1 is = ✓ R3+r3 3(R+r) which let be denoted by ; then (Mechan. prop. 50) £ . (м + m) will represent the mass equivalent R2 to the beam or lever when reduced to the point A ; while the weight equivalent to w, when referred to that point, will be w. Hence, proceeding Hence, proceeding as in the last prop. we R2 the accelerating force of P, or of w reduced to A. Multiply this by w; and, for the sake of simplifying the proand n for P + (M+m); cess, put q for P+M > 2R R2 be a quantity which varies as the effect varies, and which, indeed, when multiplied by gt, denotes the effect itself. Putting the fluxion of this equal to nothing, and reducing, we at length find Cor. When R=r, and м = m, if we'restore the values of n and q, the expression will become w = √(2p2 + 2mr + m2)-(P+m). PROPOSITION III, Given the length l and angle e of elevation of an inclined plane BC; to find the length of another inclined plane Ac, along which a given weight w shall be raised from the hori zontal line AB to the point c, in the least time possible, by means of another given weight e descending along the given plane CB: the two weights being connected by an inextensible thread PCW running always parallel to the two planes. P W Here we must, as a preliminary to the solution of this proposition, deduce expressions for the motion of bodies connected by a thread, and running upon double inclined planes. Let the angle of elevation CAD be E, while e is the elevation CBD. Then at the end of the time t, P will have a velocity v; and gravity would impress upon in the instant t following, a new velocityg sin e. t, pro H it, vided vided the weight P were then entirely free: but, by the dis position of the system, v will be the velocity which obtains in reality. Then, estimating the spaces in the direction CP, as the body w moves with an equal velocity but in a contrary sense, it is obvious that, by applying the 3d Law of Motion, the decomposition may be made as follows. At the end of the time t+t we have, for the velocity impressed on, v + v .... effective veloc, from c towards B. P...+g sin e. i, where {g sin e. i-v w.v+g sin E. t, where If, therefore, gravity impresses, during the time t, upon the masses P, W, the respective velocities g sin e.tv, and g sin E.tv, the system will be in equilibrio. The quantities of motion being therefore equal, it will be =wg sin E i +wu. Whence the effective accelerating force is found, i. e. Thus it appears that the motion is uniformly varied, and we readily find the equations for the velocity and space from which the conditions of the motion are determined: viz, But in the triangle ABC it is AC : BC :: sin B : sin ▲, that is, L::: sine sin E; hence L= sin e, and -1 = sin E; 1 m m m being a constant quantity always determinable from the data given. And becomes Now when any 1 S(P+W) m quantity, as t, is a minimum, its square is manifestly a minimum: so that substituting for s its equal L, and striking out the constant factors, we have Li(PLW) PL3i (PL-W12 =0. Here, as in all similar cases, since the fraction vanishes, its sequently 2PL - 2wlL 2w: P. numerator must be equal to 0; conPL=0, PL = 2wl, or L::: Cor. 1. Since neither sin e nor sin E enters the final equation, it follows, that if the elevation of the plane BC is not given, the problem is unlimited. Cor. Cor. 2. When sine = 1, BC coincides with the perpendicular CD, and the power P acts with al lits intensity upon the weight w. This is the case of the present problem which has commonly been considered. Scholium. P E H W This proposition admits of a neat geometrical demonstration. Thus, let CE be the plane upon which, if w were placed, it would be sustained in equilibrio by the power P on the plane CB, or the power Ρ' hanging freely in the vertical CD ; then (Mechan. prop. 23) BC: CD: CE :: P: P': w. But w is to the force with which it tends to descend along the plane CA, as CA to CD; consequently, the weight p' is to that force, as CA CE; or the weight P on the plane BC, is to the same force in the same ratio; because either of these weights in their respective positions would sustain w on CE. Therefore the excess of P above that force (which excess is the power accelerating the motions of P and w) is to P, as CA CE to CA; or, taking CH CA, as EH to CA. Now, the motion being uniformly accelerated, we have s∞ FT2, or Τ' α : consequently, the square of the time in which AC S F is described by w, will be as AC directly, and as CA2 versely; and will be least when is a minimum; that is, CE2 EH when + EH+2CE, or (because 2CE is invariable) when CE2 EH EH EH is a minimum. Now, as, when the sum of two quantities is given, their product is a maximum when they are equal to each other; so it is manifest that when their product is given, their sum must be a minimum when they are equal. But the product of and EH is CE12, and con CE2 EH EC2 sequently given; therefore the sum of and EH is least, EH when those parts are equal; that is, when EH = CE, or CA 2CE. So that the length of the plane CA is double the length of that on which the weight w would be kept in equilibrio by P acting along CB. . When CD and CB coincide, the case becomes the same as that considered by Maclaurin, in his View of Newton's Philosophical Discoveries, pa. 183, 8vo. edit. PROPOSITION PROPOSITION IV. Let the given weight ℗ descend along CB, 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, zien is romentum is a maximum, the lengths and positions of the planes being given. (See the preceding fig.). The general expression for the vel.is – F.sin e~w sin z v = gt, which, by substitut. —-1 for sin e, and -—--1 for sin E, becomes P+W 2236 (PWL-W2) += P+W 1 PWL-Wil gt. This mul. into w, gives -gt; which, by the prop, is to be a maximum. Or, striking out the constant factors, gi, then is = a max. Putting this into fluxions, and reducing, we have PL-2pWl— x2 = 0, or w = P、' ́(† + 1) − P. P÷W Cor. When the inclinations of the planes are equal, L and I are equal, and w = PV2−P = P X (√ 2 − 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 e 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 the point A is F, while w acts in a direction contrary to P, with a force = ; this therefore subducted from P, R TW RP-TW =1, for the re R leaves P duced force impelling the point A. And the inertia which resists the communication of motion to the point a will be the same as if g3w+r?W + R2P the mass R2 were concentrated in the point ▲ (Mechan. prop. 50). If the former of these be divided by the latter, the quotient R(RP-r2w) is the force accelerating a: g2w + r2w + R2P multiplying W RTP-12w multiplying this by, we have RTP2W gw++; which multi RTPW-72w2 plied into w gives As g2w+rew+ept for the momentum. this is to be a maximum, its fluxion will = O; whence we √(R1p2 + 2R2pg2w + g4w3 + PWRrg2 + P2R3r) — R2 p—§3W 19 shall obtain w R2 Cor. 2. When the pulley is a cylinder of uniform matter P. -W Cor. 3. If, in the first general expression for the mo mentum of w, a be put RP+gw, we shall have RTPW-?W? e+r2w ~ a maximum. Which, in fluxions and reduced, gives TV = Q (Q + Rrp) 72 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 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 e denoting the same as in the last proposition. WRTP-72w2 Here we suppose r to vary in the expression for the momentum of w, g2+2w+2pt. And we suppose, that by the conditions of any specified instance, we can ascertain what quantity of matter q shall make r'qw, which, in fact, r2q 'may always be done as soon as we can determine . The exRTPW -r2w? pression for the work will then become Rap+r(q+wt. The fluxion of which being made O, gives, after a little reduc P(q+W) Cor. When the inertia of the machine is evanescent, with respect to that of p+w, then is r = R√(1 + 1) −1. P W PROPOSITION |
