PROPOSITION IV. Let the given weight descend along cв, and by means of the thread PCW (running parallel to the planes) draw a weight w up the plane AC: it is required to find the value of w, rien its momentum is a maximum, the lengths and positions of the planes being given. (See the preceding fig.). The general expression for the vel. is v = P.Sin C-W sin z P+W -gt, which, by substitut. —-1 for sin e, and for sin E, becomes -L ——1 which, by the prop is to be a maximum. Or, striking out the constant factors, — gf, then is 1 PWL-W2/ =a max. Putting this into fluxions, and reducing, we have PL-2PWl— x2 = 0, or w = Pa' († + 1) − p. Cor. When the inclinations of the planes are equal, Land are equal, and w = P ¥2 − P = P × (√ ° − 1) =·4142P; agreeing with the conclusion of the lever of equal arms, or the extreme case of the wheel and axle, i. e. the pulley. PROPOSITION V. Given the radius of a wheel, and the radius r, of its axle, the weight of both, w, and the distance of the centre of gyration from the axis of motion, ; also a given power p acting at the circumference of the wheel; to find the weight w raised by a cord folding about the axle, so that its momentum shall be a maximum. - The force which absolutely impels ; this therefore subducted from P, A W munication of motion to the point a will be the same as if the mass R2 chan. prop. 50). If the former of these be divided by the RTP- -r2w g2w + r2w + R3P for the force which accelerates the weight w in its ascent. Consequently RTPW-72w2 g2w + r2w + R2p gt; which multi gt for the momentum. As this is to be a maximum, its fluxion will = 0; whence we √(R1p2+2R2Pg2w + g4w3 + PWRrg2 + P2R3r) — R2p— g3w shall obtain w Cor. 1. When Rr, as in the case of the single fixed pulley, then w=√(2¥2R3+2RPg2W+ £w2 + PWRg2)~ S2_W—P. R R2 Cor. 2. When the pulley is a cylinder of uniform matter g2=R2, and the express. becomes w=[R3(2P2+PW + w2)] Cor. 3. If, in the first general expression for the mo mentum of w, a be put R2P+ gw, we shall have etr2w a maximum. Which, in fluxions and reduced, gives W = 1 Cor. 4. If the moving force be destitute of inertia, then will ag2w, and w, as in the last corollary. PROPOSITION VI. Let a given power p be applied to the circumference of a wheel, its radius R, to raise a weight w at its axle, whose radius is r, it is required to find the ratio of R and r when w is raised with the greatest momentum; the characters w and g denoting the same as in the last proposition. Here we suppose r to vary in the expression for the momentum of w, gt. And we suppose, that by the g2w + r2w + R2P conditions of any specified instance, we can ascertain what quantity of matter q shall make r2qw, which, in fact, may always be done as soon as we can determine S. The expression for the work will then become R2p+r2(q+w)st. The fluxion of which being made = 0, gives, after a little reducRp2w2+r3(q+W)]-PW tion, r = P(q+w) RTPW-r2w2 Cor. When the inertia of the machine is evanescent, with respect to that of p+w, then is r√(1 + — ) − ì. ̧ ̧ 1. PROPOSITION 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. |