Page:The American Cyclopædia (1879) Volume XI.djvu/339

 MECHANICS 327 the lever in an oblique direction ; but in making calculations the perpendicular distance of the lines of direction from the fulcrum must be regarded instead of the actual distances on the lever, as will be x ' readily understood /''' by observing fig. 30, where the power is applied in the direc- tion dp, the long arm of the lever, c, being practical- ly reduced to the side c d of the right- angled triangle c d a. When two or more levers, of one or more kinds, are combined, the system is called a compound lever. Plat- form weighing scales, such as are shown in fig. 31, are combinations of this kind, where the beam is a lever of the first kind, and e &, a 5, and gf levers of the second kind, if we consider the weights on the beam Jem the power; but if we consider the load as a force acting on the FIG. 30. FIG. 81. platform and raising the weights at jt>, then a 5, c &, and fg become levers of the third kind. 2. The Wheel and Axle. This power is a modification of the lever, and consists of a wheel and cylinder, or of two cylinders of unequal radii, revolving about a common axis, the larger cylinder being called the wheel and the smaller the axle. The wheel and axle may have the action of either kind of lever, but usually has that of the first, as shown in fig. 32, where the power is applied at #, the ful- crum is the axis c, while the weight is sus- pended from the short end of the lever, which is equal to the radius of the axle, the long end being equal to the radius of the wheel. If, how- ever, the weight is suspended upon the same side with the power, and the latter is applied in an upward direction, as is often the case, the machine acts upon the principle of a lever of the second kind ; and by applying the power to the axle the machine may be made to act as a lever of the third kind. Indeed, prac- tically, levers are constant- ly changing in their action from one kind into another ; thus a shovel or hay fork held in the two hands may at one moment be a lever of the first kind and at the next one of the third kind, as the one or the FIG. other hand becomes fixed or movable. In the simple wheel and axle the mechanical advan- tage is in proportion to the ratio of the radius of the wheel to that of the axle ; if the former is 5 ft. while the latter is 5 in., the ratio of power to weight is as 1 to 12. This mode of increasing the efficiency of the machine is often inconvenient, and may be obviated by employing a differential axle, consisting of two parts of different diameters, as represent- ed in fig. 33, the cord winding upon' one part and off the other, and the weight being sup- FIG. 33. FIG. 84. ported by a pulley. The most common plan in machinery is to employ a system of cog wheels, as shown in fig. 34. An equilibrium of forces will obtain where the product of the power multiplied into the radii of all the wheels is equal to that of the weight multi- plied into the radii of all the pinions. Cog wheels are of three kinds, spur, crown, and bevelled. A spur wheel is shown at a, fig. 35, a crown wheel at 5, and bevelled wheels at c. They are used to change the direction of force or axis of mo- tion. The wheel and axle when applied to carriages serves a dif- ferent purpose from that in ordinary ma- chinery, and acts in a different way. The action of the carriage wheel when ascending an inclination is like that modification of the inclined plane called the toggle joint (fig. 44). When the road is a rigid and level plane, the wheel merely serves to afford successive supporting points to the load, the only resistance to be over- come being the sliding friction of the axle and the rolling fric- tion between the wheel and the road. (See FEIOTION.) 3. The Pulley. The pulley is common- ly said to act upon the princi- ple of a lever of the first kind with equal arms, but this does not explain its principle. In fig. 36 we have indeed a lever of the first kind, with equal arms a c and c 5 ; and fig. 37 shows another such lever in the FIG. 35. FIG. 36.