------------------------------------------------------------- Force due to water: FR =  - p dA  FR i =  - p dA i Assume FR is in the plus x-direction. A A For the coordinate system specified, dA = dy dz. Our integral becomes: W L FR =   - p dy dz o o Since p does not vary with z, we can simplify this to: L FR = W  - p dy o However, p does depend on y, so we must find that relationship before integrating. From the hydrostatic equation, we obtain p = Po + gh = Po + g(D + y) at some point y on the gate.

Substituting for the pressure, L L FR = - W  (Po + g(D + y) ) dy = - W[Poy + gDy + gy2/2] o o FR = - [PoWL + gDLW + gL2W/2] The first term represents the contribution due to atmospheric pressure; but this is also exerted on the other side(PoWL). We drop this term. FR = -gLW[ D + L/2 ] = - (62.4 lbf /ft3)(6 ft)(1 ft)[ 10 ft + 3 ft] The final value for the resultant force is: FR = -4870 lbf What does the negative sign indicate? What is the torque about hinge B caused by the hydrostatic force? x T FR T =  r x dF =  y j x p dA (- i) = -(j x i)  y p dA A A A Start with the general relationship for torque. Making a smart choice of coordinate system can greatly simplify the problem. What is r for this problem? How is dF described? What is element dA, perpendicular to i?

Neglecting atmospheric contribution: p = g(D + y) We know dA = dy dz Hence, L W T = k   y( D + y ) g dz dy = k g W ( yD + y2 ) dy o o o We carry out the integration. L T = k g W[ - y2 D/2 + y3/3 ]o = k g W[ - L2 D/2 + L3/3 ] T = k (62.4 lbf /ft3)(1 ft)[ (6 ft)2(10 ft)/2 + (6 ft)3/3] T = 15,725 ft-lbf k Now, where is the resultant force applied to give the same torque about hinge B? The location of the unknown point will be designated r´ where r´ = y´ j. r´ x FR = T  y´ j x FR (-i) = T k Recall j x (-i) = k r´ = y´ = T / FR  = 15,725 ft-lbf /4870 lbf = 3.2 ft Is this result reasonable?