SME
Fluid Mechanics · Problem type
Force on a submerged gate
Use this for “A gate 2 m wide and 3 m tall has its top 1 m under water. Find the force and where it acts.”
Diagram
hc
=
d
+
H
2
=
1
+
3
2
=
2.5
m.
F
=
ρg·hc·bH
=
1,000
×
9.81
×
2.5
×
2
×
3
=
147,200
N.
ycp
=
hc
+
H
2
12 hc
=
2.8
m,
below
the
centroid.
This is an example. Type your own values to start a new problem.
F (kN)
=
ρ (kg/m³)
×
9.81
×
hc (m)
×
b (m)
×
H (m)
d
Depth of the gate’s top
example
Δy
Offset below the centroid
calculated
ycp
Depth of the center of pressure
calculated
Formulas
hc
=
d
+
H
2
2.5
=
1
+
3
2
F
=
ρ
×
9.81
×
hc
×
b
×
H
147,150 = 1,000 × 9.81 × 2.5 × 2 × 3
Δy
=
H
2
12 × hc
0.3
=
3
2
12 × 2.5
ycp = hc + Δy
2.8 = 2.5 + 0.3
Formulas are worked in N. The step-by-step shows the conversions.
Assumptions
•
The fluid is at rest and has the same density at every depth (incompressible).
•
g = 9.81 m/s².
•
The gate is a vertical rectangle in the wall; the air presses on both sides, so gauge pressure is used.
Step-by-step
WE KNOW
d = 1 m, b = 2 m, H = 3 m, ρ = 1,000 kg/m³
FIND
depth of the centroid (hc), force on the gate (F), offset below the centroid (Δy), depth of the center of pressure (ycp)
Step 1 · Find depth of the centroid (hc)
Use hc = d + H ÷ 2
A rectangle’s centroid is halfway down it: add half the height to the top’s depth.
hc
=
1
+
3
2
hc = 1 + 1.5
hc = 2.5 m
Step 2 · Find force on the gate (F)
Use F = ρ × 9.81 × hc × b × H
The pressure at the centroid, ρghc, is the average over the gate: multiply it by the area bH.
F
=
1,000
×
9.81
×
2.5
×
2
×
3
F
=
9,810
×
2.5
×
2
×
3
F
=
24,525
×
2
×
3
F
=
49,050
×
3
F
=
147,150
N
Step 3 · Find offset below the centroid (Δy)
Use Δy = H² ÷ (12 × hc)
Pressure
grows
with
depth,
so
the
force
acts
below
the
centroid
by
I
hcA
=
H
2
12hc
for
a
rectangle.
Δy
=
3
2
12 × 2.5
Δy
=
3
2
30
Δy
=
9
30
Δy = 0.3 m
Step 4 · Find depth of the center of pressure (ycp)
Use ycp = hc + Δy
The center of pressure is the offset below the centroid.
ycp = 2.5 + 0.3
ycp = 2.8 m
Step 5 · Convert the answers
Change each answer back to the unit you chose.
F = 147,150 N = 147.2 kN (1 kN = 1000 N)
Check
Put the numbers back in. Both sides should match.
2.5
=
1
+
3
2
✓
147,150 = 1,000 × 9.81 × 2.5 × 2 × 3 ✓
0.3
=
3
2
12 × 2.5
✓
2.8 = 2.5 + 0.3 ✓
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