Mechanics of Fluids

where the suffixes C and Oy indicate the axes used. Therefore
(Ak 2 )Oy = (Ak 2 )C + Ax 2
The direction of the y-axis was arbitrary; hence it may be said that the
second moment of a plane area about any axis equals the sum of the second
moment about a parallel axis through the centroid and the product of
the area and the square of the perpendicular distance between the axes.
This result is frequently known as the parallel axes theorem. Moreover, by
definition,
(Ak 2 )Oy + (Ak 2 �
)Ox =
�
=
A
A
x 2 �
dA + y
A
2 dA
(x 2 + y 2 �
)dA =
Hydrostatic thrusts on submerged surfaces 59
A
r 2 dA
The last term corresponds to the second moment of the area about an axis
perpendicular to the plane of the area at the origin. Since the origin was arbitrarily
chosen we have the perpendicular axes theorem: the second moment of
a plane area about an axis meeting the plane perpendicularly at any point P
equals the sum of the second moments of that area about two axes in the
plane that intersect perpendicularly at P.
The second moment of mass about a particular axis is ∫ M z 2 dM, where z
represents the perpendicular distance of an element from the axis in question.
If the mean value of z 2 is represented by k 2 , the second moment of
mass may alternatively be written Mk 2 . The second moment of mass is also
known as the moment of inertia, and k is termed the radius of gyration.
The dimensional formula of moment of inertia is [ML 2 ] and a suitable unit
is kg · m 2 .
Unfortunately the second moment of area is sometimes referred to wrongly
as moment of inertia. Inertia is a property of matter and has nothing to do
with area. Second moment of area Ak 2 , a purely geometric quantity, and
moment of inertia Mk 2 are fundamentally different. In this book we use Ak 2
as the symbol for second moment of area, a suitable suffix indicating the axis
about which the moment is taken.
Examples of second moments about centroidal axes are given in Fig. 2.14.
2.5 HYDROSTATIC THRUSTS ON SUBMERGED SURFACES
The pressure of a fluid causes a thrust to be exerted on every part of any
surface with which the fluid is in contact. The individual forces distributed
over the area have a resultant, and determination of the magnitude, direction
and position of this resultant force is frequently important. For a plane
horizontal surface at which the fluid is in equilibrium the matter is simple:
the pressure does not vary over the plane and the total force is given by the
product of the pressure and the area. Its direction is perpendicular to the
plane – downwards on the upper face, upwards on the lower face – and its
position is at the centroid of the plane. But if the surface is not horizontal the