1 dp 1 dp 1 d[theta] [rho] 1 d[theta] 1 1 d[theta]
----- -- = --- -- - ------- -------- = ----- - ------- -------- = --- - ------- --------, (8)
[rho] dz p dz [theta] dz p [theta] dz k [theta] dz
which is positive, and the density [rho] diminishes with the ascent,
provided the temperature-gradient d[theta]/dz does not exceed
[theta]/k.
With uniform temperature, taking k constant in the gas-equation,
dp/dz = [rho] = p/k, p = p0e^(z/k), (9)
so that in ascending in the atmosphere of thermal equilibrium the
pressure and density diminish at compound discount, and for pressures
p1 and p2 at heights z1 and z2
(z1 - z2)/k = log e (p2/p1) = 2.3 log10 (p2/p1). (10)
In the convective equilibrium of the atmosphere, the air is supposed
to change in density and pressure without exchange of heat by
conduction; and then
[rho]/[rho]0 = ([theta]/[theta]0)^n, p/p0 =
([theta]/[theta]0)^(n+1), (11)
dz 1 dp p 1
-------- = ----- -------- = (n + 1)------------R, [gamma] = 1 + ---,
d[theta] [rho] d[theta] [rho][theta] n
where [gamma] is the ratio of the specific heat at constant pressure
and constant volume.
In the more general case of the convective equilibrium of a spherical
atmosphere surrounding the earth, of radius a,
dp p0 d[theta] a²
----- = (n + 1)------ -------- = - --- dr, (12)
[rho] [rho]0 [theta]0 r²
gravity varying inversely as the square of the distance r from the
centre; so that, k = p0/[rho]0, denoting the height of the homogeneous
atmosphere at the surface, [theta] is given by
(n + 1) k (1 - [theta]/[theta]0) = a(1 - a/r), (13)
or if c denotes the distance where [theta] = 0,
[theta] a c - r
-------- = --- · -----. (14)
[theta]0 r c - a
When the compressibility of water is taken into account in a deep
ocean, an experimental law must be employed, such as
p - p0 = k([rho] - [rho]0), or [rho]/[rho]0 = 1
+ (p - p0)/[lambda], [lambda] = k[rho]0, (15)
so that [lambda] is the pressure due to a head k of the liquid at
density [rho]0 under atmospheric pressure p0; and it is the gauge
pressure required on this law to double the density. Then
dp/dz = kd[rho]/dz = [rho], [rho] = [rho]0e^(z/k),
p - p0 = k[rho]0(e^(z/k) - 1); (16)
and if the liquid was incompressible, the depth at pressure p would be
(p - p0)/p0, so that the lowering of the surface due to compression is
ke^(z/k) - k - z = ½z²/k, when k is large. (17)
For sea water, [lambda] is about 25,000 atmospheres, and k is then
25,000 times the height of the water barometer, about 250,000 metres,
so that in an ocean 10 kilometres deep the level is lowered about 200
metres by the compressibility of the water; and the density at the
bottom is increased 4%.
On another physical assumption of constant cubical elasticity
[lambda],
dp = [lambda]d[rho]/[rho], (p - p0)/[lambda] = log([rho]/[rho]0), (18)
dp [lambda] d[rho] / 1 1 \ [rho]0 z
-- = -------- ------ = [rho], [lambda]( ------ - ----- ) = z, 1 - ------ = ---, [lambda] = k[rho]0, (19)
zd [rho] dz \[rho]0 [rho]/ [rho] k
and the lowering of the surface is
p - p0 [rho] / z \ z²
------ - z = k log ------ - z = k log ( 1 - --- ) - z [approx] --- (20)
[rho]0 [rho]0 \ k / 2k
as before in (17).
16. _Centre of Pressure._--A plane area exposed to fluid pressure on
one side experiences a single resultant thrust, the integrated
pressure over the area, acting through a definite point called the
centre of pressure (C.P.) of the area.
Thus if the plane is normal to Oz, the resultant thrust
_ _
/ /
R = | |pdxdy, (1)
_/_/
and the coordinates [=x], [=y] of the C.P. are given by
_ _ _ _
/ / / /
[=x]R = | | xp dx dy, [=y]R = | | yp dx dy. (2)
_/_/ _/_/
The C·P. is thus the C·G. of a plane lamina bounded by the area, in
which the surface density is p.
If p is uniform, the C·P. and C·G. of the area coincide.
For a homogeneous liquid at rest under gravity, p is proportional to
the depth below the surface, i.e. to the perpendicular distance from
the line of intersection of the plane of the area with the free
surface of the liquid.
