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Measure of it by Amontons.

in the Encycl. art. ROTATION, no 64. and fhall there

fore use it here.

parts of machinery, and to obtain fome general rules for afcertaining the quantity of what unavoidably remains. Mr Amontons, of the Royal Academy of Suppofing now this previous knowledge of all these Sciences at Paris, gave us, about the beginning of this variable circumftances which affect the motion of macentury, the chief information that we have on the fub- chines of the rotative kind, fo that, for any momentary ject. He difcovered, that the obftruction which it gave pofition of it while performing work, we know what to motion was very nearly proportional to the force by are the precife preffures acting at the impelled and which the rubbing furfaces are preffed together. Thus working points, and the conftruction of the machine, he found, that a smooth oaken board, laid on another on which depend the friction, and the momentum of fmooth board of the fame wood, requires a force nearly its inertia (expreffed in the article ROTATION by equal to one third of what preffes the furfaces toge- pr2); we are now in a condition to determine its mo, ther. Different fubftances required different proportion, or at least its momentary acceleration, competent to that pofition. Therefore,

tions.

He also found, that neither the extent of the rubbing furfaces, nor the velocity of the motion, made any confiderable variation on the obftruction to motion. These were curious and unexpected refults. Subfequent obfervations have made feveral corrections neceffary in all thefe propofitions. This fubject will be more particularly confidered in another place; but fince the deviations from Mr Amontons's rule are not very confiderable, at least in the cases which occur in this general confideration of machines, we fhall make ufe of it in the mean time. It gives us a very easy method of eftimating the effect of friction on machines. It is a certain proportion of the mutual preffure of the rubbing furfaces, and therefore muft vary in the fame proportion with this preffure. Now, we learn from the principles of ftatics, that whatever preffures are exerted on the impelled and working point of the machine, all the preffures on its different parts have the fame conftant proportion to these, and vary as thefe vary: Therefore the whole friction of the machine varies in the fame proportion. But farther, fince it is found that the friction does not fenfibly change with the velocity, the force which is juft fufficient to overcome the friction, and put the loaded machine in motion, muft be very nearly the fame with the force expended in overcoming the friction while the machine is moving with any velocity whatever, and performing work. Therefore if we deduct from the force which juft puts the loaded machine in motion that part of it which balances the reaction of the impelled point occafioned by the refiftance of the work, or which balances the refiftance of the work, the remainder is the part of the impelling power which is employed in overcoming the friction. If in deed the actual refifting preffure of the work varies with the velocity of the working point, all the pref. fures, and all the frictions in the different communicating parts of the machine, vary in the fame proportion. But the law of this variation of working refiftance be ing known, the friction is again afcertained.

We can now ftate the dynamical equilibrium of forces in the working machine in two ways. We may either confider the efficient impelling power as diminifhed by all that portion which is expended in overcoming the friction, and which only prepares the ma chine for performing work, or we may confider the impelling power as entire, and the work as increased by the friction of the machine; that is, we may fuppofe the machine without friction, and that it is loaded with a quantity of additional refiftance acting at the working point. Either of these methods will give the fame refult, and each has its advantages. We took the laft method in the flight view which we took of this fubject

It is

12

Let there be a rotative machine, fo conftructed, that Compofiwhile it is performing work, the velocity of its impelled tion of thei point is to that of its working point as m to n. formula eafy to demonftrate, from the common principles of the perexpreffing ftatics, that if a fimple wheel and axle be fubftituted formance for it, having the radius of the wheel to that of the of a maaxle in the fame proportion of m to n, and having the chine. fame momentum of friction and inertia, and actuated by the fame preffures at the impelled and working points, then the velocities of these points will be precifely the fame as in the given machine.

Let p reprefent the intenfity (which may be measured by pounds weight) of the preffure exerted in the mo ment at the impelled point; and r exprefs the preffure exerted at the working point by the refiftance oppofed by the work that is then performing. This may arife from the weight of a body to be raised, from the cohefion of timber to be fawed, &c. Any of thefe refiftances may alfo be measured by pounds weight; because we know, that a certain number of pounds hung on the faw of a faw mill, will juft overcome this cohefior, or overcome it with any degree of fuperiority. Therefore the impelling power p, and the refiftance r, however differing in kind, may be compared as mere preffures.

