Page images
PDF
EPUB

And

weight w; then P: w: CB. DE. FG: AC. BD . ef. 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: 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 THE PULLEY.

70. 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.

71. PROP. If a power sustain a weight by means of a fixed pulley: the power and weight are equal.

For through the centre c of the pulley draw the horizontal diameter AB: then will AB represent a lever of the first kind, its prop being the fixed centre c; from which the points A and B, where the power and weight act, being equally distant, the power P is consequently equal to the weight w.

A

P

B

72. Corol. Hence, if the pulley be put in motion, the power P will descend as fast as the weight w ascends. So that the power is not increased by the use of the fixed pulley, even though the rope go over several of them. It is, however, of great service in the raising of weights, both by changing the direction of the force, for the convenience of acting, and by enabling a per. son to raise a weight to any height without moving from his place, and also by permitting a great many persons at once to exert their force on the rope at P, which they could not do to the weight itself; as is evident in raising the hammer or weight of a pile-driver, as well as on many other occasions.

73. PROP. If a power sustain a weight by means of one

moveable pulley; the power is but half the weight, if the portions of the sustaining cord are parallel to each other.

[blocks in formation]

74. Corol. 1. Hence it is

evident, that, when the pulley is put in motion, the velocity of the power will be

[blocks in formation]

double the velocity of the weight, as the point p moves twice as fast as the point c and weight w rises. It is also evident, that the fixed pulley r makes no difference in the power P, but is only used to change the direction of it, from upwards to downwards.

75. Corol. 2. Hence we may estimate the effect of a combination of any number of fixed and moveable pulleys; by which we shall find that every cord going over a moveable pulley always adds 2 to the power; since each moveable pulley's rope bears an equal share of the weight: while each rope that is fixed to a pulley, only increases the power by unity.

[blocks in formation]

Note.-If the portions of the sustaining cords between the pulleys are not parallel, the forces will be reduced upon the principle of art. 31.

OF THE INCLINED PLANE.

76. THE INCLINED PLANE, is a plane inclined to the horizon, or making an angle with it. It is often reckoned one of the simple mechanic powers; and the double inclined plane makes the wedge. It is employed to advantage in raising heavy bodies in certain situations, diminishing their weights by laying them on the inclined planes.

77. PROP. The power gained by the inclined plane, is in proportion as the length of the plane is to its height. That is, when a weight w is sustained on an inclined plane BC, by a power p acting in the direction Dw, parallel to the plane; then the weight w, is in proportion to the power P, as the length of the plane is to its height; that is, w: P:: BC: AB.

D

BE

W

Р

Α

For, draw AE perp. to the plane BC, or to Dw. Then we are to consider that the body w is sustained by three forces, viz. 1st, its own weight or the force of gravity, acting perp. to ac, or parallel to BA; 2d, by the power P, acting in the direction wD, parallel to BC, or BE; and 3dly, by the re-action of the plane, perp. to its face, or parallel to the line EA. But when a body is kept in equilibrio by the action of three forces, it has been proved, (art. 30.) that the intensities of these forces are proportional to the sides of the triangle ABE, made by lines drawn in the directions of their actions; therefore those forces are to one another as the three lines AB, BE, AE; that is,

[ocr errors]

the weight of the body w is as the line the power P is as the line

[ocr errors]
[ocr errors]

AB,

BE,

and the pressure on the plane as the line AE.

But the two triangles ABE, ABC, are equiangular, and have therefore their like sides proportional; that is,

the three lines

AB, BE, AE,

are to each other respectively as the three BC, AB, ac, or also as the three

AC, AE, CE,

which therefore are as the three forces, W, P, P, which Ρ denotes the pressure on the plane. That is, w: P :: BC AB, or the weight is to the power, as the length of the plane is to its height.

See more on the Inclined Plane in the Dynamics.

78. Scholium. The Inclined Plane comes into use in some situations in which the other mechanical powers cannot be conveniently applied, or in combination with them. As, in sliding heavy weights either up or down a plank or other plane laid sloping: or letting large casks down into a cellar, or drawing them out of it. Also, in removing earth from a lower situation to a higher by means of wheelbarrows, or otherwise, as in making fortifications, &c.; inclined planes, made of boards are employed. Rail-roads, or inclined planes, serve often to convey coals from the mouth of a mine.

Of all the various directions of drawing bodies up an inclined plane, or sustaining them on it, the most favourable is where it is parallel to the plane BC, and passing through the centre of the weight; a direction which is easily given to it, by fixing a pulley at D, so that a chord passing over it, and fixed to the weight, may act or draw parallel to the plane. In every other position, it would require a greater power to support the body on the plane, or to draw it up. For if one end of the line be fixed at w, and the other end inclined down towards B, below the direction wD, the body would be drawn down against the plane, and the power must be increased in proportion to the greater difficulty of the traction. And, on the other hand, if the line were carried above the direction of the plane, the power must be also increased; but here only in proportion as it endeavours to lift the body off the plane.

If the length BC of the plane be equal to any number of times its perp. height AB, as suppose 3 times; then a power P of 1 pound, hanging freely, will balance a weight w of 3 pounds, laid on the plane; and a power P of 2 pounds, will balance a weight w of 6 pounds; and so on, always 3 times as much. But then if they be set moving, the perp. descent of the power P, will be equal to 3 times as much as the perp. ascent of the weight w. For, though the weight w ascends up the direction of the oblique plane, BC, just as fast as the power P descends perpendicularly, yet the weight rises only the perp. height AB, while it ascends up the whole length of the plane BC, which is three times as much; that is, for every foot of the perp. rise of the weight, it ascends 3 feet up in the direction of the plane, and the power r descends just as much, or 3 feet.

[blocks in formation]

80. PROP. 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 perp. to the force acting on the back, to urge the wedge forward; and the sides AC, BC are perp. to the forces acting on them; therefore the three forces are as AB, AC, BC.

AB,

ADB

[graphic]

81. Corol. The force on the back Its effect in direct. perp. to AC,

AC,

And its effect parallel to AB ;

DC,

which are per. to them.

are as the three lines

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.

SCHOLIUM.

82. 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 line AC or DC. Various other theories of the wedge are given by different authors, but they need not here be detailed, on account of the irregulari. ties introduced by friction.

VOL. II.

23

« PreviousContinue »