ON THE MOTION OF BODIES IN FLUIDS.

PROBLEM XIX.

To determine the Force of Fluids in Motion; and the Cir. cumstances attending Bodies moving in Fluids.

1. Ir is evident that the resistance to a plane, moving perpendicularly through an infinite fluid, at rest, is equal to the pressure or force of the fluid on the plane at rest, and the fluid moving with the same velocity, and in the contrary direction, to that of the plane in the former case. But the force of the fluid in motion, must be equal to the weight or pressure which generates that motion; and which, it is known, is equal to the weight or pressure of a column of the fluid, whose base is equal to the plane, and its altitude equal to the height through which a body must fall, by the force of gravity, to acquire the velocity of the fluid: and that altitude is, for the sake of brevity, called the altitude due to the velocity. So that, if a denote the area of the plane, v the velocity, and n the specific gravity of the fluid;

v2

then, the altitude due to the velocity v being the whole

2g'

resistance, or motive force m, will be a Xnx

And

g being, as we have all along assumed it, = 321 feet. hence, cæteris paribus, the resistance is as the square of the velocity.

2. This ratio, of the square of the velocity, may be other. wise derived thus. The force of the fluid in motion must be as the force of one particle multiplied by the number of them; but the force of a particle is as its velocity; and the number of them striking the plane in a given time, is also as the velocity; therefore the whole force is as v X vor v2, that is, as the square of the velocity.

3. If the direction of motion, instead of being perpendicular to the plane, as above supposed, be inclined to it in any angle, the sine of that angle being s, to the radius 1 ; then the resistance to the plane, or the force of the fluid against the plane, in the direction of the motion, as assigned above, will be diminished in the triplicate ratio of radius to the sine of the angle of inclination, or in the ratio of 1 to s3. VOL. II.

55

A

-D

For, AB being the direction of the plane, and BD that of the motion, making the angle ABD, whose sine is s; the number of particles, or quantity of the fluid striking the plane, will be diminished in the ratio of 1 to s, or of radius to the E sine of the angle в of inclination; and the force of each particle will also be diminished in the same ratio of 1 to s so that, on both these accounts, the whole resistance will be diminished in the ratio of 1 to s2, or in the duplicate ratio of radius to the sine of the said angle. But again, it is to be considered that this whole resistance is exerted in the direction BE perpendicular to the plane; and any force in the direction BE, is to its effect in the direction AE, parallel to BD, as AE to BE, that is, as 1 to s. So that finally, on all these accounts, the resistance in the direction of motion, is diminished in the ratio of 1 to s3, or in the triplicate ratio of radius to the sine of inclination. Hence, comparing this with article 1, the whole resistance, or the motive force on the plane, will be anv2s3

4. Also, if w denote the weight of the body, whose plane face a is resisted by the absolute force m; then the retarding

5. And if the body be a cylinder, whose face or end is a, and diameter d, or radius r, moving in the direction of its axis; because then s = 1, and a pr2 = pd2, where p=3.1416; the resisting force m will be npd13___npr2v3

=

8g 2g

=

and the retarding force ƒ =

npd22 npr22

=

8gw 2gw

6. This is the value of the resistance when the end of the cylinder is a plane perpendicular to its axis, or to the direc tion of motion. But were its face a conical surface, or an elliptic section, or any other figure every where equally inclined to the axis, the sine of inclination being s: then the number of particles of the fluid striking the face being still the same, but the force of each opposed to the direction of motion, diminished in the duplicate ratio of radius to the sine of inclination, the resisting force m would be

2g

But if the body were terminated by an end or face of any

other form, as a spherical one, or such like, where every part of it has a different inclination to the axis; then a further investigation becomes necessary, such as in the following proposition.

PROBLEM XX.

To determine the Resistance of a Fluid to any Body, moving in it, of a Curved End; as a Sphere, or a Cylinder, with a Hemispherical End, &c.

1. LET BEAD be a section through the axis ca of the solid, moving in the direction of that axis. To any point of the curve draw the tangent EG, meeting the axis produced in G: also, draw the perpendicular ordinates EF, ef, indefinitely near each other; and draw ae parallel to

CG.

