Cylindrical Motion

PROPOSITION 37 THEOREM 29

A cylinder moves uniformly forwards in a compressed, infinite, and non-elastic fluid, in the direction of its length.

The arising resistance is to the force by which its whole motion may be destroyed or generated, in the time that it moves 4 times its length, as the density of the medium to the density of the cylinder, nearly.

Let:

• the vessel ABDC (Plate 7. Figure 5.) touch the surface of stagnant water with its bottom CD.
• the water run our of this vessel into the stagnant water through the cylindric canal EFTS perpendicular to the horizon.
• the little circle PQ be placed parallel to the horizon any where in the middle of the canal

Produce CA to K, so that AK may be to CK in the duplicate of the ratio, which the excess of the orifice of the canal EF above the little circle PQ, bears to the circle AB.

Then ’tis manifest (by case 5. case 6. and corollary 1. proposition 36.) that the velocity of the water passing thro’ the annular space between the little circle and the sides of the vessel, will be the very same which the water would acquire by falling, and in its fall describing the altitude KC or IG.

By corollary 10, proposition 36, if the breadth of the vessel be infinite, so that the lineola HI may vanish, and the altitudes IG, HG become equal.

The force of the water that flows down, and presses upon the circle will be to the weight of a cylinder whose base is that little circle and the altitude ½IG, as EF² to EF² - ½PQ² very nearly.

For the force of the water flowing downwards uniformly thro’ the whole canal will be the same upon the little circle PQ in whatsoever part of the canal it be placed.

Let now the orifices of the canal EF, ST be closed, and let the little circle ascend in the fluid compressed on every side, and by its ascent let it oblige the water that lies above it to descend thro’ the annular space between the little circle and the sides of the canal.

Then will the velocity of the ascending little circle be to the velocity of the descending water as the difference of the circles EF and PQ is to the circle PQ; and the velocity of the ascending little circle will be to the sum of the velocities, that is, to the relative velocity of the descending water with which it passes by the little circle in its ascent, as the difference of the circles EF and PQ to the circle EF, or as EF² - PQ² to EF².

Let that relative velocity be equal to the velocity with which it was shewn above that the water would pass thro’ the annular space if the circle were to remain unmoved, that is, to the velocity which the water would acquire by falling, and in its fall describing the altitude IG.

The force of the water upon the ascending circle will be the same as before, (by corollary 5. of the laws of motion) that is, the resistance of the ascendirg little circle will be to the weight of a cylinder of water whose base is that little circle and its altitude ½IG, as EF² to EF² - ½PQ² nearly.

But the velocity of the little circle will be to the velocity which the water acquires by falling, and in its fall describing the altitude IG, as EF² - PQ² to EF².

Let the breadth of the canal be increased in infinitum; and the ratio’s between EF² - P² and EF², and between EF² and EF² - ½PQ² will become at last ratio’s of equality.

Therefore, the velocity of the little circle will now be the same which the water would acquire in falling, and in its fall describing the altitude IG; and the resistance will become equal to the weight of a cylinder whose base is that little circle, and its altitude half the altitude IG, from which the cylinder must fall to acquire the velocity of the ascending circle.

With this velocity the cylinder in the time of its fall will describe sour times its length. But the resistance of the cylinder moving forwards with this velocity in the direction of its length, is the same with the resistance of the little circle, (by lemma 4.) and is therefore nearly equal to the force by which its motion may be generated while it describes 4 times its length.

If the length of the cylinder be augmented or diminished, its motion, and the time in which it describes four times its length, will be augmented or diminished in the same ratio.

Therefore, the force by which the motion, so increased or diminished, may be destroyed or generated, will continue the same; because the time is increased or diminished in the same proportion.

Therefore, that force remains still equal to the resistance of the cylinder, because (by lemma 4.) that resistance will also remain the same.

If the density of the cylinder be augmented or diminished, its motion, and the force by which its motion may be generated or destroyed in the same time, will be augmented or diminished in the same ratio.

Therefore the resistance of any cylinder whatsoever will be to the force by which its whole motion may be generated or destroyed in the time during which it moves four times its length, as the density of the medium to the density of the cylinder, nearly. Q.E.D.

A fluid must be compressed to become continued; it must be continued and non-elastic, that all the pressure arising from its compression may be propagated in an instant; and so acting equally upon all parts of the body moved, may produce no change of the resistance.

The pressure arising from the motion of the body is spent in generating a motion in the parts of the fluid, and this creates the resistance. But the pressure arising from the compression of the fluid, be it never so forcible, if it be propagated in an instant, generates no motion in the parts of a continued fluid, produces no change at all of motion therein; and therefore neither augments nor lessens the resistance.

The action of the fluid arising from the compression cannot be stronger on the hinder parts of the body moved than on its fore parts, and therefore cannot lessen the resistance described in this Proposition.

If its propagation be infinitely swifter than the motion of the body pressed, it will not be stronger on the fore parts than on the hinder parts. But that action will be infinitely swifter and propagated in an instant, is the fluid be continued and non-elastic.

Corollary 1

The resistances made to cylinders going uniformly forwards in the direction of their lengths thro’ continued infinite mediums, are in a ratio compounded of the duplicate ratio of the velocities and the duplicate ratio of the diameters, and the ratio of the density of the mediums.

Corollary 2

If the breadth of the canal be not infinitely increased, but the cylinder go forwards in the direction of its length through an included quiescent medium, its axis all the while coinciding with the axis of the canal.

