Page:Cyclopaedia, Chambers - Supplement, Volume 2.djvu/890

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I739, 4to, 1742, 4 vols. 4to. We have alio the fciences of Statics and Mechanics treated of under the name of Pho- ronomia, by the late learned profeflbr Herman. Amft. 1716. 4to. Mr. Euler's Mechanica, five de motus fcimtia ; is fuflkiently recommended by the name of the author, well known as one of the moft eminent mathematicians in Europe. This work, however-, did not efcape the cen- fure of a late ingenious writer ; but granting the juftnefs of his remarks, it may be faid, that a few inaccurate expref- fions, owing to analogies carried too far, and to the gyle of infinitefimals, are but fmall flaws in fuch a work. As to ftatics, the fubjeft has been almoft exhaufted by Va- rignon, in his Mecanique, Paris,' 1725, 2 vol. 4to. The doctrine of machines or engines is a principal branch of Mechanics ; but has not as yet been treated as it deferves. We fhall here infert a few obfervations on this ufeful fubjecr, from a late eminent author.

In treating of machines, we fhould conhder the weight that is to be raifed, the power by which it is to be railed, and the inftrument or engine by which this effect is to be produced. There are two principal problems that ought to be refolved in treating of eacli of them. The firft is, to determine the proportion which the power and weight ought to have to each other, that they may juft fuftain one another, or be in aequilibrio. The fecond is, to determine what ought to be the proportion of the power and weight to each other, in a given machine, that it may produce the greateft effect poflible, in a given time. All the writers on Mechanics treat of the firft of thefe pro- blems, but few have confidered the fecond, tho' equally ufeful with the other.

As to the firft problem, this general rule holds in all powers : Suppofc the engine to move, and reduce the velocities of the power and weight to the refpefiive directions in which they ait ; find the proportions of thofe velocities ; then if the power be to the weight, as the velocity of the weight is to the velocity of the power, or, which amounts to the fame thing, if the power multiplied by its velocity, gives the fame product as the weight multiplied by its velocity, this is the cafe wherein the power and weight fuftain each other, and are in equilibrio ; fo that in this cafe, the one would not prevail over the other, if the engine was at reft ; and if it is in motion, it would continue to proceed uniformly, if it were not for the friftion of its parts, and other refiftances. The fecond general problem in Mechanics, is, to determine the proportion which the power and weight ought to bear to each other, that when the power prevails, and the ma- chine is in motion, the greateft effect poflible may be pro- duced by it in a given time. It is manifeft, that this is an inquiry of the greateft importance, tho' few have treated of it. When the power is only a little greater than that which is fufficient to fuftain the weight, the motion is too flow ; and tho' a greater weight is railed in this cafe, it is not fufficient to compenfate the lofs of time. When the weight is much lefs than that which the power is able to fuftain, it is raifed in lefs time ; and this may happen not to be fuffi- cient to compenfate the lofs arifing from the fmallncfs of the load. It ought, therefore, to be determined when the pro- duct of the weight multiplied by its velocity, is the greateft poffible 5 for this product meafures the effect of the en- gine in a given time, which is always the greater in propor- tion as the weight which is raifed is greater, and as the velo- city with which it is raifed is greater. We fhall therefore fubjoin fome inftances of this kind, that may be demonftrated from the common elementary geometry ; wifhing that farther improve- ments may be made in this moft ufeful part of Mechanics. When the power prevails, and the engine begins to move, the motion of the weight is at firft gradually accelerated. The action of the power being fuppofed invariable, its influ- ence in accelerating the motion of the weight decreafes, while the velocity of the weight increafes. Thus the action of a ftream of water, or air, upon a wheel, is to be eftimated only from the excefs of the velocity of the fluid above the velocity already acquired by the part of the engine which it ftrikes, or from their relative velocity. On the other hand, the weight of the load that is to be raifed, and the friction , tend to retard the motion of the engine ; and when thefe forces, viz. thofe that tend to accelerate it, and thofe that tend to retard it, become equal, the engine then proceeds with the uniform motion it has acquired.

