1. If a man weighing 170 pounds be resting upon a lever 10 feet long, what weight will he balance on the other end, the prop being one foot from the weight? Ans. 1530lbs. 2. If a weight of 1530 pounds were to be raised by a lever 10 feet long, and the prop fixed one foot from the weight, what power applied to the other end of the lever would balance it? Ans. 170lbs. 3. If a weight of 1530 pounds be placed one foot from the prop, at what distance from the prop must a power of 170 pounds be applied to balance it? Ans. 9 feet. 4. At what distance from a weight of 1530 pounds must a prop be placed, so that a power of 170 pounds, applied 9 feet from the prop, may balance it? Ans. 1 foot. 5. Supposing the earth to contain 4,000,000,000,000,000,000,000 cubic feet, each foot weighing 100 pounds, and that the earth was suspended at one end of a lever, its centre being 6000 miles from the fulcrum or prop, and that a man at the other end of the lever was able to pull, or press with a force of 200 pounds; what must be the distance between the man and the fulcrum, that he might be able to move the earth? Ans. 12,000,000,000,000,000,000,000,000 miles. 6. Supposing the man in the last question to be able to move his end of the lever 100 feet per second, how long would it take him to raise the earth one inch? Ans. 52,813,479,690y. 17d. 14h. 57m. 46 sec. THE WHEEL AND AXLE. The wheel and axle is a wheel turning round together with its axle; the power is applied to the circumference of the wheel, and the weight to that of the axle by means of cords. An equilibrium is produced in the wheel and axle, when the weight is to the power as the diameter of the wheel to the diameter of its axle. To find, therefore, how large a power must be applied to the wheel to raise a given weight on the axle, we adopt the following RULE. — As the diameter of the wheel is to the diameter of the axle, so is the weight to be raised by the axle to the power that must be applied to the wheel. 7. If the diameter of the axle be 6 inches, and the diameter of the wheel 4 feet, what power must be applied to the wheel to raise 960 pounds at the axle? Ans. 120lbs. 8. If the diameter of the axle be 6 inches, and the diameter of the wheel 4 feet, what power must be applied to the axle to raise 120 pounds at the wheel ? Ans. 960lbs. 9. If the diameter of the axle be 6 inches, and 120 pounds applied to the wheel raise 960 pounds at the axle, what is the diameter of the wheel ? Ans. 4 feet. 10. If the diameter of the wheel be 4 feet, and 120 pounds applied to the wheel raise 960 pounds at the axle, what is the diameter of the axle ? Ans. 6 inches. To find the weight that may be raised by a given power. RULE. — Multiply the power by the number of cords that support the weight, and the product is the weight. 11. What power must be applied to a rope, that passes over one movable pulley, to balance a weight of 400 pounds ? Ans. 200lbs. 12. What weight will be balanced by a power of 10 pounds, attached to a cord that passes over 3 movable pulleys ? Ans. 60lbs. 13. What weight will be balanced by a power of 144 pounds, attached to a cord that passes over 2 movable pulleys ? . Ans. 576lbs. 14. If a cord, that passes over two movable pulleys, be attached to an axle 6 inches in diameter, whose wheel is 60 inches in diameter, what weight may be raised by the pulley, by applying 144 pounds to the wheel ? Ans. 5760lbs. THE INCLINED PLANE. An inclined plane is a plane which makes an acute angle with the horizon. The motion of a body descending an inclined plane is uniformly accelerated. The force with which a body descends an inclined plane, by the force of attraction, is to that with which it would descend freely, as the elevation of the plane to its length; or, as the size of its angle of inclination to radius. To find the power that will draw a weight up an inclined plane. RULE. — Multiply the weight by the perpendicular height of the plane, and divide this product by the length. 15. An inclined plane is 50 feet in length, and 10 feet in perpendicular height; what power is sufficient to draw up a weight of 1000 pounds ? Ans. 200lbs. 16. What weight, applied to a cord passing over a single pulley at the elevated part of an inclined plane, will be able to sustain a weight of 1728 pounds, provided the plane was 600 feet long, and its perpendicular height 5 feet? Ans. 14şlbs. 17. A certain railroad, one mile in length, has a perpendicular elevation of 50ft. ; what power is sufficient to draw up this elevation a train of cars weighing 20,000lbs. ? Ans. 18941. 