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. P: W:: CA: CB.

HERE the cord, by which the power P acts, goes about 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 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.

B

P

That is,

W

Corollary. 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.

Corollary. 2. If the power do not act at right angles to the radius Cb, but obliquely; draw CD perp. to the direction of the power; then, by the nature of the lever, P: W:: CA: CD.

SCHOLIUM.

33. 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 CBD; 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.

34. And the same for all cranes, capstans, windlasses, and such like;

W

B

D

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.

35. 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 radius of the roller; or, to the radius of the roller and half the thickness of the cord, when there is but one fold.

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

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 turn the wheel Q, 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 P: W:: 13: 10s 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.

38. If ropes are used for the action of the power and weight, we must consider the forces applied to the axes of the ropes. Hence if R, r denote the radii of the wheel and axle, and T, t half the thickness of the ropes, we have P: Wr+t: R+T.

OF THE PULLEY.

39. A PULLEY is a small wheel, commonly made of wood or brass, which turns about an iron axis passing through the centre, and fixed in a block, by means of a cord passed round its circumference, which serves to draw up any weight. The pulley is either single, or combined together, to increase the power. It is also either fixed or moveable, according as it is fixed to one place or moves up and down with the weight and power.

When a power sustains a weight by means of a fixed pulley, the power and weight are obviously equal; for if through the centre of the pulley a horizontal line be drawn, it will represent a lever of the first kind, whose prop or fulcrum is the fixed centre; hence the points where the power and weight act, are equally distant from the centre, and therefore the power must be equal to the weight. No mechanical advantage, however, is gained by the fixed pulley, though it is still of great utility in the raising of weights, both by changing the direction of the force, and also by enabling several persons to exert their united forces.

PROP. VII.

40. In the single moveable pulley, and the strings parallel, the power is to the weight as 1:2; but if the strings produced make an angle = 24; then P: W::1 : 2 cos .

Through the centre of the pulley draw the vertical p line WAC, and take AC to represent the weight W, where A is the point of intersection of the strings produced. Draw CB parallel to AH; then since the string is equally stretched throughout, we have AB BC and angle BAC = 0; whence

P: W::AB: AC:: sin : sin 24 :: 1:2 cos &; and when the strings are parallel P:W: :1:2, for $=0. Cor. 1. If w= weight of the moveable block, then 2PW+w.

B

W

Cor. 2. In the system where there are two blocks of pulleys, the one fixed and the other moveable, and the same rope passing round all the pulleys, then we have simply a combination of the preceding case; and therefore

nPW+w

H

where n = number of strings at the moveable block and w, its weight. If the strings are not parallel, the cosine of the angle made with the vertical in each case must be introduced, as above.

Cor. 3. In the system where each pulley hangs by a separate string, we have merely a repetition of the single moveable pulley; and the strings being parallel, we get

2"PW+w1 + 2w2 + 22w3 +

where n is the number of moveable pulleys, and w1, w2, w3, the weights of the pulleys including the blocks respectively.

41. In the system of pulleys, where each string is attached to the weight, and the strings parallel, we have P: W:: 1:2"-1, where n is the number of pulleys.

Let w1, w2, w3, &c. be the weights of the pulleys, and let the strings passing over the pulleys w1, w2, wз, &c. be attached to the weight at the points P1, P2, P3, &c.; then we have

tension of string at p1 = weight supported at p1 = Р

and so on.

ported is

P2 =

P3

=..

P2 = 2P+ w.
P322P+2w, + w2

Hence, if n be the number of pulleys, the whole weight sup

Cor. If the weights of the pulleys be neglected, we have W = (2" — 1) P; hence it is manifest that the weights of the pulleys increase the weight supported, and the advantage is therefore on the side of the power.

ON THE INCLINED PLANE.

42. THE inclined plane assists by its reaction in sustaining a heavy body.

PROP. IX.

43. Let a weight W be supported on the inclined plane AB, by a power P acting in the direction WP; and let angle BAC=a, and angle BWP B; then

P: W sin a: cos B.

Draw WH perpendicular to the horizon, WK perpendicular to the plane AB, and HK parallel to WP; then the weight W is A kept at rest by three forces, viz. the

power

H

K

B

W

P

P in direction HK, gravity in direction WH, and the reaction of the plane AB in direction WK; hence if WH be taken to represent the weight, we have

P: W: HK: HW:: sin KWH: sin HKW

:: sin BAC: sin KWP:: sin a: cos ẞ;

because sin KWP = cos PWB, since BWK is a right angle. Cor. 1. If p represent the pressure on the plane; then we have

Sπ

P :p :: HK: KW : : sin a: sin HWP :: sin a : sin {

2

P sin a Р cos (a + B) Р cos (a + B) Cor. 2. When WP is parallel to the plane, B = 0; hence we have

clined plane, is: = × W, or the force with which it descends, or endea

BC
AB

vours to descend, is as the sine of the angle A of inclination.

FFF

Cor. 4. Hence, if there be two planes of the same height, and two bodies be laid upon them proportional to the lengths of the planes, they will have equal tendencies to descend down the planes; and, consequently, they will mutually sustain each other if they be connected by a string acting parallel to the planes.

OF THE WEDGE.

44. THE Wedge is a body of wood or metal, in form of a prism. AF or BG is the breadth of its back; CE its height; GC, BC its sides, and its end GBC is composed of two equal inclined planes, GCE, BCE.

PROP. X.

45. When a wedge is in equilibrio; the power acting against the back, is to the force acting perpendicularly against either side, as the breadth of the back AB, is to the length of the side AC or BC.

FOR, any three forces, which sustain one another in equilibrio, are as the corresponding sides of a triangle drawn perpendicular to the directions in which they act. But AB is perpendicular to the force acting on the back, to urge the wedge forward; and the sides AC, BC are perpendicular to the forces acting upon them; therefore the three forces are as AB, AC, BC.

And therefore the thinner a wedge is, the greater is its effect, in splitting any body, or in overcoming any resistance against the sides of the wedge.

46. SCHOLIUM.-But it must be observed, that the resistance, or the forces above mentioned, respect one side of the wedge only. For if those against both sides be taken in, then, in the foregoing proportions, we must take only half the back AD, or else we must take double the lines AC and DC. In the wedge, the friction against the sides is very great, at least equal to the force to be overcome, because the wedge retains any position to which it is driven; and therefore the resistance is doubled by the friction. But then the wedge has a great advantage over all the other powers, arising from the force of percussion or blow with which the back is struck, which is a force incomparably greater than any dead weight or pressure, such as is employed in other machines. And accordingly, we find it produces effects vastly superior to those of any other power; such as