Resistance of Globes Problem

Experiment 5: Four globes, weighing 1541/8 grains in air, and 21½ grains in water, being let fall several times, fell in the times of 28½, 29, 29½, and 30, and sometimes of 31, 32, and 33 oscillations, describing a height of 15 feet and 2 inches.

They ought by the theory to have fallen in the timeos 29 oscillations, nearly.

Experiment 6: Five globes, weighing 2121/8 grains in air, and 79½ in water, being several times let fall, fell in the times of 15, 15½, 16, 17, and 18 oscillations, describing a height of 15 feet and 2 inches.

By the theory they ought to have fallen in the time of 15 oscillations, nearly.

Experiment 7: Four globes weighing 2933/8 grains in air, and 35 grains7/8 in water, being let fall several times, fell in the times of 29½, 30, 30½, 31, 32, and 33 oscillations, describing a height of 15 feet and 1 inch and ½.

By the theory they ought to have fallen in the time ps 28 oscillations, nearly.

In searching for the cause that occasioned these globes of the same weight and magnitude to fall, some swifter and some slower, I hit upon this; that the globes, when they were first let go and began to fall, oscillated about their centres, that side which chanced to be the heavier descending first, and producing an oscillating motion.

By oscillating thus, the globe communicates a greater motion to the water, than if it descended without any oscillations; and by this communication loses part of its own motion with which it should descend.

Therefore as this oscillation is greater or less it will be more or less retarded. Besides the globe always recedes from that side of itself which is descending in the oscillation, and by so receding comes nearer to the sides of the vessel so as even to strike against them sometimes. And the heavier the globes are, the stronger this oscillation is; and the greater they are, the more is the water agitated by it.

Therefore to diminish this oscillation of the globes, I made new ones of lead and wax, sticking the lead in one side of the globe very near its surface; and I let fall the globe in such a manner, that as near as possible, the heavier side might be lowest at the beginning of the descent.

By this means the oscillations became much less than before, and the times in which the globes fell were not so unequal: as in the following experiments.

Experiment 8: Four globes weighing 139 grains in air and 6½ in water, were let fall several times, and fell mostly in the time of 51 oscillations, never in more than 52, or in fewer than 50; describing a height of 182 inches.

By theory, they should fall in about 51 oscillations.

Experiment 9: Four globes weighing 273¼ grains in air, and 140¼ in water, being several times let fall, fell in never fewer than 12, and never more than 13 oscillations, describing a height of 182 inches.

These globes by the theory ought to have fallen in the time of 111/3 oscillations, nearly.

Experiment 10: Four globes, weighing 384 grains in air and 119½ in water, being let fall several times, fell in the times of 17¾, 18, 18½, and 19 oscillations, describing a height of 181½ inches.

When they fell in the time of 19 oscillations, I sometimes heard them hit against the sides of the vessel before they reached the bottom.

By the theory they ought to have fallen in the time of 155/9 oscillations, nearly.

Experiment 11: Three equal globes, weighing 48 grains in the air, and 329/32 in water, being several times let fall, fell in the times of 43½, 44, 44½, 45 and 46 oscillations, and mostly in 44 and 45, describing a height of 182 inches ½, nearly.

By the theory they ought to have fallen in the time of 46 oscillations and 5/9 , nearly.

Experiment 12: Three equal globes, weighing 141 grains in air and 43/8 in water, being let fall several times, fell in the times of 61, 61, 63, 64 and 65 oscillations, describing a space of 182 inches.

By the theory they should have fallen in 64½ oscillations, nearly.

From these experiments it is manifest, that when the globes fell slowly, as in the second, fourth, fifth, eighth, eleventh, and twelfth experiments, the times of falling are rightly exhibited by the theory.

But when the globes fell more swiftly as in the sixth, ninth, and tenth experiments, the resistance was somewhat greater than in the duplicate ratio of the velocity.

For the globes in falling oscillate a little; and this oscillation, in those globes that are light and fall slowly, soon ceases by the weakness of the motion but in greater and heavier globes, the motion being strong, it continues longer; and is not to be checked by the ambient water, till after several oscillations.

Besides, the more swiftly the globes move, the less are they pressed by the fluid at their hinder parts; and if the velocity be perpetually increased, they will at list leave an empty space behind them, unless the compression of the fluid be increased at the same time.

For the compression of the fluid ought to be increased (by proposition 32 and 33.) in the duplicate ratio os the velocity, in order to preserve the resistance in the same duplicate ratio.

But because this is not done, the globes that move swiftly are not so much pressed at their hinder parts as the others, and by the defect of this pressure it comes to pass that their resistance is a little greater than in a duplicate ratio of their velocity.

So that the theory agrees with the phenomena of bodies falling in water; it remains that we examine the phænomena of bodies falling in air.

