must therefore be a

THE MECHANICAL POWERS.

the qe the difference of leverage necessary to support the weight in equilibrio. Hence, a small addition either of leverage or weight will cause the power to preponderate. o.

EXAMPLE 1. A ball weighing 3 tons is to be raised by 4 men, who can exert a force of 12 cwt. Required the proportionate length of lever.

3 tons 60 owt., and

60

12

In this example, the proportionate lengths of the lever to maintain the weight in equilibrio, are as 5 to 1. But, although the ball is sustained by a force of only one fifth of its weight, no power is gained, for the weight passes through only one fifth of the space passed through by the power.

EXAMPLE 2. A weight of 1 ton is to be raised with a lever 8 feet in length, by a man who can exert, for a short time, a force of rather more than 4 cwt. Required at what part of the lever the fulcrum must be placed.

5; . e., the weight is to the power as 5 to 1; therefore,

=1 foot and a third from the weight.

T

EXAMPLE 3. A weight of 40 lbs. is placed one foot from the ful crum of a lever. Required the power to raise the same when the length of the lever on the other side of the fulcrum is five feet.

Case 2. When the lever is of the second order, or when the fulcrum is at one end of the lever and the power at the other, with the weight between them.

RULE. As the distance between the power and the fulcrum is to the distance between the weight and the fulcrum, so is the effect to the power.

EXAMPLE 1. Required the power necessary to raise 120 lbs. when the weight is placed six feet from the power and two feet from the fulcrum.

As 8: 2 :: 120: 30 lbs., the power.

EXAMPLE 2. A beam 20 feet in length, and supported at both ends, bears a weight of two tons at the distance of eight feet from one end. Required the weight on each support.

40 cwt. x 8 feet 20 feet

weight; and

16 cwt, on the support that is furthest from the

40 x 12

20 feet

24 cwt. on the support nearest to the weight.

Case 3. When the lever is of the third order, or the weight is at one end of the lever, the fulcrum at the other, and the power is applied

between them.

RULE. As the distance between the power and the fulcrum is to the length of the lever, so is the weight to the power.

EXAMPLE. The length of the lever being eight feet, and the weight at its extremity 60 lbs., required the power to be applied six feet from the fulcrum to raise it.

As 6 8: 60: 80 lbs., Ans.

The Pulley.

Pulleys are of two kinds, fixed and movable.

The fixed pulley affords no economy of power, but merely changes its direction. The movable pulley changes its position with that of the weight, and effects a saving equal to half the power. An equilibrium is preserved between the power and weight, when the weight is equal to the product of the power and twice the number of movable pulleys.

RULE. Divide the weight to be raised by twice the number of pulleys in the lower block; the quotient will give the power neces sary to raise the weight.

EXAMPLE. Required the power to raise 600 lbs. when the lower block contains six pulleys.

The wheel and axle act as a revolving lever; and in order to obtain an equilibrium between the power acting on the circumference of the wheel, and the weight or resistance acted on by the circumference of the axle, the power must be to the weight as the radius of the axle is to that of the wheel. One or more radii of the wheel, or winches, are often substituted for the wheel in the simple machine; and in compound machines the action is communicated by teeth or cogs, forming wheel-and-pinion work.

RULE. As the radius of the wheel is to the radius of the axle, so is the effect to the power.

EXAMPLE. A weight of 50 lbs. is exerted on the periphery of a wheel whose radius is 10 feet. Required the weight raised at the extremity of a cord wound round the axle, the radius being 20 inches.

The Inclined Plane.

The inclined plane acts as a mechanical power by sustaining a portion of the weight to be raised, while the direction of the applied force is changed from the perpendicular to one more or less horizontal, and the weight moves upwards on it in a diagonal between them. Equilibrium is sustained when the power is to the weight as the perpendicular height of the inclined plane is to its inclined length or hypothenuse, when the power acts in a direction parallel to the inclination of the plane; but as the height is to the base when in a direction parallel to the base.

RULE. As the length of the plane is to its height, so is the weight to the power.

EXAMPLE. Required the power necessary to raise 540 lbs. up an inclined plane 5 feet long and 2 feet high.

As 5: 2 :: 540: 216 lbs., the power.

The length, in the above rule, must represent that of the inclined surface, or of the base, accordingly as the power acts parallel to either of these surfaces.

