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170. Corol. 2. If any force › be applied to a lever at ́A; its effect on the lever, to turn it about the centre of motion c, is as the length of the lever CA, and the sine of the angle of direction CAE. For the perp. CE is as CA × s. 4 A.

171. Corol. 3. Because the product of the extremes is equal to the product of the means, therefore the product of the power by the distance of its direction, is equal to the product of the weight by the distance of its direction.

That is, P X CE W X CD.

172. Corol. 4. If the lever, with the weight and power fixed to it, be made to move about the centre c; the momentum of the power will be equal to the momentum of the weight; and their velocities will be in reciprocal proportion to each other. For the weight and power will describe circles whose radii are the distances CD, CE; and since the circumferences or spaces described, are as the radii, and also as the velocities, therefore the velocities are as the radii CD, CE; and the momenta, which are as the masses and velocities, are as the masses and radii; that is, as P X CE and w x CD, which are equal by cor. 3.

173. Corol. 5. In a straight lever, kept in equilibrio by a weight and power acting perpendicularly; then, of these three, the power, weight, and pressure on the prop, any one is as the distance of the other two.

174. Corol. 6. If several weights P, Q, R, s, act on a straight lever, and keep it in equilibrio; then the sum of the products on one side of the prop, will be equal to

P

E

the sum on the other side, made by multiplying each weight by its distance; namely,

PX AC +QX BC= R X DC + S X EC.

For, the effect of each weight to turn the lever, is as the weight multiplied by its distance; and in the case of an equilibrium, the sums of the effects, or of the products on both sides, are equal.

175. Corol. 7. Because, when

two weights and R are in equilibrio, Q R :: CD : CB;

therefore, by composition, a + R: Q :: BD : CD,

R

VOL. II.

and,

Q+RR :: BD: CB,
N

That

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It, That the points of spension of the scales and the centre of motion of the beam, A, B, C, should be in a vraight line: 24, That the arms 18, BC, be of an equal length: 34, That the centre of gravity be in the centre of motion E, or a little below it: 42, That they be in equilibrio when empty : C6, That there be as little friction as possible ar the centre B. A defect in any of these properties, makes the scales either imperfect or false. But it often happens that the one side of the beam is made shorter than the other, and the defect covered by making that scale the heavier, by which means the scales hang in equilibrio when empty; but when they are charged with any weights, so as to be still in equilibrio, those weights are not equal; but the deceit will be detected by changing the weights to the contrary sides, for then the equilibrium will be immediately destroyed.

177. To find the true weight of any body by such a false balance:-First weigh the body in one scale, and afterwards weigh it in the other; then the mean proportional between these two weights, will be the true weight required. For, if any body b weigh w pounds or ounces in the scale D, and only w pounds or ounces in the scale E: then we have these two equations, namely, AB . b = BC w.

and BC b = AB

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the product of the two is AB. Bc. b2 = AB. гc hence then

and

f2 = ww,
b = √ww,

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the mean proportional, which is the true weight of the body b.

178. The Roman Statera, or Steelyard, is also a lever, but of unequal brachia or arms, so contrived, that one weight only may serve to weigh a great many, by sliding it back

ward

ward and forward, to different distances, on the longer arm of the lever; and it is thus constructed:

AB

H

E

Let AB be the steelyard, and c its centre of motion, whence the divisions must commence if the two arms just balance each other: if not, slide the constant moveable weight I along from B towards c, till it just balance the other end without a weight, and there make a notch in the beam, marking it with a cipher O. Then hang on at A a weight w equal to 1, and slide 1 back towards till they balance each other; there notch the beam, and mark it with 1. Then make the weight w double of 1, and sliding 1 back to balance. it, there mark it with 2. Do the same at 3, 4, 5, &c,.by making w equal to 3, 4, 5, &c, times 1; and the beam is finished. Then, to find the weight of any body b by the steelyard; take off the weight w, and hang on the body b at A; then slide the weight 1 backward and forward till it just balance the body b, which suppose to be at the number 5; then is b equal to 5 times the weight of 1. So, if I be one pound, then bis 5 pounds; but if I be 2 pounds, then bis 10 pounds; and so on.

OF THE WHEEL AND AXLE.

PROPOSITION XXXII.

179. In the Wheel-and-Axle; the Weight and Power will be in Equilibrio, when the Power P is to the Weight w, Reciprocally as the Radii of the Circles where they act; that is, as the Radius of the Axle CA, where the Weight hangs, to the Radius of the Wheel CB, where the Power acts. That is,

PW:: CA: CB.

HERE the cord, by which the power r acts, goes about N 2

the

the circumference of the wheel, while that of the weight w goes round its axle, or another smaller wheel, attached to the larger, and having the same axis or centre c. So that BA is a lever moveable about the point c, the power P acting always at the distance BC, and the weight w at the distance CA; therefore P: W:: CA: CB.

BD

P

P

180. Cerol. 1. If the wheel be put in motion; then, the spaces moved being as the circumferences, or as the radii, the velocity of w will be to the velocity of P, as CA to CB; that is, the weight is moved as much slower, as it is heavier than the power; so that what is gained in power, is lost in time. And this is the universal property of all machines and engines.

181. Corol. 2. If the power do not act at right angles to the radius cb, but obliquely; draw CD perpendicular to the direction of the power; then, by the nature of the lever,'

PW:: CA: CD.

SCHOLIUM.

182. To this power belong all turning or wheel machines, of different radii. Thus, in the roller turning on the axis or spindle cE, by the handle CED; the power applied at B is to the weight w on the roller, as the radius of the roller

is to the radius CB of the handle.

W

B

C

183. And the same for all cranes, capstans, windlasses, and such like; the power being to the weight, always as the radius or lever at which the weight acts, to that at which the power acts; so that they are always in the reciprocal ratio of their velocities. And to the same principle may be referred the gimblet and augur for boring holes.

184. But all this, however, is on supposition that the ropes or cords, sustaining the weights, are of no sensible thickness. For, if the thickness be considerable, or if there be several folds of them, over one another, on the roller or barrel; then we must measure to the middle of the outermost rope, for

the

the radius of the roller; or, to the radius of the roller we must add half the thickness of the cord, when there is but one fold.

185. The wheel-and-axle has a great advantage over the simple lever, in point of convenience. For a weight can be raised but a little way by the lever; whereas, by the continual turning of the wheel and roller, the weight may be raised to any height, or from any depth.

186. By increasing the number of wheels too, the power may be multiplied to any extent, making always the less wheels to turn greater ones, as far as we please: and this is commonly called Tooth and Pinion Work, the teeth of one circumference working in the rounds or pinions of another, to turn the wheel. And then, in case of an equilibrium, the power is to the weight, as the continual product of the radii of all the axles, to that of all the wheels. So, if the

power P

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turn the wheel a, and this turn the small wheel or axle R, and this turn the wheel s, and this turn the axle T, and this turn the wheel v, and this turn the axle x, which raises the weight w; then P : w: CB. DE. FG AC. BD . EF. And in the same proportion is the velocity of w slower than that of P. Thus, if each wheel be to its axle, as 10 to 1; then Pw:: 13: 103 or as 1 to 1000. So that a power of one pound will balance a weight of 1000 pounds; but then, when put in motion, the power will move 1000 times faster than the weight.

OF

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