a vertical dock gate is 5m wide and is hinged at 1m from the base of the dock.
the dock has sea water with density of 1025 kg /m^3 to a depth of 7.5 on one side and
fresh water to depth of 3m on the other side
a) find the resultant horizontal force
b) Determine the overtunning moment that the gate hinge has to withstand
To find the resultant horizontal force on the vertical dock gate, we need to calculate the pressure difference between the two sides of the gate.
a) Resultant Horizontal Force:
The pressure at any point in a fluid is given by the equation P = ρgh, where P is pressure, ρ is density, g is gravity, and h is the height or depth.
On the side of the gate with seawater, the pressure is:
P_sea = ρ_sea * g * h_sea
On the side of the gate with fresh water, the pressure is:
P_fresh = ρ_fresh * g * h_fresh
Since the height (depth) of seawater is 7.5m and that of fresh water is 3m, and the gravity (g) is 9.8 m/s^2, the pressures are:
P_sea = 1025 * 9.8 * 7.5
P_fresh = 1000 * 9.8 * 3
The resultant horizontal force is given by the equation: F = P * A, where F is force, P is pressure, and A is the area.
The area of the gate is the width multiplied by the effective height. Since the hinge is 1m from the base, the effective height is from the surface of the water to the hinge point. For the seawater side, the effective height is 7.5m - 1m = 6.5m, and for the fresh water side, the effective height is 3m - 1m = 2m.
So, the areas are:
A_sea = 5 * 6.5
A_fresh = 5 * 2
Now we can calculate the resultant horizontal forces:
F_sea = P_sea * A_sea
F_fresh = P_fresh * A_fresh
The resultant horizontal force is the difference between the two horizontal forces:
Resultant Horizontal Force = F_sea - F_fresh
b) Overturning Moment:
The overtur
To find the resultant horizontal force on the vertical dock gate, we need to consider the pressure exerted by the water on each side of the gate. The formula for pressure is:
Pressure = Density × Gravity × Height
a) Resultant Horizontal Force:
1. Calculate the pressure exerted by the sea water side:
Pressure_sea = Density_sea × Gravity × Height_sea
= 1025 kg/m^3 × 9.8 m/s^2 × 7.5 m
2. Calculate the pressure exerted by the fresh water side:
Pressure_fresh = Density_fresh × Gravity × Height_fresh
= Density_fresh × 9.8 m/s^2 × 3 m
= Density_sea × 9.8 m/s^2 × 3 m (since it's not given)
3. Calculate the resultant horizontal force:
Resultant Force = Pressure_sea × Area_sea - Pressure_fresh × Area_fresh
= Pressure_sea × (Width × Height_sea) - Pressure_fresh × (Width × Height_fresh)
= (1025 kg/m^3 × 9.8 m/s^2 × 7.5 m) × (5 m) - (Density_sea × 9.8 m/s^2 × 3 m) × (5 m)
b) Overtuning Moment:
To determine the overtuning moment that the gate hinge has to withstand, we need to calculate the torque exerted by the resultant horizontal force around the hinge.
1. Torque = Resultant Force × Distance from Hinge
= Resultant Force × 1 m (since the hinge is located 1 m from the base)
Now you can substitute the calculated values into the formulas to find the answers.
I assume you want force on gate, not on the wall to the bottom
force salt = 8*1025 g integral z dz from 0 to 6.5 (z is down from surface)
= 8*1025*9.81 (6.5)^2/2 =1.7 *10^6 Newtons on gate from salt water
force fresh on gate =8*1000g int zdz from 0 to 2
=8000*9.81* 2^2/2 = 157*10^3 =.157*10^6
net force = (1.7-.157)10^6 = 1.54*10^3
moment from salt = 1.7*10^6 (1/3)(6.5)
(centroid of triangle 1/3 of distance from base)
= 3.68*10^6 Nm
for fresh .157*10 ^6 *(1/3)(2)
= .105*10^6 Nm
net moment = (3.68-.105)10^6 =3.57*10^6Nm