Its resistance will be to the force by which its whole motion in the time in which it describes four times its length, may be generated or destroyed, in a ratio compounded of the ratio of EF² to EF² - ½PQ² once, and the ratio of EF² to EF² - PQ² twice, and the ratio of the density of the medium to the density of the cylinder.

Corollary 3

The same things supposed, and that a length L is to the quadruple of the length of the cylinder in a ratio compounded of the ratio EF² - ½PQ² to EF² once, and the ratio of EF² - PQ² to EF² twice; the resistance of the cylinder will be to the force by which its whole motion, in the time during which it describes the length L, may be destroyed or generated, as the density of the medium to the density of the cylinder.

Scholium

In this proposition we have investigated that resistance which arises from the magnitude of the transverse section of the cylinder.

We neglect that part of the same which may arise from the obliquity of the motions.

For as in Case 1. of Proposition 36. the obliquity of the motions with which the parts of the water in the vessel converged on every side to the hole EF, hindered the efflux of the water thro’ the hole; so in this proposition, the obliquity of the motions, with which the parts of the water, pressed by the antecedent extremity of the cylinder, yield to the pressure and diverge on all sides, retards their passage, thro’ the places that lie round that antecedent extremity, towards the hinder parts of the cylinder, and causes the fluid to be moved to a greater distance; which increases the resistance, and that in the same ratio almost in which it diminished the efflux of the water out of the vessel, that is, in the duplicate ratio of 25 to 21, nearly. And as in Case 1. of that Proposition, we made the parts of the water pass thro’ the hole EF perpendicularly and in the greatest plenty, by supposing all the water in the vessel lying round the cataract to be frozen, and that part of the water whose motion was oblique and useless to remain without motion; so in this proposition, that the obliquity of the motions may be taken away, and the parts of the water may give the freest passage to the cylinder, by yielding to it with the most direct and quick motion possible, so that only so much resistance may remain as arises from the magnitude of the transverse section, and which is incapable of diminution, unless by diminishing the diameter of the cylinder; we must conceive those parts of the fluid whose motions are oblique find useless, and produce resistance, to be at rest among themselves at both extremities of the cylinder, and there to cohere, and be joined to the cylinder. Let ABCD (Plate 7. Figure 6.) be a rectangle, and let AE and BE be two parabolic arcs, described with the axis AB, and with a latus rectum that is to the space HG, which must be described by the cylinder in falling in order to acquire the velocity with which it moves, as HG to ½AB. Let CF and DF be two other parabolic arcs described with the axis CD, and a latus rectum quadruple of the former; and by the convolution of the figure about the axis EF let there be generated a solid, whose middle part ABDC is the cylinder we are here speaking of, and the extreme parts ABE and CDF contain the parts of the fluid, at rest among themselves, and concreted into two hard bodies, adhering to the cylinder at each end like a head and tail. Then if this solid EACFDB move in the direction of the length of its axis FE towards the parts beyond E, the resistance will be the same which we have here determined in this proposition, nearly; that is, it will have the same ratio to the force with which the whole motion of the cylinder may be destroyed or generated in the time that it is describing the length 4.AC with that motion uniformly continued, as the density of the fluid has to the density of the cylinder, nearly. And (by corollary 7. proposition 36.) the resistance must be to this force in the ratio of 2 to 3, at the least.

Lemma V. If a cylinder, a sphere, and a sphæroid, of equal breadths be placed successively in the middle of a cylindric canal, so that their axes may coincide with the axis of the canal; these bodies will equally hinder the passage of the water thro’ the canal.

For the spaces, lying between the sides of the canal, and the cylinder, sphere, and sphæroid, thro’ which the water passes, are equal; and the water will pass equally thro’ equal spaces.

This is true upon the supposition that all the water above the cylinder, sphere, or sphæroid, whose fluidity is not necessary to make the passage of the water the quickest possible, is congealed, as was explained above in Corollary 7. Proposition 36.

Lemma VI. The same supposition remaining, the forementioned bodies are equally acted on by the water flowing thro’ the canal.

This appears by Lemma 5. and the third law. For the water and the bodies act upon each other mutually; and equally.

Lemma VII. If the water be at rest in the canal, and these bodies move with equal velocity and the contrary way thro’ the canal, their resistances will be equal among themselves.

This appears from the last Lemma, for the relative motions remain the same among themselves.

Scholium

The case is the same of all convex and round bodies whose axes coincide with the axis of the canal. Some difference may arise from a greater or less friction; but in these lemmata we suppose the bodies to be perfectly smooth, and the medium to be void of all tenacity and friction; and that those parts of the fluid which by their oblique and superfluous motions may disturb, hinder and retard the flux of the water thro’ the canal, are at rest amongst themselves; being fixed like water by frost, and adhering to the fore and hinder parts of the bodies in the manner explained in the Scholium of the last Proposition. For in what follows, we consider the very least resistance that round bodies described with the greatest given transverse sections can possibly meet with.

Bodies swimming upon fluids, when they move straight forwards, cause the fluid to ascend at their fore parts and subside at their hinder parts, especially if they are of an obtuse figure; and thence they meet with a little more resistance than if they were acute at the head and tail. And bodies moving in elastic fluids, if they are obtuse behind and before, condense the fluid a little more at their fore parts, and relax the same at their hinder parts; and therefore meet also with a little more resistance than if they were acute at the head and tail. But in these lemma’s and propositions we are not treating of elastic, but non-elastic fluids; not of bodies floating on the surface of the fluid, but deeply immersed therein. And when the resistance of bodies in non-elastic fluids is once known, we may then augment this resistance a little in elastic fluids, as our air; and in the surfaces of stagnating fluids, as lakes and seas.