Let x reprefent the quantity of inert matter which must be urged by the impelling power p, with the fame velocity as the impelled point, in order that this preffure p may really continue to be exerted on that point. Thus, if the impelling power is a quantity of water in the bucket of an overfhot wheel, acting by its weight, this weight cannot impel the wheel except by impelling the water. In this way, may be confidered as reprefenting the inertia of the impelling power, while preprefents its preffure on the machine. In like manner, let y reprefent the quantity of external inert matter which is really moved with the velocity of the working point in the execution of the task performed by the machine.

Whatever be the momentum of the inertia of the machine, we can always afcertain what quantity of mat. ter, attached to the impelled point, or the working point of the wheel and axle, will require the fame force to give the wheel the fame angular motion; that is, which fhall have the fame momentum of inertia. Let the quantity a, attached to the working point, give this momentum of inertia a n2.

Laftly, fuppofing that the wheel and axle have no friction, let ƒ be fuch a refiftance, that if applied to the working point, it fhall give the fame obftruction as the friction of the machine, or require the fame force at the impelled point to overcome it. M 2

Thefe

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led point.

15

Velocity of Velocity of working point =

the work

ing point.

16 Abfolute

them.

(II.)

(III.)

pmn—r+fn2. x m2 + a + y n2 In order to obtain a clear conception of thefe velocities, we must compare them with motions with which we are well acquainted. The propofition being univerfally true, we may take a cafe where gravity is the fole power and refittance; where, for example, p and r are the weights of the water in the bucket of a wheel, and in the tub that is raifed by it. In this cafe, px, and We may alfo, for greater fimplicity, fuppofe the machine without inertia and friction. The velocity pm2. -rmn of p is now

pm2 + rn2

Let g be the velocity which gravity generates in a measure of fecond. Then it will generate the velocity gi in the moment i. Letv be the velocity generated during this moment in p, connected as it is with the wheel and axle, and with r. This connection produces a change of condition giv. For, had it fallen freely, it would have acquired the velocity gt, whereas it only acquires the velocity v. In like manner, had r fallen freely, it would have acquired the velocity gt. But, inftead of this, it is raised with the velocity

m

n •

-v. The

m

change on it is therefore = gi + 1v. These changes of mechanical condition arife from their connection with the corporeal machine. Their preffures on it bring into action its connecting forces, and each of the two external forces is in immediate equilibrium with the force exerted by the other. The force excited at the impelled point, by r acting at the working point, may be called the momentum or energy of r. Thefe ener gies are precifely competent to the production of the changes which they really produce, and muft therefore be conceived as having the fame proportions. They are therefore equal and oppofite, by the general laws obferved in all actions of tangible matter; that is, they are fuch as balance each other. Thus, and only thus, the remaining motions are what we observe them to be. That is, pxgx m = rx gt+v x n

n

m

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Orpm2-rmn × g t = p m2 + r n2 X That is, the denominator of the fraction, expreffing the ve That is, p m2+rn2 : pm2 -rmn=gt:v locity of the impelled point, is to the numerator as the ve locity which a heavy body would acquire in the moment t, by falling freely, is to the velocity which the impelled point acquires in that moment. The fame thing is true of the velocity of the working point.

This reafoning fuffers no change from the more complicated nature of the general propofition. Here the impelling power is ftill p, but the matter to be accelerated by it at the working point is a +y, while its reaction, diminishing the impelling power, is only r. We have only to confider, in this cafe, the velocity with which a+y would fall freely when impelled, not by ay, but only by r. The refult would be the fame; gt would ftill be to v as the denominator of the fame fraction to its numerator.

Thus have we discovered the momentary acceleration of our machine. It is evident, that if the preffures p and r, and the friction and inertia of the machine, and the external matter, continue the fame, the acceleration will continue the fame; the motion of rotation will be uniformly accelerated, and p m2 + a + y n2 will be to pm2-r+fmn as the spaces, through which a heavy body would fall in any given time t, is to the space through which the impelled point will really have moved in the fame time. In like manner, the space through which the working point moves in the fame m n − r + ƒn2 time is = p m2 + a + y n1

2

I

4

S.

