PROPOSITION VII.

In any machine whose motion accelerates, the weight will be moved with the greatest velocity, when the velocity of the power is to that of the weight, as 1 + P(+)to1; the inertia of the machine being disregarded.

W

For any such machine may be considered as reduced to a lever, or to a wheel and axle whose radii are R and 7: in

which the velocity of the weight raw gt (prop. 1) is to

R2p+r2w

be a maximum, r being considered as variable. Hence then, following the usual rules, we find PR = r(w+w+PW). From which, since the velocities of the power and weight are respectively as R and r, the ratio in the proposition immediately flows.

Cor. When the weight moved is equal to the power, then is Rr 1 +√2: 1 :: 24142:1 nearly.

PROPOSITION VIII.

If in any machine whose motion accelerates, the descent of one weight causes another to ascend, and the descending weight be given, the operation being supposed continually repeated, the effect will be greatest in a given time when the ascending weight is to the descending weight, as 1 to 1.618, in the case of equal heights; and in other cases, when it is to the exact counterpoise in a ratio which is always between 1 to 1 and 1 to 2.

Let the space descended be 1, that ascended s; the descending weight 1, the ascending weight: then would the equilibrium require ws; and i will be the force act

w

reduced to the point at which

32

==== ; consequently the

w

พ

to 1+, and the relative

-) =

But, the space bethe root of the accelerating force

force is (1 — —-—-) ÷ (1 + —-—-_-) ing given, the time is as inversely, that is, as

and the whole effect in a given

time, being directly as the weight raised, and inversely as

the time of ascent, will be as; which must be a

maximum.

maximum. Consequently its square

w-s

w3+ s2w2

must be a max.

likewise. This latter expression, in fluxions and reduced,

gives w = [√(s2 + 10s + 9)—a +3].

1+√5 :

Here if s = 1,w=1+5; but if s be diminished without

2

limit, w = s; if it be augmented without limit, then will √(s2 + 10s+9) approach indefinitely near to s+ 5, and consequently = 25. Whence the truth of the proposition is manifest.

PROPOSITION IX.

Let & denote the absolute effort of any moving force, when it has no velocity; and suppose it not capable of any effort when the velocity is w; let be the effort answering to the velocity v; then, if the force be uniform, I will be =

For it is the difference between the velocities w`and v which is efficient, and the action, being constant, will vary as the square of the efficient velocity. Hence we shall have this analogy, F: (w ́— 0)2 : (w-v)2; consequently, F = P(W — V)2 = q(1 — ——)2.

W

W

Though the pressure of an animal is not actually uniform during the whole time of its action, yet it is nearly so so that in general we may adopt this hypothesis in order to approximate to the true nature of animal action. On which supposition the preceding prop. as well as the remaining one, in this chapter, will apply to animal exertion.

Cor. Retaining the same notation, we have w = This, applied to the motion of animals, gives this theorem: The utmost velocity with which an animal not impeded can move, is to the velocity with which it moves when impeded by a given resistance, as the square root of its absolute force, to the difference of the square roots of its absolute and efficient forces.

PROPOSITION X.

To investigate expressions by means of which the maximum effect, in machines whose motion is uniform, may determined.

be

I. It follows, from the observations made in art. 1 and the definitions in this chapter, that when a machine, whether simple or compound, is put into motion, the velocities of the

impelled

232

imed and working points, are inversely as the forces which are equbrio, when applied to those points in the direcDo of their motion. Consequently, if ƒ denote the resistwhen reduced to the working point, and v its velocity; while and v denote the force acting at the impelled point, its velocity; we shall have Fvfu, or introducing t the me, jut. Hence, in all working machines which ezequired a uniform motion, the performance of the mache is equal to the momentum of impulse.

II. Let F be the effort of a force on the impelled point of a machine when it moves with the velocity v, the velocity being w when F = 0, and let the relative velocity w¬v=u. (), the momentum of inThen since (prop. 1X) F = 4(

W

22

W2

=W - u = w

-

=

pulse rv will become vo (—)2 = 9. — (w – u); because w-u. Making this expression for FV a maximum, Fv or, suppressing the constant quantities, and making u2(w --u) a max. or its flux. 0, when u is variable, we find 2w=3u, or uw. Whence v w jw. Consequently, when the ratio of v to v is given, by the construction of the machine, and the resistance is susceptible of variation, we must load the machine more or less till the velocity of the impelled point, is one-third of the greatest velocity of the force; then will the work done be a maximum.

Or, the work done by an animal is greatest, when the velocity with which it moves, is one-third of the greatest velocity with which it is capable of moving when not impeded.

