lengths are 60 inches, and in the first and second 10 inches from weight to prop, the third being 10 inches from weight to power. EXAMPLE 4-What power is necessary to raise a weight of 620 lbs. by a lever of the first order, 72 inches long, and the prop placed 12 inches from the weight? 72 12 = 60 inches to power. Then 620 × 12 60 = 124 lbs. EXAMPLE 5.--A weight of 620 lbs. is to be lifted by a power of 124 lbs. applied to the end of a lever of the first order, 72 inches long; required at what distance from the weight the prop must be placed. EXAMPLE 6.-A beam 20 feet long, and supported at both ends, bears a weight of 73 cwt. 4 feet 6 inches from one end; required the proportion of weight upon each support. And 73 x 4.5 20 73 x 15.5 20 = 16.425 cwt. on the farthest support. = 56.675 cwt. on the nearest support. EXAMPLE 7.-A weight of 300 lbs. is fixed on the end of a lever 6 feet long; required the power, applied 24 feet from the prop, to raise the weight. 300 × 6 2.5 720 lbs. power. Wheel and Axle. Here the velocity of the power is to the velocity of the weight as the circumference of the wheel is to the circumference of the axle; hence, divide the velocity of the power by the velocity of the weight, and the quotient is the weight that the power is equal to. EXAMPLE 1.-A power equal to 30 lbs. is applied to the winch of a crane whose length is 15 inches; the pinion contains 10 teeth, the wheel 120, and the barrel is 9 inches diameter; required the weight raised. 15 × 2 × 3.1416 94.248 circumference of the circle described by the winch, or handle. 120 ÷ 10 = 12 revolutions of the pinion for one of the wheel, and 3.1416 x 9 28.2744 the barrel's circumference, then, 94.248 x 12 x 30 28.2744 = 1200 lbs. raised by this crane. EXAMPLE 2.-What would be the increase of power, in the last example, if a wheel of 150 teeth, and a pinion of 15, were added to the crane? 150 ÷ 15 = 10, that is, the velocity of the weight is diminished, while the velocity of the power is the same; then, 94.248 x 12 x 10 x 30 =12000 lbs. raised, the power being increased ten times. EXAMPLE 3.-What power is requisite to raise 42 tons 60 feet high in 10 minutes, the velocity of the power being 20 feet per minute? 60 10 = 6, and 40 × 6 20 The Pulley. = 12 tons power. A single pulley that only turns on its axis, and does not move out of its place, serves only to change the direction of the power, but gives no mechanical advantage. The advantage gained is always as twice the number of moveable pulleys, without taking any notice of the fixed pulleys necessary to compose the system of pulleys; hence, divide the weight to be raised by twice the number of moveable pulleys, and the quotient is the power required to raise the weight. EXAMPLE 1.-What power is requisite to raise 250 lbs. with a pair of four-shieved blocks, the one block moveable and the other fixed? 4 x 28, and 250 = 31.25 lbs. power. EXAMPLE 2.-What weight will a power of 120 lbs. raise, when applied to a three and four-shieved block, the three being moveable and the other fixed? 3 x 2 = 6, and 120 × 6 = 720 lbs. raised. The Inclined Plane. The advantage gained by the inclined plane is as great as its length exceeds its perpendicular height; 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. EXAMPLE 1-Required the power capable of moving a weight of 300 lbs. up an inclined plane 50 feet long and 16 feet high. As 50 16: 300: 96 lbs. power. That is, 300 x 16 48005096. EXAMPLE 2.-A power of 120 lbs. with a velocity of 50 feet per minute, is to be applied to move a weight up an inclined plane at the rate of 30 feet per minute; the plane is 25 feet long and 8 feet high; required the weight that the power is equal to. 120 × 50 = 6000, and 30 x 8 240; then As 240 25: 6000 625 lbs. The Wedge. = As the wedge is seldom used without being driven, the force of the blow is not easily ascertained; of course, in practice it is not worth taking into account, with respect to calculation. The Screw. The advantage gained by the screw is as much as the circumference of a circle, described by the lever or handle, exceeds the interval or distance between the spirals of the screw; hence, as the circumference of the circle described by the handle is to the pitch of the screw, so is the weight to the power. EXAMPLE.-What power is necessary to raise a weight of 6000 lbs., the length of the lever being 20 inches, and the screw pitch. 20 x 240 x 3.1416 125.6 inches, then, As 125.6 .75 :: 6000: 35.8 lbs. power required. N.B.-There are few machines but what, on account of the friction of the parts against one another, will require a third part more power to work them, when loaded, than is requisite to constitute a balance between power and weight. The following Table shows the estimated power of man or horse as applied to machinery. OF FALLING BODIES. In bodies falling freely by their own weight, their velocities are as the times, and the spaces, as the square of the times; therefore, if the times be as the numbers, .. ... And the spaces for each time, as the odd numbers, .... 1, 2, 3, 4, &c., 1, 2, 3, 4, &c., 1, 4, 9, 16, &c. 1, 3, 5, 7, &c. It has been ascertained by experiment that a body falling freely from rest, will descend through 16 feet in the first second of time, and will then have acquired a velocity which being continued uniformly, will carry it through 322 feet in the next second, consequently, if the first series of numbers be expressed in seconds. 3μ &c. 1" Velocities in feet will be......32 2" 64, 96 &c. 64, 144 &c. And the spaces for each second 162 48 80 &c. To find the Velocity a falling Body will acquire in any given time. RULE.-Multiply the time in seconds by 32.166, and the product will be the velocity acquired in feet per second. EXAMPLE. Required the velocity in 7 seconds. 32.166 × 7 = 225.162 feet, velocity acquired. To find the Velocity a Body will acquire by falling from any given height. RULE.-Multiply the space in feet by 64.33, and the square root of the product will be the velocity acquired in feet per second. |