17

Thus are the motions of the working machine determined. We may illuftrate it by a very fimple example. Suppofe a weight p of five pounds, defcending from a pulley, and dragging up another weight r of three m and n are equal, and pounds on the other fide. P each may be called 1. The formula becomes p+r or 53, or,s. Therefore, in a fecond, the 5+3 weight p will defcend th of 16 feet, or 4 feet; and will acquire the velocity of 8 feet per fecond. Having obtained a knowledge of the velocity of eve- Perform! ry point of the machine, we can easily ascertain its per- ance of the quan. machine. formance. This depends on a combination of the tity of refiftance that is overcome at the working point, and the velocity with which it is overcome. raifing water, it depends on the quantity (proportional velocity with which it is lifted up. to the weight) of water in the bucket or pump, and the This will be had by multiplying the third formula by r, or by rgi, or by rs. Therefore we obtain this expreffion, Work done = pmrn-r+fra' pm2+a+yn' formance of the machine, including every circumftance Such is the general expreffion of the momentary perwhich can affect it. But a variation of thofe circumftances produces great changes in the refults. These muft. be diftinctly noticed.

Thus, in

gi. (IV.)

Cor. 1. If pmrn be equal to r+fr n2, there will be no work done, because the numerator of the fraction is annihilated. There is then no unbalanced force, and the

the natural power is only able to balance the preffure propagated from the working point to the impelled point.

2. In like manner, if no, no work is done altho' the machine turns round. The working point has no motion. For the fame reason, if m be infinitely great, although there is a great prevalence of impelling momentum, there will not be any fenfible performance during a finite time. For the velocity which can imprefs is a finite quantity, and the impelled point cannot move faster than x would be moved by it if detached from the machine. Now when the infinitely remote impelled point is moved through any finite space, the motion of the working point muft be infinitely lefs, or nothing, and no work will be done.

Remark. We fee that there are two values of n, viz.

v, and m × 2, which give no performance. But in all other proportions of m and n fome work is done. Therefore, as we gradually vary the proportion of m to n, we obtain a series of values expreffing the performance, which must gradually increase from nothing, and then decrease to nothing. There muft therefore be fome proportion of m to n, depending on the proportion of tor+, and of x to a+y, which will give the greateft poffible value of the performance. And, on the other hand, if the proportion of m to n be already determined by the conftruction of the machine already erected, there must be some proportion of p to r+f, and of x to a +y, by which the greateft performance of the machine may be enfured. It is evident, that the determination of these two proportions is of the utmoft importance to the improvement of machines. The well informed reader will pardon us for endeavouring to make this appear more forcibly to thofe who are lefs in ftructed, by means of fome very fimple examples of the first principle.

Suppofe that we have a ftream of water affording three tons per minute, and that we want to drain a pit which receives one ton per minute, and that this is to be done by a wheel and axle ? We wish to know the beft proportion of their diameters m and n. Let » be taken = 6; and suppose, 1. That n = 5. pm r n

Then

=

r2 n2 3.6.1.5—1.25_65 pm2 + rn2 3.36+1.36

2. Let n be

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6. The formula is = 0,5.

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(√

p2a+y r+f2 x

2a+y

1)gi. This expreffes the

x + velocity of the working point in feet per fecond, and therefore the actual performance of the machine.

But the proper proportion of m to n, afcertained by this procefs, varies exceedingly, according to the nature both of the impelling power, and of the work to be performed by the machine.

1. It frequently happens that the work exerts no contrary ftrain on the machine, and confifts merely in impelling a body which refifts only by its inertia. This is the cafe in urging round a millftone or a heavy fly; in urging a body along a horizontal plane, &c. in this cafer does not enter into the formula, which now be√ x2ƒ3 + p2 x (a + y *f. If the friccomes m X p(a + y)

-

=0,4887 tion be infignificant we may take n = m

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/p2 x (a + 3)

p2 (a + y)2

The velocity of the working point is

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In this cafe, it will be found

that the velocity acquired at the end of a given time will be nearly in the proportion of the power applied to 13' the machine.

10X 4-16 10+ 4

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10X5-25 3. Let r be 5. The formula is = 10+5 = 1,6666. Here it appears, that more work is done when r is 4 than when it is 5 or 3.