Experiment 13: From the top of St. Paul’s Church Jn London in June 1710 there were let fall together two glass globes, one full of quicksilver, the other of air.

In their fall they described a height of 220 English feet.

A wooden table was suspended upon iron hinges on one side, and the other side of the same was supported by a wooden pin.

The 2 globes lying upon this table were let fall together by pulling out the pin by means of an iron wire reaching from thence quite down to the ground; so that, the pin being removed, the table, which had then no support but the iron hinges, fell downwards.

Turning round upon the hinges, gave leave to the globes to drop off from it. At the same instant, with the same pull of the iron wire that took out the pin, a pendulum oscillating to seconds was let go, and began to oscillate. The diameters and weights of the globes, and their times of falling, are exhibited in the following table.

But the times observed must be corrected; for the globes of mercury (by Galileo’s theory) in 4 seconds of time, will describe 257 English feet, and 220 feet in only 3" 42"’. So that the wooden table, when the pin was taken out, did not turn upon its hinges so quickly as it ought to have done; and the slowness of that revolution hindered the descent of the globes at the beginning.

For the globes lay about the middle of the table, and indeed were rather nearer to the axis upon which it turned, than to the pin. And hence the times of falling were prolonged about 18"'.

Therefore, it should be corrected by subducting that excess, especially in the larger globes, which, by reason of the largeness of their diameters, lay longer upon the revolving table than the others.

This being done, the times in which the six larger globes fell, will come forth 8"12"’, 7"42"’, 7"42"’, 7"57"’, 8"12"’, and 7"42"'.

Therefore the fifth in order among the globes that were full of air, being 5 inches in diameter, and 483 grains in weight, fell in 8"12"’, describing a space of 220 feet.

The weight of a bulk of water equal to this globe is 16600 grains; and the weight of an equal bulk of air is 16600/860 grains, or 19 3/10 grains; and therefore the weight of the globe in vacuo is 5023/10 grains.

This weight is to the weight of a bulk of air equal to the globe as 5023/10 to 193/10 and so is 2.F to 2/3 of the diameter of the globe, that is, to 131/3 inches. Whence 2.F becomes 28 feet 11 inches.

A globe falling in vacuo with its whole weight of 5023/10 grains, will in one second of time describe 1931/3 inches as above; and with the weight of 483 grains will describe 185,905 inches; and with that weight 483 grains in vacuo will describe the space F or 14 feet 5½ inches, in the time of 57"’ 58"", and acquire the greatest velocity it is capable of descending with in the air. With this velocity the globe in 8"12"’ of time will describe 245 feet and 51/3 inches.

Subduct 1,3863.F or 20 feet and ½ an inch, and there remain 225 feet 5 inches. This space therefore the fafling globe ought by the theory to describe in 8"12"’. But by the experiment it described a space of 220 feet. The difference is insensible.

By like calculations applied to the other globes full of air, I composed the following table.

Experiment 14: In July 1719, Dr. Desaguliers made some experiments of this kind again, by forming hogs bladders into sphærical orbs; which was done by means of a concave wooden sphere, which the bladders, being wetted well first, were put into.

After that, being blown full of air, they were obliged to fill up the sphærjcal cavity that contained them; and then, when dry, were taken out. These were let fall from the lantern on the top of the cupola of the same church; namely, from a height of 272 feet; and at the same moment of time there was let fall a leaden globe whose weight was about 2 pounds Troy weight.

And in the mean time some persons standing in the upper part of the church where the globes were let fall, observed the whole times of falling; and others standing on the ground observed the differences of the times between the fall of the leaden weight, arid the fall of the bladder. The times were measured by pendulums oscillating to half seconds.

And one of those that stood upon the ground had a machine vibrating four times in one second; and another had another machine accurately made with a pendulum vibrating four times in a second also.

One of those also who stood at the top of the church had a like machine. And these instruments were so contrived, that their motions could be stopped or renewed at pleasure. Now the leaden globe fell in about four seconds and ¼ of time; and from the addition of this time to the difference of time above spoken of, was collected the whole time in which the bladder was falling.

The times which the five bladders spent in sailing after the leaden globe had reached the ground were the first time, 14¾", 12¾", 145/8", 17¾", and 167/8"; and the second time 14½", 14¼", 14", 19" and 16¾".

Add to these 4¼", the time in which the leaden globe was falling, and the whole times in which the five bladders fell, were, the first time 19", 17", 187/8", 22" and 211/8" and the second time, 18¾", 18½, 18¼", 23¼", 21". The times observed at the top of the church were, the first time, 191/8", 17¾", 18¼", 221/8", and 215/8"; and the second time, 19", 185/8", 181/8", 24" and 21¼".

