Page:Cyclopaedia, Chambers - Volume 2.djvu/618

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From the given Celerity of the Body moved, the Height of the Liquid Cylinder is found, as alio the Weight of it from the known fpecific Gravity of the Liquid, and Diameter of the Bo- dy. — Let a Ball, for Inltance, of three Inches Diameter be mo- ved in Water with a Celerity wherewith it would go fixteen Foot in a Second : FromExperiments on falling Bodies, and o- thers made on Pendulums, it has been .found that this is the Celerity which a Body acquires in falling from a Height of four Foot, therefore the Weight of a Cylinder of Water, of three Inches Diameter, and two Foot high, that is, a Weight of a- bout fix Pound and three Ounces, is equal to the Reffience of die aforefaid Ball. See Descent.

Let the Refifieace fo difcovered be divided by the Weight of the Body, which determines its Quantity or" Matter, and you will have the Retardation.

Resistence of fluid Mediums to the Motion of Pendulums.

The Arch defcribed by a Pendulum ofcillating in vacuo, with the Celerity it has acquir'd in descending, is equal to the Arch defcribed by the Defcent ; but the fame does not happen in a Fluid, and there is a greater Difference between thole Arches the greater the Reffience is; that is, the greater the Arch is which is deicrib'd in the Defcent.

Let the Reffience of the Liquid be in Proportion to the Ve- locity j and let two Pendulums, entirely alike, ofcillating in a Cy- cloid, perform unequal Vibrations, and begin to fall the fame Moment ; they here begin to move by Forces that are as the Arches to be defcribed. If thole Impreffions alone, which are made the firft Moment, be conhdered ; after a given Time the Celerities will be in the fame Ratio, as in the Beginning; for the Retarda- tions which are as the Velocities themfclves, cannot change their Proportions, the Ratio between Quantities not being changed by the Addition and Subftradion of Quantities in the fame Ratio. Therefore in equal Times, however the Celerities of Bodies are changed in their Motion by the Reffience, the Spaces gone through, are as the Forces in the Beginning ; that is, as the Arches to be defcribed by the Defcent ; therefore after any time the Bodies are in the correfpondent Point of thofe Arches. Butinthefe Points the Forces generated are in the fame Ratio as in the Be- ginning, and the Proportion of the Celerities, which is not vari- ed by the Reffience, fuffers no Change from the Gravity. In the Afcent, Gravity retards the Motion of the Body ; but in corref- pondent Points, its Actions are in the lame Ratio as in Defcents. And therefore every wherein correfpondent Points, the Celerities are in die lame Ratio. But as in the fame Moments the Bodies are in their correfpondent Points, it follows that the Motion of both is deftroy'd in the fame Moment, that is, they hnifh their Vibrations in the fame Time. The Spaces run through in the Time of one Vibration, arc as the Forces by which they are run thro 5 ; that is, the Arches of the whole Vibrations are as the Arches defcribed by the Defcent, the doubles whereof arc the Arches to be defcribed in Vacuo. The Defects of the Arches to be defcri- bed in Liquids from the Arches to be defcribed in Vacuo, are the Differences of Quantities in the fame Ratio, and are as the Arches delcrib'd by the Defcent. See Pendulum.

Resistence of Fluid Mediums to the Motion of falling Bodies.

The Refflences are as the Squires of the Celerities, and therefore every where in correfpondent Points, as the Squares of the Arches defcribed by the Defcent, in which Ratio alio, the Retardations arej but as each of them keep the fame Proporti- on in correfponding Points, the Sums of them all will be in the fame Proportion \ that is, the whole Retardations, which are the Defects of the Arches defcribed in the Liquid, from the Arches to be defcribed in Vacuo, or which is the fame, the Difference between the Arches defcribed in the Defcent, and the next Af- ' cent. Therefore thefe Differences, if the Vibrations are not ve- ry unequal, are nearly as the Squares of the Arches defcribed by the Defcent : Which is alfo confirm'd by Experiments in greater Vibrations j for in thefe the Proportion of Reffience, here conil- dered, obtains.

A Body freely defcending in a Fluid is accelerated by the re- fpedive Gravity of the Body, which continually ads upon it, yet not equably, as in a Vacuum: the Reffience of the Liquid oc- cafions a Retardation, that is, a Diminution of Acceleration, which Diminution increafts with the Velocity of the Body. Now there is a certain Velocity, which is the greateft a Body can acquire by falling,- for if its Velocity be fuch that the Reffience arUmg from it becomes equal to the refpedive Weight of the Body, its Motion can be no longer accelerated ; for the Motion here continually generated by the refpedive Gravity, will be deftroy'd by the Reffience, and the Body forced to go on equably. A Body continually comes nearer and nearer to this greateft Celerity, but can never attain to it.