25 ==

15'

2. On the other hand, and more frequently, the inertia of the external matter which muft be moved in performing the work need not be regarded. Thus, in the grinding of grain, fawing of timber, boring of cylinders, &c. the quantity of motion communicated to the flour, to the faw duft, &c. is too infignificant to be taken into the account. In this cafe, y vanishes from the formula, which becomes extremely fimple when the fric tion and inertia of the machine are inconfiderable. We fhall

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city of the working point is

4 x (r+ƒ) +

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a p2 4 (r+f) But it is rare that machines of this kind have a small inertia. They are generally very ponderous and powerful; and the force which is neceffary for generating even a very moderate motion in the unloaded machine (that is, unloaded with any work), bears a great proportion to the force neceffary for overcoming the refiftance oppofed by the work. The formula muft therefore be used in all the terms, because a is joined with y. It would have been fimpler in this particular, had a been joined with x in the expreffion of the angular velocity.

3. In fome cafes we need not attend to the inertia of the power, as in the fteam engine. In this cafe, if taken strictly, n appears to have no value, becaufe x is a factor of every term of the numerator. But the for. mula gives this general indication, that the more infig. nificant the inertia of the moving power is fuppofed, the larger fhould m be in proportion to n; provided always, that the impelling power is not, by its nature, greatly diminished, by giving fo great a velocity to the impelled point. This circumftance will be particularly

confidered afterwards.

This muft fuffice for a very general view of the first problem.

19

II. The next queftion is not less momentous, namely, Beft proto determine for a machine of a given conftruction that portion of proportion of the refistance at the working point to the the power impelling power which will enfure the greatest perform- and work. ance of the machine; that is, the proportion of m to n being given, to find the beft proportion of p to r.

This is a much more complicated problem than the other; for here we have to attend to the variations both of the preffures p and r, and alfo of the external matters x and y, which are generally_connected with them. It will not be fufficient therefore to treat the queftion by the ufual fluxionary procefs for determining the maximum, in which is confidered as the only varying quantity. We muft, in this curfory difcuffion, reft fatisfied with a comprehenfion of the circumstances which moft generally prevail in practice.

4. If the inertia of the power and the refiftance be proportional to their preffures, as when the impelling power is water lying in the buckets of an overfhot wheel, and the work is the raifing of water, minerals, or other heavy body, acting only by its weight; then p and r1t muft either happen, that when r changes, there is may be fubftituted for x and y, and the formula expreffing the value of n, when the performance is a maximum, becomes

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I.

2

no change (that is, of moment) in the mafs of external matter which must be moved in performing the work, or that there is alfo a change in this circumftance. If no change happens, the denominator of the fourth formula, expreffing the performance, remains the fame; and then the formula attains a maximum when the nu. merator pr mnr + frn2 is a maximum. Also, we may include f without complicating the procefs, by the confideration, that f is always in nearly the fame ratio to r; and therefore r +ƒ may be confidered as a certain multiple of r, fuch as br. We may therefore omit f in the fluxionary equations for obtaining the maximum, and then, in computing the performance, divide the whole by b. Thus if the whole friction be

21

21

th of the refifting preffure r, we have r+f= 20 of r, and b = Having afcertained the best value for r, we put this in its place in the fourth formula, and

20

20

take of this for the performance. This will never differ much from the truth.

Or, making = 1, we have n = Thefe very fimple expreffions are of confiderable ufe, even in cafes where the inertia of the machine is very confiderable, provided that it have no reciprocating motions. A fimple wheel and axle, or a train of good wheelwork, have very moderate friction. The general results, therefore, which even very unlettered readers can deduce from thefe fimple formule, will give notions that are useful in the cafes which they cannot fo tho-= roughly comprehend. Some fervice of this kind may be derived from the following little table of the buit proportions of m to n, correfponding to the proportions of the power furnished to the engineer, and the refiftance which must be overcome by it. The quantity ie always = 10, and m = 1.

r

21

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; and if we farther fimplify the process, by making p = 1, and m = 1, we have r = -; a molt fim

I 211

ple expreffion, directing us to make the refiftance one half of what would balance the impelling power by the intervention of the machine.

This will evidently apply to many very important cafes,

cafes, namely, to all thofe in which the matter put in motion by the working point is but trifling.

