The same force which accelerates a falling body, acting in an ofposite direction on a body thrown upwards, must retard it; and since the action of gravitation is uniform, in whatever time it generates any velocity in a falling body, it must, in the same time, destroy the same velocity in a rising body: therefore a body will descend from any height in an equal space of time, and with equal force to that with which it was projected to that height.

The force with which a body descends to the earth, is as its quantity of matter multiplied into its acquired velocity, in the last second of its descent.

EXAMPLE.

Suppose a bullet, 7 lb. wt. be projected from a piece of ordnance perpendicular to the horizon, and that the time of ascent and descent be 1' 20": Required the height to which the ball was thrown, and the force with which it descends to the earth.

First. 1.' 20."240" time of ascent or descent.

Then as 1,2 16.1 :: 40,2: 25760 feet high; and as it descends so far in 40", the ratio of descent for the last second is (79 x 16.1) 1271.9 feet: this, multiplied by the quantity of matter, viz. 7 lb. gives 8903.3 lb. the force or momentum with which it strikes the ground.

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Mechanical Powers.

OF THE LEVER.

The lever is a bar movable about a fixed point, which is called its fulcrum or prop. In theory, it is considered as an inflexible

line, without weight.

The lever is of three orders.

In the first, the fulcrum or prop is between the weight and the power.

In the second, the weight is between the prop and the power; and,

In the third, the power is between the weight and the prop.

FIRST ORDER.

PROPOSITION 1. A power and weight, acting on the arms of a lever, will balance each other, when the distance of the power from the prop is to the distance of the prop from the weight, as the weight is to the power reciprocally.

Let the lever, a, b, c, be supported by the prop b; if the wt. d, multiplied into a b, be equal to the power c, multiplied into b c, the weight and power will balance each other: Or, a b will be to bc, as the weight is to the power reciprocally.

EXAMPLE.

What weight will a man be able to raise, who presses with a force of 11⁄2 cwt. on the arm of an equipoised handspike, 100 inches long, which rests on a fulcrum, 71⁄2 inches from the other end?

100-71-92 longer arm.

Then reciprocally.

in. crt. in. cwt.

As 924 : 14 :: 71⁄2 : 181⁄2 ans.

PROPOSITION 2. The statera, or steelyard, is of this order. By this machine the different weights of bodies are ascertained, by one single weight, which acts as the power.

Let the bar, s x, be divided into any number of equal parts, and exactly poised or balanced at d; let the power, f, equal to 1 lb. be applied on any part of the longer arm, d x, it shall counterpoise as many lb. suspended at w, as are equal to the number of parts counted from d to p.

d

1 2 3 4 5 6 7 8 9 10 11 12

For if equals 1 lb. and placed at the first division in the arm, x, it would balance 1 lb. at w; but being placed at 8, it counterpoises 8 lb. at w.

What weight, applied 70 inches from the centre of motion, will equipoise a hhd. of tobacco, wt. 94 cwt. freely suspended from a steelyard, at 2 inches from the said centre? Ans. 302 lb.

The shorter arm of a balance is 27 inches, the longer arm 36 inches: How many lb. suspended on the longer end, will equipoise 20 lb. on the other? Ans.-15 lb.

The beam of a balance measures 63 inches, and 20 lb. on one end equipoises 15 lb. on the other: Required the length of each Ans. 27 and 36 inches.

arm.

SECOND ORDER OF LEVERS.

A lever of the second order has the weight between the prop and the power.

In this, as well as the former, the advantage gained is, as the distance of the power from the prop, is to the distance of the

weight from the prop; for the respective velocities of the power and weight are in that proportion.

PROPOSITION 3. Thus, if a b be a lever on which the weight, w, 5 lb. hangs, at the distance of 1 inch from the prop, 8 and 1, equal to 1 lb. 5 inches from the prop, the power will just support the weight; and a small addition to the power will raise the weight 1 inch for every 5 inches the power ascends.

This lever explains why two men, carrying a load on a pole between them, bear unequal shares of the weight, in the inverse proportion of their distance from it; for the nearer either is to the weight, the greater part he bears; and if he goes directly under it, he bears the entire.

This may be applied to two horses of unequal strength, (in drawing weighty drafts, or ploughing) by dividing the beam they pull, so that the point of attraction may be as much nearer to the stronger horse than the weaker, as the strength of the one exceeds that of the other.

