tension of string at p1 = weight supported at

and so on. ported is

Pi = P

P2 = 2P+ w1
P32P+2w1 + w,

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

W = (1 + 2 + 22 + 23 +

+ (1 + 2 + 22 + 23 +

= (2"−1) P + (2a−1—1)w, + (2n−2—1) w2 +

Cor. If the weights of the pulleys be neglected, we have W = (2a — 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=8; 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, kept at rest by three forces, viz. the power

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

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

ог

=

P sin a

=

cos B

P

sin a

p cos (a + B) P cos (a + B)

Cor. 2. When WP is parallel to the plane, 8 = 0; hence we have
W AB W AB Р

urges a body W down the in× W, or the force with which it descends, or endea

Cor. 3. The power or relative weight that clined plane, is = BC AB vours to descend, is as the sine of the angle A of inclination.

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.

Corollary. The force on the back,

Its effect in direct. perp. to AC,

AB,

And its effect parallel to AB,

Are as the three lines,

AC,

DC,

which are perp.
to them.

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

the splitting and raising the largest and hardest rocks, the raising and lifting the largest ship, by driving a wedge below it, which a man can do by the blow of a mallet; and thus it appears that the small blow of a hammer, on the back of a wedge, is incomparably greater than any mere pressure, and will overcome it.

OF THE SCREW.

47. THE Screw is one of the six mechanical powers, chiefly used in pressing or squeezing bodies close, though sometimes also in raising weights.

The screw is a spiral thread or groove cut round a cylinder, and everywhere making the same angle with the length of it. So that if the surface of the cylinder, with this spiral thread on it, were unfolded and stretched into a plane, the spiral thread would form a straight inclined plane, whose length would be to its height, as the circumference of the cylinder is to the distance between two threads of the screw; as is evident by considering, that, in making onė round, the spiral rises along the cylinder the distance between the two threads.

PROP. XI.

48. The force of a power applied to turn a Screw round, is to the force with which it presses upwards or downwards, setting aside the friction, as the distance between two threads is to the circumference where the power is applied.

THE Screw being an inclined plane, or half wedge, whose height is the distance between two threads, and its base the said circumference; and the force in the horizontal direction, being to that in the vertical one, as the lines perpendicular to them, namely, as the height of the plane, or distance of the two threads, is to the base of the plane, or circumference at the place where the power is applied; therefore the power is to the pressure, as the distance of two threads is to that circumference.

Corollary. When the screw is put in motion; then the power is to the weight which would keep it in equilibrio, as the velocity of the latter is to that of the former; and hence their two momenta are equal, which are produced by multiplying each weight or power by its own velocity. So that this is a general property in all the mechanical powers, namely, that the momentum of a power is equal to that of the weight which would balance it in equilibrio: or that each of them is reciprocally proportional to its velocity.

49. SCHOLIUM.-Hence we can easily compute the force of any machine turned by a screw. Let the annexed figure represent a press driven by a screw, whose threads are each a quarter of an inch asunder; and that the screw is turned by a handle of 4 feet long from A to B; then, if the natural force of a man, by which he can lift, pull, or draw, be 150 pounds; and it be re

quired to determine with what force the screw will press on the board at D, when the man turns the handle at A and B with his whole force. The diameter AB of the power being 4 feet or 48 inches, its circumference is 48 X 3.1416 or 1504 nearly; and the distance of the threads being of an inch; therefore the power is to the pressure, as 1 to 6034: but the power is equal to 150 lb.; therefore as 1: 603 150: 90,480; and conse

quently the pressure at D is equal to a weight of 90,480 pounds, independe14 of friction.

50. Again, if the endless screw AB be turned by a handle AC of 20 inches, the threads of the screw being distant half an inch each; and the screw turn a toothed wheel E, whose pinion L turns another wheel F, and the pinion M of this another wheel G, to the pinion or barrel of which is hung a weight W; it is required to determine what weight the man will be able to raise, working at the handle C; supposing the diameters of the wheels to be 18 inches, and those of the pinions and barrel 2 inches; the teeth and pinions being all of a size.

Here 20 X 3.1416 X 2 = 125.664, is the circumference of the power.

And 125 664 to, or 251-328 to 1, is the force of the screw alone.

Also, 18 to 2, or 9 to 1 being the proportion of the wheels to the pinions; and as there are three of them, therefore 9 to 1, or 729 to 1 is the power gained by the wheels.

Consequently 251-328 X 729 to 1, or 1832184 to 1 nearly, is the ratio of the power to the weight, arising from the advantage of both the screw and the wheels.

But the power is 150 pounds; and therefore 150 x 183218, or 27482716 pounds, is the weight the man can sustain, which is equal to 12269 tons weight.

But the power has to overcome, not only the weight, but also the friction of the screw, which is very great, in some cases equal to the weight itself, since it is sometimes sufficient to sustain the weigh, when the power is taken off.

EXAMPLES ON THE PRINCIPLES OF THE MECHANICAL POWERS.

ON THE LEVER.

51. Ex. 1. The arms of a bent lever are to each other as 4 to 5, and arc inclined at an angle of 135°. The lever rests upon a fulcrum at its angular point, and weights are suspended from the extremities of the two arms, such that the shorter arm rests in a horizontal position; what is the ratio of the weights? Ans. 8 5/2 or 1:·8838835.

Ex. 2. The difference of the lengths of the arms of a lever is (a) inches; the same weight weighs (w) pounds at one end, and (w) ounces at the other; find the lengths of the arms.

a 4a Ans. and 3 3

Ex. 3. A lever three feet in length weighs 6lb.; what weight on the shorter arm will balance 12lb. on the longer, the fulcrum being one foot from the end?

Ans. 27lb.

Ex. 4. The compound lever DK is composed of three levers of the first kind, DA, AB, BK, acting upon one another. The arms DC, CA of the first lever are respectively 8 and 6 inches; those of the second, AO, OB, are 12 and 2, and those of the third, BH, HK are 16 and 3; find the ratio of P, the power at D, to W, the weight suspended at K.

Ans. P: W:: 3 : 128.

Ex. 5. Suppose AB is a squared beam, or lever of oak, 30 feet long, each end being one foot square; what weight W at the end A would keep it in a horizontal position on a fulcrum C, 3 feet from that end, each cubic foot of the beam weighing 54lb.? Ans. 6480lb.

Ex. 6. AB is a uniform straight lever, 20 feet in length, and weighing 40lb.; and HBK, a flexible chain of the same length, and weight 130lb., is laid upon the lever in such a manner that it is kept in equilibrium on a fulcrum C, which is five feet from the end B; how much of the chain overhangs the end B?

Ans. 20

30

1326, or 8.233032 feet.

ON THE WHEEL AND AXLE.

52. Ex. 1. In a combination of four wheels and axles, each of the radii of the wheels is to each of the radii of the axles as 5 to 1; what power will balance a weight of 1875 pounds? Ans. 3 pounds.

Ex. 2. A power of 6lb. keeps in equilibrium a weight of 240lb., by means of a wheel and axle: the diameter of the axle is 6 inches; what is the radius of the wheel?

Ans. 10 feet.