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EXAMPLE.-Required the velocity a ball will acquire in descending through 201 feet.

64.33 X 201 = 113.7 feet.

To find the Space through which a Body will fall in

any given time. Rule.-Multiply the square of the time in seconds by 16.083, and the product will be the space in feet.

EXAMPLE.- Required the space fallen through in 7 seconds.

16.083 x 49 = 788.067 feet. NOTE.—The velocity acquired by a body in falling from rest, through a given height, is the same whether it fall freely or descend through a plane any way inclined.

The diameter of a circle perpendicular to the horizon, and any chord terminating at either extremity of that diameter, are fallen through in the same time.

And the velocities which bodies acquire by descending along chords of the same circle are as the lengths of those chords.

ON PENDULUMS.

A pendulum that vibrates seconds, or 60, in the latitude of London, is 39.1393 inches long ; and ✓39.1393 x 60 = 375.36, which serves as a constant number for other pendulums : thus, 375.36 divided by the square root of the pendulum's length, gives the number of vibrations per minute; and divided by the vibrations per minute, gives the square root of the length of pendulum.

EXAMPLE 1.-Required the number of vibrations a pendulum of 25 inches long will make per minute. 375.36

= 75.072 vibrations per minute. 725 EXAMPLE 2.-Required the length of a pendulum to make 80 vibrations per minute. 375.36

= 4.6922 = 22.014864 inches long. 80

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 contain 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 wbeels, the result will be the same; because their circumferences, describing equal spaces,

render their revolutions unequal: from this the rule is derived, namely,

Multiply the velocity of the driver by the number of teeth it contains, and divide by the velocity of the driven; the quotient will be the number of teeth it ought to contain.-Or, Multiply the velocity of the driver by its diameter, and divide by the velocity of the driven; the quotient will be the diameter of the driven.

EXAMPLE 1.-If a wheel that contains 75 teeth makes 16 revolutions per minute, required the number of teeth in another to work in it, and make 24 revolutions in the same time. 75 x 16

= 50 teeth. 24 EXAMPLE 2.-A wheel 64 inches diameter, and making 42 revolutions per minute, is to give motion to a shaft at the rate of 77 revolutions in the same time; required the diameter of a wheel suitable for

that purpose.

64 x 42

= 34.9 inches nearly. 77 Example 3.-Required the number of revolutions per minute made by a wheel or pulley 20 inches diameter, when driven by another of 4 feet diameter, and making 46 revolutions per minute. 48 x 46

= 110.4 revolutions. 20 EXAMPLE 4.-A shaft at the rate of 22 revolutions per minute is to give motion, by a pair of wheels, to another shaft at the rate of 157, the distance of the shafts from centre to centre is 452 inches; the diameters of the wheels at the pitch lines is required. 45.5 x 15.5

= 18.81 radius of the driving wheel. 22 + 15.5

45.5 x 22 And

= 26.69 radius of the driven wheel. 22 + 15.5

Example 5.-Suppose a drum making 20 revolutions per minute, required the diameter of another to make 58 revolutions in the same time.

58 = 20 = 2,9, that is, their diameters must be as 2.9 to 1; thus, if the one making 20 revolutions be called 30 inches, the other will be

30 = 2.9 = 10.345 inches diameter nearly. Example 6 –Required the diameter of a pulley, to make 124 revolutions in the same time as one of 32 inches making 26. 32 x 26

= 66.56 inches diameter. 12.5

EXAMPLE 7.-A shaft at the rate of 16 revolutions per minute, is to give motion to a piece of machinery at the rate of 81 revolutions in the same time; the motion is to be communicated by means of two wheels and two pulleys with an intermediate shaft; the driving wheel contains 54 teeth, and the driving pulley is 25 inches diameter; required the number of teeth in the other wheel, and the diameter of the other pulley

✓81 x16=36, the mean velocity between 16 and
16 x 54

36 x 25
81;-then = 24 teeth ;- and
36

81 11.11 inches diameter of pulley. EXAMPLE 8.-Suppose in the last example the revolutions of one of the wheels being given, the number of teeth in both, and likewise the diameter of each pulley, to find the revolutions of the last pulley. 16 x 54

= 36, velocity of the intermediate 24

36 x 25 shaft ;-and, = 81 the velocity of the

11.11 machine.

A TABLE

For finding the radius of a wheel when the pitch is given, or the pitch of a wheel when the radius is given, that shall contain from 10 to 150 teeth, and any pitch required.

Num.

of Teeth

Radius.

Num.

of Teeth

Radius.

Num.

of Teeth

Radius.

Num.

of Teeth

Radius.

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 14 45

1.618 46
1.774 47
1.932 48
2.089 49
2.247 50
2.405 51
2.563 52
2.721 53
2.879 54
3.038 55
3.196 56
3.355 | 57
3.513 58
3.672 59
3.830

60
3.989
4.148
4.307 63
4.465 64
4.624 65
4.788 66
4.942 67
5.101 68
5.260 69
5.419 70
5.578 71
5.737 72
5.896
6.055 74
6.214
6.373 76
6.532 77
6.691 78
6.850

79
7.009 80
7.168

7.327 7.486 7.645 7.804 7.963 8.122 8.281 8.440 8.599 8.758 8.917 9.076 9.235 9.394 9.553 9.712 9.872 10.031 10.190 10.349 10.508 10.667 10.826 10.985 11.144 11.303 11.463 11.622 11.781 11.940 12.099 12.258 12.417 12.576 12.735

81 12.895 116 18.464 82 13.054 117 18.623 83 13.213 118 18.782 84 | 13.370 119 18.941 85 13.531 120 19.101 86 | 13.690 121 19.260 87 13.849 122 19.419 88 14.008 123 19.578 89 14.168 124 | 19.737 90 14.327 125 19.896 91 14.486 126 20.055 92 14.645 127 20.214 93 14.804 128 20.374 94 | 14.963 129 20.533 95 | 15.122 130 20.692 96 | 15.281 131 | 20.851 97 | 15.440 132 21.010 98 | 15.600 133 21.169 99 | 15.759 134 21.328 100 15.918 135 21.488 101 16.077 136 21.647 102 16.236 137 21.806 103 16.395 138 21.965 104 | 16.554 139 22.124 105 | 16.713 140 22.283 106 | 16.873 141 22.442 107 17.032 142 22.602 108 17.191 143 22.761 109 17.350 144 22.920 110 17.509 145 23.079 111 17.668 146 23.238 112 17.827 147 23.397 113 17.987 148 23.556 114 | 18.146 149 23.716 115 18.305 150 23.875

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