If a lever be 100 inches long, what weight, lying 7 inches from the end, may be moved with a power of 168 lb. lifting at the other end?

in.

100-71-921

Then reciprocally.
lb. in. lb.

As 92: 168 :: 71⁄2 : to 2072 ans.

Oars, rudders of ships, cutting-knives fixed at one end, doors moving on hinges, &c. &c. belong to this order.

THIRD ORDER.

In this lever, we suppose the power and the weight to change. places, so that the power may be between the weight and the prop. That there may be a balance between the power and the weight, the intensity of the power must exceed the intensity of the weight, just as much as the distance of the weight from the prop exceeds the power.

It is evident, that the force of the power is increased by the two first kinds of lever, being adapted to produce slow motion by a swift one: but this order serves to produce a swift motion of the weight, by a slow motion of the power. This order may be applied to the muscles of animals, particularly a man's arm.

a

e

PROPOSITION 4.

Let e be the prop of the lever; a, b and w, a weight of 1 lb. placed three times as far from the prop as the power, , which acts at f, by the cord going over the fixed pulley, d. In this case, the power must be equal to 3 lb. in order to support the weight of 1 lb.

A weight of 1 lb. placed on a man's shoulder, is no more than its absolute weight: What difference will he feel when the wt. is applied at his elbow, being 12 inches, and in the palm of his hand, being 28 inches from his shoulder; and how much must his muscles draw to support it at right angles, that is extended right out? Ans. 18 lb. at the elbow, 42 lb. at the palm, dif. 24.

The second mechanical power is the wheel and axis. In this machine, the power is applied to the circumference of the wheel, and the weight to the axis, by means of a cord winding round it.

PROPOSITION. An equilibrium is produced in this machine when the weight is to the power, as the diameter of the wheel is to the diameter of the axis: For,

Let db be the diameter of the wheel; c a, the diameter of the axis; w, the wt.; and, the power. When the wheel has performed one revo- e lution, by means of the power, 12, the wt. , is raised through a space equal to the circumference of the axis; therefore, the velocity of the power

d

exceeds the velocity of the weight, as the circumference of the wheel exceeds that of the axis.

It is plain, that the wheel and axis is a lever of the first order, of which the centre of motion is the line, és, in the centre of the

axis; the weight, w, is applied at the distance of half c a from the centre of motion. If, therefore, the radius, or semi-diameter of the wheel, exceed the semi-diameter of the axis, as the weight does the power, the momenta will be equal, and the power and weight will balance each other. To the wheel and axis we may refer cranes, mills, the capstan, winch, &c.

The third mechanical power is a pulley, or system of pullies.

A single pulley that only moves on its axis, may serve to change the direction of the weight or power, but can give no mechanical advantage, being only as the beam of a balance whose arms are of equal length and weight.

Thus, a is a single pully supporting the equal weights w and h: The cord bb, to which b they are appended, will be equally drawn, and the pulley a sustain both the weights, or be drawn with a force equal to twice p.

A combination, or system of pullies, whereof two, or more, as a and b, run in a fixed block x, and two others, c and d, in the movable block which raises the weight, w, by pulling the cord at 12, which goes successively round the pullies, a, d, b, c, and is fastened to the fixed block at s.

The purchase of this machine is known by considering that the cord is equally stretched throughout, by applying two such weights as will counterpoise each other. Thus, if be 4 lb. it will counterpoise w, 16, because is sustained by a single cord, and w by 4 folds of the same; therefore a man who pulls at p with a force of 100 lb. will poise a weight of 400 lb. or raise it by a small additional force.

A system of pullies can be applied in many cases where other engines cannot; particularly in raising great weights.

p

If in the solid block a, grooves be cut, whose radi are 1, 3, 5, 7, &c. and in the block b other groves be cut, whose radi are 2, 4, 6, 8, &c. and a cord be placed round the grooves, they will P answer the purpose of as many distinct pulliesevery point in each, moving with the velocity of the cord, in contact with it the whole friction will be removed to the two centres of motion in the blocks a and b, which is a great advantage over common pullies; therefore, a man pulling at f with a force of 100 lb. will poise a weight of 1200 at w, because

b