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ON THE MOTION OF BODIES IN FLUIDS.

PROBLEM XIX.

To determine the Force of Fluids in Motion; and the Circum

stances attending Bodies Moving in Fluids. 1. It 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 Auid 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 snown, 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;

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

4g

anva resistance, or motive force m, will be a xnx

4g 48 g being 1644 feet. And 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, inust be as the force of one particļe 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 v or v?, that is, as the square of the velocity.

3. If the direction of motion, instead of being perpendi. cular 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

3

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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 53. For, AB being the direction of the plane, and Bd that of the motion, making the angle ABD, whose sine is s; the number

D 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 sine of the angle B 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 s’, or in the duplicate ratio of radius to the sine of the said angle. But again, it is to be considered that this whole resitance 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 I to s. So that finally, on all these accounts, the resistance in the direction of motion, is diminished in the ratio of 1 to s', 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

any's..

4g

m

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

anv?s3 retarding force f, or will be

4gτυ 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 thens = 1, and a = pr?

pd, where p= 3.1416; the resisting force m will be npd ́v? npr m2

npdo_npr?q? and the retarding force

f = 16g 4g

16gw

4gw :

6. This is the value of the resistance when the end of the cylinder is a plane perpendicular to its axis, or to the direction 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 zle same, but the force of each, opposed to the direction

of

of motion, diminished in the duplicate ratio of radius to the sine of inclination, the resisting force m would be mpd'vosi _ nprivs ?

16g 4g

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

BO E axis ca of the solid, moving in the direction of that axis. To any point of the curve draw the tangent EG, meeting the C axis produced in G: also, draw the per

IfAG pendicular ordinates EF, ef, indefinitely near each other; and draw ae parallel to CG,

Putting cf = X, EF =, BE = %, s= sine < Ġ 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 about the axis CA; and 2py * 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 nu??

pnu`s3 x 2pyž or x yż is the resistance on that ring, 4g

2g or the fluxion of the resistance to the body, whatever the figure of it may be. And the Quent of which will be the resistance required. 2. In the case of a spherical form : putting the radius CA

EF or cb = r, we have y = n god yż, or EF X Ee = CE X seri; therefore the general x ryż becomes

pnu?

**** r* = * x3x3 2g 2g

the

CF

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sand

EG

CE

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Auxion pnum

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2gr2

the fluent of which, or

png?

x4, is the resistance to the

8gr? spherical surface generated by be. And when x or CF is =r

pnv en? or ca, it becomes for the resistance on the whole

8g hemisphere, which is also equal to

pnvéd

where d = 2r the diameter.

328

m

х

W

16gndi which is also =

N

and s =

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12

3. But the perpendicular resistance to the circle of the same diameter d or bd, by art. 5 of the preceding problem, is pnu*d?

; which, being double the former, shows that the 16g 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.

4. Since fpdis the magnitude of the globe; if n denote its density or specific gravity, its weight w will be = ipd,

pnvida 6 and therefore the retardive force for

32g

pnd3 3 nu?

02

hy art. 8 of the general 4gs

3n 1 theorems in page 342; hence then

And x 4d; which is the space that would be described by the globe, while its whole motion is generated or destroyed by à 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 describi.g the space $d, or of its diameter, by that 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 will 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 = 1pd3 (N - 11) is the

pnuod?

N

n

weight by which it is urged ; also m = pnv da

is the resistance ; consequently = tpd (N n) 32g

32g when the velocity becomes uniform; from which equation is found v = v(4g · d.

"), for the said uniform or greatest velocity.

And, by comparing this form with that in art. 6 of the general theorems in page 342, 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 velocity generated by gravity in freely describing the space

x 4d. If n = 2n, or the specific gravity of the

Nn

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n

N

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n

N - n

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globe be double that of the fluid, then =l= 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.

34

2500 3

.

50

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 1lf, and 1, and zoo Then v=v4.167774.101 = 131.193 = 8.5944 feet, is the greatest velocity per second the ball can acquire by descending in water.

And v =

✓4. YZ? • ਨ nearly

= 259.82 is the greatest velocity it can acquire in air.

But if the globe were only too of an inch diameter, the greatest velocities it could acquire, would be only to 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 subduplicate ratio of the diameter of the ball.

V 34'197

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. Vol. II.

1. LET

BE

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