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the lever. And because this power applied to the point accelerates the masses P and w, the mass to be substituted

T2

for w, in the point. a, must bew, (Mechan. prop. 50) in order that this mass at the distance R may be equally accelerated with the mass w at the distance R. Hence the power

P

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R

T2

R2

- w will accelerate the quantity of matter P+W; and

the accelerating force F = (P ——w) ÷ (P+ ·w)=

w;
PR2- -RTW
PR2+ T2W

R

T2
R2

But (vol. ii. p. 335) α Ft or is = gtr (g being = 32% feet);

which in this case =

R2P-RTW
R2p+r2w

.gt, the velocity of r. And,

because veloc. of p: veloc. of w :: K: r, therefore veloc. of

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Cor. 1. The space described by the power in the time t,

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Cor. 2. If Rr::n 1, then will the force which acce

lerates A be =

Cor. 3. force p be

A will be =

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If at the same time the inertia of the moving 0, as in muscular action, the force accelerating

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Cor. 4. If the mass moved have no weight, but possesses inertia only, as when a body is moved along a horizontal plane, the force which accelerates A will be = either of these values may be readily introduced into the investigation.

Pn2 + W

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Cor. 5. The work done in the time t, if we retain the original notation, will be =

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R2P+r2wgt × W= Cor. 6. When the work done is to be a maximum, and we wish to know the weight when P is given, we must make the fluxion of the last expression = 0. Then we shall have ¿aR3p2 — 2r2R2PW—r1w2=0 and w = PX [√(+3) − = 2]. Cor. 7. If nr::n: 1, the preceding expression will become w = P × [√ (na + 223) — n2].

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Cor. 8. When the arms of the lever are equal in length, that is, when n = 1, then is WPX (√2—1) ·414214P, or nearly of the moving force.

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t

1

2444

Scholium.

are in like manner investigate the formulæ relating to expressions correspond exactly. Hence it follows, that when merion on the axis in peritrochio, it will be seen that the obtain a maximum effect on the wheel and axle, (the weight at is required to proportion the power and weight so as to conclusions of cors. 6 and 7 of this prop. And in the exof the machinery not being considered), we may adopt the treme case where the wheel and axle becomes a pulley, the expression in cor. 8 may be adopted. The like conclusions expr be applied to machines in general, if R and 7 represent the distances of the impelled and working points from the 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.

PROPOSITION II.

r

Given Rundr, the arms of a straight lever, м and m their respective weights, and p 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 4m is the weight of

the shorter end reduced to B, and

mr

conseq. 2R

is the weight which,

W

applied at A, would balance the shorter end: therefore

mr

2R

r

+ w, would sustain both the shorter end and the

R

weight w in equilibrio. But PM is the power really acting at the longer end of the lever; consequently

P+M

mr

(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:

In a circular wheel of uniform thickness

In the periphery of a circle revolving about the diam.
In the plane of a circle

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rad.

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In a plane ring formed of circles whose radii are R, †,
revolving about centre

R4

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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

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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

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> 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

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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

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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

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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

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=wg sin E i

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+wu.

Whence the effective accelerating force is found, i. e.

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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,

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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)

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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

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=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

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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.

.

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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

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