If the equation of this line, referred to new coordinate axes in the
plane area, is written
x cos [alpha] + y sin [alpha] - h = 0, (3)
_ _
/ /
R = | | [rho](h - x cos [alpha] - y sin [alpha]) dx dy, (4)
_/_/
_ _
/ /
[=x]R = | | [rho]x(h - x cos [alpha] - y sin [alpha]) dx dy, (5)
_/_/
_ _
/ /
[=y]R = | | [rho]y(h - x cos [alpha] - y sin [alpha]) dx dy.
_/_/
Placing the new origin at the C.G. of the area A,
_ _ _ _
/ / / /
| | xd x dy = 0, | | y dx dy = 0, (6)
_/_/ _/_/
R = [rho]hA, (7)
_ _ _ _
/ / / /
[=x]hA = -cos [alpha] | | x² dA - sin [alpha] | | xy dA, (8)
_/_/ _/_/
_ _ _ _
/ / / /
[=y]hA = -cos [alpha] | | xy dA - sin [alpha] | | y² dA. (9)
_/_/ _/_/
Turning the axes to make them coincide with the principal axes of the
area A, thus making [int][int] xy dA = 0,
[=x]h = -a² cos [alpha], [=y]h = -b² sin [alpha], (10)
where
_ _ _ _
/ / / /
| | x² dA = Aa², | | y² dA = Ab², (11)
_/_/ _/_/
a and b denoting the semi-axes of the momental ellipse of the area.
This shows that the C.P. is the antipole of the line of intersection
of its plane with the free surface with respect to the momental
ellipse at the C.G. of the area.
Thus the C.P. of a rectangle or parallelogram with a side in the
surface is at 2/3 of the depth of the lower side; of a triangle with a
vertex in the surface and base horizontal is ¾ of the depth of the
base; but if the base is in the surface, the C·P. is at half the depth
of the vertex; as on the faces of a tetrahedron, with one edge in the
surface.
The _core_ of an area is the name given to the limited area round its
C.G. within which the C·P. must lie when the area is immersed
completely; the boundary of the core is therefore the locus of the
antipodes with respect to the momental ellipse of water lines which
touch the boundary of the area. Thus the core of a circle or an
ellipse is a concentric circle or ellipse of one quarter the size.
The C.P. of water lines passing through a fixed point lies on a
straight line, the antipolar of the point; and thus the core of a
triangle is a similar triangle of one quarter the size, and the core
of a parallelogram is another parallelogram, the diagonals of which
are the middle third of the median lines.
In the design of a structure such as a tall reservoir dam it is
important that the line of thrust in the material should pass inside
the core of a section, so that the material should not be in a state
of tension anywhere and so liable to open and admit the water.
[Illustration: FIG. 3.]
17. _Equilibrium and Stability of a Ship or Floating Body. The
Metacentre._--The principle of Archimedes in § 12 leads immediately to
the conditions of equilibrium of a body supported freely in fluid, like
a fish in water or a balloon in the air, or like a ship (fig. 3)
floating partly immersed in water and the rest in air. The body is in
equilibrium under two forces:--(i.) its weight W acting vertically
downward through G, the C.G. of the body, and (ii.) the buoyancy of the
fluid, equal to the weight of the displaced fluid, and acting vertically
upward through B, the C.G. of the displaced fluid; for equilibrium these
two forces must be equal and opposite in the same line.
The conditions of equilibrium of a body, floating like a ship on the
surface of a liquid, are therefore:--
(i.) the weight of the body must be less than the weight of the total
volume of liquid it can displace; or else the body will sink to the
bottom of the liquid; the difference of the weights is called the
"reserve of buoyancy."
(ii.) the weight of liquid which the body displaces in the position of
equilibrium is equal to the weight W of the body; and
(iii.) the C.G., B, of the liquid displaced and G of the body, must lie
in the same vertical line GB.
18. In addition to satisfying these conditions of equilibrium, a ship
must fulfil the further condition of stability, so as to keep upright;
if displaced slightly from this position, the forces called into play
must be such as to restore the ship to the upright again. The stability
of a ship is investigated practically by inclining it; a weight is moved
across the deck and the angle is observed of the heel produced.
Suppose P tons is moved c ft. across the deck of a ship of W tons
displacement; the C.G. will move from G to G1 the reduced distance
G1G2 = c(P/W); and if B, called the centre of buoyancy, moves to B1,
along the curve of buoyancy BB1, the normal of this curve at B1 will
be the new vertical B1G1, meeting the old vertical in a point M, the
centre of curvature of BB1, called the _metacentre_.
If the ship heels through an angle [theta] or a slope of 1 in m,