Mechanics of Fluids

62 Fluid statics
of the free (i.e. atmospheric) surface divided by the first moment of the area
about the intersection of its plane with that of the free surface.
The centre of pressure is always lower than the centroid (except when the
surface is horizontal) as the following calculation demonstrates.
From the parallel axes theorem:
so eqn 2.16 becomes
(Ak 2 )Ox = (Ak 2 )C + Ay 2
y ′ = (Ak2 )C + Ay 2
Ay
= y + (Ak 2 )C/Ay (2.17)
Since a second moment of area is always positive it follows that y ′ > y. QED.
We see also that the more deeply the surface is submerged, that is, the
greater the value of y, the smaller is the contribution made by the last term
in eqn 2.17 and the closer is the centre of pressure to the centroid. This
is because, as the pressure becomes greater with increasing depth, its variation
over a given area becomes proportionately smaller, so making the
distribution of pressure more uniform. Thus where the variation of pressure
is negligible the centre of pressure may be taken as approximately at the
centroid. This is justifiable in gases, because in them the pressure changes
very little with depth, and also in liquids provided the depth is very large
and the area small.
The expressions 2.16 and 2.17, it is re-emphasized, give the distance to
the centre of pressure measured down the plane from the level of the free
surface and not vertically.
The x-coordinate of the centre of pressure may be determined by taking
moments about Oy. Then the moment of δF is ϱgy sin θδAx and the total
moment is
Fx ′ �
�
= ϱgxy sin θdA = ϱg sin θ xydA
so
A
x ′ =
�
A xydA
Ay
A
(2.18)
When the area has an axis of symmetry in the y direction, this axis may
be taken as Oy and then �
A xydA is zero, that is, the centre of pressure lies
on the axis of symmetry. It will be noted from eqns 2.16 and 2.18 that the
position of the centre of pressure is independent of the angle θ and of the
density of the fluid. However, a constant value of ϱ was used; the relations
are therefore valid only for a single homogeneous fluid.
For the plane lamina of negligible thickness illustrated in Fig. 2.15, the
force on one face would exactly balance the force on the other if both faces
were in contact with the fluid. In most cases of practical interest, however,
there is no continuous path in the fluid from one face of the plane to the
other and therefore the pressures at corresponding points on the two faces
are not necessarily the same. For example, the surface may be that of a plate
covering a submerged opening in the wall of a reservoir, or a canal lock-gate