Page:The American Cyclopædia (1879) Volume XI.djvu/340

328 MECHANICS stationary pulley, and also a lever of the second kind in the movable pulley, where the power is applied at /, the weight sus- pended at J, while the fulcrum, which is rhaiigoabK' as the pulley ascends, is at a. This vould seem to account for the mechan- Fio. 87. FIG. 83. ical advantage, which is in the ratio of 2 to 1 ; hut in place of the pulleys there may be substituted smooth cylinders which shall not revolve, and if they are well lubricated a similar mechanical advantage is obtained as when revolving pulleys are used. In this case there is evidently no use of a lever, so that it must be concluded that the mechani- cal advantage which is obtained is derived from the use of the cord. The system of pul- leys shown in fig. 38 may include any number, the mechanical advantage being doubled with every additional movable pulley; for it is evident that the cord a sustains half of the weight W, and the cord c half of this, or one fourth of the whole; and further, that the cord e or f sustains one eighth of the whole. Several pulleys may be placed in one block. In- stead of having one fixed pulley and the others movable, they may be placed in two frames or blocks, as shown in fig. 39, one block being fixed and the other movable. In this case one cord goes round all the pul- leys, and therefore the weight is divided equally between the parts of the cord in the lower block, which parts are equal to twice the number of pulleys in the block. In this arrangement, where there are three pulleys in each block, the power will there- fore be to the weight in the ratio of 1 to 6. The pulley is a very portable and efficient power, the cord allowing great freedom in changing the direction in which the power is applied, by the employment of fixed single pulleys, -hirh may be fastened to the ground, allowing of the application of horse or steam Fio. 89. power, or of any desired number of men. The rigging of ships is almost entirely managed by means of the pulley, and the hoisting of build- ing material is to a great extent effected by the same machine. 4. The Inclined Plane. This power depends for its efficiency in the elevation of great weights upon the nearness with which the plane approaches to a hori- zontal surface. The power re- quired to pro- duce the equi- librium of forces on an inclined 'plane may be deter- mined in the following man- ner: We will suppose two cases, the first in which the FIG. 40. power is ap- plied in a direction parallel to the plane, and the second in which it is applied in a hori- zontal direction, or parallel to the base. Let m, fig. 40, be the centre of gravity of a freely moving body resting on a plane whose length is a 5, and whose height is & c. Let the per- pendicular me fall from the centre of grav- ity upon the plane ; also draw m d perpen- dicular to the plane. Let m e represent the force of gravity, then will m d represent the pressure perpendicular to the plane, and d e or ml will represent the force in quantity and direction with which the body tends to move downward along the plane. An equal force acting in the opposite direction will therefore produce equilibrium. Since the triangle m e d is similar to a & e, ed : em :: & e : a j. Con- sequently, when the power is applied in a di- rection parallel to the plane, equilibrium will exist where p : w : : height of plane : length of plane, or p : w : : sin a : rad. In the sec- ond case, where the power is applied in a di- rection parallel with the base, produce m d to A, and draw e Ti parallel to the base ; then will e h or m Ic of the parallelogram mehlc repre- sent the force necessary to produce equilib- rium, and m h will represent the pressure per- pendicular to the plane. But in this case e h : e m : : I c : a c. Therefore, power : weight : : height of plane : length of base of plane, or p : w : : sin a : cos a, 5. TheWedge,ft%.4:, is a double inclined plane. It is used for forcing asunder bod- ies which offer great resistance, such FIG. 41. fibres of wood and the seams of rocks. It is usually propelled by percussion, which is applied to the head in the direction of its length, from a to &. The forces will be in equilibrium where the proportion of power to resistance is the same as that of thickness of