When the Denfities of a Liquid Body are given, the re- fpedive Weight of the Body may be known ; and by knowing the Diameter of the Body, it may be found from what Height a Body falling in Vacuo, can acquire fuch a Velocity, as that the Reffience in a Liquid (Kail be equal to that refpedive Weight,

which will be that greateft Velocity above mentioned —If the Body be a Sphere, it is known that a Sphere is equal to a Cylin- der of the fame Diameter, whofe Height is two third Parts of thatDiameter; whichHeight istobeiiicreifed in the Ratio where- in the refpedive weight of the Body exceeds the weight of the Liquid, in order to have the Height of a Cylinder of the Liquid, whofe weight is equal to the refpedive weight of the Body; but if you double this Height, you will have a Height from which a Body falling in Vacuo, acquires fuch a Velocity 3 as generates a Reffience equal to this refpedive weight, and which therefore is the greateft Velocity which a Body can acquire fall- ing in a Liquid from an infinite Height. Lead is eleven times heavier than Water, wherefore its refpedive weight is as to the weight of Water as 10 to I ; therefore a leaden Ball, as it ap* pears from what has been laid, cannot acquire a greater Velocity in falling in Water, than it would acquire in falling in Vacuo, from an Height of i 3 f of its Diameters.

A Body lighter than a Liquid, and afcending in it by the Acti- on of the Liquid, is moved exadly by the fame Laws as an hea- vier Body failing in the Liquid. Wherever you place the Body, it is fuftained by the Liquid, and carried up with a Force equal to the Difference of the weight of the Quantity of the Liquid, of the fame Bulk as the Body, from the weight of the Body. Therefore you have a Force that continually ads equably up- on the Body, by which not only the Adion of the Gravity of the Body is deitroyed fo as that it is not to be conlidered in this Cafe, but the Body is alfo carried upwards by a Moti- on equably accelerated, in the fame manner as a Body heavier than a Liquid defcends by its refpedive Gravity j bu: the equabi- lity of the Acceleration is deitroyed in the fame Manner by the Reffience^ in the Afcent of a Body lighter than the Liquid, as is is deftroyed in the Defcent of a Body heavier.

When a Body fpecirically heavier than a Fluid is thrown up in it, it is retarded upon a double Account; on Account of the Gravity of the Body, and on Account of the Reffience of the Liquid; confequently, a Body rifes to a lefs Height than it would rile in Vacuo with the fame Celerity. But the Defects of the Height in a Liquid from the Heights to which a Body would rife in Vacuo with the fame Celerities, r have a greater Proporti- on to each other than the Heights themfelves ; and in lefs Heights the Defeds are neaily as the Squares of the Heights in Vacuo.

Resistence of the Air-) in Pneumaticks, is the Force where- with the Motion of Bodies, particularly Projediles, is retarded by the Reffience of the Air or Atmofphere. See Air and Pro- jectile.

The Air being a Fluid, the general Laws of the Reffience of Fluids obtain therein; only the different Degrees of Denfity in the different Stages or Regions of the Atmofphere, occafion feme irregularity. See Atmosphere.

The different Resistence of the funs Mediums to Bodies of diffs~ rent Figures.

Sir Jfaac Newto?i mews. That if a Globe and a Cylinder of equal Diameters, be moved with equal Velocity in a thin Me-" dium, continuing of equal Particles, difpofed at equal Diftances, according to the Diredion of the Axis of the Cylinder; theKe-

fence of the Globe will be lefs by half than that of the Cy- der.

Solid of the leaf Resistence.— From the lad Propofition the fame Author deduces the Figure of a Solid, which fhall have the leafi Reffience of any containing the fame Quantity of Matter and Surface. See Solid.

The Figure is this. — Suppofe DNFB (Tab. Mechanicks, Fig. 57.J to be fuch a Curve, as, that if from any Point N, be lee fall a Perpendicular NM, to the Axis AB; and from a given Point G, be drawn a right Line GR, parallel to a Tangent to the Fi- gure in N, and cut the Axis when continued, in R : A Solid defcribed by the Revolution of this Figure about its Axis AB, moving in a Medium from A towards B, is lefs reffied than in any other circular Solid of the fame Area, &c. Newt. Princ. p. 327.

The Reffience of a Globe, perfedly hard and in a Medium, whofe Particles are fo too ; is to the Force wherewith the whole Motion may be cither deftroy'd or generated which it h«s when at the time, when it has defcribed four thirds of its Diameter ; as the Denfiry of the Medium to the Denfity of the Globe. — Hence alio, the fame Author infers that the Reffience of a Globe* is, asteris paribus, in a duplicate Ratio of its Velocity. Or its Reffience is ceteris paribus, in a duplicate Ratio of its Diameter. Or, ceteris paribus, as the Denfity of the Medium. Laftly, that the adual Reffience of a Globe is in a Ratio compounded of the duplicate Ratio of the Velocity, and of the duplicate Ratio of the Diameter, and of the Ratio of the Denuty of the Me- dium.

In thefe Articles the Medium is fuppofed to be difcon tin uo us, as Air probably is : If the Medium be continuous, as Water, Mercury, $>c. where the Globe does not ftrike immediate- ly on all the Particles of the Fluid generating the Refflente^ but only on thofe nrxt it, and thofe again on others, &c. the Reffience will be lefs by half. And a Globe in fuch a Medium undergoes a Reffience which is to the Force

where-