But it alfo happens in many important cafes, that the change is at leaft equally confiderable in the inertia of the work. In this cafe it is very difficult to obtain a general folution. But we can hardly imagine fuch a change, without fuppofing that the inertia of the work varies in the fame proportion as the preffure excited by it at the working point of the machine; for fincer continues the fame in kind, it can rarely change but by a proportional change of the matter with which it is connected. Yet fome very important cafes occur where this does not happen. Such is a machine which forces water along a long main pipe. The refiftance to motion and the quantity of water do not follow nearly the fame ratio. But in the cafes in which this ratio is obferved, we may represent y by any multiple br of r, which the cafe in hand gives us; b being a number, integer, or fractional. In the farther treatment of this cafe, we think it more convenient to free r from all other combinations; and instead of fuppofing the force f (which we made equivalent with the friction of the machine) to be applied at the working point, we may apply it at the impelled point, making the effective power q=p-f. For the fame reafons, inftead of making the momentum of the machine's inertia an2, we may make it a m2, and make a+x=%. Now, fuppofing q, or pf, = 1, and alfo m = 1, our formula expreffing the perrn —— p2 122 formance becomesThis is a maximum when x + b r n2 z 2 +z bn — Z b n2

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Cor. 1. If the inertia of the work is always equal to its preffure, as when the work confifts wholly in raifing a weight, fuch as drawing water, &c. then b = 1, and the formula for the maximum performance becomes √ zn+z2

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.

neous ma

properly adjufted to the preffures, the inertia, and the friction of the machine, we do not derive all the advantages which we might from our fituation. Hence alfo we learn the falfity of the maxim which has been received as well founded, that the augmentation of intenfity of any force, by applying it to the long arm of General a lever, is always fully compenfated by a lofs of time; but erro or, as it is ufually expreffed, "what we gain by a machine in force we lofe in time." If the proportion of m to n is well chofen, we fhall find that the work done, when it refits by its inertia only, increases nearly in the proportion of the power employed; whereas when the inertia of the work is but a small part of the refiftance, it increafes nearly in the duplicate ratio of the power employed.

It was remarked, in the fetting out in the prefent problem, that the formule do not immediately exprefs the velocity of any point of the machine, but its momentary acceleration. But this is enough for our purpofe; because, when the momentary acceleration is a maximum, the velocity acquired, and the space defcribed, in any given time, is alfo a maximum. We allo fhewed how the real velocities, and the spaces defcribed, may be afcertained in known meafares. We may fay in general, that if g reprefent the preffure of gravity on any mafs of matter w, then is to pmn-rf n

w

as

a m2 + a + y n2 16 feet to the space defcribed in a fecond by the working point in a fecond, or as 32 feet per fecond is to the velocity acquired in that time.

xim.

20

celerate.

A remark now remains to be made, which is of the Caufes why the whole of the preceding doctrines. It appears, from do not con greatest confequence, and gives an unexpected turn to machines must be uniformly accelerated, and that any point will all that has been faid, that the motion of a machine tinually ac defcribe spaces proportional to the fquares of the times; for while the preffures, friction, and momentum of inertia remain the fame, the momentary acceleration must alfo be invariable. But this feems contrary to all experience. Such machines as are properly conftructed, and work without jolts, are obferved to quicken their pace for a few feconds after ftarting; but all of them, in a very moderate time, acquire a motion that is fenfibly uniform. Is our theory erroneous, or what are the cir

cumftances which remain to be confidered, in order to make it agree with obfervation? The fcience of machines is imperfect, till we have explained the caufes of this deviation from the theory of uniform accele ration.

Thefe caufes are various.

1. In fome cafes, every increase of velocity of the r. Increase machine produces an increase of friction in all its com- of friction. municating parts. By thefe means, the accelerating force, which is pm — r + ƒn, or p-fm-rn, is diminished, and confequently the acceleration is diminifhed. But it feldom happens that friction takes away or employs the whole accelerating force. We are not yet well inftructed in the nature of friction. Most of the kinds of friction which obtain in the communicating parts of machines, are fuch as do not fenfibly increafe by an increase of velocity; fome of them really diminish. Yet even the most accurately conftructed machines, unloaded with work, attain a motion that is feafibly uniform. If we take off the pallets from a pen

dulum

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