Question

Can someone solve this with neat writing!! And a free body diagram as well thank you!

h - 2 m Oil SG =0.8 m Question 1 (20 points): Calculate the value of FH needed to maintain the gate as shown in Figure 1 where a layer of oil rests on top of water. The width of the gate is b=6 m into the page. Neglect the weight of the gate and use specific weight of water yw 9800 N/m a 10 m h - 6 m Water = 60 hinge Note: b afor a rectangular plate. Figure 1

Answer 1

Hydraulic pressure at a depth h from the surface P= hdg.

consider a width dx of the plate at depth h from. Pressure P acts vertically down.

component normal to the plate = PCos($\alpha$)

plate width = 6m

area of the strip dA = 6dx

Force on the strip = PdA = hdgCos($\alpha$) 6dx

x - the distance of the strip from the hinge, then

(10- h)/x = sin$\alpha$

h = 10-xsin($\alpha$)

Force on the strip = 6gdCos($\alpha$) *(10-xsin$\alpha$) dx

torque on the strip about the hinge = 6gdCos($\alpha$) *(10-xsin$\alpha$) dx *x

= 6*9.8*d*0.5 *(10- 0.87x) x dx

= 294d xdx - 25.58d x² dx

d - density of the liquid

we have 2 layers of liquid

water with d = 1000 kg/cu.m

oil = 800 kg/cu.m

water x varies from 0 to 6.93

oil x varies 6.93 to 9.24

Total torque on the plate by oil and water

294000.cdr - 25580.cºd.: +1 J 6.93 235200.cdr - 20464.cºd.c Jo

= 5503506.5 N-m

Component of F_h normal to the gate = F_hCos(30)

Torque by F_h on the gate = 10*F_hCos(30) = 5503506.5

F_h = 635490 N - force required to maintain the gate in position.