There are other species of lever, such as the bent lever, the curvilinear lever, &c. The mode of action and the theory of all are the same. The crowbar, pincers, tongs, nut-crackers, chaff-cutters, hand-barrows, a door turning on its hinges, steelyards and other weighing machines, are examples of levers. The common claw-hammer, when used for drawing nails, is an example of the bent lever. In the use of the lever, the relative distances through which the power and weight move are in exact proportion to their distance from the fulcrum. Hence, by the law above stated, The power and weight will be at rest when the product of the power multiplied by its distance from the fulcrum is equal to the product of the weight multiplied by its distance from the fulcrum. NOTE 1. The pupil, before attempting to solve any question in Mechanic Powers, should draw upon his slate a figure in which the relative position, &c., of the weight and power should be indicated. He will, by this means, get a clearer idea of the nature of the question, and at the same time be cultivating a talent which is too apt to be neglected. 1. If a lever 50 inches long have its fulcrum 4 inches from the weight, what power will be needed to balance a weight of 644 pounds? SOLUTION.-644 X 4-2576, is the product of the weight multiplied by its distance from the fulcrum; it is, therefore, the product of the power multiplied by its distance from the fulcrum; and, therefore, (97,) 644X4 56 pounds, the an swer. 2. If a man weighing 150 pounds rest upon one end of a lever 12 feet long, what weight will he balance at the other end, the fulcrum being 11 feet from the weight? 150 X 101 = 1050 lb. 3. A lever 15 feet long rests on a fulcrum 14 feet from the end; how large must the power be to balance a weight of 2000 pounds? How large a weight will 2 men balance, one weighing 150 lb., and the other 175 lb., by resting upon the end of the longer arm? 4. If the weight be 2500 lb., and the power 150 lb., where must the fulcrum be placed, under a lever 16 feet long, so as to have the power and weight balance each other? NOTE 2. The fulcrum must be placed as much nearer the weight than the power, as the weight is heavier than the power. The lever has to sustain 2500+ 150-2650 lb. The weight must, therefore, be of 16 ft. from the fulcrum, and the power 8 of 16 ft. from the fulcrum. 5. A weight of 30 lb. and a power of 31 lb. are to be so adjusted to a lever 3 feet long as to balance each other. Where shall the prop be placed? 6. Three men are to carry a stick of timber, 15 feet long, and of uniform size from end to end, one by lifting at one end of the stick, and the other two by a bar placed under the stick. Where should the bar be placed, that they may each carry an equal part of the weight? NOTE 3. The bar that is to sustain two thirds of the weight must be placed twice as near centre of gravity, or the middle of the stick, as that which is to sustain only one third; and the same for any other proportion. 7. In a lever of the second kind, 10 feet long, what power will be necessary to balance a weight of 500 lb., the weight being suspended 2 feet from the fulcrum? 8 feet from the fulcrum? 5 ft.? 31 ft.? 8. In a lever of the second kind, 5 feet long, how large a weight 1 feet from the fulcrum will a power of 18 pounds balance? Where shall a weight of 100 lb. be placed, to balance a power of 18 lb. ? 9. In a lever of the third kind, suppose the weight to be 100 lb., 10 feet from the fulcrum, what power 5 feet from the fulcrum will balance it? Where must a power of 300 lb. be placed to balance it? 196. THE WHEEL AND AXLE. The WHEEL AND AXLE is a modification of the lever; it is, indeed, a continuously acting lever. It consists of a cylinder, C, revolving upon an axis, and having a wheel, A B, of larger diameter, immovably affixed to it. The power is applied to the circumference of the wheel, and the weight to that of the axle. The radius of the wheel is to be regarded as the longer arm of a lever, and that of the axle as the shorter arm. Equilibrium, therefore, takes place when the product of the power, multiplied by the radius of the wheel, equals the product of the weight multiplied by the radius of the axle. NOTE. The diameter or the circumference may be substituted for the radius in the above formula. 1. The wheel for raising goods in a grain store is 4 feet in diameter; the axle 6 inches in diameter. What power applied to the rope passing over the wheel, will it take to balance a bag of wheat weighing 180 lb., attached to a rope passing over the axle ? 2. If the radius of a wheel is 6 feet, and that of the axle 5 inches, what weight upon the axle would be balanced by a power of 145 lb. upon the wheel? 3. The radius of a wheel being 4 feet, what must be the radius of the axle, in order that a power of 25 lb. may balance a weight of 375 lb. ? 4. The arm of a windlass used in raising water from a well is 15 inches long, measuring from the centre of motion; the axle on which the rope is wound is 6 inches in diameter. What power at the crank will it take to balance a bucket of water weighing 50 lb. ? 