Question

A channel is closed by a rectangular gate shown in the Figure below. The gate which is hinged at the red circle in the figure is 7 m high and 1 m wide. It has seawater to the left to a depth of 7 m above the baseline and on the right has freshwater to a depth of 9.5 m above the baseline. The density of seawater is 1025 kg/m2, and density of freshwater is 1000 kg/m. Calculate the positive net force required on the bottom of the gate to keep the gate closed. [25 Marks] freshwater seawater seawater depth gate freshwater depth height Fur N

Answer 1

Given Information :

• Gate is hinged at the red circle :
  • height of gate :
  • width of gate :
    1 m
    
• Sea water :
  • density: $\rho_s =1025\ kg/m^3$
  • depth :
    -dz 7 m
    
• Fresh water:
  • density :
    Pj = 1000 kg/m
    
  • depth :

To Find : Net Force at the bottom of the gate,to keep the gate closed.

Solution :

The horizontal force acting on the gate is given by :

$F_h=\rho*g*A*\bar x$

Where
$A=$ Projected area of gate
$\bar x=$ Distance of the centroid of the projected area from the free surface.

The point of application of Horizontal force is given by : The distance is from the

h = c+ Ig AT

Where
$I_g=$ Second Moment of Inertia of the projected area of the gate.

Calculating the Horizontal Force acting on the gate from the sea water side :

$F_{hs}=\rho_s*g*A_s*\bar x_{s}$

Substitute the values :

$F_{hs}=1025*9.81*7*1*3.5$

Calculating the point of application of Horizontal Force acting on the gate from the sea water side :

$h_s=\bar x_s+\frac{I_g}{A_s\bar x_s}$

$h_s=3.5+\frac{\frac{1*7^3}{12}}{7*1*3.5}$

Calculating the Horizontal Force acting on the gate from the fresh water side :

$F_{hf}=\rho_f*g*A_f*\bar x_{f}$

Substitute the values :

$F_{hf}=1000*9.81*9.5*1*4.75$

Calculating the point of application of Horizontal Force acting on the gate from the fresh water side :

$h_f=\bar x_f+\frac{I_g}{A_f\bar x_f}$

$h_f=4.75+\frac{\frac{1*9.5^3}{12}}{1*9.5*4.75}$

Calculating the distance from the hinge to the point of application of horizontal forces :

For the sea water side :

Since the depth of water is equal to the distance between the hinge and the baseline :

For the fresh water side :

$h_{sn}=h_s-(9.5-7)$

$h_{sn}=6.33-(9.5-7)$

Calculating the Net force that should be acting at the bottom of the gate, to keep it closed.

Let the force denoted by $F_{net}$ , be acting from the sea water side.

Taking moments about the hinge of the gate :

$F_{net}*7+F_{hs}*4.66=F_{hf}*3.83$

$F_{net}*7+246.35*4.66=442.67*3.83$

Therefore the net force that must be acting at the bottom of the gate to keep the gate closed is equal to :