10 feet. Now to find the distance, f d, which is the versed sine: RULE.-Find the square of a b, and from it subtract the square of a f; the remainder find the square root of, then the product; subtract from a b, and the remainder will give the distance, df. 100 400 A TABLE of the Fractional Parts of an Inch, when divided into thirty-two parts; likewise a foot of twelve inches, reduced to decimals. The great utility of the above table is to facilitate the multiplication and division of parts of an inch; also in calculations. For example, suppose a sheet of iron to be 20ğ inches long, 12 and inches broad, and and of an inch in thickness, what number of cubic inches does it contain? 32 32 20.625 12.78 165000 144375 41250 20625 263.58750 3d. W. 263.58 •84 THE LEVER. 1. A LEVER is an inflexible bar, either straight or bent, supposed capable of turning round a fixed point, called the fulcrum. F. 105432 According to the relative positions of the weight, power, and fulcrum, on the lever, it is said to be of three kinds, viz. when the fulcrum is somewhere betwixt the weight and power, it is of the first kind; when the weight is between the power and the fulcrum, it is of the second kind; and when the power is between the weight and the fulcrum, it is of the third kind, thus: 1st. 2d. F. 210864 221-4072 cubic inches of iron. F W. P. P. P. W 2. In the first and second kinds there is an advantage of power, but a proportionate loss of velocity; and in the third kind there is an advantage in velocity, but a loss of power. 3. When the weight × its distance from the fulcrum = the power × its distance from the fulcrum, then the lever will be at rest, or in equilibrio; but if one of these products be greater than the other, the lever will turn round the fulcrum in the direction of that side whose product is the greater. 4. In all the three kinds of levers, any of these quantities, the weight, or its distance from the fulcrum, or the power or its distance from the fulcrum, may be found from the rest, such, that when applied to the lever, it will remain at rest, or the weight and power will balance each other. Weight X its distance from fulc. Dist. of power from fulc. Power X its distance from fulc. 5. 6. 7. 8. = power. weight. dist. power from fulc. Power. Power X dist. power from fulc. Weight. 9. In the first kind of lever, the pressure upon the fulcrum = sum of weight and power; in the second and third = their difference. dist. weight from fule. 10. If there be several weights on both sides of the fulcrum, they may be reckoned powers on the one side of the fulcrum, and weights on the other. Then, if the sum of the product of all the weights their distances from the fulcrum be to the sum of the products of all the powers × their distances from the fulcrum, the lever will be at rest; if not, it will turn round the fulcrum in the direction of that side whose products are greatest. 11. In these calculations the weight of the lever is not taken into account; but if it is, it is just reckoned like any other weight or power acting at the centre of gravity. 12. When two, three, or more levers act upon each other in succession, then the entire mechanical advantage which they give, is found by taking the product of their separate advantages. 13. It is to be observed in general, before applying these observations to practice, that if a lever be bent, the distances from the fulcrum must be taken, as perpendiculars drawn from the lines of direction of the weight and power of the fulcrum. Example. In a lever of the first kind, the weight is 16, its distance from the fulcrum 12, and the power is 8; therefore by No. 7 of this chapter, 16 × 12 24 the distance of power from the 8 fulcrum. |