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119
HYDROMECHANICS


and the lowering of the-surface ia, ' ' ~, f .. U th- ... e; - (J) .g gf

z klogpu z klogr E, z 2-E (zo)

as beforem 17). . a - A-16.

Centre of Pressure.—A plane area exposed to fluid pressure on one side experiences asingle 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 resultantthrust R=ff1>dxdy, (F).

and the co-ordinates x, y of the are iven by * &R =fé'xpdxdy, 5»R =f§ 'ypdxdy. ' (2) The C.P. is thus the .G. of aplairelamitia 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 ine of intersection of the plane of the area with the free surface of the liquid. f-If

the equation of this line, referred to. new coordinate axes in the plane area, is written

xcos a-{-ysin iz.—h=0, (3)

R=f p(h-xcos a-y sin'a)dxdy, - " (4)

5cR= px(h-<;x-cos a—ysina/)dxdy, (5)

5/R = fpy(h -~x cos a.-y sin u)dxdy.

Placing the new ori in at the C.G. of the area A, ' " |'§ :¢dxdy=0, fJydxdy=0, ' i 'f(6)

R=Ph» »; (7)

xhA= -cos affx'dA- sin affxydA, (8)

yhA= 4-cos af{xydA >~sin ¢ffy°dA.1 < ' (§)f Turning the axes to make t em coincide with the principal axesof the area A, thus making ffxydA=0, 1 = ' ' 2 aEh=v-ia"cos»a., yh=-b'sin-a; . i » (io) < where »

f {x'dA=A4', ffy'dA=Ab', (II)

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 5 of the depth of the lower side; of triangle with a vertex in the surface and base horizontal is % of the depth o the basei 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 'tolthe 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 oran ellilpse is a concentric circle or ellipse of one quarter the Size: he C.P. of water lines passing through a fixed point lies on a straight line, the anti polar 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. sln the design of a structure such as a tall reservoir dam 1t 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. 17. Equilibrium and Stability of a, Ship or Floating Body. The M etacenlrer'-The. principle of Archirnedes, in § 12 leads Mintmediately.. to the

conditions of equilibrium

of a. body supported

freely in fluid,

like a iishiu water or

a balloon in the air

or like a ship (fig, gl

floating partly im-Jl

G 61 .

f

I-1"

t . mersed inwater and

the rest in a, ir. The(

body is in equilibrium

under two

forces:-(i.) its

weight W acting

verticallyvdownward

through G, the C.G. of the body, and (il.) 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;

FIG. 3.

for equilibrium V-these two; forces must be equaland opposite in the same' line; ' ~ A

The conditions. of equilibrium of a body, floating like aship on the surface of a liquid, are therefore:-(i.) the weight of thebody must be less than the weight of the total volume of liquid it can displace; or else the body will sink -to the bot-tom of the liquid; the difference of the weights is called the "' reserve of buoyancy.”

(ii.) the weight of liq1iid'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 liein the same vertical line GB; 18. In addition to satisfying these conditions of equilibrium, a ship must fulfil the further condition of stability, soas to keep upright; if displaced slightly from this position, the forces called into play must besuch as to restore the ship to the upright again. The stability of a shipis investigated practically by inclining it; a weight is moved across the deck and the angle is observed of the heel produced. ~ A

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 G-fthe' reduced distance G»Cu=c(P/W);-and' if B, called the centre of buoyancy, moves to Bi .along the curve of buoyancy BB1, the normal of this curve at B1 will be the new vertical B1Gly meeting the old vertical in a point M, the centre of curvature of BB1, called the meta centre. < 'If the'ship'heelsfthrqugh an angle 0 or a slope of I in m, - ' ' 'GM=GGicot6=mc(P/W), ' ' (1)

and GM is called the' meta centric hei ht; and the shi' 'must be ballasted, so that G lies below M. If 5 was above M, the tangent drawn (rpm G to the évolute of B, and normal to the curve of buoyanc, would: give the vertical in a new position of equilibrium. ¢"I'hus<lh M.S. ' ' Achilles T' of,9000 tons displacement i twas found that movirig~2o'tons' across the deck, a distance of 42 ft., causedtlie bob ofla pendulum 20lf1i:'l0l'l'g' to move 'through' to in., so that * » ' *-V iGMf%>§ 42>§ % =2'~24ft.i ~ iii) 1°

alS9 ' . - A I f

y cot a ==24, 0 =2°24/. (3), ,

Ina diagram it isconduciveto clearness to draw the.ship in one position, and to incline the wa.ter-line; and the pagecan be turned if it is desired to bring the new water-line horizontal. Suppose the ship turns about an axis.through F in-thewater-line area, perpendicular to the plane of the paper; denoting by y the distance of an element-dA 1f'the water-line area from the axis of rotation; the change of displacement is 2ydA tan 6, so that there is no change of displacement if 2ydA=jo, , that 1s, 'if the axis passes through the 'C.G. of the watéréline area, which we denote by F and call the centre of flotation.

The righting couple of the wedges of immersion and 'emersion willbe - i » “

~ EwydA tan'9Ly sw tan 0Zy”dA =-iw tan 6.Ak2 ft. tons, ' (4) w denoting the density of water in tons/ft.3, and W=wV, x for a displacement of V ft.3 <

his couple, combined with the original buoyancy*W through B, is equivalent to the new buoyancy through B, so that. W.BB;l='wAk2tan6, A » ' (5)

»BM=BB, cot6=Ak?-/V, (6)

giving the radius of curvature BM of the curve ofibuoyancy B, in terms of the displacement V, and Akz the moment of inertia of the water-line area about 'an axis -th-rough F, perpendicular to the plane of displacement., - »= . ,

An inclining couple due to moving a weight about in a ship -wgill-heel the, ship aboutan axis perpendicular to the plane of the coupletouly when this axis is a 'principal axis at F of themomental ellipse. of the water-li ne area A. For if the ship turns through a smallvangleid about the l1ne Fl, then bl, bzythe C.G. of the .wedge of .immersion and emersion, will be the C.P. with respect 'to F F ' of the two partsof the water-line area, so that bib; will be conjugate to FF' with respect to the momental ellipse at F. Q =;

The naval architect distinguishes between the stability of, form, represented by the righting couple W.BM, and the stability»of'balla§ ling, represented by W, BG. Ballasted with G at B, the righting couple when the ship is heeled through 0 is given by W.BM. tan 6; but if weights inside the ship are raised to bring Q above B, the -righting couple is diminished by W§ BGJ tan 0, so t 'at the resultant righting couple is W.GM. tan P, Provided the ship' is designed to float upright at the smallest draft with no load on board, the stability at any other draft of water can be arranged by the stowage of .the weight, high or low. '

19. Proceeding as in § 16 for the determination of the C.P. of an

area, the same argument will show that an inclining couple due to