Please show all work. Thank you! Refer to the figure below, assume that z=0 at t
ID: 717693 • Letter: P
Question
Please show all work. Thank you!
Refer to the figure below, assume that z=0 at the surface of the lighter (lower SG) fluid and increases in a downward direction, and answer the following:
a. Develop functions for pressure in each fluid as a function of vertical distance z, making sure to clearly indicate ranges of z over which the functions are valid.
b. What is the vertical force acting on the bottom of the gate? Assume a 2 meter wide gate. Report your answer in Newtons.
45° SG = 0.86 50 cm A Hinge 10111 80 cm SG = 1.23 40 cmExplanation / Answer
a) for z= 0 to 0.5 m , Pressure= atmospheric pressure + density of lighter oil x acc. due to garvity x z height=101325 +1000 x 0.86 x 9.81 x z
=101325 +8436.6 x z Pascal
for z=0.5 to 1.3 m , pressure = total pressure at 0.5 m + density of heavier oil x acc. due to gratvity x z height
=101325+8436.6 x 0.5 + 1230 x 9.81 x z=105543.3+12066.3 x z Pascal
b) vertical force on the bottom gate CB= area x pressure=( 0.4 x 2) x (105543.3 + 12066.3 x 1.3)=96983.592 Newton
for wedge hinged at A, area=(0.4/(sin 45 degree))x 2=0.565 metre square
( width of wedge taken 2 m )
force= area x total pressure
total pressure = density x acc. due to gravity x centroid of triangle
centrod of trainagle is at 2 x 0.4 /3 from above =0.26 from above
total pressure=2193.516
vertical force on wedge= 1239.33 N ( it was not asked in problem)
c)pressure from top layer= 4218.3 N ( not considering atmospheric pressure)
4218.3= 1230 x 9.81 x h
it implies, h=0.35 m
hence we can replce the top layer with bottom layer with heavier oil with height of 0.35 m
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