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so as to transmit and accumulate a mechanical advantage, whether the communication be by means of cords and belts, or of teeth and pinions, the weight will be to the power, as the continual product of the radii of the wheels to the continual product of the radii of the axles.
Thus, if the radii of the axles a, b, c, d, e, be each 3 inches, while the radii of the wheels A, B, C, D, E, be 9, 6, 9, 10 and 12 inches respectively: then W:P :: 9x6 x 9 x 10 x 12:3x3 x 3 x 3 x3, or as 240 : 1. putation, however, in which the effect of friction is disregarded.
A train of wheels and pinions may also serve for the augmentation of velocities. Thus, in the preceding example, whatever motion be given to the circumference of the axle e, the rim of the wbeel A will move 240 times as fast.
And if a series of 6 wheels and axles, each having their diameters in the ratio of 10 to 1, were employed to accumulate velocity, the produced would be to the producing velocity as 10% to 1; that is, as 1,000,000 to 1.
Note. — A man's power producing the greatest effect, is 31 lbs. at a velocity of 2 feet per second, or 120 feet per minute.
The Rule to find the power of Cranes is, viz.
Divide the product of the driven by the product of the drivers, and the quotient is the relative velocity, as 1 :v, which multiplied by the length of winch, and by the power applied (in lbs.) and divided by the radius of the barrel, the quotient will be the weight raised.
A weight of 94 tops is to be raised 360 feet in 15 minutes, by a power, the velocity of wbich is 220 feet per minute :- What is the power required ? 360
=24 feet per minute, velocity of weight. 15 24 x 94--2256
=10.2545 tons power required. 220
A stone weighing 986 lbs. is required to be lifted : what power must be applied, when the power is to the weight as 9 is to 2 ? 986 x 2 1972
=2195 tons power. 9
A power of 18 lbs. is applied to the winch of a crane, the length of wbich is 8 inches; the pinion makes 12 revolutions for 1 of the wheel, and the barrel is 6 inches diameter. 8 x 2 x 22
=50.28 circumference of the winch's circle. 7 50.28 x 12=603.36 inches velocity of power on winch to 1 revolution of the barrel. 603.36 x 18=10860.48
=571.604 lbs. weight. 6 x 22
18.857 . . 19. 7 that can be raised by a power of 18 lbs. on this crane.
There are two kinds of pulleys, the fixed and the moveable. From the fixed pulley no power is derived; it is as a common beam used in weighing goods, having the two ends of equal weight, and at the same distance from the centre of motion; the only advantage gained by the fixed pulley, is in changing the direction of the power.
From the moveable pulley power is gained; it operates as a lever of the second order; for if one end of a string be fixed to an immoveable stud, and the other end to å moveable power, the string doubled and the ends parallel, the pulley that hangs between is a lever; the fixed end of the string being the fulcrum, and the other the moveable end of the lever : hence the power is double the distance from the fulcrum, than is the weight hung at the pulley; and therefore the power is to the weight as 2 is to 1. This is all the advantage gained by one moveable pulley; for twc, twice the advantage; for three, thrice the advantage; and 80 on for every additional moveable pulley.
From this the following rule is derived : - Divide the weight to be raised by twice the number of moveable pulleys, and the quotient is the power required to raise the weight.
What power is requisite to lift 100 lbs. when two blocks of three pulleys or sheives each, are applied, the one block moveable and the other fixed ? 100
=164 lbs. the power required, 3 shieves X 2=6. 6
What weight will a power of 80 lbs. lift, when applied to a 4 and 5 sheived block, and tackle, the 4 sheived block being moveable ?
80 x 8 = 640 lbs. weight raised.
When a body is drawn up a vertical plane, the whole weight of the body is sustained by the power that draws or lifts it up: bence the power is equal to the weight.
When a body is drawn along an horizontal (truly level) plane, it takes no power to draw it, (save the friction occasioned by the rubbing along the plane.)
From these two hypotheses, if a body is drawn up an inclined plane, the power required to raise it is as the inclination of the plane; and hence when the power acts parallel to the plane, the length of the plane is to the weight, as the height of the plane is to the power; for the greater the angle, the greater the height.
What power is requisite to move a weight of 100 lbs. up an inclined plane, 6 feet long and 4 feet high?
If 6 : 4:: 100 : 623 lbs. power.
A power of 68 lbs. at the rate of 200 feet per minute, is applied to pull a weight up an inclined plane, at the rate of 50 feet per minute-When the plane is 37 feet long and 12 feet high, how much will be the weight drawn? As 12 : 37 :: 68 x 200 : 50 x 8384 68 x 200 x 37 503200
-=838 lbs. weight. 12x50 600
The wedge is a double inclined plane, and therefore subject to the same rules; or the following rule, which is particularly for the wedge, but drawn from its near connection to the inclined plane, is,-If the power acts perpendicularly upon the head of the wedge, the power is to the pressure which it exerts perpendicularly on each side of the wedge, as the head of the wedge is to its side: hence, it is evident, that the sharper or thinner the wedge is, the greater will be the
power. But the power of the wedge being not directly according to its length and thickness, but to the length and width of the split or rift in the wood to be cleft, the rule therefore is of little use in practice; besides, the wedge is very seldom used as a power; for these reasons, the nature of its properties and effects need not be here discussed.
The screw is a cord wound in a spiral direction round the periphery of a cylinder, and is therefore an inclined plane, the length being the circumference of the cylinder, and the height, the distance between two consecutive cords or threads of the
screw, hence, the rule is derived ; -As the circumference of the screw is to the pitch, or distance between the threads, so is the weight to the power. When the screw turns, the cord or thread runs in a continued ascending line round the centre of the cylinder, and the greater the radius of the cylinder, the greater will be the length of the plane to its height, consequently, the greater the power. A lever fixed to the end of the screw will act as one of the second order, and the power gained will be as its length, to the radius of the cylinder; or the circumference of the circle described by it, to the circumference of the cylinder; hence, an addition to the rule is produced, which is,-If a lever is used, the circumference of the lever is taken for, or instead of, the circumference of the screw.
What is the power requisite to raise a weight of 8000 lbs. by a screw of 12 inchæ circumference and 1 inch pitch
As 12:1:: 8000 : 6668 lbs. = power at the circumference of the screw.
How much would be the power if a lever of 30 inches were applied to the screw?
Circumference of 30 inches 1884. As 1884 :1:: 8000 : 42 1536 lbs.=power with a lever of 30 inches long.
VELOCITY OF WIIEELS.
Wheels are for conveying motion to the different parts of a machine, at the same, or at a greater or less velocity, as may be required. When two wheels are in motion, their teeth act on one another alternately, and consequently, if one of these wheels has 40 teeth, and the other 20 teeth, the one with 20 will turn twice upon its axis for one revolution of the wheel with 40 teeth. From this the rule is taken, which is -As the velocity required is to the num