Resistance of Globes Problem

Find by phænomena the resistance of a globe moving through a perfectly fluid compressed medium.

Let A be the weight of the globe in vacuo, B its weight in the resisting medium, D the diameter of the globe, F a space which is to 4/3D as the density of the globe to the density of the medium, that is, as A to A - B, G the time in which the globe falling with’ the weight B without resistance describes the space F, and H the velocity which the body acquires by that fall.

Then H will be the greatest velocity with which the globe can possibly descend with the weight B in; the resisting medium, by corollary 2, proposition 38; and the resistance which the globe meets with, when descending with that velocity, will be equal to its weight B: and the resistance it meets with, in any other velocity, will be to the weight B in the duplicate ratio of that velocity to the greatest velocity H, by corollary 1, proposition 38.

This is the resistance that arises from the inactivity of the matter of the fluid. That resistance which arises from the elasticity, tenacity, and friction of its parts, may be thus investigated.

Let the globe be let fall so that it may descend in the fluid by the weight B; and let P be the time of falling, and let that time be expressed in seconds, if the time G be given in seconds.

Find the absolute number N agreeing to the logarithm 0,4342944819 2P/G and let L be the logarithm of the number (N + 1) / N: and the velocity acquir’d in sailing will be (N - 1)/(N + 1) H, and the height described will be 2PF / G - 1,3862943611 F + 4,605170186 LF.

If the fluid be of a sufficient depth, we may neglect the term 4,605170186 LF; and 2PF / G - 1,3862943611 F will be the altitude described, nearly. These things appear by proposition 9. book 2. and its corollaries, and are true upon this supposition, that the globe meets with no other resistance but that which arises from the inactivity of matter. Now if it really meet with any resistance of another kind, the descent will be slower, and from the quantity of that retardation will be known the quantity of this new resistance.

That the velocity and descent of a body falling in a fluid might more easily be known, I have composed the following table; the first column of which denotes the times of descent, the second shews the velocities acquir’d in falling, the greatest velocity being 100.000.000, the third exhibits the spaces described by falling in those times, 2.F being the space which the body describes in the time G with the greatest velocity, and the fourth gives the spaces described with the greatest velocity in the same times.

The numbers in the fourth column are 2.P/G, and by subducting the number 1,3862944 - 4,6051702.L, are found thenumbers in the third column; and these numbers must be multiplied by the space F to obtain the spaces described in falling.

A fifth column is added to all these, containing the spaces described in the same times by a body falling in vacuo with the force of B its comparative weight.

Scholium

In order to investigate the resistances of fluids from experiments, I procured a square wooden vessel, whose length and breadth on the inside was 9 inches English measure, and its depth 9 foot ½.

This I filled with rain-water: and having provided globes made up of wax, and lead included therein, I noted the times of the descents of these globes, the height through which they descended being 112 inches.

A solid cubic foot of English measure contains 76 pounds Troy weight of rain-water; and a solid inch contains 19/36 ounces Troy weight or 2531/3 grains; and a globe of water of one inch in diameter contains 132,645 grains in air, or 132,8 grains in vacuo; and any other globe will be as the excess of its weight in vacuo above its weight in water.

Experiment 1: A globe whose weight was 156¼ grains in air, and 77 grains in water, described the whole height of 112 inches in 4 seconds.

Upon repeating the experiment, the globe spent again the very same time of 4 seconds in falling.

The weight of this globe in vacuo is 15613/38 grains; and excess of this weight above the weight of the globe in water is 7913/38 grains. Hence the diameter of the globe appears to be 0,84224 parts of an inch.

Then it will be, as that excess to the weight of the globe in vacuo, so is the density of the water to the density of the globe; and so is 2/3 parts of the diameter of the globe (viz,. 2,24597 inches) to the space 2.F, which will be therefore 4,4256 inches.

A globe falling in vacuo with its whole Weight of 15613/38 grains in one second of time will describe 1931/3 inches; and falling in water in the same time with the weight of 77 grains without resistance, will describe 95,219 inches; and in the time G which is to one second of time in the subduplicate ratio of the space F, or of 2,2128 inches to 95,219 inches, will describe 2,2128 inches, and will acquire the greatest velocity H with which it is capable of descending in water Therefore the time G is o,“15244.

In this time G with that greatest velocity H, the globe will describe the space 2.F, which is 4,4256 inches; and therefore in 4 seconds will describe a space of 116,1245 inches.

Subduct the space 1,3862944.F or 3,0676 inches, and there will remain a space of 113,0569 inches, which the globe falling thro’ water in a very wide vessel will describe in 4 seconds.

But this space, by reason of the narrowness of the wooden vessel beforementioned, ought to be diminished in a ratio compounded of the subduplicate ratio of the orifice of the vessel to the excess of this orifice above half a great circle of the globe, and of the simple ratio of the same orifice to its excess above a great circle of the globe, that is, in a ratio of 1 to 0,9914.

