Question

A long, vertical wall separates seawater from freshwater. If the seawater stands at a depth of 7m, what depth of freshwater is required to give a zero resultant force on the wall? When the resultant force is zero will the moment due to the Auld forces be zero? Explain

Answer 1

the forces acting on the wall from both the sides are:

1/2*ρ1gh1*h1 from the left side where h1 = 7m (given) ata position h1/3 from the base.

1/2*ρ2gh2*h2 from the right side where h2 = h at aposition h2/3 from the base.

where ρ1 and ρ2 are the densities of seawater and fresh water
Note : we have not considered the force caused atmospheric pressurebecause, it will cancel out from both the sides.

so, to get a force equilibrium,

both these force should cancel each other

so, 1/2*ρ1*g*h1² =1/2*ρ2*g*h2²

or, h2 = h1√(ρ1/ρ2)

or, h = 7√(ρ1/ρ2)

net moment M = 1/2*ρ1*g*h12*h1/3 -1/2*ρ2*g*h22*h2/3      in clockwisedirection

= g/6(ρ1*h1³ - ρ2*h2³)

= g/6(ρ1*7³ -ρ2*7³*ρ1/ρ2*√(ρ1/ρ2))

=(g/6)*ρ1*7³(1-√(ρ1/ρ2))

we can put the values of ρ1= 1025 kg/m3 and ρ2 =1000 kg/m3

h = 7*√(1025/1000) = 7.08 m

M = (9.81/6)*1025*7³*(1-√(1025/1000)) =-7140.95 N-m

i.e. 7140.95 N-m in counterclockwise direction.

Answer 2

Draw a long vertical wall separating the sea water and fresh water YF Freshwater Seawater Im Consider that the resultant force acting on the wall to be zero RSFRF hos is the vertical distance from the surface of sea water to the centroid of the area, As is the cross sectional area on which the forceFs is acting Consider the length of the wall to be 1 nm Properties of fresh water and sea water 10.1xx(7x1) 9.80xx(hx1) 247.45 4.9h M 50.5 h7.1063 m Location of the resultant force(F) of the sea water xc ус xds 1x73 12 x(7x1) (2 - 4.6666m The distance (Fs)from the bottom is (7-4.6666)m 2.3334nm Location of the resultant force(Fr)of the fresh water 1x7.10633 12 7.1063 R7.1063 7 063x1) C 2 -4.7375 m The distance from the bottom is (7.1063-4.7375)m 2.3688m Since the resultant forces acting on the wall are non collinear, the moment will not be zero for zero resultant force