197. THE PULLEY. The PULLEY is a wheel round the rim of which a groove is cut in which a cord can work, and the centre of which moves on a pivot in a block. The wheel is sometimes called a sheave. Pulleys are either movable or fixed. By a fixed pulley we mean one that only revolves on its axis, without changing its place. It gives no mechanical advantage, its use being only to change the direction of the weight. Thus, in No. 1, the power and weight pass over equal spaces in equal times; their force must, therefore, be equal.* In No. 2, the pulley is movable, and the *The pulley, in No. 1, may be regarded as a lever of the first kind, with equal arms, the pivot being the fulcrum, and the radius of the wheel the length of each arm. power moves through twice the distance that the weight does. The advantage gained is, therefore, as 2 to 1.* In Nos. 3 and 4, the power moves through 4 times the distance of the weight; the advantage gained is, therefore, as 4 to 1. When several movable pulleys act as in No. 3, the power and weight balance each other when the power is to the weight as ǹ is to that power of 2 which equals the number of movable pulleys. Thus, if, as in figure 3, there are two movable pulleys, the power is to the weight as I to 22; that is, as 1 to 4. If three movable pulleys were arranged in this manner, the proportion would be as i to 23, or as 1 to 8. When several movable and fixed pulleys are employed as in No. 4, equilibrium is produced when the power equals the weight divided by twice the number of movable pulleys; or, when the weight equals the power multiplied by twice the number of movable pulleys. 1. If a power of 160 pounds be applied to a rope connecting a system of 4 movable pulleys, arranged as in No. 3, what weight will the power balance? 1.2160 2560 pounds. What power would be required to balance a weight of 640 lb. ? 2. What weight will be balanced by a power of 15 pounds attached to a cord that passes over 3 movable pulleys arranged as in No. 4? What power will it take to balance a weight of 1500 pounds? P W 198. THE INCLINED PLANE. AN INCLINED PLANE is an unyielding plane surface, inclined at an acute angle to the horizon. When a body is placed on such a plane, a part of the weight is supported by the plane, and the remainder, which urges the weight down the plane, is the part to be supported by the power. When the power acts in the direction of the plane, equilibrium ensues when the power multiplied by the length of the PLANE is equal to the weight multiplied by the height of the plane; that is, when the power is to the weight, as the perpendicular of the plane is to its length. If the power acts in the direction of the base, equilibrium is obtained when the product of the power multiplied by the length of the BASE Cquals the product of the weight multiplied by the height of the plane; or, when the power is to the weight, as the perpendicular of the plane is to the length of the base. *The pulley in No. 2 is a lever of the 2d kind upon a movable fulcrum, in which the diameter of the wheel is the longer arm, and the radius of the wheel the shorter arm. 1. An inclined plane is 10 feet long, and its perpendicular height 2 feet. What weight will a power of 10 pounds balance, if it act parallel to the plane? What weight will it balance, if it act parallel to the base? (163.) 2. A hill rises 8 feet in 100. How much more power must a horse exert to draw a load of 1000 pounds up the hill than on level ground, the friction in the two cases being equal? B 199. THE WEDGE. THE WEDGE may be regarded as two inclined planes, laid base to base. It is not usually employed by the agency of continued pressure, but by that of percussion, as the blows of a hammer or mallet upon its back. Its power increases as the thickness of its back, compared with the length of its sides, is diminished. When acted on by continued pressure, the condition of equilibrium is, that the product of the power multiplied by the length of the wedge be equal to the product of the weight multiplied by half the thickness of the head. 200. THE SCREW. THE SCREW is a cylinder with a spiral line cut round it. This spiral line is called the worm or thread of the screw. The distance from one thread to another is called the breadth of the In most cases, the screw requires a corresponding cavity, called a nut, in which it may work. Sometimes the screw moves in the nut, and sometimes the nut moves on the screw. The power is applied to the end of a lever attached to the movable part; the weight or resistance is applied to the end of the screw. In the screw, equilibrium is obtained, when the product of the weight multiplied by the breadth of the worm, equals the product of the power multiplied by the circumference of the circle described by it. Or, If the screw is advancing through a nut, when the product of the weight multiplied by the distance through which the head of the screw advances, is equal to the product of the power multiplied by the circumference it describes. Great allowance is to be made for friction in the practical application of the screw. 1. If the lever of a screw is 8 feet long, and the distance between the threads 1 inches, what power will balance a weight of 4000 pounds, making no allowance for friction? |