The wedge may be regarded as two inclined planes, united by a common base, acting on two weights or resistances at once, or on a fulcrum and a weight, between which it moves, generally, in practice, by the impulse of successive blows.

As in the inclined plane, equilibrium consists in the power being to the resistance as the back of the wedge is to its length, or to the length of its side, accordingly as the resistance acts perpendicularly to the central line of length or to that of the side.

Case 1. When two bodies are forced from one another by means of a wedge, in a direction parallel to its back.

RULE. As the length of the wedge is to half its back or head, so is the resistance to the power.

EXAMPLE. The breadth of the back or head of the wedge being 3 inches, and the length of either of its inclined sides 10 inches, required the power necessary to separate two substances with a force of 150 lbs.

As 10 1 150: 223 lbs., the power.

Case 2. When only one of the bodies is movable.

RULE. As the length of the wedge is to its back or head, so is the resistance to the power.

EXAMPLE, The breadth, length, and force, the same as in the last example.

As 10 3: 150: 45 lbs., the power.

:

The Screw.

The screw is an inclined plane, and may be supposed to be generated by wrapping a triangle, or an inclined plane, round a cylinder. The base of the triangle is the circumference of the cylinder; its height, the distance between two consecutive cords or threads; and the hypothenuse forms the spiral cord or inclined plane.

RULE. To the square of the circumference of the screw, add the square of the distance between two threads, and extract the square root of the sum this will give the length of the inclined plane. Its height is the distance between two consecutive cords or threads. When a winch or lever is applied to turn the screw, the power of

1 the screw is as the circle described by the handle of the winch, or lever, to the internal or distance between the spirals.

Case 1. When the weight to be raised is given, to find the

RULE

power.

Multiply the weight by the distance between two threads of the screw, and divide the product by the circumference of the circle described by the lever. The quotient is the power.

EXAMPLE. Required the power to be applied to the end of a lever three feet long, to raise a weight of five tons with a screw of 1 inch between the threads.

11200 lbs. x 1.25:

36 inches x 2 x 3.1416

61.9 lbs., the power.

Case 2. When the power is given, to find the weight it will raise. RULE Multiply the power by the circumference of the circle described by the lever, and divide the product by the distance between two threads of the screw the quotient will be the weight. The example is the converse of that in the former case.

To HARDEN AND POLISH ALABASTER. 1. Take a strong solution of alum, strain it, and put it into a wooden trough sufficiently large to contain the figure, which must be suspended in it by means of a thread of silk; let it rest until a sufficient quantity of the salt is crystallized on the cast, then withdraw it, and polish it with a clean cloth and water.

2. Take white wax, melt it in a convenient vessel, and dip the cast or figure into it; withdraw, and repeat the operation of dipping until the liquid wax rests upon the surface of the cast; then let it cool and dry, when it must be polished with a clean

brush.

TOOTHED WHEELS.

The pitch (or the distance between the centres of two contiguous teeth) of cog-wheels is measured on the pitch-line, or extreme circumference of the wheel; and the distance between that line and the centre of the circle is reckoned as the radius of the wheel.

The following rules have been laid down for the diameters and number of teeth for wheels and pinions.

RULE 1.

As the number of teeth in the wheel + 2.25,
Is to the diameter of the wheel,

So is the number of teeth in the pinion + 15,
To the diameter of the pinion.

210, the

EXAMPLE. Given the number of teeth in the wheel diameter of the wheel 25 inches, and the number of teeth in the pinion 30, to find the diameter of the pinion. As 210+ 2.25: 25: 30+ 1.5: 3·7102,

the diameter of the

pinion.

RULE 2.

As the number of teeth in the wheel + 2:25,

Is to the diameter of the wheel,

So is (No. of teeth in pinion + No. of teeth in wheel) ÷ 2,
To the distance of their centres.

On the Velocity of Wheels, Drums, Pulleys, &c.

When wheels are applied to communicate motion from one part of a machine to another, their teeth act alternately on each other; consequently, if one wheel contains 60 teeth and another 20, the one containing 20 teeth will make three revolutions, while the other makes but one; and if drums or pulleys are taken in place of wheels, the result will be the same, because their circumferences, describing equal spaces, render their revolutions unequal; from this the rule is derived, namely,