SME
Calculus II · Problem type
Force on an upright rectangular gate
Use this for “A rectangular gate 2 m wide and 3 m tall stands upright with its top 1 m under the water. Find the force on it 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
=
147,150
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.
h
c
=
d (m)
+
H (m)
÷
2
=
hc (m)
F
=
ρ
gh
c
bH
=
F (N)
ρ
=
ρ (kg/m³)
,
b
=
b (m)
ycp
Depth of the center of pressure
worked out (not typed) · m
Formulas
hc
=
d
+
H
2
2.5
=
1
+
3
2
F
=
ρ
×
9.81
×
b
×
H
×
hc
147,150 = 1,000 × 9.81 × 2 × 3 × 2.5
ycp
=
hc
+
H
2
12 × hc
2.8
=
2.5
+
3
2
12 × 2.5
Assumptions
•
Water pressure at depth y is ρgy and pushes square to the gate; g = 9.81 m/s² and water is 1000 kg/m³.
•
A strip at depth y, dy tall, carries force ρgy·b dy, so F = ∫ from d to d + H of ρgyb dy.
•
Air presses on both faces of the gate, so it cancels and only the water’s pressure counts.
•
Deeper strips push harder, so the force acts at the center of pressure ycp, below the centroid.
Step-by-step
WE KNOW
b = 2 m, H = 3 m, d = 1 m, ρ = 1,000 kg/m³
FIND
depth of the centroid (hc), force on the gate (F), depth of the center of pressure (ycp)
Step 1 · Find depth of the centroid (hc)
Use hc = d + H ÷ 2
The 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 × b × H × hc
Integrate
ρgyb
from
d
to
d
+
H
:
ρgb(
(
d
+
H
)
2
−
d
2
)
÷
2
=
ρgbH(
d
+
H
2
),
and
d
+
H
2
is
hc.
F
=
1,000
×
9.81
×
2
×
3
×
2.5
∫
from
1
to
4
of
1,000
×
9.81
×
2
×
y
dy
=
[1,000
×
9.81
×
2
×
y
2
2
]
from
1
to
4
=
147,150
F
=
147,150
N
Step 3 · Find depth of the center of pressure (ycp)
Use ycp = hc + H² ÷ (12 × hc)
ycp
=
∫y·ρgyb
dy
F
.
For
the
rectangle
this
works
out
to
hc
plus
H
2
12hc
,
below
the
centroid.
ycp
=
2.5
+
3
2
12 × 2.5
ycp
=
2.5
+
3
2
30
ycp
=
2.5
+
9
30
ycp = 2.5 + 0.3
ycp = 2.8 m
Check
Put the numbers back in. Both sides should match.
2.5
=
1
+
3
2
✓
147,150 = 1,000 × 9.81 × 2 × 3 × 2.5 ✓
2.8
=
2.5
+
3
2
12 × 2.5
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.