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obtain a maximum effect on the wheel and axle, (the weight of the machinery not being considered), we may adopt the conclusions of cors. 6 and 7 of this prop. And in the extreme case where the wheel and axle becomes a pulley, the expression in cor. 8 may be adopted. The like conclusions may be applied to machines in general, if R and r represent the distances of the impelled and working points from the axis of motion; and if the various kinds of resistance arising from friction, stiffness of ropes, &c. be properly reduced to their equivalents at the working points, so as to be comprehended in the character w for resistance overcome.
Given R and r, the arms of a straight lever, s and m their respective weights, and r, the power acting at the extremity of the arm R; to find the weight raised at the extremity of the other arm when the effect is a maximum.
In this case im is the weight of the shorter end reduced to B, and
applied at A, would balance the shorter end; therefore
+w, would sustain both the shorter end and the weight 2r R
w in equilibrio. Put + is the power really acting at the longer end of the lever; consequently
x + 1 m - (2 +w), is the absolute moving power. Now 2r R
the distance of the centre of gyration of the beam from *
* The distance of R, the centre of gyration, from c the centre or axis of motion, in some of the most useful cases, is as below;
is = √ prop. 50)
which let be denoted by e; then (Mechán.
(M+m) will represent the mass equivalent 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 as in the last prop. we
w) ÷ 5, (x+m) + e +
of P, or of w reduced to A. Mul-
ánd n for p + (Mm), then
r2 be a quantity which varies as the effect
varies, and which, indeed, when multiplied by gt, dénotes
Cor. When R=r, and м=m, if we restore the values of n and 9, the expression will become w⇒✓✓(2p2+2mr+‡m2)—(P+m).
Given the length l and angle e of elevation of an inclined plane
will have a velocity v; and gravity would impress upon it in the instant i following, a new velocityg sin e. t, provided the weight p were then entirely free but, by the disposition of the system, will be the velocity which obtains in reality. Then, estimating the spaces in the direction cr, 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 1+ we have, for the velocity in ressed on,
* . . . v†g sín e. t, where Sv + v
effective veloc. from c towards B. g sin e.t v,.... velocity destroyed.
effective veloc. from c towards A..
ŵ. ~~ v4 g sín E. t, where+g sin E. t,.. velocity destroyed.
If, therefore, gravity impresses, during the time upon the
pg sin e.t Po wg sin E i + w v
Whence the effective accelerating force is found, i. e.
è sin e -w sin 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 A, that is,
any quantity, as t, is a minimum, its square is manifestly a minimum: so that substituting for its equal , and striking out
the constant factors, we have
2LL(PL — Wl—PL2 L
= a min. or its fluxion
=0. Here, as in all similar cases, since
the fraction vanishes, its numerator must be equal to 0; consequently 2PL2-2wll—PL2=0, PL=2wl, or L :/:: 2w : r.
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. 2. When sin e=1, Bc coincides with the perpendicu lar CD, and the power p acts with all its intensity upon the weight w, This is the case of the present problem which has commonly been considered.
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 e on the plane CB, or the power r hanging freely in the vertical co; 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 the same force in the same ratio; because either of these weight in their respective positions would sustain w on CE, Therefore the excess of r above that force (which excess is the power accelerating the motions of r and w) is to P, as CA - CE to ca; or taking CH=CA, as En to CA. Now, the motion being uniformly accelerated, we have s¤ FT2, or т2 α = ; consequently, the square of the time in which ac' is described by w, will be as ac directly, and as —
inversely; and will be least when is a minimum; that is
when +En+2CE, or (because 2ce is invariable) when Ex 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 CE2, and consequently given ;
therefore the sum of
and EH is least 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 r acting along ce.
When co and Cв coincide, the case becomes the same as that considered by Maclaurin, in his View of Newton's Philosophical Discoveries, pa. 183, 8vo. edit.
Let the given weight e descend along CB, and by means of the
The general expression for the vel. in v—
P sin e-w sin É
gt, which, by substitut. - L for sin e, and – 7 for sin E, becomes
gt. This mul. into w, gives
which, by the prop. is to be a maximum. Or, striking out the
a max. Putting this into fluxions, and reducing, we have PL-2pwl –w3l=0,
or w=p√/ (~+1)—p.
Cor. When the inclinations of the planes are equal, L, and 7 are equal, and w P/2-P-X(2-1)=4142P agreeing with the conclusion of the lever of equal arms, or the extreme ease of the wheel and axle, i. e. the pulley.
Given the radius R 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 e; 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
And the inertia which resists the com-
were concentrated in a point a (Mechan.
prob. 50). If the former of these be divided by the latter,