But the bladders did not always fall directly down, but sometimes fluttered a little in the air, and waved to and fro as they were descending. And by these motions the times of their falling were prolonged, and increased by half a second sometimes, and sometimes by a whole second.

The second and fourth bladder fell most directly the first time, and the first and third the second time. The fifth bladder was wrinkled, and by its wrinkles was a little retarded.

I found their diameters by their circumferences measured with a very fine thread wound about them twice. In the following table I have compared the experiments with the theory; making the density of air to be to the density of rain-water as 1 to 860, and computing the spaces which by the theory the globes ought to describe in falling.

Our theory therefore exhibits rightly, within a very little, all the resistance that globes moving either in air or in water meet with; which appears to be proportional to the densities of the fluids in globes of equal velocities and magnitudes.

In the scholium subjoined to the sixth section, we shewed by experiments of pendulums, that the resistances of equal and equally swift globes moving in air, water, and quicksilver, are as the densities of the fluids.

We here prove the same more accurately by experiments of bodies falling in air and water. For pendulums at each oscillation excite a motion in the fluid always contrary to the motion of the pendulum in its return.

The resistance arising from this motion, as also the resistance of the thread by which the pendulum is suspended, makes the whole resistance of a pendulum greater than the resistance deduced from the experiments of falling bodies.

For by the experiments of pendulums described in that scholium, a globe of the same density as water in describing the length of its semidiameter in air would lose the 1/3342 part of its motion.

But by the theory delivered in this seventh section, and confirmed by experiments of falling bodies, the same globe in describing the same length would lose only a part of its motion equal to supposing the density of water to be to the density of air as 860 to 1.

Therefore the resistances were found greater by the experiments of pendulums (for the reasons just mentioned) than by the experiments of falling globes; and that in the ratio of about 4 to 3.

But yet since the resistances of pendulums oscillating in air, water, and quicksilver, are alike increased by like causes, the proportion of the resistances in these mediums will be rightly enough exhibited by the experiments of pendulums, as well as by the experiments of falling bodies. And from all this it may be concluded, that the resistances of bodies, moving in any fluids whatsoever, tho’ of the most extreme fluidity, are, ceteris paribus, as the densities of the fluids.

These things being thus established, we may now determine what part of its motion any globe projected in any fluid whatsoever would nearly lose in a given time. Let D be the diameter of the globe, and V its velocity at the beginning of its motion, and T the time in which a globe with the velocity V can describe in vacuo a space that is to the space 8/3D as the density of the globe to the density of the fluid; and the globe projected in that fluid will, in any other time t, lose the part t.V/(T + t), the part T.V/(T + t) remaining; and will describe a space, which may be to that described in the same time in vacuo with the uniform velocity V, as the logarithm of the number (T + t)/T multiplied by the number 2,302585093 is to the number t/T, by corollary 7. proposition 35. In slow motions the resistance may be a little less, because the figure of a globe is more adapted to motion than the figure of a cylinder described with the same diameter. In swift motions the resistance may be a little greater, because the elasticity and compression of the fluid do not increase in the duplicate ratio of the velocity. But these little niceties I take no notice of.

Air, water, quicksilver, and the like fluids, by the division of their parts in infinitum, are subtilized and become mediums infinitely fluid.

Yet, the resistance they would make to projected globes would be the same.

For the resistance consider’d in the preceding propositions, arises from the inactivity of the matter; and the inactivity of matter is essential to bodies, and always proportional to the quantity of matter. By the division of the parts of the fluid, the resistance arising from the tenacity and friction of the parts may be indeed diminished;

But the quantity of matter will not be at all diminished by this division; and if the quantity of matter be the same, its force of inactivity will be the same; and therefore the resistance here spoken of will be the same, as being always proportional to that force. To diminish this resistance, the quantity of matter in the spaces thro’ which the bodies move must be diminished. And therefore the celestial spaces, thro’ which the globes of the Planets and Comets are perpetually passing towards all parts, with the utmost freedom, and without the least sensible diminution of their motion, must be utterly void of any corporeal fluid, excepting perhaps some extremely rare vapours, and the rays of light.

Projectiles excite a motion in fluids as they pass through them.

This motion arises from the excess of the pressure of the fluid at the foreparts of the projectile above the pressure of the same at the hinder parts; and cannot be less in mediums infinitely fluid, than it is in air, water, and quicksilver, in proportion to the density of matter in each. Now this excess of pressure does, in proportion to its quantity, not only excite a motion in the fluid, but also acts upon the projectile so as to retard its motion: and therefore the resistance in every fluid is as the motion excited by the projectile in the fluid; and cannot be less in the most subtile aether in proportion to the density of that aether, than it is in air, water, and quicksilver, in proportion to the densities of those fluids.