นว W

#2

W2

=

III. Since F = 0 = P(P(), in the case of the maximum, we have FV = 40v 40. w = 10w, for the momentum of impulse, or for the work done, when the machine is in its best state. Consequently, when the resistance is a given quantity, we must make v :v::9f: 46; and this structure of the machine will give the maximum effect

V

ບ

V

IV. If we enquire the greatest effect on the supposition that only is variable, we must make it infinite in the above expression for the work done, which would then become WF, or w f, or wft, including the time in the formula. Hence we see, that the sum of the agents employed to move a machine may be infinite, while the effect is finite: for the variations of, which are proportional to this sum, do not influence the above expression for the effect.

Scholium.

Scholium.

The propositions now delivered contain the most material principles in the theory of machines. The manner of applying several of them is very obvious: the application of some, being less manifest, may be briefly illustrated, and the chapter concluded with two or three observations.

The last theore.n may be applied to the action of men and of horses, with more accuracy than might at first be supposed. Observations have been made on men and horses drawing a lighter along a canal, and working several days together. The force exerted was measured by the curvature and weight of the track-rope, and afterwards by a spring steelyard. The product of the force thus ascertained, into the velocity per hour, was considered as the momentum. In this way the action of men was found to be very nearly as (w-v): the action of horses loaded so as not to be able to trot was nearly as (w - v)17, or as (w-v). Hence the hypothesis we have adopted may in many cases be safely assumed.

According to the best observations, the force of a man at rest is on the average about 70 pounds; and the utmost velocity with which he can walk is about 6 feet per second, taken at a medium. Hence, in our theorems, = 70, and w= 6. Consequently == 313 lbs. the greatest force a man can exert when in motion: and he will then move at the rate of w, or 2 feet per second, or rather less than a mile and a half per hour.

The strength of a horse is generally reckoned about 6 times that of a man; that is, nearly 420 lbs. at a dead pull. His utmost walking velocity is about 10 feet per second. Therefore his maximum action will be of 420 = 1863 lbs. and he will then move at the rate of of 10, or 34 feet, per second, or nearly 24 miles per hour. In both these instances we suppose the force to be exerted in drawing a weight along a horizontal plane; or by raising a weight by a cord running over a pulley, which makes its direction horizontal.

2. The theorems just given may serve to show, in what points of view machines ought to be considered, by those who would labour beneficially for their improvement.

The first object of the utility of machines consists in furnishing the means of giving to the moving force the most commodious direction; and, when it can be done, of causing its action to be applied immediately to the body to be moved. These can rarely be united: but the former can be accomplished in most instances; of which the use of the simple

lever,

ever, puiler, and wheel and axle, furnish many examples. The second object gained by the use of machines, is an accuation of the velocity of the work to be performed, to the velocity with which alone a natural power can act. Thus, whenever the natural power acts with a certain velocity which cannot be changed, and the work must be performed with a greater velocity, a machine is interposed moveable round a fixed support, and the distances of the impelled and working points are taken in the proportion of the two given

velocities.

But the essential advantage of machines, that, in fact, which properly appertains to the theory of mechanics, consists in augmenting, or rather in modifying, the energy of the moving power, in such manner that it may produce effects of which it would have been otherwise incapable. Thus a man might carry up a flight of steps 20 pieces of stone, each weighing 50 pounds (one by one) in as small a time as he could (with the same labour) raise them all together by a piece of machinery, that would have the velocities of the impelled and working points as 20 to 1, and, in this case, the instrument would furnish no real advantage, except that of saving his steps. But if a large block of 20 times 30, or 600 lbs. weight, were to be raised to the same height, it would far surpass the utmost efforts of the man, without the intervention of some such contrivance.

The same purpose may be illustrated somewhat differently; confining the attention all along to machines whose motion is uniform. The product fu represents, during the unit of time, the effect which results from the motion of the resistance; this motion being produced in any manner whatever. If it be produced by applying the moving force immediately to the resistance, it is necessary not only that the products FV and fu should be equal; but that at the same time F=ƒ, and v=v: if, therefore, as most frequently happens, ƒ be greater than F, it will be absolutely impossible to put the resistance in motion by applying the moving force immediately to it. Now machines furnish the means of disposing the product FV in such a manner that it may always be equal to fv, however much the factors of FV may differ from the analogous factors in fu; and, consequently, of putting the system in motion, whatever is the excess of ƒ over F.

Or, generally, as M. Prony remarks (Archi. Hydraul. art. 504), machines enable us to dispose the factors of Fvt in such a manner, that while that product continues the same, its factors may have to each other any ratio we desire. If, for instance, time be precious, the effect must be produced in a very

short