fluid_mechanics

60 Chapter 2 ■ Fluid Statics
The resultant fluid
force does not pass
through the centroid
of the area.
and, therefore,
x R
xy dA
A
I xy
y c A y c A
where I xy is the product of inertia with respect to the x and y axes. Again, using the parallel axis
theorem, 1 we can write
x R I xyc
y c A x c
(2.20)
F Rleft
c
Gate
F Rright
where I xyc is the product of inertia with respect to an orthogonal coordinate system passing through
the centroid of the area and formed by a translation of the x–y coordinate system. If the submerged
area is symmetrical with respect to an axis passing through the centroid and parallel to either the
x or y axes, the resultant force must lie along the line x x c , since I xyc is identically zero in this
case. The point through which the resultant force acts is called the center of pressure. It is to be
noted from Eqs. 2.19 and 2.20 that as y c increases the center of pressure moves closer to the centroid
of the area. Since y c h csin u, the distance y c will increase if the depth of submergence, h c ,
increases, or, for a given depth, the area is rotated so that the angle, u, decreases. Thus, the hydrostatic
force on the right-hand side of the gate shown in the margin figure acts closer to the centroid
of the gate than the force on the left-hand side. Centroidal coordinates and moments of inertia
for some common areas are given in Fig. 2.18.
––
b
2
––2
b
c
y
x
––2
a
––2
a
A = ba
I –––
1 xc = ba 3
12
I yc = –––
1
ab 3
12
I xyc = 0
R
c
y
x
A = π R 2
I –––––
π R 4
xc = I yc =
4
I xyc = 0
(a) Rectangle
(b) Circle
R
c
y
x
R
4R
–––
3π
A = –––––
π R2
2
I xc = 0.1098R 4
I yc = 0.3927R 4
I xyc = 0
a
d
b + d
–––––––
3
c
y
b
x
A = –––
ab I 2 xc = –––-–
ba 3
36
I xyc = –––––
ba 2
(b – 2d)
72
a
–– 3
(c) Semicircle
(d) Triangle
4R
–––
3π
c
R
y
4R
–––
3π
x
A = –––––
π R2
4
I xc = I yc = 0.05488R 4
I xyc = –0.01647R 4
F I G U R E 2.18
(e) Quarter circle
Geometric properties of some common shapes.
1 Recall that the parallel axis theorem for the product of inertia of an area states that the product of inertia with respect to an orthogonal
set of axes 1x–y coordinate system2 is equal to the product of inertia with respect to an orthogonal set of axes parallel to the original set
and passing through the centroid of the area, plus the product of the area and the x and y coordinates of the centroid of the area. Thus,
I xy I xyc Ax c y c .