Putting CF = x, EF = Y, BE = z, s = sine o to radius 1, and p 3.1416: then 2py is the circumference whose radius is EF, or the circumference described by the point E, in revolving upon the axis ca; and 2py X Ee or 2pyż is the fluxion of the surface, or it is the surface described by Ee, in the said revolution about ca, and which is the quantity represented by a in art. 3 of the last problem: hence nv2s3 pnv83 g

X 2pyż or Xyż is the resistance on that ring,

2g or the fluxion of the resistance to the body, whatever the figure of it may be. And the fluent of which will be the re

sistance required.

2. In the case of a spherical form: putting the radius ca

or CB = r, we have y = √(r2—x3), s =

yż or EF X Ee = CE X ae = r; therefore the general

fluxion X s3yż becomes X

pnv
g

g

pnv2 X ri = gr2

4g, is the resistance to the

And when x or CF is r

for the resistance on the whole

spherical surface generated by be.
pnv22
or CA, it becomes
4g

hemisphere; which is also equal to

the diameter.

pnv❜d
16g

where d 2r

3. But the perpendicular resistance to the circle of the same diameter d or BD, by art. 5 of the preceding problem,

is

pnv'd2 Sg resistance to the sphere, is just equal to half the direct resistance to a great circle of it, or to a cylinder of the same diameter.

; which, being double the former, shows that the

4. Since pd' is the magnitude of the globe; if N denote its density or specific gravity, its weight w will be = pd'N, m pnv2d2 6 X

and therefore the retardive force f or

w 16g PNd3

theorems in page 401; hence then

3n
= and s =
?
4Nd S

Xd; which is the space that would be described by the globe, while its whole motion is generated or destroyed by a constant force which is equal to the force of resistance, if no other force acted on the globe to continue its motion. And if the density of the fluid were equal to that of the globe, the resisting force is such, as, acting constantly on the globe without any other force, would generate or destroy its motion in describing the space 4d, or of its diameter, by the accelerating or retarding force.

5. Hence the greatest velocity that a globe will acquire by descending in a fluid, by means of its relative weight in the fluid, will be found by making the resisting force equal to that weight. For, after the velocity is arrived at such a degree, that the resisting force is equal to the weight that urges it, it can increase no longer, and the globe will afterwards continue to descend with that velocity uniformly. Now, N and n being the separate specific gravities of the globe and fluid, N-n will be the relative gravity of the globe in the fluid, and therefore w = pd3 (N-n) is the weight by which it is urged; also m =

is found v= √(2g. ‡d.), for the said uniform or great

est velocity.

And, by comparing this form with that in art. 6 of the general theorems in page 400, it will appear that its greatest velocity is equal to the velocity generated by the accelerating force in describing the space d, or equal to the

N-n
n

velocity generated by gravity in freely describing the space

Xd. If N = 2n, or the specific gravity of the

globe be double that of the fluid, then

Nn
n

= 1 = the

natural force of gravity; and then the globe will attain its greatest velocity in describing d or of its diameter.-It is further evident that if the body be very small, it will very soon acquire its greatest velocity, whatever its density may be.

EXAM. If a leaden ball, of 1 inch diameter, descend in water, and in air of the same density as at the earth's surface, the three specific gravities being as 11, and 1, and Then (4. 16.. 10)=√(31. 193)=8-5944 feet, is the greatest velocity per second the ball can acquire by descending in water. And v √(4.193.. ༢、.25,༠༠) nearly 34193=259-82 is the greatest velocity it can acquire in air.

But if the globe were only of an inch diameter, the greatest velocities it could acquire, would be only of these, namely of a foot in water, and 26 feet nearly in air. And if the ball were still further diminished, the greatest velocity would also be diminished, and that in the subdupli. cate ratio of the diameter of the ball.

PROBLEM XXI.

To determine the Relations of Velocity, Space, and Time, of a Ball moving in a Fluid, in which it is projected with a given Velocity.

1. Let a the first velocity of projection, r the space described in any time t, and v the velocity then.

art. 4 of the last problem, the accelerative force ƒ

Now, by

=

3nv2

8g Nd'