18. An inclined plane is 300 feet in length, and 30 feet in perpendicular height; what power is sufficient to draw up a weight of 2000 pounds ? Ans. 200lbs. 19. An inclined plane is 1000 feet in length, and 100 feet in perpendicular height; what power is sufficient to draw up this elevation a weight of 5000 pounds ? Ans. 500lbs. 20. What weight applied to and passing over a single pulley, at the elevated part of an inclined plane, will be able to sustain a weight of 7000lbs., provided the plane is 300 feet long and its perpendicular height 30 feet? Ans. 700lbs. THE SCREW. The screw is a cylinder, which has either a prominent part or a hollow line passing round it in a spiral form, so inserted in one of the opposite kind that it may be raised or depressed at pleasure, with the weight upon its upper, or suspended beneath its lower surface. In the screw the equilibrium will be produced, when the power is to the weight as the distance between the two contiguous threads, in a direction parallel to the axis of the screw, to the circumference of the circle described by the power in one revolution. To find the power that should be applied to raise a given weight. Rule. — As the distance between the threads of the screw is to the circumference of the circle described by the power, so is the power to the weight to be raised. NOTE. - One third of the power is lost in overcoming friction. 21. If the threads of a screw be 1 inch apart, and a power of 100 pounds be applied to the end of a lever 10 feet long, what force will be exerted at the end of the screw ? Ans. 75,398.20+lbs. 22. If the threads of a screw be 1 an inch apart, what power must be applied to the end of a lever 100 inches in length to raise 100,000 pounds ? Ans. 79.57747-lbs. 23. If the threads of a screw be 4 an inch apart, and a pow. er of 79.5774+ pounds be applied to the end of a lever 100 inches in length, what weight will be raised ? Ans. 100,000lbs. 24. If a power of 79.5774+ pounds be applied to the end of a lever 100 inches long, and by this force a weight of 100,000 pounds be raised, what is the distance between the threads of the screw ? Ans. } an inch. 25. If a power of 79.5774+ pounds be applied to the end of a lever, raising by this force a weight of 100,000 pounds, what must be the length of the lever, if the threads of the screw be an inch apart ? Ans. 100 inches. · WEDGE. The wedge is composed of two inclined planes, whose bases are joined. When the resisting forces and the power which acts on the wedge are in equilibrio, the weight will be to the power as the height of the wedge to a line drawn from the middle of the base to one side, and parallel to the direction in which the resisting force acts on that side. To find the force of the wedge. RULE. - As half the breadth or thickness of the head of the wedge is to one of its slanting sides, so is the power which acts against its head to the force produced at its side. 26. Suppose 100 pounds to be applied to the head of a wedge that is 2 inches broad, and whose slant is 20 inches long, what force would be affected on each side ? Ans. 2000lbs. 27. If the slant side of a wedge be 12 inches long, and its head 14 inches broad, and a screw whose threads are of an inch asunder be applied to the head of this wedge, with a power of 200 pounds at the end of the lever, 16 feet long, what would be the force exerted on the sides of the wedge ? Ans. 5147184.3+Ibs. SECTION LXXXI. To find the specific gravity of a body. Rule. - Weigh the body both in water and out of water, and note the difference, which will be the weight lost in water; then, as the weight lost in water is to the whole weight, so is the specific gravity of water to the specific gravity of the body. But if the body whose specific gravity is required is lighter than water, affix to it another body heavier than water, so that the mass compounded of the two may sink together. Weigh the dense body and the compound mass separately, both in water and out of it; then find how much each loses in water by subtracting its weight in water from its weight in air; and subtract the less of these remainders from the greater ; then say, as the last remainder is to the * The specific gravity of a body is its weight compared with water; the water being considered 1000. |