57:020 Mechanics of Fluids and Transport
Fall 2010
EXAM3
Solutions
Problem 1: Hydrostatic Forces on Plane Surface (Chapter 2)
Information and assumptions
•
Circular gate with diameter
D
= 2 m;
R
=
D
/2 = 1 m
•
Gate orientation = 60
°
•
Vertical centroid location
h
c
= 3 m from the water surface
•
γ
= 9.8 kN/m
3
for water
•
I
xc
=
π
R
4
/4 for a circle of radius
R
Find
•
Water force acting on the gate
F
R
and the center of pressure
y
R
.
Solution
(a)
Water force:
𝐹
𝑅
=
𝑝
𝑐
𝐴
=
𝛾ℎ
𝑐
𝐴
(+3 points)
Thus,
𝐹
𝑅
=
�
9.8
kN
m
3
�
(3 m)
�
𝜋
4
(2 m)
2
�
= 92.4 kN
(+2 points)
(b)
Center of pressure:
𝑦
𝑅
=
𝐼
𝑥𝑐
𝑦
𝑐
𝐴
+
𝑦
𝑐
where,
𝐼
𝑥𝑐
=
𝜋𝑅
4
4
=
𝜋
(1 m)
4
4
=
𝜋
4
m
4
𝑦
𝑐
=
ℎ
𝑐
cos 30
∘
=
3 m
cos 30
∘
= 3.464 m
(+4 points)
Hence,
𝑦
𝑅
=
𝜋
4
m
4
(3.464 m)
�
𝜋
4
(2 m)
2
�
+ 3.464 m = 3.536 m
(+1 point)

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