I^t A B (Fig. 1.) reprefent the velocity of the ftream, A C, the velocity of the part of the engine which it ftrikes,

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effect of the engine depends. It is known that the action of a fluid, upon a given plane, is as the fquare of this rela- tive velocity ; confequently the weight railed by the engine, when its motion becomes uniform, being equal to this ac- tion, it is likewife as the fquare of C B. Let this be multiplied by AC, the velocity of the part of the engine impelled by the fluid ; and the effect of the engine in a given time will be proportional to A C X C B 2 = {^- pofing C B to be biflected inD) ACx iCDx2.DB = 4ACxCDxl^B; confequently, the effect of the en- gine is greateft, when the product of A C, C D, and D B is greateft. But it is eafy to fee, that this product is great- eft, when the parts AC, CD, and D B, are equal j for if you defcribe a femicircle upon A D, and the perpendicular C E meet the circle in E, then A C < C D = C h% and is greateft, when C is the center of the circle j fo that in order that ACxCDxDB may be the greateft poffible, A D muft be biflected in C j and C B having been biflected in D, it follows, that AC, CD, D B, muft be equal ; or that A C, the velocity of the part of the engine impelled by the fheam, ought to be but one third of A B, the velocity of the ftream. In this cafe, when (abftracting from friction) the engine acts with the utmoft advantage i the weight raifed by it is to the weight that would juft fuftain the force of the ^. ftream, as the fquare of C B, the relative velocity of the engine and ftream, to the fquare of A B, which would be the relative velocity, if the engine was quiefcent ; that is, as 2x2 to 3X 3) or 4 to 9. Therefore, that the engine may have the greateft effect poffible, it ought to be loaded with no more than £ of the weight, which is juft able to fuftain the efforts of the ftream. See Mac Laurin's Account of Sir Ifaac Newton's Difcov. p. 171. feq. and Fluxions., "Art. 908.

Again fuppofe that a given weight P, (Fig.- 2.) defending by its gravity in the vertical line, raifes a greater weight VV like- wife given, by the rope PMW (that panes over the fixed

pully M) along the inclined plane B D, the height of which B A is given ; and let it be required to find the pofition of this plane, along which VV will be raifed in :he lead: time, from the horizontal line AD to B, Let B C be the plane upon which if W was placed, it would he exactly fuftained by P j in which cafe, P is to W as A B to B C. But W is to the force with which it tends to defcend along the plane B D, as B D to A B ; confequently the weight P is to that force, as B D to B C. Therefore the excefs of P above that force (which excefs is the power that accelerates the mo- tions of P and W) is to P, as B D — B C to B D ; or taking B H upon B C equal to B D, as C H to B D. But it is known that the fpaces defcribed by motions uniformly accelerated, are in the compound ratio of the forces which produce them and the fquares of the times j or that the fquare of the time is directly as the fpace defcribed in that time, and inverfely as the force ; confequently, ihe fquare of the time, in which B D is defcribed by W, will be directly

as B D, and inverfely as. ■ * and will be leaft when— - - B D, C H

B C a is a minimum ; that is, when -7^—^ -f C H -}■ 2 B C, or

C H

BC :

when the motion of the machine becomes uniform ; and C B will reprefent their relative velocity, upon which the |

(becaufe 2 B C is invariable) when I— .. -f- C H is a mini-

C H mum. Now as when the fum of two quantities is given, their product is a maximum, when they are equal to each other ; fo it is manifeft, that, when their product is given, their fum muft be a minimum, when they are equal. Thus it is evident, that as in Fig. 1, the rectangle or product of the equal parts A C and C D was equal to CE 1 ; fo the rectangle of any two unequal parts, into which A D may be divided, is lefs than CE 2, and A D is the leaft fum of any two quantities, the product of which is equal to C £ a. But

B C * the product of — - -, - and CH is B C% and confequently C H

up j

given 3 therefore the fum of — ^= and C H is leaft, when

C rl

thefe