This done, we have a space of 112,08 inches, which a globe falling thro’ the water in this wooden vessel in 4 seconds of time ought nearly to describe by this theory: but it described 112 inches by the experiment.

Experiment 2: Three equal globes, whose weights were severally 761/3 grains in air, and 51/16 grains in water, were let fall successively; and every one fell thro’ the water in 15 seconds of time, describing in its fall a height of 112 inches.

By computation, the weight of each globe in vacuo 765/12 grains; the excess of this weight above the weight in water, is 71 grains 17/48; the diameter of the globe 0,81296 of an inch: 8/3 parts of this diameter 2,16789 inches; the space 2.F is 2,3217 inches.

The space which a globe of 51/16 grains in weight would describe in one second without resistance, 12,808 inches, and the time G 0”,301056. Therefore the globe with the greatest velocity it is capable of receiving from a weight of 51/16 grains in its descent thro’ water, will describe in the time 0",301056 the space of 2,3217 inches; and in 15 seconds the space 115,678 inches.

Subduct the space 1,3862944.F or 1,609 inches, and there remains the space 114,069 inches; which therefore the falling globe ought to describe in the same time, if the vessel were very wide.

But because our vessel was narrow, the space ought to be diminished by about 0,895 of an inch. And so the space will remain 113,174 inches, which a globe falling in this vessel ought nearly to describe in 15 seconds by the theory. But by the experiment it described 112 inches. The difference is not sensible.

Experiment 3: Three equal globes, whose weights were severally 121 grains in air, and 1 grain in water, were successively let fall.

They fell through the water in the times 46", 47", and 50", describing a height of 112 inches.

By the theory these globes ought to have fallen in about 40". Now whether their falling more slowly were occasion’d from hence, that in slow motions the resistance arising from the force of inactivity, does really bear a less proportion to the resistance arising from other causes; or whether it is to be attributed to little bubbles that might chance to stick to the globes, or to the rarefaction of the wax by the warmth of the weather, or of the hand that let them fall;

or, lastly, whether it proceeded from some insensible errors in weighing the globes in the water, I am not certain. Therefore the weight of the globe in water should be of several grains, trrat the experiment may be certain, and to be depended on.

Experiment 4: I procured a wooden wooden vessel, whose breadth on the inside was 82/3 inches, and its depth 15 feet and 1/3.

Then I made 4 globes of wax, with lead included, each of which weighed 139¼ grains in air, and 71/8 grains in water.

These I let fail, measuring the times of their falling in the water with a pendulum oscillating to half seconds.

The globes were cold, and had remained so sometime, both when they were weighed and when they were let fall; because warmth rarefies the wax, and by rarefying it diminishes the weight of the globe in the water.

Wax, when rarefied, is not instantly reduced by cold to its former density.

Before they were let fall, they were totally immersed under water, lest, by the weight of any part of them that might chance to be above the water, their descent should be accelerated in its beginning.

Then, when after their immersion they were perfectly at rest, they were let go with the greatest care, that they might not receive any impulse from the hand that let them down.

They fell successively in the times of 47½, 48½, 50 and 51 oscillations, describing a height of 15 feet and 2 inches. But the weather was now a little colder than when the globes were weighed, and therefore I repeated the experiment another day.

Then the globes fell in the times of 49, 49½, 50 and 53 ; and at a third trial in the times of 49½, 50, 51 and 53 oscillations.

By making the experiment several times over, I found that the globes fell mostly in the times of 49½ and 50 oscillations. When they fell slower, I suspect them to have been retarded by striking against the sides of the vessel.

Computing from the theory, the weight of the globe in vacuo is 1392/5 grains.

The excess of this weight above the weight of the globe in water 13219/40 grains, the diameter of the globe 0,99868 of an inch, 1/3 parts of the diameter 2,66315 inches, the space 2.F 2,8066 inches, the space which a globe weighing 71/8 grains falling without resistance describes in a second of time 9,88154 inches, and the time G 0",376843.

Therefore the globe with the greatest velocity with which it is capable of descending thro’ the water by the force of a weight of 71/8 grains will in the time 0",376843 describe a space of 2,8066 inches, and in one second of time a space of 7,44766 inches, and in the time 25", or in 50 oscillations the space 186,1915 inches.

Subduct the space 1,386294.F or 1.9454 inches, and there will remain the space 184,2461 inches, which the globe will describe in that time in a very wide vessel.

Because our vessel was narrow, let this space be diminished in a ratio compounded of the subduplicate ratio of the orifice of the vessel to the excess of this orifice above half a great circle of the globe, and of the simple ratio of the same orifice to its excess above a great circle of the globe; and we shall have the space of 181,86 inches, which the globe ought by the theory to describe in this vessel in the time of 50 oscillations, nearly. But it described the space of 182 inches, by experiment, in 49½ or 50 oscillations.