CENTRE OF PRESSURE.

703. Def. The centre of pressure of a surface pressed by a fluid, is that point, at which if a force equal to the whole pressure were applied, but in a contrary direction, it would keep the surface at rest.

PROPOSITION CLXXV.

704. If a plane surface which is pressed by a fluid, be produced to the surface of the fluid, and their common intersection be made the axis of suspension, the centre of percussion will be the centre of pressure.

Let ABCD (fig. 217.) be the surface of the fluid which presses. on the plane VW; produce this plane till it meets the surface in the line mSC, and let Obe the centre of pressure; from any point x in the pressed surface, erect the perpendicular av, and, in the surface AC, draw vm perdendicular to the intersection mS.

The pressure on the point x is as xxxv, and its effect to turn the plane about mS, is as x ×xv xmS; also its effect to turn the plane about SL, will be as xxxv×mS.

In like manner if the plane VW be supposed to revolve about the axis mS, and to strike an obstacle at O; the percussive force of the particle x, by which it endeavours to turn the plane about mS, will be as xxxm2, or as xxxm×xv; and its force to turn the plane about SL, will be as x ×xm ×mS, or as xxxvxmS. The percussive forces, therefore, of all the particles, whereby they endeavour to move the plane in the two directions, are the same with the forces of pressure; and therefore the centres of pressure and percussion are coincident.

Consequently, the theorems given in art. 700, &c. may be applied to the determination of the centre of pressure. From what is done above, it is plain that the centres of pressure and percussion are always coincident, but the centres of pressure and oscillation, do not always coincide.

On the discharge of fluids through apertures in the bottoms and sides of vessels, &c.

(CONTINUED)

PROPOSITION CLXXVI.

705. To find the time in which a quantity of water (or any other non-elastic fluid) contained in a vessel of known form, will be evacuated through an orifice in the bottom, or in the side contiguous to the bottom, the altitude of the orifice being very small compared with the altitude of the fluid.

Put the area of the orifice a, that of the upper surface of the fluid A, the height of the fluid above the orifice=h, the vertical space descended by the upper surface in any time t=r, and 16=g; then, h-x is the altitude of the fluid at the end of the time t, and the velocity of the fluid at the orifice4g(h-¿). This velocity evidently varies every instant, but for the indefinitely small time t, we may regard it as constant; therefore, in the time t, there will flow through the orifice a prism of fluid which has the area of the orifice a for its base, and4g(h—x) for its altitude; thus the quantity of fluid which runs out during the time i=ei√4g(h—x).

But during this time the upper surface has descended through the vertical space x, and the vessel has lost a prism of the fluid whose base is A and altitudex, or a prism whose capacity is Ax; hence, by making these two values equal we have Ax=at√4g(h—a); from whence,

As the area A will be given in terms of x, by the form of the vessel, the second member of the above equation may be considered as containing only the variable quantity x; and by taking the fluents, we shall obtain the successive depressions and evacuations of the fluid in the vessel.

EXAMPLES.

Er. 1. Let the vessel be an upright prism or cylinder.

Here the area A is the same at all altitudes, or it is constant;

But when the time t is nothing, the depression x of the superior surface of the fluid A, is nothing also; thus when to, x=0; this condition determines the constant quantity 2A C=- h, and gives for the time of depressing the upper av4g

We can now easily find the time in which the vessel will be entirely empty; for we need only make a=h, in the preceding expression, and we have t This time is evidently

A

double of that found in art. 553, where the vessel was kept constantly at the same altitude.

Ex. 2. Let the vessel be a solid generated by any curve revolving round its axis, the axis being vertical.

Here A will be the area of a circle which has for its radius the ordinate y of the generating curve; that is, A=py2; (where p=3.141, &c.) and the equation (a) above becomes i__?

a√4g

yż (h

Er. 3. Let the solid of rotation; that is, the vessel, be a paraboloid with its vertex downwards, (fig. 218.)

Let BC be the upper surface of the fluid at first; put CAh, and when the parameter is b, the equation of the

parabola is y2=br; let the origin be at A, the point where we suppose the orifice to be; then, if we transfer the origin from A to C, the equation is y2=b×(h—x), and by substituting this for y2 in the equation (c) above, we have

and determining C by the condition that when to, ao also,

fluid at the commencement of the exhaustion, and h the whole height,

PROPOSITION CLXXVII.

706. To determine with what velocity air will rush into a void space or vacuum when impelled only by its own weight.

By art. 250, when the moving force is proportional to the body, the velocity is constant; that is, when fxbv, and fxb, then vol, or v is constant; hence, this problem is analagous to art. 548, from whence it follows, that air rushes from the atmosphere into a vacuum, with the velocity which a heavy body would acquire by falling from the altitude of a homogeneous atmosphere.

Therefore, put the altitude of a homogeneous atmosphere= 27600 feet=h; then (art. 284.) v=√(4gh)=1333 feet nearly, is the velocity per second, as required.

707. If, therefore, a ball from a cannon or musket, move with a velocity greater than about 1333 per second, it will leave a vacuum behind it; and of course it must sustain the whole pressure of the atmosphere on its fore part, beside the resistance which it meets with from the inertia of the particles of which the atmosphere is composed,

PROPOSITION CLXXVIII.

708. Suppose the force of gravity to vary as the nth power of the distance from the earth's centre, and the compressive r force of the air to vary as its density; to determine the relation between the density of the air, and its altitude above the surface of the earth.

Let the earth's radius=1, the distance of any point from its centrer, the density of the air at that point=d, the density at the surface being unity, and h the height of a homogeneous atmosphere.

Since the compressive force is as the density, the fluxion of the compressive force is proportional to the fluxion of the density, and this ratio is the same at all altitudes.

But at any distance x from the earth's centre, the fluxion of the compressive force must vary as the force of gravity, the density, and the fluxion of the altitude, conjointly; therefore, xdx has a constant ratio to-d where d is negative because d

decreases as r increases.

Now since (art. 577.) h denotes the compressive force of the air at the earth's surface, where the density is 1; we have h: 1 :: xdx:-d, and r"——h×d÷d; therefore

hxhyp. log. v+C.

But when x=1, d=1, and this equation becomes

n+1

n+1

1

n+

1

n+1

hence, the correct fluent is =-hxhyp. log.d+

therefore,

hxhyp. log. d; which is a general

expression for the relation between the density and altitude. 709. Cor. 1. Cor. 1. When the force of gravity varies inversely as the square of the distance, we have n=-2; and the above equation becomes 1÷÷x-1=h× hyp. log. d.

710. Cor. 2. If the force of gravity be supposed constant, then no; and 1-x=hxhyp, log. d.